Fluent-Adv Turbulence 15.0 L02 Rans Models

7/18/2019 Fluent-Adv Turbulence 15.0 L02 Rans Models

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1 © 2014 ANSYS, Inc. April 23, 2014 ANSYS Confidential

Lecture 2:

RANS Turbulence Models in A

Turbulence Modeling Using A




Overview

• RANS models in ANSYS-Fluent

–

Spalart-Allmaras model – k -e family of models

– k -w family of models

• Includes transition models (covered in L-3)

• Includes SAS model (covered in L-4)

–Reynolds stress models




Turbulence Models Available in ANSYS F

RANS based

models

One-Equation Model

Spalart-Allmaras

Two-Equation Models

Standard k –ε

RNG k –ε

Realizable k –ε

Standard k –ω

SST k –ω

4-Equation v2f *

Intermittency Transition Modelk –kl –ω Transition Model

Transition SST Model

Reynolds Stress Model

Scale-Adaptive Simulation

Detached Eddy Simulation

Large Eddy Simulation

Comp

P




Spalart-Allmaras Model Equations

Modified turbulent viscosity

normal dist

the wall

damping functio

2

1

2

2~

1

~

~~

~1~~

~

d f c

xc

x xS c

Dt D ww

j

b

j j

b

~,,~

3

1

3

3

11t

cv

vv f f

1

2222 11,

~~

v

vv f

f f d

S S

22

6

2

6/1

6

3

6

6

3~

~,g,

1

d S r r r cr

g g f w

w

ww

cc

0~: conditionboundaryWall


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Default definition uses rotation rate tensor only:

Alternative formulation also uses strain rate tensor:

− reduces turbulent viscosity for vortical flows

− more correctly accounts for the effects of rotation

Spalart-Allmaras Production Term

i

j

j

i

xU

xU S

ijijij 2

1;2

)-min(0, ijij prodij S C S



Spalart-Allmaras Model• Spalart-Allmaras model developed for unstructured codes in

aerospace industry

–

Increasingly popular for turbomachinery applications – “Low-Re” formulation by default

– can be integrated right down to the viscous sublayer

– Fluent’s implementation can also use the wall function (law of the wal

– Economical and accurate for:

– wall-bounded flows

–

flows with mild separation and recirculation

– Weak for:

– massively separated flows

– free shear flows

– simple decaying turbulence



Two-Equation Models

• Two transport equations are solved, giving two independfor calculating

t

–

Virtually all use the transport equation for the turbulent kinetic – Several transport variables have been proposed, based on dime

arguments, and used for second equation

– Kolmogorov, w :

t k /w

l

k 1/2 / w k e / w

•w

is specific dissipation rate

• defined in terms of large eddy scales that define supp

– Chou, e :

t k 2 / e l k 3/2 / e

– Rotta, l :

t k 1/2 l e k 3/2 / l

– Boussinesq relation still used for Reynolds Stresses



e e e

e e e

e 2

2t1

t C S C k x x Dt

D

j j

Standard k -e Model Equations

production dissipation

inverse time scale

Empirical constants determined from benchmar

experiments of simple flows using air and water

k – transport equation

e

transport equation

coefficients

Finally, turbulent viscosity

ijijt jk j S S S S x

k

x Dt

Dk

2;

2t

e

2,,, e e e C C ik

e

2

k

C t




Closure Coefficients

• Simple flows render simpler model equations

– Coefficients can be isolated and compared with experiment

–e.g.,

• Uniform flow past grid

– Standard k -e equations reduce to just convection and dissipati

• Homogeneous Shear Flow

• Near-Wall (Log layer) Flow

k

C

x

U

x

k U

2

2

d

d;

d

d e e e e




• High-Reynolds number model

– (i.e., must be modified for the near-wall region)

• The term “standard” refers to the choice of coefficie

• Sometimes additional terms are included

– production due to buoyancy

• unstable stratification (g·T > 0) supports k production

–

dilatation dissipation due to compressibility• added dissipation term, prevents overprediction of spreading rate in com

Buoyancy

production

Di

Dis

x g S

x

k

x Dt

Dk

it

t it

jk

t

j

2Pr

2

e

Standard k -e Model




Standard k -e Model Pros & Cons

• Strengths:

– robust

–

economical – reasonable accuracy for a wide range of flows

• Weaknesses:

– overly diffusive for many situations• flows involving strong streamline curvature, swirl, rotation, separating flows,

–

cannot predict round jet spreading rate

• Variants of the k -e model have been developed to addresdeficiencies

– RNG and Realizable




RNG k -e Model Equations

• Derived using renormalization group theory

– scale-elimination technique applied to Navier-Stokes equatequations to specific flow regimes)

• k – equation is similar to standard k -e model

• Additional strain rate term in e - equation

– most significant difference between standard and RNG k -e models

• Analytical formula for turbulent Prandtl numbers

• Differential-viscosity relation for low Reynolds numbers

– Boussinesq model used by default

wheree e e

e

e e e

*

2

2

1eff C S C

k x x Dt

Dt

j j

t

k S

C

C C

eff

0

2

*

2

,

e

e e

coeare

e transport equation




RNG k -e Model Pros & Cons

• For large strain rates:

– where > 0 e is augmented, and therefore k and

t are

• Option to modify turbulent viscosity to account for

• Buoyancy and compressibility terms can be included

• Improved performance over std. k -e model for

–

rapidly strained flows – flows with streamline curvature

• Still suffers from the inherent limitations of an isotrviscosity model




Realizable k -e Model: Motivation

• Standard k -e model could not ensure:

– Positivity of normal stresses

– Schwarz’s inequality of shear stresses

• Modifications made to standard model

– k − equation is same; new formulation for t and e

– C

is variable

– e − equation is based on a transport equation for the mean

vorticity fluctuation

02 u

uuuu 22

2




Realizable k -e Model: Realizability

• How can normal stresses become negative?

– Standard k -e Boussinesq viscosity relation:

– Normal component:

– Normal stress will be negative if:

3

2 -

2

ij

i

j

j

i

ji k x

U

x

U k C uu

e

23

2

22

x

U k

C k u

e

3.73

1

e C x

U k




Realizable k -e Model: C

• C

is not a constant, but varies as a function of mean velocity field and

(0.09 in log-layer S k / e = 3.3, 0.05 in shear layer of S k / e = 6)

C

contours for 2D backward-facing step

C

along bottom-w




Realizable k -e Model Equations

where

e transport equation

turbulent viscosity

e

e e

e

e

e

k

C S C

x x Dt

D

j

t

j

2

21

0.1 ,/,5

,43.0max 21

C k S C e

e e

k U A A

C k

C

s

t *

0

2

1,

W A A s 6cos3

1,cos6,04.4 1

0

ijij

ki jk ij

ijijijij S S S

S

S S S W S S U

~,~ ,*




Realizable k -e Model Pros & Cons

• Performance generally exceeds the standard k -e mo

• Buoyancy and compressibilty terms can be included• Good for complex flows with large strain rates

– recirculation, rotation, separation, strong

P

• Resolves the round-jet/plane jet anomaly

–

predicts the spreading rate for round and plane jets

• Still suffers from the inherent limitations of an isotroeddy-viscosity model




Standard and SST k -w Models

• k-w models are a popular alternative to k -e

– w ~ e / k

–

t k / w

• Can be used in the near-wall region without modificatio

• Wilcox’s original model was found to be quite sensitive tfar-field boundary values of w

• Latest version contains several refinements: – reduced sensitivity to boundary conditions

– modification for the round-jet/plane-jet anomaly

– compressibility effects

– low-Re (near wall) effects




Standard k -w

Model

• Standard k -w model in Fluent is Wilcox’ 1998 model

j

t

j j

i

ij

jk

t

j j

iij

t

x x f

x

U

k Dt

D

x

k

xk f

x

U

Dt

Dk

k

w

w

w

w

w

w

w

2

*

*

*




Standard k -w

Turbulent Viscosity

• Turbulent viscosity is computed from:

• The dependency of * upon ReT was designed to recovecorrect asymptotic values in the limiting cases. In particu

that:

w

k t

*

0.1 ,Re,6

125

9,

3,

Re1

Re

*

*

0

*

0**

w

k R

R

R

T k

ii

k T

k T where

turbulent(fully T Reas1*




Standard k -w Turbulent Kinetic Energ

• Note the dependence upon ReT , M t , andk

• “Dilatation” dissipation is accounted for via M t term

• The cross-diffusion parameter ( k

) is designed to improve free shea

k of Diffusionk of ratenDissipatio

*

k of production

*

jk

t

j j

iij

x

k

xk f

x

U

Dt

Dk

w

09.0 ,5.1 ,0.2

8,Re1

Re154

1

**

4

4

**

***

k

T

T

i

t i

R R

R

M F

d-cross

k

k

k

k

k

t t

t t t t

t t

t

f

RT a M a

k M

M M M M

M M M F

2

2

02

2

0

2

0

2

0

,0

4001

6801

01

,4

1,

2

0

*




Standard k -w Specific Dissipation Equa

• Note the dependence upon ReT , M t , and

w

• Vortex-stretching parameter (w

) designed to remedy the plane/rou

j

t

j j

iij

x x

f

x

U

k Dt

D w

w

w

w

w

2

i

j

j

iij

i

j

j

iij

ki jk ij

t

i

ii

T

T

xU

xU

xU

xU S

S f M F

R R

R

*

3*

**

00

*

21,

21

5.1 ,,801

701,1

0.2,95.2,9

1,

25

13,

Re1

Re

w

w

w

w

w w

w

w




Standard k -w Sub-models & Options (• “Low-Re Corrections” option – Corresponds to all terms involving ReT

terms in the model equations

–Deactivated by default

– Can benefit low-Re flows where the meshcan support good near-wall resolution

• “Compressibility Effects” option – Takes effects via F(Mt )

–

Accounts for “dilatation” dissipation

k k

k d d s

x

u

x

u

e e e e

3

4,

– Available with ideal-gas option only and is turned off by defaul

– Improves predictions for high-Mach number free shear flows −

reduces sprea




Standard k -w Sub-models & Options (

• “Shear-Flow Corrections” option – activated by default

Controls both cross-diffusion and vortex-stretching terms

–

Cross-diffusion term (in k − equation)

• Designed to improve the model performance for free shear flows without afflayer flows

–Vortex-stretching term• Designed to resolve the round/plane-jet anomaly

• Takes effects for axi-symmetric and 3-D flows but vanishes for planar 2-D flo

3*,

801

701

w

w

w

w

ki jk ij S

f

parameter diffusion-cross

j j

k

k

k

k

k

x x

k f

w

w

3

2

21

,0

4001

6801

01

*




Menter’s SST k -w

Model: Background

• Many people, including Menter (1994), have noted

– Wilcox’ original k -w model is overly sensitive to the freestr

value (BC) of w, while k -e model is not prone to such probl

– The k -w model has many good attributes and performs mu

better than k -e models for boundary layer flows

– Most two-equation models, including k -e models, over-pre

turbulent stresses in the wake (velocity-defect) region, whto poor performance of the models for boundary layers un

adverse pressure gradient and separated flows




Menter’s SST k -w Model: Main Compon

• The SST k -w model consists of

– Zonal (blended) k -w / k -e equations (to address item 1 and 2

previous slide)

– Clipping of turbulent viscosity so that turbulent stresses sta

is dictated by the structural similarity constant. (Bradshaw,

addresses item 3 in the previous slide

Inner layer

(sublayer, log-layer) Wilcox’ original k-w model

e

e

23

k

Wall

Outer layer (wake and

outward)

k -w model transformed

from standard k -e model

Modified Wilcox k -w model




Menter’s SST k -w

Model: Inner Layer

• The k -w model equations for the inner layer are tak

Wilcox original k -w

model with some constants mo

j

t

j j

iij

t

jk

t

j j

iij

x x x

U

Dt

D

x

k

xk

x

U

Dt

Dk

w

w

w

w

w 1

2

11

1

*

41.0,,09.0

0.2,176.1,075.0

1

*2*

11

*111

w

w k

w

w

w

22

2

22

21

1

500,

09.0

2maxarg,argtanh

,amax

y y

k F

F

k at




Menter’s SST k -w

Model: Outer Layer

• The k -w model equations for the outer layer are obtainetransforming the standard k -e equations via change-of-v

• Turbulent viscosity computed from:

j j

j

t

j j

iij

t

jk

t

j j

iij

x x

k

x x x

U

Dt

D

x

k

xk

x

U

Dt

Dk

w

w

w

w

w

w

w

w

12 2

2

2

22

2

*

41.0,,09.0

168.1,0.1,0828.0

2

*2*

22

*

222

w

w k

w k

t




Menter’s SST k -w: Blending the Equatio

• The two sets of equations and the model constants are blended in su

the resulting equation set transitions smoothly from one equation to

*1

1

2max

maxminarg

tan

k CD

k

F

w

ww

w ,,,where

1

1

2111

outer

1

inner

1

k

F F

Dt Dk F

Dt Dk F

in

in

0

1

1

1

F

F Wilcox’ original k-w model

e

e

23

k

Wall

k-w model transformedfrom std. k-e model

Modified Wilcox k-w model




Menter’s SST k -w: Blended k -w Equatio

• The resulting blended equations are:

Wall

j j

j

t

j j

iij

t

jk

t

j j

iij

x x

k F

x x x

U

Dt

D

x

k

x

k

x

U

Dt

Dk

w

w

w

w

w

w

w

w

112

21

2

*

w ,,,,1 2111 k F F




Menter’s SST k -w: Turbulent Viscosity• Honors the “structural similarity” constant for boundary layer

(Bradshaw, 1967)

• Turbulent stress implied by turbulence models can be written

– In many flow situations (e.g. adverse pressure gradient flows), production TKE can be much larger than dissipation ( P k >> e ), which leads to predicteturbulent stress larger than what is implied by the structural similarity con

– Turbulent stress can be limited by clipping the turbulent viscosity such tha

1967(Bradshaw,11 ak

vuk avu

ε

P k a

y

U k t t 1

k at 1




Menter’s SST k -w

Model Clipping t

• Turbulent viscosity for the inner layer is computed f

• Remarks – F

2 is equal to 1 inside boundary layer and goes to zero far from the wall an

shear layers

– The name SST (shear-stress transport) originates from this

– Note that the vorticity magnitude is used (strain-rate magnitude could also

magnitude)(vorticity2

500,

09.0

2maxarg,argtanh

,min,amax

22

2

22

2

1

21

1

ijij

t

y y

k F

F

k ak

F

k a

w

w

w

w




Menter’s SST k -w: Submodels & Optio• SST k -w model comes with: – Low-Re Corrections option

(Off by default)

–

Compressibility Effects option when ideal-

gas option is selected

(Off by default)

• The original SST k -w model in theliterature does not have any of these

options – These submodels are being borrowed from

Wilcox’ 1998 model - should be used withcaution

– Do not activate any options to recover theoriginal SST model

d l d d




k -w Models: Boundary Conditions

• Wall boundary conditions – The Enhanced Wall Treatment (EWT) is the sole near-wall optio

models. Neither the standard wall functions option nor the non

equilibrium wall functions option is available for k-w models inFluent• The blended laws of the wall are used exclusively

• w values at wall adjacent cells are computed by blending the wall-limiti→ 0 ) and the value in the log-layer

– The k -w

models can be used with either a fine near-wall mesh onear-wall mesh

• For other BCs (e.g., inlet, free-stream), the following relais used internally, whenever possible, to convert to and fdifferent turbulence quantities:

09.0, ** w e k

d l &




k -w Models: Pros & Cons

• Use of w instead of e allows for better resolution of bounlayers, especially under adverse pressure gradients

• The Standard k-w model is sensitive to inlet and far field – The SST k-w model does not suffer from this

• Sometimes more difficult to converge than any of the k-e

• The best RANS models for wall bounded aerodynamic flo

airfoils, compressors, turbines, and flows with separationadverse pressure gradient

– Use SST to prevent boundary condition sensitivity

• Often the best models for heat transfer predictions

F lt i th B i A ti




Faults in the Boussinesq Assumption

• Boussinesq: R i j = 2 tS i j

– Is simple linear relationship sufficient?

• R i j is strongly dependent on flow conditions and history

• R i j changes at rates not entirely related to mean flow processes

– R i j is not strictly aligned with S i j for flows with:

• sudden changes in mean strain rate

• extra rates of strain (e.g., rapid dilatation, strong streamline curv

• rotating fluids

• stress-induced secondary flows

• Modifications to two-equation models cannot be generaarbitrary flows

R ld St M d l




• Starting point is the exact transport equations for th

transport of Reynolds stresses, R ij

–

six transport equations in 3D

• Equations are obtained by Reynolds-averaging the p

of the exact momentum equations and a fluctuating

• The resulting equations contain several terms that mmodeled

0)()(

i j ji u NS uu NS u

Reynolds Stress Models

Reynolds Stress Transport Equations




Reynolds Stress Transport Equations

k

ijk

ijijij

ij

x

J P

Dt

DR

e

Generation

k

ik j

k

j

k iij x

U uu

x

U uu P

i

j

j

iij

x

u

x

u p

k

j

k

iij

x

u

xu

e 2

Pressure-Strain

Redistribution

Dissipation

Turbulent

Diffusion

(modeled)

(related to e)

(computed)

(incompressible fl

forces)

Reynolds Stress

Transport Eqns.

( ji

k

k jiik j jk iijk uu x

uuuu pu p J

Pressure/velocity

fluctuations

Turbulent

transport

Molecular

transport

Di i ti M d li




• Dissipation rate is predominantly associated with sm

eddy motions

–

Large scale eddies affected by mean shear – Vortex stretching process breaks eddies down into continual

scales

• The directional bias imprinted on turbulence by mean flow is gradually lo

• Small scale eddies assumed to be locally isotropic

•e

is calculated with its own (or related) transport equation

• Compressibility effects can be included

Dissipation Modeling

e e ijij3

2

Turbulent Diffusion




Turbulent Diffusion

• Most closure models combine the pressure diffusion

triple products and use a simple gradient diffusion h

– Overall performance of models for these terms is generally i

based on isolated comparisons to measured triple products

– DNS data indicate that above p terms are negligible

lk s

k

ji

k

jik ikjk ji

k uu

k C

xuu

xuu

puuu

x e

'

k

k C

x e

2

Or even a simpler model

Pressure Strain Modeling




Pressure-Strain Modeling• Pressure-strain term of same order as production

• Pressure-strain term acts to drive turbulence towards an

isotropic state by redistributing the Reynolds stresses• Decomposed into parts

• Model of Launder, Reece & Rodi (1978)

i

j

j

i

i

j

i

ii x

u

x

u

x

uu

x x

p

21

ij

m

l ml

l

il j

l

j

l iijij x

U uu

x

U uu

x

U uucbc

3

221

i

j

i

ii x

U

x

u

x x

p

2

1 2

“Rapid” Part “Slow” Part

ijbwhere

wijijijij ,2,1,

ij p

Pressure Strain Modeling Options




Pressure-Strain Modeling Options

• Wall-reflection effect

– contains explicit distance from wall

–

damps the normal stresses perpendicular to wall – enhances stresses parallel to wall

• SSG (Speziale, Sarkar and Gatski) Pressure Strain Mo

–

Expands the basic LRR model to include non-linear (quadrati – Superior performance demonstrated for some basic shear flo

• plane strain, rotating plane shear, axisymmetric expansion/contr

Stress Model




Stress-w Model

• Uses the w equation from the standard k -w model rae

from the k -e model

• Selecting “Low-Re Corrections” in the viscous modelafter selecting “Stress-Omega” activates the same teactivated in the standard k -w model and also low Remodifications of the pressure strain model

• Potentially beneficial compared to other RSM optionwhere the viscous sublayer must be resolved in ordeproduce an accurate solution

– Predictions of k -w models are generally superior to those ofsuch flows

Characteristics of RSM




Characteristics of RSM

• Effects of curvature, swirl, and rotation are directly a

for in the transport equations for the Reynolds stres

–

when anisotropy of turbulence significantly affects the meanconsider RSM

• More cpu resources (vs. k -e models) is needed

–

50-60% more CPU time per iteration and 15-20% additional m

• Strong coupling between Reynolds stresses and the

– number of iterations required for convergence may increase

Heat Transfer




Heat Transfer

• The Reynolds averaging process produces an additio

in the energy equation: (q

is the fluctuating co

of static temperature) – Analogous to the Reynolds stresses, this is called the turbule

• It is possible to model a transport equation for the heat flux, but

common practice

• Instead, a turbulent thermal diffusivity is defined proportional to

turbulent viscosity – The constant of proportionality is called the turbulent Prandtl

– Generally assumed that Pr t ~ 0.85-0.9

• Applicable to other scalar transport equations

q iu

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Fluent-Adv Turbulence 15.0 L02 Rans Models