Fluent.5.Turbulence Imp

© Fluent Inc. 04/21/23D1

Fluent Software TrainingTRN-98-006

Modeling Turbulent Flows



Unsteady, aperiodic motion in which all three velocity components fluctuate mixing matter, momentum, and energy.

Decompose velocity into mean and fluctuating parts:

Ui(t) Ui + ui(t)

Similar fluctuations for pressure, temperature, and species concentration values.

What is Turbulence?

Time

U i (t)

Ui

ui(t)



Why Model Turbulence?

Direct numerical simulation of governing equations is only possible for simple low-Re flows.

Instead, we solve Reynolds Averaged Navier-Stokes (RANS) equations:

where (Reynolds stresses)

Time-averaged statistics of turbulent velocity fluctuations are modeled using

functions containing empirical constants and information about the mean flow. Large Eddy Simulation numerically resolves large eddies and models small

eddies.

(steady, incompressible flow w/o body forces)

jiij uuR

j

ij

jj

i

ik

ik x

R

xx

U

x

p

x

UU

2



Is the Flow Turbulent?

External Flows

Internal Flows

Natural Convection

5105 xRe along a surface

around an obstacle

where

UL

ReL where

Other factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow.

L = x, D, Dh, etc.

108 1010 Ra 3TLg

Ra



How Complex is the Flow?

Extra strain rates Streamline curvature Lateral divergence Acceleration or deceleration Swirl Recirculation (or separation) Secondary flow

3D perturbations Transpiration (blowing/suction) Free-stream turbulence Interacting shear layers



Choices to be Made

Turbulence Model&

Near-Wall Treatment

Flow Physics

AccuracyRequired

Computational Resources

Turnaround TimeConstraints

Computational Grid



Zero-Equation Models

One-Equation Models Spalart-AllmarasTwo-Equation Models Standard k- RNG k-Realizable k-Reynolds-Stress Model

Large-Eddy Simulation

Direct Numerical Simulation

Turbulence Modeling Approaches

IncludeMorePhysics

IncreaseComputationalCostPer Iteration

Availablein FLUENT 5

RANS-basedmodels



RANS equations require closure for Reynolds stresses.

Turbulent viscosity is indirectly solved for from single transport equation of modified viscosity for One-Equation model.

For Two-Equation models, turbulent viscosity correlated with turbulent kinetic energy (TKE) and the dissipation rate of TKE.

Transport equations for turbulent kinetic energy and dissipation rate are solved so that turbulent viscosity can be computed for RANS equations.

Reynolds Stress Terms in RANS-based Models

Turbulent Kinetic Energy:

Dissipation Rate of Turbulent Kinetic Energy:

2kCt Turbulent Viscosity:

Boussinesq Hypothesis:(isotropic stresses)

i

j

j

itijjiij x

U

x

UkuuR

3

2

2/iiuuk

i

j

j

i

j

i

x

u

x

u

x

u



Turbulent viscosity is determined from:

is determined from the modified viscosity transport equation:

The additional variables are functions of the modified turbulent viscosity and velocity gradients.

One Equation Model: Spalart-Allmaras

21

2

2~

1

~~~~1~~~

dfc

xc

xxSc

Dt

Dww

jb

jjb

3

13

3

/~/~

~

ct

~

Generation Diffusion Destruction



One-Equation Model: Spalart-Allmaras

Designed specifically for aerospace applications involving wall-bounded flows.

Boundary layers with adverse pressure gradients turbomachinery

Can use coarse or fine mesh at wall Designed to be used with fine mesh as a “low-Re” model, i.e., throughout

the viscous-affected region. Sufficiently robust for relatively crude simulations on coarse meshes.



Two Equation Model: Standard k- Model

Turbulent Kinetic Energy

Dissipation Rate

21, ,, CCk are empirical constants

(equations written for steady, incompressible flow w/o body forces)

Convection Generation DiffusionDestruction

ikt

ii

j

j

i

i

jt

ii x

k

xx

U

x

U

x

U

x

kU )(

DestructionConvection Generation Diffusion

kC

xxx

U

x

U

x

U

kC

xU

it

ii

j

j

i

i

jt

ii

2

21 )(



Two Equation Model: Standard k- Model

“Baseline model” (Two-equation) Most widely used model in industry Strength and weaknesses well documented

Semi-empirical k equation derived by subtracting the instantaneous mechanical energy

equation from its time-averaged value equation formed from physical reasoning

Valid only for fully turbulent flows Reasonable accuracy for wide range of turbulent flows

industrial flows heat transfer



Two Equation Model: Realizable k- Distinctions from Standard k- model:

Alternative formulation for turbulent viscosity

where is now variable

(A0, As, and U* are functions of velocity gradients)

Ensures positivity of normal stresses;

Ensures Schwarz’s inequality;

New transport equation for dissipation rate, :

2kCt

kUAA

C

so

*

1

0u2i

2j

2i

2ji u u)uu(

bj

t

j

Gck

ck

cScxxDt

D

31

2

21

GenerationDiffusion Destruction Buoyancy



Shares the same turbulent kinetic energy equation as Standard k- Superior performance for flows involving:

planar and round jets boundary layers under strong adverse pressure gradients, separation rotation, recirculation strong streamline curvature

Two Equation Model: Realizable k-



Two Equation Model: RNG k-Turbulent Kinetic Energy

Dissipation Rate

Convection DiffusionDissipation

ik

it

ii x

k

xS

x

kU eff

2

Generation

j

i

i

jijijij x

U

x

USSSS

2

1,2

where

are derived using RNG theory 21, ,, CCk


Additional termrelated to mean strain& turbulence quantities

Convection Generation Diffusion Destruction

RkC

xxS

kC

xU

iit

ii

2

2eff2

1



Two Equation Model: RNG k- k- equations are derived from the application of a rigorous statistical

technique (Renormalization Group Method) to the instantaneous Navier-Stokes equations.

Similar in form to the standard k- equations but includes: additional term in equation that improves analysis of rapidly strained flows the effect of swirl on turbulence analytical formula for turbulent Prandtl number differential formula for effective viscosity

Improved predictions for: high streamline curvature and strain rate transitional flows wall heat and mass transfer



Reynolds Stress Model

k

ijkijijij

k

jik x

JP

x

uuU

Generationk

ikj

k

jkiij x

Uuu

x

UuuP

i

j

j

iij x

u

x

up

k

j

k

iij x

u

x

u

2

Pressure-StrainRedistribution

Dissipation

TurbulentDiffusion

(modeled)

(related to )

(modeled)

(computed)


Reynolds StressTransport Eqns.

Pressure/velocity fluctuations

Turbulenttransport

)( jikijkkjiijk uupuuuJ



Reynolds Stress Model RSM closes the Reynolds-Averaged Navier-Stokes equations by

solving additional transport equations for the Reynolds stresses. Transport equations derived by Reynolds averaging the product of the

momentum equations with a fluctuating property Closure also requires one equation for turbulent dissipation Isotropic eddy viscosity assumption is avoided

Resulting equations contain terms that need to be modeled. RSM has high potential for accurately predicting complex flows.

Accounts for streamline curvature, swirl, rotation and high strain rates Cyclone flows, swirling combustor flows Rotating flow passages, secondary flows



Large Eddy Simulation Large eddies:

Mainly responsible for transport of momentum, energy, and other scalars, directly affecting the mean fields.

Anisotropic, subjected to history effects, and flow-dependent, i.e., strongly dependent on flow configuration, boundary conditions, and flow parameters.

Small eddies: Tend to be more isotropic and less flow-dependent More likely to be easier to model than large eddies.

LES directly computes (resolves) large eddies and models only small eddies (Subgrid-Scale Modeling).

Large computational effort Number of grid points, NLES Unsteady calculation

2Reu



Comparison of RANS Turbulence Models



Near-Wall Treatments

Most k- and RSM turbulence models will not predict correct near-wall behavior if integrated down to the wall.

Special near-wall treatment is required.

Standard wall functions Nonequilibrium wall functions Two-layer zonal model

Boundary layer structure



Standard Wall Functions

/

2/14/1

w

PP kCUU

)(ln1

Pr

)(Pr

**

**

Tt

T

yyPEy

yyyT

PP ykC

y2/14/1

q

kCcTTT PpPw

2/14/1)(*

Mean Velocity

Temperature

where

where and P is a function of the fluid and turbulent Prandtl numbers.

thermal sublayer thickness

EyU ln1



Nonequilibrium Wall Functions Log-law is sensitized to pressure gradient for better

prediction of adverse pressure gradient flows and separation.

Relaxed local equilibrium assumptions for TKE in wall-neighboring cells.

Thermal law-of-wall unchanged

ykC

EkCU

w

2/14/12/14/1

ln1/

~

y

k

yyyy

k

ydxdp

UU vv

v

v2

2/12/1ln

21~

where



Two-Layer Zonal Model Used for low-Re flows or

flows with complex near-wall phenomena.

Zones distinguished by a wall-distance-based turbulent Reynolds number

High-Re k- models are used in the turbulent core region. Only k equation is solved in the viscosity-affected region. is computed from the correlation for length scale. Zoning is dynamic and solution adaptive.

yk

Rey

200yRe



Comparison of Near Wall Treatments

Strengths Weaknesses

Standard wallFunctions

Robust, economical,reasonably accurate

Empirically based on simplehigh-Re flows; poor for low-Reeffects, massive transpiration,p, strong body forces, highly3D flows

Nonequilibriumwall functions

Accounts for p effects,allows nonequilibrium:

-separation-reattachment-impingement

Poor for low-Re effects, massivetranspiration, severe p, strongbody forces, highly 3D flows

Two-layer zonalmodel

Does not rely on empiricallaw-of-the-wall relations,good for complex flows,applicable to low-Re flows

Requires finer mesh resolutionand therefore larger cpu andmemory resources



Computational Grid GuidelinesWall Function

ApproachTwo-Layer Zonal Model Approach

First grid point in log-law region

At least ten points in the BL.

Better to use stretched quad/hex cells for economy.

First grid point at y+ 1.

At least ten grid points within buffer & sublayers.

Better to use stretched quad/hex cells for economy.

50050 y



Estimating Placement of First Grid Point

Estimate the skin friction coefficient based on correlations either approximate or empirical:

Flat Plate-

Pipe Flow-

Compute the friction velocity:

Back out required distance from wall: Wall functions • Two-layer model

Use post-processing to confirm near-wall mesh resolution

2.0Re0359.02/ Lfc

2.0Re039.02/ Dfc

2// few cUu

y1 = 250/u y1 = / u



Setting Boundary Conditions

Characterize turbulence at inlets & outlets (potential backflow) k- models require k and Reynolds stress model requires Rij and

Several options allow input using more familiar parameters Turbulence intensity and length scale

length scale is related to size of large eddies that contain most of energy. For boundary layer flows: l 0.499

For flows downstream of grids /perforated plates: l opening size Turbulence intensity and hydraulic diameter

Ideally suited for duct and pipe flows Turbulence intensity and turbulent viscosity ratio

For external flows: Input of k and explicitly allowed (non-uniform profiles possible).

10/1 t



GUI for Turbulence Models

Define Models Viscous...

Turbulence Model options

Near Wall Treatments

Inviscid, Laminar, or Turbulent

Additional Turbulence options



Example: Channel Flow with Conjugate Heat Transfer

adiabatic wall

cold airV = 50 fpmT = 0 °F

constant temperature wall T = 100 °F

insulation

1 ft

1 ft

10 ft

P

Predict the temperature at point P in the solid insulation



Turbulence Modeling Approach Check if turbulent ReDh

= 5,980

Developing turbulent flow at relatively low Reynolds number and BLs on walls will give pressure gradient use RNG k- with nonequilibrium wall functions.

Develop strategy for the grid Simple geometry quadrilateral cells Expect large gradients in normal direction to horizontal walls fine mesh

near walls with first cell in log-law region. Vary streamwise grid spacing so that BL growth is captured. Use solution-based grid adaption to further resolve temperature gradients.



Velocitycontours

Temperaturecontours

BLs on upper & lower surfaces accelerate the core flow

Prediction of Momentum & Thermal Boundary Layers

Important that thermal BL was accurately resolved as well

P



Example: Flow Around a Cylinder

wall

wall

1 ft

2 ft

2 ft

airV = 4 fps

Compute drag coefficient of the cylinder

5 ft 14.5 ft



Check if turbulent ReD = 24,600

Flow over an object, unsteady vortex shedding is expected, difficult to predict separation on downstream side, and close proximity of side walls may influence flow around cylinder use RNG k- with 2-layer zonal model.

Develop strategy for the grid Simple geometry & BLs quadrilateral cells. Large gradients near surface of cylinder & 2-layer model

fine mesh near surface & first cell at y+ = 1. Use solution-based grid adaption to further resolve pressure

gradients.

Turbulence Modeling Approach



Grid for Flow Over a Cylinder



Prediction of Turbulent Vortex Shedding

Contours of effective viscosity eff = + t

CD = 0.53 Strouhal Number = 0.297

U

DSt

where



Summary: Turbulence Modeling Guidelines

Successful turbulence modeling requires engineering judgement of: Flow physics Computer resources available Project requirements

Accuracy Turnaround time

Turbulence models & near-wall treatments that are available Begin with standard k- and change to RNG or Realizable k- if

needed. Use RSM for highly swirling flows. Use wall functions unless low-Re flow and/or complex near-wall

physics are present.

Documents

Fluent.5.Turbulence Imp