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Modeling of Turbulent Viscosity Fluid property – often called laminar viscosity Flow property – turbulent viscosity MVM: Mean velocity models TKEM: Turbulent kinetic energy equation models Additional models: LES: Large Eddy simulation models RSM: Reynolds stress models
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Lecture Objectives:
• Define 1) Reynolds stresses and 2) K-ε turbulence models
• Analyze General CFD (Transport) Equation– Discretization of Transport Equation
• Introduce Boundary Conditions
Modeling of Turbulent Viscosity
μtμ
Fluid property – often called laminar viscosity
Flow property – turbulent viscosity
......
-k-k-k
Re
321
Re
-k
Eq.Two
Eq.-One
TKEM
constantMVM
μon based Models
t
t
fkkll
CurvatureBuoyancyLow
LayerLayerLayer
boundedwall
Free
High
lengthmixing
MVM: Mean velocity modelsTKEM: Turbulent kinetic energy equation models
LES: Large Eddy simulation modelsRSM: Reynolds stress models
Additional models:
One equation models:Prandtl Mixing-Length Model (1926)
yVl ρ x2
tμ
-Two dimensional model-Mathematically simple
-Computationally stable
-Do not work for many flow types
Characteristic length (in practical applications: distance to the closest surface)
l
Vx
y
x
There are many modifications of Mixing-Length Model:- Indoor zero equation model: t = 0.03874 V l
Air velocity Distance to the closest surface
Kinetic energy and dissipation of energyKolmogorov scaleEddy breakup and decay to smaller length scales where dissipation appear
andk
model
Two equation turbulent model
Kinetic energy Energy dissipation
εkC ρ
2
μtμ
constant
We need to model
Two additional equations:
ρ-2k grad/μμdiv))Vdiv(kτkρ( μt ijijt EE
kinetic energy
k
ρ2-2 grad/μμdiv))Vdiv(τ
ρ(2
2211t
fCEEfC ijijt
dissipation
-k
From dimensional analysis
Reynolds Averaged Navier Stokes equations
xy
ty
ty
tx
zx
yx
xx S]
zV
)μμ[(z
]y
V)μμ[(
y]
xV
)μμ[(xx
P)z
VVy
VVx
VVτ
Vρ(
yy
ty
ty
ty
zy
yy
xy S]
zV
)μμ[(z
]y
V)μμ[(
y]
xV
)μμ[(xy
P)z
VV
yV
Vx
VV
τV
ρ(
zz
tz
tz
tz
zz
yz
xz S]
zV)μμ[(
z]
yV)μμ[(
y]
xV)μμ[(
xzP)
zVV
yVV
xVV
τVρ(
0z
Vy
Vx
V zyx
Momentum:
Continuity:
1)
2)
3)
4)
S)graddiv(ΓVdiv ρτ
ρ eff,
General format:
General CFD Equation
S)graddiv(ΓVdiv ρτ
ρ eff,
Values of , ,eff and S
Equation ,eff S
Continuity 1 0 0
x-momentum V1 + t -P/x+Sx
y-momentum V2 + t -P/y-g(T∞-Twall)+Sy
z-momentum V3 + t -P/z+Sz
T-equation T /l + t/t ST
k-equation k (+ t)/k G- +GB
-equation (+ t)/ [ (C1G-C2)/k] +C3GB(/k)
Species C (+ t)/c SC
Age of air + t
t =Ck2/ , G= t (Ui/xj +Uj/xi) Ui/xj , GB=-g(/CP)( t/T,t) T/ xi
C1=1.44, C2=1.92, C3=1.44, C=0.09 , t=0.9, k =1.0, =1.3, C=1.0
Discretization - Computers do not solve a partial differential equation (computers do only 1 + 1, 1 + 0, and, 0 + 1 )
- Convert partial differential equation into algebraic equations
obtain finite number of numerical values instead of continuous solution
- Available methods: - finite volume method
- finite difference method- finite element method
Finite Volume Method- Conservation of for the finite volume
w e
wel
hn
s
P EW
x
x x
- Finite volume is a fixed space in the flow domain with imaginary boundaries that allow the fluid to flow in and out.
- Integral conservation of the quantities such as mass, momentum and energy.
Divide the whole computation domain into sub-domains
One dimension:
General Transport Equation -3D problem steady-state
W E
N
S
H
L
P
Equation for node P in the algebraic format:
fΦaΦaΦaΦaΦaΦaΦa LLHHNNSSWWEEPP
S)gradΓ(div)Vdiv( ρ eff,
A 1D example of discretization of general transport equation
SgraddivΓVdiv ρτ
ρ eff,
Φeff Φ,x S)xΦ(Γ
xρΦρV
x
Steady state 1dimetion (x):
Point W and E represent the cell center of the west and east neighbors of cell P and w, e the neighboring surfaces. Integrating with Gaussian theorem on this control volume gives:
ew
P EW
x
xw xe
dAndVdivAV
e
wΦ
weff Φ,
eeff Φ,wxex dxS}
xΦΓ
xΦΓ{ρΦρVΦρV
To obtain the equations for the value at point P, assumptions are used to convert the surface values to the center values.
0dxS}xΦΓ
xΦΓ{ΦρVΦρV
e
wΦ
weff Φ,
eeff Φ,wxex
Steady–state 1D example
ew
P EW
x
xw xeWw ΦΦ If Vx > 0,
If Vx < 0, Ee ΦΦ
X direction
Pe ΦΦ
Pw ΦΦ
and
and
Diffusion term:
Source term: xSdxS Φ
e
wΦ
Assumption:Source is constant over the control volume
a)
b)
c)
I)
Convection term - Upwind-scheme:
WxPxwxex ΦρVΦρVΦρVΦρV
PxExwxex ΦρVΦρVΦρVΦρV
x
ΦΦ2ΦΓ x
ΦΦx
ΦΦΓxΦΓ -
xΦΓ
:thenx
ΦΦΓxΦΓ and
xΦΦΓ
xΦΓ
WPEeff Φ,
w
WP
e
PEeff Φ,
weff Φ,
eeff Φ,
w
WPeff Φ,
weff Φ,
e
PEeff Φ,
eeff Φ,
When mesh is uniform:X = xe = xw
1D example
Φeff Φ,x S)xΦ(Γ
xΦρV
x
0xSx
ΦΦ2ΦΓΦΦρV ΦWPE
eff Φ,P)W(or E)P(or x
ΦWEPeff Φ,
P)W(or E)P(or x SΦΦΦ2
xxΓ
ρ ΦΦx
ρV
After substitution a), b) and c) into I):
same
We started with partial differential equation:
and developed algebraic equation:
We can write this equation in general format:
fcΦbΦaΦ WPE
Equation coefficients
Φ Unknowns
General Transport Equation -3D problem steady-state
W E
N
S
H
L
PEquation in the algebraic format:
fΦaΦaΦaΦaΦaΦaΦa LLHHNNSSWWEEPP
Wright this equation for each discretization volumeof your discretization domain
x =
FΦA
7-diagonal matrix
60,000 cells (nodes)N=60,000
60,000 elements
60,0
00 e
lem
ents
This is the system for only one variable ( )Φ When we need to solve p, u, v, w, T, k, , C
system of equation is larger
S)gradΓ(div)Vdiv( ρ eff,
Boundary conditionsin CFD application in indoor airflow
Real geometry
Model geometry
Where are the boundary Conditions?
CFD ACCURACY
Depends on airflow in the vicinity ofBoundary conditions
1) At air supply device 2) In the vicinity of occupant 3) At room surfaces
Detailed modeling- limited by computer power
Surface boundaries
Wall surface
W use wall functions to model the micro-flow in the vicinity of surfaceUsing relatively large mesh (cell) size.
0.01-20 mmfor forced convection
thickness
Airflow at air supply devices
Complex geometry - Δ~10-4mWe can spend all our computing power for one small
detail
momentum sources
Diffuser jet properties
High Aspiration diffuser
D
L
D
L
How small cells do you need? We need simplified models for diffusers
Peter V. Nielsen
Simulation of airflow in In the vicinity of occupantsHow detailed should we make the geometry?
AIRPAK Software