Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Theory and Modeling
Hyon Kook MYONG
School of Mechanical Engineering
httpcfdkookminackr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Contents
서론
기초방정식
난류의 발생
난류의 생성 및 소산
와동(Vortex) Dynamics난류 스케일
상관
난류 스펙트럼
난류모델
난류 현상의 예측 예
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Modeling
단순화
亂 流
어지러울 난
模 型 化
흐를 류
TurbulenceTurbulence ModelingModeling++
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-1
15~16 15~16 CenturyCentury
L da Vinci (1452-1519) Visual and descriptive (available)
Mechanistic theory of nature + () Math modelNote no mathematical model available to describe flow motions
17~18 Century17~18 Century
I Newton (1643-1727) L Euler (1707-1783)D Bernoulli (1700-1782) J DrsquoAlembert (1717-1783)
Continuum inviscid Newtonrsquos law (available)
Viscous fluid model
Note no mathematical model available to describe viscous flows
( )+minusminus=i
ii
xPg
DtDU
partpartρρ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-2
1919thth CenturyCentury
L M H Navier (1785-1836) J B Fourier (1768-1830)B de Saint Venant (1797-1886) G G Stokes (1819-1903)
Continuum viscous Stokes postulations on and Fourier postulations on (available)
Turb flow model
Turb heat flux model
Note no mathematical model available to describe turbulent flow and heat transfer
τ ijqi
( )++minusminus=j
ij
ii
i
xxPg
DtDU
partτpart
partpartρρ
( )ρ τpartpart
partpart
CDTDt
Ux
DPDt
qxp ij
i
j
i
i= + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-3
19~20 Century19~20 Century
O Reynolds (1842-1912) T von Karman (1881-1963)L Prandtl (1875-1953) G I Taylor (1880-1975)
Continuum averaging viscous turbulent postulations
Note Development of turbulence model
( )
( )ρ ρpartpart
partτpart
partτpart
DUDt g
Px x x
ii
i
ij
j
tij
j= minus + + + +
( )
( )ρ τpartpart
partpart
partpart
CDTDt
Ux
DPDt
qx
qxp ij
i
j
i
i
ti
i= + + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model Postulations
Before we take on the postulation let us ask ifMomentum diffusion
(N-S)Heat diffusion
Mass diffusion
then shouldTurb M diffusion
Turb H diffusion
( )ρUi iij UGrad~τ
( )ρC Tp q GradTi ~
( )ρC M GradCi ~
( )ρu ui j
( )ρ θC up i
( )u u up
Grad u ui j i j+ρ
~ ~
u up
Grad ui iθθρ
θ+ ~ ~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Scale
Small Scale
UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena
Large Scale
Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy
Effective in transport phenomena
u t η υ τ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Contents
서론
기초방정식
난류의 발생
난류의 생성 및 소산
와동(Vortex) Dynamics난류 스케일
상관
난류 스펙트럼
난류모델
난류 현상의 예측 예
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Modeling
단순화
亂 流
어지러울 난
模 型 化
흐를 류
TurbulenceTurbulence ModelingModeling++
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-1
15~16 15~16 CenturyCentury
L da Vinci (1452-1519) Visual and descriptive (available)
Mechanistic theory of nature + () Math modelNote no mathematical model available to describe flow motions
17~18 Century17~18 Century
I Newton (1643-1727) L Euler (1707-1783)D Bernoulli (1700-1782) J DrsquoAlembert (1717-1783)
Continuum inviscid Newtonrsquos law (available)
Viscous fluid model
Note no mathematical model available to describe viscous flows
( )+minusminus=i
ii
xPg
DtDU
partpartρρ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-2
1919thth CenturyCentury
L M H Navier (1785-1836) J B Fourier (1768-1830)B de Saint Venant (1797-1886) G G Stokes (1819-1903)
Continuum viscous Stokes postulations on and Fourier postulations on (available)
Turb flow model
Turb heat flux model
Note no mathematical model available to describe turbulent flow and heat transfer
τ ijqi
( )++minusminus=j
ij
ii
i
xxPg
DtDU
partτpart
partpartρρ
( )ρ τpartpart
partpart
CDTDt
Ux
DPDt
qxp ij
i
j
i
i= + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-3
19~20 Century19~20 Century
O Reynolds (1842-1912) T von Karman (1881-1963)L Prandtl (1875-1953) G I Taylor (1880-1975)
Continuum averaging viscous turbulent postulations
Note Development of turbulence model
( )
( )ρ ρpartpart
partτpart
partτpart
DUDt g
Px x x
ii
i
ij
j
tij
j= minus + + + +
( )
( )ρ τpartpart
partpart
partpart
CDTDt
Ux
DPDt
qx
qxp ij
i
j
i
i
ti
i= + + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model Postulations
Before we take on the postulation let us ask ifMomentum diffusion
(N-S)Heat diffusion
Mass diffusion
then shouldTurb M diffusion
Turb H diffusion
( )ρUi iij UGrad~τ
( )ρC Tp q GradTi ~
( )ρC M GradCi ~
( )ρu ui j
( )ρ θC up i
( )u u up
Grad u ui j i j+ρ
~ ~
u up
Grad ui iθθρ
θ+ ~ ~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Scale
Small Scale
UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena
Large Scale
Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy
Effective in transport phenomena
u t η υ τ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Modeling
단순화
亂 流
어지러울 난
模 型 化
흐를 류
TurbulenceTurbulence ModelingModeling++
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-1
15~16 15~16 CenturyCentury
L da Vinci (1452-1519) Visual and descriptive (available)
Mechanistic theory of nature + () Math modelNote no mathematical model available to describe flow motions
17~18 Century17~18 Century
I Newton (1643-1727) L Euler (1707-1783)D Bernoulli (1700-1782) J DrsquoAlembert (1717-1783)
Continuum inviscid Newtonrsquos law (available)
Viscous fluid model
Note no mathematical model available to describe viscous flows
( )+minusminus=i
ii
xPg
DtDU
partpartρρ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-2
1919thth CenturyCentury
L M H Navier (1785-1836) J B Fourier (1768-1830)B de Saint Venant (1797-1886) G G Stokes (1819-1903)
Continuum viscous Stokes postulations on and Fourier postulations on (available)
Turb flow model
Turb heat flux model
Note no mathematical model available to describe turbulent flow and heat transfer
τ ijqi
( )++minusminus=j
ij
ii
i
xxPg
DtDU
partτpart
partpartρρ
( )ρ τpartpart
partpart
CDTDt
Ux
DPDt
qxp ij
i
j
i
i= + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-3
19~20 Century19~20 Century
O Reynolds (1842-1912) T von Karman (1881-1963)L Prandtl (1875-1953) G I Taylor (1880-1975)
Continuum averaging viscous turbulent postulations
Note Development of turbulence model
( )
( )ρ ρpartpart
partτpart
partτpart
DUDt g
Px x x
ii
i
ij
j
tij
j= minus + + + +
( )
( )ρ τpartpart
partpart
partpart
CDTDt
Ux
DPDt
qx
qxp ij
i
j
i
i
ti
i= + + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model Postulations
Before we take on the postulation let us ask ifMomentum diffusion
(N-S)Heat diffusion
Mass diffusion
then shouldTurb M diffusion
Turb H diffusion
( )ρUi iij UGrad~τ
( )ρC Tp q GradTi ~
( )ρC M GradCi ~
( )ρu ui j
( )ρ θC up i
( )u u up
Grad u ui j i j+ρ
~ ~
u up
Grad ui iθθρ
θ+ ~ ~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Scale
Small Scale
UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena
Large Scale
Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy
Effective in transport phenomena
u t η υ τ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-1
15~16 15~16 CenturyCentury
L da Vinci (1452-1519) Visual and descriptive (available)
Mechanistic theory of nature + () Math modelNote no mathematical model available to describe flow motions
17~18 Century17~18 Century
I Newton (1643-1727) L Euler (1707-1783)D Bernoulli (1700-1782) J DrsquoAlembert (1717-1783)
Continuum inviscid Newtonrsquos law (available)
Viscous fluid model
Note no mathematical model available to describe viscous flows
( )+minusminus=i
ii
xPg
DtDU
partpartρρ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-2
1919thth CenturyCentury
L M H Navier (1785-1836) J B Fourier (1768-1830)B de Saint Venant (1797-1886) G G Stokes (1819-1903)
Continuum viscous Stokes postulations on and Fourier postulations on (available)
Turb flow model
Turb heat flux model
Note no mathematical model available to describe turbulent flow and heat transfer
τ ijqi
( )++minusminus=j
ij
ii
i
xxPg
DtDU
partτpart
partpartρρ
( )ρ τpartpart
partpart
CDTDt
Ux
DPDt
qxp ij
i
j
i
i= + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-3
19~20 Century19~20 Century
O Reynolds (1842-1912) T von Karman (1881-1963)L Prandtl (1875-1953) G I Taylor (1880-1975)
Continuum averaging viscous turbulent postulations
Note Development of turbulence model
( )
( )ρ ρpartpart
partτpart
partτpart
DUDt g
Px x x
ii
i
ij
j
tij
j= minus + + + +
( )
( )ρ τpartpart
partpart
partpart
CDTDt
Ux
DPDt
qx
qxp ij
i
j
i
i
ti
i= + + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model Postulations
Before we take on the postulation let us ask ifMomentum diffusion
(N-S)Heat diffusion
Mass diffusion
then shouldTurb M diffusion
Turb H diffusion
( )ρUi iij UGrad~τ
( )ρC Tp q GradTi ~
( )ρC M GradCi ~
( )ρu ui j
( )ρ θC up i
( )u u up
Grad u ui j i j+ρ
~ ~
u up
Grad ui iθθρ
θ+ ~ ~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Scale
Small Scale
UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena
Large Scale
Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy
Effective in transport phenomena
u t η υ τ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-2
1919thth CenturyCentury
L M H Navier (1785-1836) J B Fourier (1768-1830)B de Saint Venant (1797-1886) G G Stokes (1819-1903)
Continuum viscous Stokes postulations on and Fourier postulations on (available)
Turb flow model
Turb heat flux model
Note no mathematical model available to describe turbulent flow and heat transfer
τ ijqi
( )++minusminus=j
ij
ii
i
xxPg
DtDU
partτpart
partpartρρ
( )ρ τpartpart
partpart
CDTDt
Ux
DPDt
qxp ij
i
j
i
i= + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-3
19~20 Century19~20 Century
O Reynolds (1842-1912) T von Karman (1881-1963)L Prandtl (1875-1953) G I Taylor (1880-1975)
Continuum averaging viscous turbulent postulations
Note Development of turbulence model
( )
( )ρ ρpartpart
partτpart
partτpart
DUDt g
Px x x
ii
i
ij
j
tij
j= minus + + + +
( )
( )ρ τpartpart
partpart
partpart
CDTDt
Ux
DPDt
qx
qxp ij
i
j
i
i
ti
i= + + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model Postulations
Before we take on the postulation let us ask ifMomentum diffusion
(N-S)Heat diffusion
Mass diffusion
then shouldTurb M diffusion
Turb H diffusion
( )ρUi iij UGrad~τ
( )ρC Tp q GradTi ~
( )ρC M GradCi ~
( )ρu ui j
( )ρ θC up i
( )u u up
Grad u ui j i j+ρ
~ ~
u up
Grad ui iθθρ
θ+ ~ ~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Scale
Small Scale
UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena
Large Scale
Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy
Effective in transport phenomena
u t η υ τ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Historical Development of Turbulence Modeling-3
19~20 Century19~20 Century
O Reynolds (1842-1912) T von Karman (1881-1963)L Prandtl (1875-1953) G I Taylor (1880-1975)
Continuum averaging viscous turbulent postulations
Note Development of turbulence model
( )
( )ρ ρpartpart
partτpart
partτpart
DUDt g
Px x x
ii
i
ij
j
tij
j= minus + + + +
( )
( )ρ τpartpart
partpart
partpart
CDTDt
Ux
DPDt
qx
qxp ij
i
j
i
i
ti
i= + + + +
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model Postulations
Before we take on the postulation let us ask ifMomentum diffusion
(N-S)Heat diffusion
Mass diffusion
then shouldTurb M diffusion
Turb H diffusion
( )ρUi iij UGrad~τ
( )ρC Tp q GradTi ~
( )ρC M GradCi ~
( )ρu ui j
( )ρ θC up i
( )u u up
Grad u ui j i j+ρ
~ ~
u up
Grad ui iθθρ
θ+ ~ ~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Scale
Small Scale
UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena
Large Scale
Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy
Effective in transport phenomena
u t η υ τ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model Postulations
Before we take on the postulation let us ask ifMomentum diffusion
(N-S)Heat diffusion
Mass diffusion
then shouldTurb M diffusion
Turb H diffusion
( )ρUi iij UGrad~τ
( )ρC Tp q GradTi ~
( )ρC M GradCi ~
( )ρu ui j
( )ρ θC up i
( )u u up
Grad u ui j i j+ρ
~ ~
u up
Grad ui iθθρ
θ+ ~ ~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Scale
Small Scale
UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena
Large Scale
Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy
Effective in transport phenomena
u t η υ τ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Scale
Small Scale
UniversalVery short lifetimeIsotropicMost dissipation of energyIneffective in transport phenomena
Large Scale
Largely depend on geometric bc Long lifetimeDirectionalMost turbulent energy
Effective in transport phenomena
u t η υ τ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Turbulence Model
Definition of an Ideal Turbulence Model
ldquoAn ideal model should introduce the minimum amount of complexitywhile the essence of the relevant physicsrdquo
How Complex does a Turbulence Model have to be
ldquoOnce the question of how detail we need is answered the level of complexity of the model follows qualitatively speakingrdquo
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Inviscid Estimate for Dissipation Rate
Rate of energy supply (=Production rate)
Production rate = Dissipation rate
Viscous dissipation of energy can be estimated from the large-scale dynamics which do not involve viscosity
32 υυυ =asymp
3~ υε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Fundamental Equations
Navier Stokes equation
Continuity equation Time averaging
Reynolds stresses
jj
i
ij
ij
i
xxU
xP
xUU
tU
partpartpart
ρμ
partpart
ρpartpart
partpart 21
+minus=+
0=i
i
xU
partpart
( ) iii
t
t ii uUUdttUtt
U +equivminus
equiv int 1 1
001
u ui j
minus
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
ρu u u u uu u u u uu u u u u
12
1 2 1 3
2 1 22
2 3
3 1 3 2 32
⎟⎟⎠
⎞⎜⎜⎝
⎛minus+minus=+
=
jij
i
jij
ij
i
i
i
uuxU
xxP
xUU
tU
xU
ρpartpartμ
partpart
ρpartpart
ρpartpart
partpart
partpart
11
0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation forReynolds Stresses
( )Uu ux x
u u uu px
u px
u uUx
u u Ux
p ux
ux
ki j
kConvection
kk i j
Diffusion
j
i
i
j
Diffusion
i kj
kj k
i
kGeneration
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
1 2
1
⎠⎟
+ minus
minusPressure strain
i j
k kViscous diffusion
i
k
j
kViscous dissipation
u ux x
ux
ux
νpartpart part
ν partpart
partpart
2
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Kinetic Energy
U kx x
uu u p u u U
x
kx x
ux
ux
ii
Convectioni
ii j
Turbulent diffusion
i ji
j
oduction
i iMolecular diffusion
j
i
j
iDissipation
partpart
partpart ρ
partpart
ν partpart part
νpartpart
partpart
= minus +⎛⎝⎜
⎞⎠⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪minus
+ minus
2
2
Pr
( )k u u u= + +12 1
222
32
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Transport Equation of Turbulent Dissipation Rate
ndissipatioViscous
lk
i
Diffusion
l
i
lij
ik
k
Generation
l
k
l
i
k
i
Generation
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
Convection
ii
xxu
xu
xp
xxuu
xxu
xu
xu
xuu
xxU
xu
xu
xu
xu
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε ν partpart
partpart
=ux
ux
i
l
i
l
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time averaged model
Integral methodMixing length modelOne-equation model (Energy equation model)Two-equation model--- k-ε model k- model
k-kl model k- modelAlgebraic stress modelReynolds stress model
Structural model
Large eddy simulation
Turbulence Models
ω
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Time Averaged Models
Models Moment Eq L k μ t u ui j Mixing length pde asm --- ale --- One-equation pde asm pde ale --- Two-equation pde pde pde ale ---
Algebraic stress pde pde or asm pde --- pde Reynolds stress pde pde or asm pde --- pde
pde partial differential equation ale algebraic equation asm assumption
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-1
Characterization of local state of turbulence by only few parameters
V = velocity scale (intensity of fluctuation)L = length scale (size of turbulence elements)
(or alternatively LV = time scale)
Task of turbulence model
1) Relate and to the parameters chosen2) Determine the variation of the parameters over the flow
uui j uiϕ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-2
Boussinesq eddy viscositydiffusivity conceptFor general flows
- These quantities are not fluid properties but depend strongly on the state of turbulence
- Main problem is determination of and - Most models employ Reynolds analogy between heatmass transfer and
momentum transfer
ydiffusiviteddyorturbulentityviseddyorturbulent
t
t
)(cos)(
=Γ
=ν
νt Γ t
Γtt
t
t turbulent prandtl or Schmidt number
=
=
νσ
σ
minus = +⎛
⎝⎜
⎞
⎠⎟ minus
minus =
u u Ux
Ux
k
ux
i j ti
j
j
ii j
i ti
ν partpart
partpart
δ
ϕ partφpart
23
Γ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Basic Concepts in Turbulence Models-3
From dimensional analysis
Flow region exist where the shear stress and the velocity gradient have opposite sign -- would have to be negative
-- Eddy viscositydiffusivity concept breaks down-- Such region are important in geophysical flows but usually not in engineering
flowsIsotropic and not always realistic Algebraic stress or anisotropic model for introducing directional influence on turbulence
( )τν 2ˆˆ VorLVt prop
υρ uminus part partU yνt
νt Γtεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-1
In 1925 Prandtlrsquos proposal
This yields
V L Uy
=partpart
ν partpartt mUy
= 2
κ λκ λ= = rarr= = rarr
0435 009041 0085
Patankar and SpaldingCrawford and Kays
λ κy δ
m
m = λδm y= κ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-2
Mixing length has to be prescribed empiricallyVery close to the wall
Cebeci-Smith model Uses in outer layer
For general shear layers
( )[ ]m y y A= minus minus + +κ 1 exp
( )ν α αt U U dy= minus =infininfinint0 0 0168
( )ν α δ δt U U U dy= prop = minus infininfinint 10
m bprop
bκ =
=+
+
von Karman constantvon Driests damping factor
function ofA
A dp dx~
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-3
Baldwin-Lomax modelInner layerOuter layer
Here Fmax is the maximum value of
and ymax is the y value at that time
μ ρ ω ωti l vorticity= 2
( )μ ρκto cp wake klebC F F y=
( ) ( )[ ]F y y y A= minus minus + +ω 1 exp
[ ]F y F C y U Fwake wake dif= min max max max max2
( ) ( )[ ]F y C y ykleb kleb= +minus
1 55 6 1 max
Klebanoffrsquos intermittency function
κ = = = =0 0168 1 6 0 25 0 3 C C Ccp wake kleb
Ddif = difference between the max and min values
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-4(Effects of Buoyancy and Rotation in MLM)
Buoyancy effect is characterized by the gradient Richardson number
For (Monin-Oboukhov relation)
For (KEYPS formula)
Bradshaw(1969) - demonstrate the close analogy between buoyancy and curvature effects on turbulence
(=ratio of centrifugal to inertial forces)
( )R g P y
U yi = minus
ρpart partpart part 2
Ri gt 0m
miR
0
1 5 101 1= minus =β β ~
( )m
miR
0
1 1421 4
2= minus congminusβ β
R U RU nis c
s=part part
β1 6 14= ~
Ri lt 0
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-5 (Discussion on Mixing Length Model)
Lack of universality of the empirical input
Based on implied assumption that turbulence is in a local state of equilibrium (production = dissipation)
Examples1) grid turbulence
model yields
2) Channel flow model yields
at symmetry plane where
-- The model is not very suitable when convective and diffusive transport and history effects are important
In complex flows is difficult to prescribe empirically
νt t= =Γ 0
part partU y = 0
m
k
νt t= =Γ 0tμ
from MLM
U
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Mixing Length Model-6 (Mixing Length Model Assessment)
AdvantagesEasy to implement and cheap in terms of computing resourcesGood predictions for thin shear layers jets mixing layers wakes and boundary layersWell established
DisadvantagesCompletely incapable of describing flows with separation and recirculationOnly calculates mean flow properties and turbulent shear stress
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -1(Energy Equation Model-1)
Transport and history effects are accounted for by transport equation for velocity scale V
Physically most meaningful scale is kinetic energy of the turbulent motion
k- equation at high Reynolds numbers
k where k uui i =12
ndissipatioviscous
j
i
j
i
ndestructioproductionbouyantG
ii
shearbyproductionP
j
iji
transportdiffusive
jji
i
transportconvective
ii
changeofrate
xu
xu
ug
xU
uupuuu
xxkU
tk
==
=
minusminus
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=+
ε
partpart
partpart
νϕβ
partpart
ρpartpart
partpart
partpart
partpart
2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -2 (Energy Equation Model-2)
Model approximations have to be introduces for unknown correlations in diffusion and dissipation terms
With eddy viscositydiffusivity relations for and the k- equation reads
Eddy viscosity formBradshaw et al(1967) converted k- equation into -equation by assuming
or boundary layers-- Eddy viscosity concept not used-- Diffusion flux of bulk velocity
Length scale L needs to be prescribed empirically
minus +⎛⎝⎜
⎞⎠⎟ = =u
u u p kx
C kLi
j j t
k iD2
3 2
ρνσ
partpart
ε
u ui j uiϕpartpart
partpart
partpart
νσ
partpart
ν partpart
partpart
β νσ
partφpart
ε
ktU k
x
xkx
Ux
Ux
gx
C kL
ii
i
t
k it
i
j
j
i
P
it
k iG
D
+
=⎛⎝⎜
⎞⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟ + minus
3 2
ν μt c kL= primeuv
uv kprop( )uv k= 03
k prop
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
One-Equation Model -3(S-A (Spalart-Allmaras) model)
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-1
The dependent variable of the length-scale-determining equation must not be the length scale L itself
1 Diffusion2 Source interaction with mean motion3 Sink self interaction
Additional diffusion usually Additional source or sink
for k-ε model
eg)
model
1011-105-115
Z k La b=
ZZttZ
t SDZk
CyU
kZC
yZ
yDtDZ prime+prime+minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
3
2
2
2
1
εpartpartν
partpart
σν
partpart
primeDZ
primeSZprime =SZ 0 prime =DZ 0
y
x
u
a bεminusk
kk minusωminusk
minusk
kC
yU
kC
yyDtD
ttt
2
2
2ε
partpartνε
partεpart
σν
partpartε
ε
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
2
ωpartpartνω
partωpart
σν
partpartω
ω
CyU
kC
yyDtD
ttt minus⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= )( ωε k=lArr
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Second Equation-2(ε-Equation)
- equation
-- 주류에 의한 생성항은 high Re수에서 다른 항에 비해 매우 작다
--확산항 구배 확산 가정
minus minus⎛
⎝⎜⎜
⎞
⎠⎟⎟ = minus minus2 2
22
1 2
2
νpartpart
partpart
partpart
νpartpart part
ε partpart
ελε ε
Ux
Ux
Ux
Ux x
Ckuu U
xC
kl
k l
k
l
i
k li j
i
j
⎟⎟⎠
⎞⎜⎜⎝
⎛=minus⎟
⎟⎠
⎞⎜⎜⎝
⎛minus
i
t
il
i
lij
ik
k xxxU
xP
xxU
Ux part
εpartσν
partpart
partpart
partpart
partpart
ρν
partpart
partpartν
ε
2
ndestructioviscous
lk
i
transportdiffusive
l
i
lij
ik
k
stretchingvortextodueproduction
l
k
l
i
k
i
production
l
ik
lk
i
k
l
i
l
l
k
l
i
k
i
transportconvective
ii
xxU
xU
xP
xxUU
xxU
xU
xU
xUU
xxU
xU
xU
xU
xU
xU
xU
22
2
2
22
22
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus⎟
⎟⎠
⎞⎜⎜⎝
⎛minusminus
minus⎟⎟⎠
⎞⎜⎜⎝
⎛+minus=
partpartpartν
partpart
partpart
partpart
ρν
partpart
partpartν
partpart
partpart
partpartν
partpart
partpartpartν
partpart
partpart
partpart
partpart
partpartν
partεpart
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-1(Standard k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
Cμ C1ε C2ε C3ε kσ
σε σt
009 144 192 0-02 1when Glt0 when Ggt0
1 13 05-07 09free shear near-wall
layers flows
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-2(RNG k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
30
3
22 1)1(
βηηηη
μεε +minus
+rArr CCC
εη kSequiv 21)
21( ijij SSS equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-3(Realizable k-ε Model)
( )
εν
εεpartεpart
σν
partpart
partεpart
partεpart
εpartφpart
σνβ
partpart
partpart
partpartν
partpart
σν
partpart
partpart
partpart
μ
εεεε
2
2
231
kC
kCGCP
kC
xxxU
t
xg
xU
xU
xU
xk
xxkU
tk
t
Pi
t
iii
G
it
ti
P
j
i
i
j
j
it
ik
t
iii
=
minus++⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
minusε
ε
μ kUAAC
s()
0
1
+rArr
ijijijij WWSSU minusequiv()
As = 6cosφ
φ = acos ( 6W)SijS jkSkiW = -------------( 2S)3
A0 = 40
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-4(Wilcoxrsquos Model)
ων
βωωαpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ω
k
Pkxxx
Ut
kxU
xU
xU
xk
xxkU
tk
t
i
t
iii
j
i
i
j
j
it
ik
t
iii
=
minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
2
kωβε =
ωminusk
ωβ
k=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-5(Shear-Stress Transport Model)
)max(
11)1(2
21
1
21
2
Faka
xxkFP
kxxxU
t
kxU
xU
xU
xk
xxkU
tk
t
iii
t
iii
j
i
i
j
j
it
ik
t
iii
Ω=
partpart
partpart
minus+minus+⎟⎟⎠
⎞⎜⎜⎝
⎛=+
minus⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛=+
ων
ωωσ
βωωγpartωpart
σν
partpart
partωpart
partωpart
ωβpartpart
partpart
partpart
νpartpart
σν
partpart
partpart
partpart
ωε
εω φφφ minusminus minus+= kk FFconModel )1( 11
]4)500090
min[max(arg)tanh(arg 221411 yCD
kyy
kwhereFkw ωωσ
ρω
νω
==
)5000902max(arg)tanh(arg 22
222 yy
kwhereFω
νω
==
ωminusk
Hybrid k-ε Model Model+ ωminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Wall functionWall function
Bridging of viscous sublayer by
Assume Local equilibriumUniversal logarithmic laws
Resulting
PPP
PP
yu
CukEuy
uU
κε
νκτ
μ
ττ
τ
32
ln1==⎟
⎠⎞
⎜⎝⎛=
( )PP
P
yu
Cuk
κε τ
μτ
3
2
=
=
Two-Equation Models-6(Near-Wall Treatment ndash Wall Function Method(1)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
ProcedureProcedure⑴ Solve k - equation up to yP
neglecting wall-ward diffusion⑵ Get ⑶ Use universal logarithmic laws
Merits Easy to use Small mesh number
Weakness Questionable for complex flow must begt12 for all region
Two-Equation Models-7(Near-Wall Treatment ndash Wall Function Method(2)
k U yP Pminus minus +
( )( )Pr+++
+++
=
=
PP
PP
yTT
yUU
yP+
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
TwoTwo--layer methodlayer method
Away from the wall --- k-ε modelNear-wall viscosity affected layer --- a simpler model involving a length scale prescription
-- Iacovides and Launder (1987) Mixing length model
-- Chen and Patel (1987) Yap (1987) Energy equation model of Wolfstein
-- Rodi (1988) Norris-Reynoldsrsquo energy equation model
Two-Equation Models-8(Near-Wall Treatment ndash Two-Layer Method)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-9 (Low-Reynolds k-ε Model-1)
tt kTfC μμν =
DxUuu
xk
xDtDk
j
iji
jk
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
= εσνν ~
ET
fCxUuu
TfC
xxDtD
tj
iji
tj
t
j
+minuspartpart
minus⎥⎥⎦
⎤
⎢⎢⎣
⎡
partpart
⎟⎟⎠
⎞⎜⎜⎝
⎛+
partpart
=εε
σννε
εεε
~12211
2
2~⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
minus=minus=jxkD νεεε
( ) ( )1993amp 21 ShihYangbykexceptkTt ενεε +=
where
etcRRyRoffunctionsf yt εμ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
( )ν
νενννε ε
τ yRykRyuykR yt
412
equivequivequivequiv +
2 etcRRyRoffunctionsf yt ε+
011 etcRRoffunctionsorf yt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-10(Low-Reynolds k-ε Model-2)
Before the review paper by Patel et alBefore the review paper by Patel et al (1985)Jones amp Launder (1972) Int J Heat Mass TransferLaunder amp Sharma (1974) Lett Heat Mass Transfer
After the review paper by Patel et alAfter the review paper by Patel et al (1985)Nagano amp Hishida (1987) J Fluid Eng
Myong amp Kasagi (1990) JSME Int JNagano amp Tagawa (1990) J Fluid EngSpeziale et al (1990) model AIAA paperWilcox (1994) model AIAA JAbe et al (1994) Int J Heat Mass Transfer
( ) ( ) ( ) ( )2033 ykyyuyt ΟpropΟpropΟpropminusΟprop ευν
( )kk εω equivminus
( )( )ννεε 21 yRy equivrarr+
( )ετ kk equivminus
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-11(Low-Reynolds k-ε Model-3)
After DNS data by Kim et alAfter DNS data by Kim et al (1987)Shih amp Mansour (1990) Eng Turb Modelling and ExpSo et al (1991) AIAA JKawamura amp Hada (1992) pressure termNagano amp Shimada (1993) pressure term 9th TSFRodi amp Mansour (1993) JFMintroduce a modelled and the effect of intothe sink termYang amp Shih (1993) NASA TM
Sakar (1995) MS Thesis Arizona State Univ
Use
( )( )ννεε 21 yRy equivrarr+
3εP εkP
( ) ⎟⎠⎞
⎜⎝⎛equiv= jiijt SSkSwhereSRfuncf
εμ
+⎟⎟⎠
⎞⎜⎜⎝
⎛
partpartpart
=+=2
2
33jk
it xx
UCEP ννεε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Two-Equation Models-12(k-ε Model Assessment)
AdvantagesSimplest turbulence model for which only initial andor boundary conditions need to be suppliedExcellent performance for many industrially relevant flowsWell established the most widely validated turbulence model
DisadvantagesMore expensive to implement than mixing length model (two extra PDEs)Poor performance in a variety of important cases such as(i) Some unconfined flows(ii) Flows with large extra strains (eg curved bl swirling flows)(iii) Rotating flows(iv) Fully developed flows in non-circular duct
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -1
Launder et al(1977)
-- It works quite well for flow along curved walls but it exaggerates the curvature effects on free shear layers and produces a correction in the wrong direction in swirling jets
Hanjalic and Launder (1980)
-- The modification is beneficial in retarded boundary layers but acts in the wrong direction for curved shear layers
Leschziner-Rodi (1981)
( ) ( )C C R where R k Ur
Urr
Cc it it cε εpartpart2
2
2 21 0 25rarr minus = cong
( )P C uv Uy
C u v Ux k
C CrArr minus + minus⎡
⎣⎢
⎤
⎦⎥ ltε ε ε ε
partpart
partpart
ε1 3
2 21 3
Ck U
nUR
UR
s s
c
s
c
μ
εpartpart
=+ +
⎛⎝⎜
⎞⎠⎟
0 09
1 0 572
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -2
Leschziner and Rodi (1981)
-- The modification yields a reduction of eddy viscosity in curved shear layers bordering recirculation zonesHanjalic and Launder (1981)
--- The modification works well when the extra rate of strain is large but the model does not produce sufficient response to curvature when the curvature effects are relatively smallMyong (1996)
( )PkC P C Sk t nsε ε ε
ε ν= prime minus primeprime1 1
P Ck
Uxti
jε ε
ε ν partpart
=⎛
⎝⎜
⎞
⎠⎟1
2
( ) ( )C C M where M k S Cc ft ft ij ij cε ε2
2
22 21 0 20rarr minus = minus congΩ
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
=i
j
j
iij
i
j
j
iij x
UxU
xU
xUS
21
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Modification for Curvature -3
Use of Invariant TermsUse of Invariant Terms
Rodi (1972)
Cotton et al (1992)
where are constants or functions of Yakhot et al (1992) additional term in eq
Kato amp Launder (1993)
( )εμ kPfuncC =
jiij SSkSwithS
Cεβ
αμ equiv
+= 21
βα εkPε
( )kS
SSSCR
2
30
3
11 ε
βμ
+
minusminus=
jiijkkwithSfCSfCP ΩΩequivΩΩrarr=ε
εε μμμμ2
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Additive term in -equation
Original Launder amp Sharma model
Myong model(1989)
(Example) In 2-D riblet flow
(Ref) Additive term in k-equation in 2-D riblet flow
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
it xx
UPpartpart
partννε
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
22
3 2nm
imnt xx
UPpartpart
partδννε
222
2
22
2
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛yx
WyW
xW
xxU
nm
i
partpartpart
partpart
partpart
partpartpart
δpart
part partpartpart
partpartmn
i
m n
Ux x
Wx
Wy
2 2 2
2
2 2
2
2⎛
⎝⎜
⎞
⎠⎟ =
⎛
⎝⎜
⎞
⎠⎟ +
⎛
⎝⎜
⎞
⎠⎟
( )2 21 2 22 2
ν part part νpartpart
partpart
k x kx
kyj
=⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
ε
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-1
SpezialeSpeziale et al (1987)et al (1987)
where
Test-- high-Re secondary flow in a square duct and a pipe U-bend backward facing flow
εδ
δδ23
2
2
31
31
32
kCwithDDC
DDDDCDkkuu
ijmmijE
ijmnmnmjimDijijji
=⎟⎠⎞
⎜⎝⎛ minusminus
⎟⎠⎞
⎜⎝⎛ minusminusminus=
ki
k
jkj
k
iijij D
xU
DxU
DtDDD
part
partminus
partpart
minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
==i
j
j
iijij
xU
xUSD
21
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-2
NishizimaNishizima amp amp YoshizawaYoshizawa (1987)(1987)
where
Test-- Low-Re channel flow Couette flow secondary flow in a square duct
⎟⎠⎞
⎜⎝⎛ minus+minus= sum
=ijijijtijji SSCkSkuu δ
ενδ βααβ
ββ 3
1232 3
12
3
21 21 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
partpart
part
part+
part
part
part
part=
part
part
partpart
=γ
γ
γ
γ
γγ xU
xU
xU
xU
SxU
xUS i
j
j
iij
jiij
jiij x
Ux
US
part
part
part
part= γγ
3
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-3
Myong amp Myong amp KasagiKasagi (1990)(1990)
where
Test--- Low-Re secondary flow in a square ductrotating channel flow bl flow channel
( )
( )mnWx
kk
kCSSkC
SSSSkCSkuu
ijn
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
32
31
312
32
2
32
1
⎟⎟⎠
⎞⎜⎜⎝
⎛
partpart
+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
εν
δεν
εν
δεννδ
kijki
j
j
iij
i
j
j
iij x
UxU
xU
xUS Ωminus⎟
⎟⎠
⎞⎜⎜⎝
⎛
part
partminus
partpart
=Ω⎟⎟⎠
⎞⎜⎜⎝
⎛
part
part+
partpart
= ε21
21
jmimjninijijW δδδδδ 4+minusminus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Non-linear k-ε Model-4
Craft et al (1996 1997)Craft et al (1996 1997)
where
Test-- pipe flow impinging jet flow around turbine blade transitional flow
( )
( )
kkijt
kkijt
ijnmnmmjmimjmit
kikjjkit
ijkkkjikt
kijkkjikt
ijkkkjikt
ijtijji
SkCSSSkC
SSSkC
SSSkC
kCSSkC
SSSSkCSkuu
ΩΩ++
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+ΩΩ+
Ω+Ω+
⎟⎠⎞
⎜⎝⎛ ΩΩminusΩΩ+Ω+Ω+
⎟⎠⎞
⎜⎝⎛ minus+minus=
2
2
72
2
6
2
2
5
2
2
4
32
1
32
31
312
32
εν
εν
δενεν
δεν
εν
δεννδ
( )Ω= SfuncCμ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-1
Reynolds stress equations
( )Uu ux x
u u uu Px
u Px
u uUx
u u Ux
P ux
ux
ki j
k kk i j
j
i
i
j
i kj
kj k
i
k
i
j
j
i
partpart
partpart ρ
partpart
partpart
partpart
partpart ρ
partpart
partpart
대류항 확산항 확산항
응력 생산항 압력
= minus minus +⎛
⎝⎜⎜
⎞
⎠⎟⎟
minus minus + +⎛
⎝⎜
⎞
⎠⎟
1 2
1
minus
+ minus
변형률 상관항
점성확산항 점성소산항
νpartpart part
ν partpart
partpart
2
2u ux x
ux
ux
i j
k k
i
j
j
i
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Reynolds Stress Equation Model-2(Reynolds Stress Equation Model Assessment)
AdvantagesPotentially the most general of all classical turbulence model Only initial andor boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets asymmetric channel and non-circular duct flows and curved flows
DisadvantagesVery large computing costs (seven extra PDEs)Not as widely validated as the mixing length and modelsPerforms just as poorly as the model in some flows owing to identical problem with the ε-equation modeling
εminuskεminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-1
Replacement of simple isotropic eddy viscositydiffusivity relation by more general algebraic stress relationDerivation of these by simplifying - model transport equationModel approximations must be introduced for conservative and diffusive transport term eg
Incorporation into model - equation
P and G are stress and buoyant production of k
u ui j
( ) ( )
( )ϕϕβ
partpart
partpart
ε
εδ
εεδ
εδ
ijjiji
l
jlj
l
jliji
jiji
jiji
jiji
ugugG
xU
uuxU
uuPwhere
GPC
GCCPP
Ckuu
+minus=
minusminus=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
minus+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛minusminus+⎟⎟
⎠
⎞⎜⎜⎝
⎛minusminus
+=1
321
321
32
1
32
u ui j
( ) ( ) ( )εminus+=⎟⎟⎠
⎞⎜⎜⎝
⎛minus=minus GP
kuu
kDifftDkD
kuu
uuDifftDuuD jiji
jiji
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Algebraic Stress Model-2(Algebraic Stress Model Assessment)
AdvantagesCheap method to account for Reynolds stress anisotropyPotentially combines the generality of approach of the RSM with the economy of the modelSuccessfully applied to isothermal and buoyant thin shear layersIf convection and diffusion terms are negligible the ASM performs as well as the RSMOnly slightly more expensive than the model (two PDEs and a system of algebraic equations)
DisadvantagesNot as widely validated as the mixing length and modelsSame as disadvantages as RSM applyModel is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply ndash validation is necessary to define the performance limits
εminusk
εminusk
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Explicit Algebraic Stress Model
Pope (1975)Generalize the non-linear constitute relation using the Cayley-Hamilton theorem
Taulbee (1992) and Gatski amp Speziale (1993)Propose elaborate coefficients for the 3-D flow form of Pope
Test-- homogeneous flows high Re flows
Abe et al (1997)Propose a low-Re nonlinear model with satisfying realizability
Test--- homogeneous flows channel flow backward facing step flow
εminusk
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-1(Concept of LES)
Only the influence of the small eddies has to be modeled by a subgrid modelwhereas the large energy-carrying eddies are computed directly
Small eddies are more universal random homogeneous and isotropic which simplifies the development of appropriate models
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-2(LES and DNS)
Schematic representation of turbulent motion
Time dependence of a velocity component
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-3(Numerical and Modeling Aspects in LES)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-4(Filtering Operation)
Any dependent variable f of the flow is split into a GS(grid scale) part and a SGS(subgrid scale) part
A filtered (or resolved or large-scale) variable denoted by an overbar is defined as
If G is a function of x-xrsquo only differentiation and the filtering operation commute
int Δ=D
xdxxGxfxf )()()(
)()()( txftxftxf +=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-5 (Typical Filter Functions)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-6(Governing Equations for Filtered Variable)
0xU
i
i =partpart
j
ij
jj
i2
iji
j
i
xxxU
xP1UU
xtU
partτpart
partpartpart
ρμ
partpart
ρpartpart
partpart
minus+minus=+ )(
jijiij UUUU minus=τ
Non-resolvable subgrid scale Reynolds stresses describe the influence of the small-scale structures on the larger eddies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-7(Subgrid Scale (SGS) Models-1)
ijijtij kS δντ322 +minus=
Boussinesqrsquos concept
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
part
part+
partpart
=i
j
j
iij x
UxU
21S
iik τ21equiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Smagorinsky model (1963)
ij2
St SC Δ=ν
ijijij SSS = ( ) 31
zyx ΔΔΔ=Δ
)( ijij2
k SSCk Δ=
23ijij
223k
23
SSCCkC
)(Δ=Δ
= εεε
20202C 2k ==π
501Cwith0420C232C K
23K2S )( asympasymp= minus
π
S23
k CC2C minus=ε
LES(Large Eddy Simulation)-8(Subgrid Scale (SGS) Models-2)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-9(Problem of SGS model near a Wall)
sC reduces from 004 to 0004 near a wall
)( ++minusminus= Ay0ss e1CC
The SGS model should depend solely on the local properties of the flow and it is difficult to see how the distance from the wall qualifies in this regard
Near a wall the flow structure is very anisotropic( ) 3
1zyx ΔΔΔ=Δ rarr ( ) 2
1222 zyx Δ+Δ+Δ=Δ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-10(Grid Parameter R)
de
eRminusΔminus
=
)( ε23e kscaleenergeticthe=
413d scalendissipatiothe )( εν=
1R0 ltlt
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
LES(Large Eddy Simulation)-11(Dynamic SGS Model)
Grid filter with Test filter with
Δ
Δ
GG
jijiij UUUU minus=τ
jijiij UUUUT minus=
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Principal Challenges of Turbulence Predictions
(I) Growth and separation of the boundary layer
(II) Momentum transfer after separation
Challenge (I)
- Include the prediction of skin friction and boundary layer thickness (which dictates the skin drag in the absence of separation) along with separation (which creates pressure drag)
- It is simpler but makes very high accuracy demands and appears to give turbulence models of higher complexity little advantage
Challenge (II) - now the arena for complex RANS (Reynolds Averaged Navier-Stokes) models
LES and the new methods- time-dependent three-dimensional simulations
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Numerical Challenges
Progress in numerical methods and computers is intensifying the challenge for turbulence treatments to provide a useful level of accuracy in slightly or massively separated flows over fairly complex geometries at very high Reynolds numbersThe technical fields such as the airliner turbine engine and automobile industries are more than ready for this capabilityThe RANS models are useful for Challenge (I) since they are built and calibrated mainly in shear flows with shallow or no separation But it is unlikely that the RANS model even complex and costly will provide the accuracy needed in the variety of separated and vortical flows we need to predict aiming at Challenge (II) Away from boundaries LES is generally successful even in flows with free shear or separation given a sensible number of grid points A serious problem is that SGS-free LES cannot deal with the wall region of the boundary layer (except by approaching DNS)
- Resolution required to resolve the outer layer of a growing boundary layer is proportional to Re04 whereas for the viscous sublayer (which in aeronautical applications only accounts for approximately 1 of the boundary layer thickness) the number of points needed increases at least like Re18 (Chapman 1979)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-1
The idea is to entrust the whole boundary layer (populated with ldquoattachedrdquoeddies) to a RANS model and only separated regions (ldquodetachedrdquo eddies) to LES (Spalart et al 1997)It is aimed primarily at external flowsIt is consistent with the two positions that Challenge (I) is a reasonable one for RANS models whereas Challenge (II) is not and that LES is well understood away from the walls
- For the boundary layer the user directs the model to operate in that mode by creating a ldquoRANS gridrdquo with a large spacing parallel to the wall compared with the boundary layer thickness
- In the separated regions good accuracy is expected once the grid spacing in all directions is far smaller than the size of the region DES offers RANS in the boundary layers and LES after massive separation within a single formulation it is not zonal
δgtgtΔ||
LltltΔ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES (Detached Eddy Simulation)-2
A typical application of DES is to a wing with a spoiler or a landing gear- Large areas of boundary layers are treated efficiently with quasi-steady RANS- The model accounts for all the turbulent stresses across the whole boundary
layer just like in a simple RANS run- Behind the spoiler the momentum transfer is dominated by large unsteady
eddies which are where LES is most desirable since they are not as numerous as the ldquohorseshoerdquo vortices in the outer part of a boundary layer and geometry-specificAn additional benefit of DES is its unsteady information
- Though useless for many purposes such as the range of airplane it will sooner or later be of great use for structural or noise studies
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
S-A (Spalart-Allmaras) model
2)(2 jijiijijtji xUxUSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21
~~~~1~~~⎥⎦⎤
⎢⎣⎡νminusνnabla+νnablaν+νsdotnabla
σ+ν=
νd
fccScDtD
wwbb
12222 1
1~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
17230)1(
41062203213550
13222
11
21
===σ++κ=
=κ==σ=
υcccccc
cc
wwbbw
bb
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Hybrid RANSLES (or DES) Method
- The S-A turbulence model contains a destruction term for its eddy viscosity which is proportional to where is the distance to the closest wall
- When balanced with the production term this term adjusts the eddy viscosity to scale with the local deformation rate and
- Now in LES the Smagorinsky model scales its SGS eddy viscosity with and the grid spacing
- Thus the S-A model with replaced by a length proportional to can be a SGS model
- If we now replace in the S-A destruction term with defined by
we have a single model that acts as S-A when and a SGS model when
ν~2)~( dν d
dS 2~ SdpropνS
Δ 2Δpropν SSGS
d Δ
d d~
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
ΔltltdΔgtgtd
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
DES model
2)(2jijiijijtji xuxuSwhereSuu partpart+partpartequivν=minus
νν
equivχ+χχ
=ν=νυ
υυ
~~
31
3
3
11 cfft
( )( ) ( )[ ]2
12
21 ~~~~~1~~~⎥⎦⎤
⎢⎣⎡minusnabla+nabla+sdotnabla+=d
fccScDtD
wwbbννννν
σνν
12222 1
1~~~
υυυ χ+
χminus=
κν
+equivf
ffd
SS
226
2
61
63
6
63 ~~
~)(1
dSrrrcrg
cgcgf w
w
ww κ
νequivminus+=⎥
⎦
⎤⎢⎣
⎡++
=
65017230)1(
41062203213550
13222
11
21
====σ++κ=
=κ==σ=
υ DESwwbbw
bb
Ccccccc
cc
)max()min(~ zyxwithCdd DES ΔΔΔequivΔΔequiv
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-1
The present application of DES in a ldquoLES gridrdquo with is unnatural but has the following motives
- The first is to predict the behavior of DES in the recovery of a thickened boundary layer such that comes to exceed
- The second is the fact that the formula of DES provide a viable SGS model on a LES grid with built-in wall modeling since its only adjustable constant to date was set in homogeneous turbulence (Shur et al 1999)To predict whether wall bounded turbulence would be sustained or how accurate the solution would be depending on the grid
Ref) Myong(2002)
δltltΔ||
||Δδ
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-2(Numerical Method)
Test for 4 cases (RANS model DES LES and no model) with same condition-- Grid no 65 x 65 x 33 (same as the case of Nikitin et al 2000) - Calculation domain 2 x 2 x (same as the case of Nikitin et al 2000) -
Calculation Code (extended version of Dr Choi)
Ref) Myong(2002)
j
ij
j
i
ji
i
xxu
xxP
DtDu
part
partminus
partpart
partpart
+partpart
minus=τ
Re1
0=partpart
i
i
xu
RANSforS
LESforS
ijtij
ijSGSij
νminus=τ
νminus=τ
)52000(Re2000Re asympν==τ HUmm
π π750)265(00037501010|| asympΔcongΔΔ=ΔcongΔ=Δ +
wwclwcl yyyyy
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-3(Skin friction coefficient)
Calculated error in skin friction coefficient ref
)2000Re( =Δ τatCf22)Re(log123750 minus= mfC
- 60 - 57 - 20 - 3
No modelLESDES (hybrid RANSLES)RANS
method fCΔ
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-4(Time History of Mean Shear Stresses)
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-5(Mean Velocity and Turbulent Viscosity Profiles)
y+
100 101 102 103 104
U+
0
10
20
30
40
50no modelLESRANSDES
U+ = lny+041 + 52
yH5 1500 10 20
ν t+
0
50
100
150
200
250
LESRANSDES
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Application of DES to Channel Flow-6(Shear Stresses)
RANS
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
DES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
LES
yH25 50 75000 100
τ+
2
4
6
8
00
10
viscousmodeledresolved
Ref) Myong(2002)
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Constraint to the Spread of CFD
In addition to a substantial investment outlay an organization needs qualified people
to run the codes and communicate their results and briefly consider the modeling skills required by CFD users
The next constraint to the further spread of CFD amongst the industrial community could be a scarcity of suitably trained personnel instead of availability andor cost of hardware and software
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator
CFD amp Turbulence Lab httpcfdkookminackr
School of Mechanical Engineering Copyright 2009 Hyon Kook MYONG
Problem Solving with CFD
In solving fluid flow problems we need to be aware that the underlying physics is complex andthe results generated by a CFD code areat best as good as the physics (and chemistry) embedded in it and at worstas good as its operator