Adv Turb v62 01 Overview

8/9/2019 Adv Turb v62 01 Overview

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Advanced Fluent Training

Turbulence Apr 2005

Overview of Turbulence Modeling


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Turbulence Apr 2005

Outline

Background

Characteristics of Turbulent Flow

Scales

Eliminating the small scales

Reynolds Averaging

Filtered Equations

Turbulence Modeling Theory RANS Turbulence Models in FLUENT

Turbulence Modeling Options in Fluent

Near wall modeling, Large Eddy Simulation (LES)

Turbulent Flow Examples Comparison with Experiments and DNS

Turbulence Models

Near Wall Treatments


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Turbulence Apr 2005

What is Turbulence?

Unsteady, irregular (aperiodic) motion in which transported quantities

(mass, momentum, scalar species) fluctuate in time and space Fluid properties exhibit random variations

statistical averaging results in accountable, turbulence related transport

mechanisms

Contains a wide range of eddy sizes (scales)

typical identifiable swirling patterns

large eddies carry small eddies


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Turbulence Apr 2005

Turbulent boundary layer on a flat plateTurbulent boundary layer on a flat plate

Homogeneous, decaying, gridHomogeneous, decaying, grid--generated turbulencegenerated turbulence

Two Examples of Turbulence


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Turbulence Apr 2005

Energy Cascade

Larger, higher-energy eddies, transfer energy to smaller eddies via

vortex stretching Larger eddies derive energy from mean flow

Large eddy size and velocity on order of mean flow

Smallest eddies convert kinetic energy into thermal energy via viscous

dissipation

Rate at which energy is dissipated is set by rate at which they receive

energy from the larger eddies at start of cascade


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Turbulence Apr 2005

Vortex Stretching

Existence of eddies implies vorticity

Vorticity is concentrated along vortex lines or bundles Vortex lines/bundles become distorted from the induced velocities of

the larger eddies

As end points of a vortex line randomly move apart

vortex line increases in length but decreases in diameter

vorticity increases because angular momentum is nearly conserved

Most of the vorticity is contained within the smallest eddies

Turbulence is a highly 3D phenomenon


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Turbulence Apr 2005

Smallest Scales of Turbulence

Smallest eddy (Kolmogorov) scales:

large eddy energy supply rate ~ small eddy energy dissipationrate = -dk/dt k (u2+v2+w2) is (specific) turbulent kinetic energy [l2/t2]

is dissipation rate ofk[l2/t3]

Motion at smallest scales dependent upon dissipation rate, , andkinematic viscosity, [l2/t]

From dimensional analysis:

= (3/)1/4; = (/)1/2; v = ()1/4


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Turbulence Apr 2005

Small scales vs. Large scales Largest eddy scales:

Assume lis characteristic of larger eddy size

Dimensional analysis is sufficient to estimate order of large eddy supplyrate ofkas k /turnover

turnoveris a time scale associated with the larger eddies the order ofturnovercan be estimated as l/ k1/2

Since ~ k /turnover, ~ k3/2/l orl~ k3/2/ Comparing lwith ,

where ReT = k1/2l / (turbulence Reynolds number)

4/34/3

4/12/3

4/13Re)/(

)/(T

lklll = 1>>l


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Turbulence Apr 2005

Implication of Scales

Consider a mesh fine enough to resolve smallest eddies and large

enough to capture mean flow features Example: 2D channel flow

Ncells~(4l /)3

or

Ncells ~ (3Re)9/4

where

Re = uH /2

ReH = 30,800 Re = 800 Ncells = 4x107 !

H4/13 )/(

ll

l


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Turbulence Apr 2005

Direct Numerical Simulation

DNS is the solution of the time-dependent Navier-Stokes

equations without recourse to modeling

Numerical time step size required, t ~ For 2D channel example

ReH = 30,800

Number of time steps ~ 48,000

DNS is not suitable for practical industrial CFD

DNS is feasible only for simple geometries and low turbulent

Reynolds numbers

DNS is a useful research tool

+

=

+

j

i

kik

ik

i

x

U

xx

p

x

UU

t

U

u

Ht ChannelD

Re

003.02


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Turbulence Apr 2005

Removing the Small Scales

Two methods can be used to eliminate need to resolve small scales:

Reynolds Averaging Transport equations for mean flow quantities are solved

All scales of turbulence are modeled

Transient solution t is set by global unsteadiness

Filtering (LES) Transport equations for resolvable scales

Resolves larger eddies; models smaller ones

Inherently unsteady, t set by small eddies

Both methods introduce additional terms that must be modeled for

closure


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Turbulence Apr 2005

l = l/ReT

3/4

Prediction Methods


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Turbulence Apr 2005

RANS Modeling - Velocity Decomposition

Consider a point in the given flow field:

( ) ( ) ( )txutxUtxu iii ,,,rrr

+=

u'i

Ui ui

time

u


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Turbulence Apr 2005

RANS Modeling - Ensemble Averaging

Ensemble (Phase) average:

Applicable to nonstationary flows such as periodic or quasi-periodic flows

involving deterministic structures

( ) ( )( )=

=

N

n

n

iN

i txuN

txU1

,1

lim,rr

U


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Turbulence Apr 2005

( )

+

+

+=

+++

+

j

ii

jik

iikk

ii

x

uU

xx

pp

x

uUuU

t

uU )()()(

)(

.,0;0;;0; etc=+=

Deriving RANS Equations

Substitute mean and fluctuating velocities in instantaneous Navier-

Stokes equations and average:

Some averaging rules:

Given = + and = +

Mass-weighted (Favre) averaging used for compressible flows


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Turbulence Apr 2005

RANS Equations

Reynolds AveragedNavier-Stokes equations:

New equations are identical to original except :

The transported variables, U, , etc., now represent the mean flow

quantities

Additional terms appear:

Rij

are called the Reynolds Stresses

Effectively a stress

These are the terms to be modeled

( )j

ji

j

i

jik

ik

i

xuu

xU

xxp

xUU

tU

+

+

=

+

jiij uuR =

ji

j

i

j

uux

U

x

(prime notation dropped)


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Turbulence Apr 2005

Turbulence Modeling Approaches

Boussinesq approach

isotropic relies on dimensional analysis

Reynolds stress transport models

no assumption of isotropy

contains more physics

most complex and computationally expensive


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Turbulence Apr 2005

The Boussinesq Approach

Relates the Reynolds stresses to the mean flow by a turbulent (eddy)

viscosity, t

Relation is drawn from analogy with molecular transport of momentum

Assumptions valid at molecular level, not necessarily valid at

macroscopic level

t is a scalar (Rij aligned with strain-rate tensor, Sij) Taylor series expansion valid iflmfp|d

2U/dy2|


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Turbulence Apr 2005

Modeling t Oh well, focus attention on modeling t anyways

Basic approach made through dimensional arguments Units oft = t/are [m2/s]

Typically one needs 2 out of the 3 scales:

velocity - length - time

Models classified in terms of number of transport equations solved,e.g.,

zero-equation

one-equation

two-equation


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Turbulence Apr 2005

Zero Equation Model

Prandtl mixing length

model:

Relation is drawn from same analogy with molecular transport of

momentum:

The mixing length model:

assumes that vmix is proportional to lmix& strain rate:

requires that lmixbe prescribed

lmix must be calibrated for each problem Very crude approach, but economical

Not suitable for general purpose CFD though can be useful where a

very crude estimate of turbulence is required

+

==i

j

j

iijijijmixt

x

U

x

USSSl

2

1;22

mfpthv2

1l= mixmixv

2

1lt =

ijijSSl 2v mixmix


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Turbulence Apr 2005

Other Zero Equation Models

Mixing length observed to behave differently in flows near solidboundaries than in free shear flows

Modifications made to the Prandtl mixing length model to account fornear wall flows

Van Driest- Reduce mixing length in viscous sublayer (inner boundarylayer) with damping factor to effect reduced mixing

Clauser- Define appropriate mixing length in velocity defect (outerboundary) layer

Klebanoff- Account for intermittency dependency

Cebeci-Smith and Baldwin-Lomax

Accounts for all of above adjustments in two layer models Mixing length models typically fail for separating flows

Large eddies persist in the mean flow and cannot be modeled from localproperties alone


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Turbulence Apr 2005

One-Equation Models

Traditionally, one-equation models were based on transport equation

fork(turbulent kinetic energy) to calculate velocity scale, v = k1/2

Circumvents assumed relationship between v and turbulence length scale

(mixing)

Use of transport equation allows history effects to be accounted for

Length scale still specified algebraically based on the mean flow very dependent on problem type

approach not suited to general purpose CFD


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Turbulence Apr 2005

+

=

+

jjii

jjj

i

ijj

j upuuux

k

xx

U

Rx

k

Ut

k

'2

1

unsteady &

convective

production

dissipation

molecular

diffusion

turbulent

transport

pressure

diffusion

k

i

k

i

x

u

x

u

=

Turbulence Kinetic Energy Equation

Exact kequation derived from sum of products of Navier-Stokes

equations with fluctuating velocities (Trace of the Reynolds Stress transport equations)

where (incompressible form)


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Turbulence Apr 2005

Modeled Equation fork The production, dissipation, turbulent transport, and pressure

diffusion terms must be modeled

Rij in production term is calculated from Boussinesq formula

Turbulent transport and pressure diffusion:

= CDk3/2/l from dimensional arguments

t = CDk2/ (recall t k1/2l)

CD, k, and lare model parameters to be specified

Necessity to specify llimits usefulness of this model

Advanced one equation models are complete

solves for eddy viscosity

jk

jjiix

kupuuu

=+

t'2

1 Using t/kassumes kcan be transported by

turbulence as can U


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Turbulence Apr 2005

Spalart-Allmaras Model Equations

~

,f,~ 31

3

3

v11t += cvvf

1v2222 1

1f,~~

vv

ff

dSS

+=+

( )22

6

2

6/1

6

3

6

6

3~

~,g,

1

dSrrrcr

ggf w

w

ww

cc

+=

+

+=

( )2

1

2

2~

1

~

~~~1~~~

+

+

+=

dfc

xc

xxSc

Dt

Dww

j

b

jj

b

0~:conditionboundaryWall =

modified turbulent viscositymodified turbulent viscosity

distance from walldistance from wall

damping functionsdamping functions


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Turbulence Apr 2005

=

i

j

j

i

x

U

x

US2

1;2 ijijij

)-Smin(0,C ijijprodij +S

Spalart-Allmaras Production Term

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


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Turbulence Apr 2005

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 through log layer and viscous sublayer to wall

Fluents implementation can also use law-of-the-wall Economical and accurate for:

wall-bounded flows

flows with mild separation and recirculation

Weak for:

massively separated flows

free shear flows

simple decaying turbulence


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Turbulence Apr 2005

Two-Equation Models

Two transport equations are solved, giving two independent scalesfor calculating t Virtually all use the transport equation for the turbulent kinetic

energy, k

Several transport variables have been proposed, based ondimensional arguments, and used for second equation

Kolmogorov, : t k /, l k1/2/, k / is specific dissipation rate defined in terms of large eddy scales that define supply rate ofk

Chou, : t k2/, l k3/2/ Rotta, l: t k1/2l, k3/2/l

Boussinesq relation still used for Reynolds Stresses


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Turbulence Apr 2005

Standard k- Model Equations

ijijtjkj

SSSSxk

xDtDk 2;2t =+

+=

( )

2

2t1

t CSCkxxDt

D

jj

+

+

=

kk--transport equationtransport equation

--transport equationtransport equationproductionproduction dissipationdissipation

2,,, CCik

coefficientscoefficients

turbulent viscosityturbulent viscosity

2

k

Ct =

inverse time scaleinverse time scale

Empirical constants determined from benchmark

experiments of simple flows using air and water.


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Turbulence Apr 2005

Simple flows render simpler model equations

Coefficients can be isolated and compared with experiment e.g.,

Uniform flow past grid

Standard k- equations reduce to just convection and dissipation terms

Homogeneous Shear Flow

Near-Wall (Log layer) Flow

kC

xU

x

kU

2

2d

d;

d

d ==

Closure Coefficients

l S i C Ad d l i i


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Turbulence Apr 2005

BuoyancyBuoyancyproductionproduction

DilatationDilatationDissipationDissipation

RT

k

xgS

x

k

xDt

Dk

it

tit

jkj

2

Pr

2t

+

+

=

Standard k- Model

High-Reynolds number model

(i.e., must be modified for the near-wall region) The term standard refers to the choice of coefficients

Sometimes additional terms are included

production due to buoyancy

unstable stratification (gT >0) supports kproduction dilatation dissipation due to compressibility

added dissipation term, prevents overprediction of spreading rate in

compressible flows

Fl U S i C Ad d Fl T i i


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Turbulence Apr 2005

Standard k- 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, low-Re flows

cannot predict round jet spreading rate

Variants of the k- model have been developed to address itsdeficiencies

RNG and Realizable

Fl t U S i C t Ad d Fl t T i i


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Turbulence Apr 2005

RNG k- Model Equations Derived using renormalization group theory

scale-elimination technique applied to Navier-Stokes equations

(sensitizes equations to specific flow regimes)

kequation is similar to standard k- model

Additional strain rate term in equation most significant difference between standard and RNG k- models

Analytical formula for turbulent Prandtl numbers

Differential-viscosity relation for low Reynolds numbers

Boussinesq model used by default

( ) ( ) whereCSCkxxDt

Dt

jj

*2

21eff +

=

t

kS

C

CC

+=

=

+

+=

eff

0

3

0

3

2

*

2

tscoefficienare,

1

1

--transport equationtransport equation

Fl t U S i C t Ad d Fl t T i i


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Turbulence Apr 2005

RNG k- Model Pros &Cons

For large strain rates:

where > 0, is augmented, and therefore kand t are reduced

Option to modify turbulent viscosity to account for swirl

Buoyancy and compressibility terms can be included

Improved performance over std. k- model for rapidly strained flows

flows with streamline curvature

Still suffers from the inherent limitations of an isotropic eddy-

viscosity model

Fluent User Services Center Advanced Fluent Training


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Turbulence Apr 2005

Standard k- model could not ensure: Positivity of normal stresses

Schwarzs inequality of shear stresses

Modifications made to standard model

kequation is same; new formulation fort and C is variable

equation is based on a transport equation for the mean-square vorticityfluctuation

02 u

( ) uuuu 222

Realizable k- Model: Motivation



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Turbulence Apr 2005

How can normal stresses become negative?

Standard k- Boussinesq viscosity relation:

Normal component:

Normal stress will be negative if:

3

2- ij

2

kx

U

x

UkCuu

i

j

j

iji

+

=

23

2

22

x

UkCku

=

3.73

1 >

Cx

Uk

Realizable k- Model: Realizability



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Turbulence Apr 2005

Realizable k-Model: C

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

turbulence (0.09 in log-layerSk/= 3.3, 0.05 in shear layer ofSk/= 6)

C contours for 2D backward-facing step

C along

bottom-wall



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Turbulence Apr 2005

Realizable k- Model Equations

kU

AA

Ck

C

s

*

0

2

t

1,

+

==

where

ijijkijkij

ijijijij SSSS

SSSWSSU ==+ ~,~,*

( )WAA s 6cos3

1,cos6,04.4 10

===

0.1,/,5

,43.0max 21 ==

+= CSkC

++

+

= kCSCxxDt

D

jj

2

21

t

--transport equationtransport equation

turbulent viscosityturbulent viscosity



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Turbulence Apr 2005

Realizable k- Model Pros & Cons

Performance generally exceeds the standard k- model

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 speading rate for round and plane jets

Still suffers from the inherent limitations of an isotropic eddy-viscosity

model



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Turbulence Apr 2005

Standard and SST k- Models k-models are a popular alternative to k-

~ / k

t k /

Wilcoxs original model was found to be quite sensitive to

inlet and far-field boundary values of

Can be used in near-wall region without modification Latest version contains several refinements:

reduced sensitivity to boundary conditions

modifcation for the round-jet/plane-jet anomaly

compressibility effects

low-Re (transitional) effects



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Turbulence Apr 2005

Standard k-Model The most well-known Wilcox k-model until recently was his 1988

model (will be referred to as Wilcox original k-model)

Fluent v6 Standard k- model is Wilcox 1998 model

Wilcox original k-is a subset of the Wilcox 1998 model, and can be

recovered by deactivating some of the options and changing some of the

model constants

+

+

=

+

+

=

=

j

t

jj

i

ij

jk

t

jj

iij

t

xxf

x

U

kDt

D

xk

xkf

xU

DtDk

k

2

*

*

*



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Turbulence Apr 2005

Standard k-Turbulent Viscosity

Turbulent viscosity is computed from:

The dependency of* uponReTwas designed to recover the correct

asymptotic values in the limiting cases. In particular, note that:

kt *=

0.1,Re,6

125

9,

3,

Re1

Rewhere

*

*

0

*

0**

===

==

+

+=

kR

R

R

Tk

ii

kT

kT

turbulent)(fullyas TRe1*



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U S C

www.fluentusers.com

g

Turbulence Apr 2005

Standard k-Turbulent Kinetic Energy

Note the dependence uponRe , Mt , andk

Dilatation dissipation is accounted for via Mt term

The cross-diffusion parameter (k) is designed to improve free shear

flow predictions

444 3444 2143421

321

kofDiffusion

kofratenDissipatio

kofproduction

+

+

=jk

t

jj

iij

x

k

xkf

x

U

Dt

Dk

**

( )( )

( )09.0,5.1,0.2

8,Re1

Re154

1

**

4

4

**

***

===

=+

+=

+=

k

T

T

i

ti

RR

R

MF( )

44 344 21parameterdiffusion-cross

jj

k

k

k

k

k

tt

tttt

tt

t

xx

kf

RTaMa

kM

MMMM

MMMF

=

>++

=

===

>

=

3

2

2

022

0

2

0

2

0

1,

04001

6801

01

,4

1,

2

0

*



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g

Turbulence Apr 2005

Note the dependence upon Re , Mt , and

Vortex-stretching parameter () designed to remedy the plane/round-jetanomaly

Standard k-Specific Dissipation Equation

+

+

=j

t

jj

iij

xxf

x

U

kDt

D

2

( )

( )

=

+

=

=

=+

+=

+=

====+

+=

i

j

j

iij

i

j

j

iij

kijkij

t

i

ii

T

T

x

U

x

U

x

U

x

US

SfMF

RR

R

2

1,

2

1

5.1,,801

701,1

0.2,95.2,9

1,

25

13,

Re1

Re

*

3*

**

00

*



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g

Turbulence Apr 2005

Standard k-Model Sub-models & Options (I) Transitional flow option

Corresponds to all terms involving ReTterms in the model

equations

Deactivatedby default Can benefit low-Re flows where the extent of the

transitional flow region is large

Compressibility Effects option

Takes effects viaF(Mt) Accounts for dilatation dissipation

Available with ideal-gas option only and is turned onby

default

Improve high-Mach number free shear and boundary

layer flow predictions - reduces spreading rates

kk

kdds

x

u

x

u

=+=

3

4,



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g

Turbulence Apr 2005

Standard k-Model Sub-models & Options (II) Shear-Flow Corrections option

Controls both cross-diffusion and vortex-stretching

terms - Activatedby default

Cross-diffusion term (in k-equation)

Designed to improve the model performance for free

shear flows without affecting boundary layer flows

Vortex-stretching term

Designed to resolve the round/plane-jet anomaly

Takes effects for axisymmetric and 3-D flows but

vanishes for planar 2-D flows

( )3*,801701

=++

=kijkij S

f

44 344 21 parameterdiffusion-cross

jj

k

k

k

k

k

xx

kf

=

>

+

+

=

3

2

21

,0

4001

6801

01

*



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www.fluentusers.comTurbulence Apr 2005

Menters SST k-ModelBackground

Many people, including Menter (1994), have noted that:

Wilcox original k-model is overly sensitive to the freestream value(BC) of, while k-model is not prone to such problem

k-model has many good attributes and perform much better than k-models forboundary layer flows

Most two-equation models, including k-models, over-predict turbulentstresses in the wake (velocity-defect) region, which leads to poor

performance of the models for boundary layers under adverse pressure

gradient and separated flows



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Menters SST k-Model Main Components The SST k-model consists of

Zonal (blended) k-/k-equations (to address item 1 and 2 in the previous

slide)

Clipping of turbulent viscosity so that turbulent stresses stay within what

is dictated by the structural similarity constant. (Bradshaw, 1967) -

addresses item 3 in the previous slide

Inner layer

(sublayer,

log-layer)Wilcox original k-model

l

23

k

=

Wall

Outer layer

(wake and

outward)

k-model transformed

from std. k-model

Modified Wilcox k-model



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Menters SST k-ModelInner Layer The k-model equations for the inner layerare taken from the Wilcox

original k- model with some constants modified

++=

+

+

=

j

t

jj

iij

t

jk

t

jj

iij

xxxU

DtD

x

k

xk

x

U

Dt

Dk

1

21

1

1

*

( )41.0,,09.0

0.2,176.1,075.0

1

*2*

11

*

111

===

===

k

( )

( )

==

=

22

2

22

21

1

500,

09.0

2maxarg,argtanh

,amax

yy

kF

F

kat



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Menters SST k-Model Outer Layer The k-model equations for the outer layerare obtained from by transforming

the standard k-equations via change-of-variable

Turbulent viscosity computed from:

jj

j

t

jj

i

ijt

jk

t

jj

iij

xx

k

xxx

U

Dt

D

x

k

xk

x

U

Dt

Dk

+

+

+

=

++

=

12 2

2

2

2

2

2

*

( ) 41.0,,09.0168.1,0.1,0828.0

2

*2*

22

*

222

===

===

k

kt =



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Menters SST k-ModelBlending the Equations The two sets of equations and the model constants are blended in such

a way that the resulting equation set transitions smoothly from one

equation to another.

( )

=

=

=

202

2

2

2*1

4

11

10,1

2max

4,

500,maxminarg

argtanh

jj

k

k

xx

kCD

yCD

k

yy

k

F

( )

( )

,,,where

1

1

2111

outer

1

inner

1

k

FF

Dt

DkF

Dt

DkF

=

+=

++

+

layeroutlerthein

layerinnerthein

0

1

1

1

=

F

F

Wilcox original k-model

l

23

k=

Wall

k-model transformed

from std. k-model

Modified Wilcox k-model


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Menters SST k- ModelBlended k-Equations The resulting blended equations are:

Wall

( )jj

j

t

jj

iij

t

jk

t

jj

i

ij

xxkF

xxx

U

Dt

D

x

k

xk

x

U

Dt

Dk

+

+

+

=

+

+

=

112 21

2

*

( ) ,,,,1 2111 kFF =+=


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Menters SST k-Model Turbulent Viscosity Honors the structural similarity constant for boundary layers

(Bradshaw, 1967)

Turbulent stress implied by turbulence models can be written as:

In many flow situations (e.g. adverse pressure gradient flows), productionof TKE can be much larger than dissipation (Pk>> ), which leads to

predicted turbulent stress larger than what is implied by the structural

similarity constant How can turbulent stress be limited? - A simple trick is to clip turbulent

viscosity such that:

PkayU ktt 1 ===

1967)(Bradshaw,11 ak

vukavu ==

kat 1


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Menters SST k-Model Clippingt

Turbulent viscosity for the inner layeris computed from:

Remarks

F2 is equal to 1 inside boundary layer and goes to zero far from the wall

and free shear layers The name SST (shear-stress transport) is a big word for this simple trick

Note that the vorticity magnitude is used (strain-rate magnitude could alsobe used)

( )

( )

magnitude)(vorticityijij

t

yy

kF

Fkak

Fka

==

=

=

2

500,

09.0

2maxarg,argtanh

,min,amax

22

2

22

2

1

21

1


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Menters SST k-Model Submodels & Options SST k-model comes with:

Transitional Flows option (Off by

default) Compressibility Effects option when

ideal-gas option is selected (On bydefault)

The original SST k-model in theliterature does not have any of theseoptions

These submodels are being borrowedfrom Wilcox 1998 model - should beused with caution

Do not activate any options to recover theoriginal SST model


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k-ModelsBoundary Conditions Wall boundary conditions

The enhanced wall treatment (EWT) is the sole near-wall option fork-

models. Neither the standard wall functions option nor the non-equilibriumwall functions option is available fork-models in FLUENT 6

The blended laws of the wall are used exclusively

values at wall adjacent cells are computed by blending the wall-limiting

value (y->0) and the value in the log-layer The k-models can be used with either a fine near-wall mesh or a coarse

near-wall mesh

For other BCs (e.g., inlet, free-stream), the following relationship is used

internally, whenever possible, to convert to and from different turbulencequantities:

09.0, ** == k


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Faults in the Boussinesq Assumption

Boussinesq:Rij = 2tSij Is simple linear relationship sufficient?

Rij is strongly dependent on flow conditions and history

Rij changes at rates not entirely related to mean flow processes

Rij is not strictly aligned with Sij for flows with:

sudden changes in mean strain rate

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

curvature)

rotating fluids

stress-induced secondary flows

Modifications to two-equation models cannot be generalized for

arbitrary flows


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0)()( =+ ijji uNSuuNSu

Reynolds Stress Models

Starting point is the exact transport equations for the

transport of Reynolds stresses,Rij

six transport equations in 3d

Equations are obtained by Reynolds-averaging the product

of the exact momentum equations and a fluctuating velocity.

The resulting equations contain several terms that must be

modeled


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Reynolds Stress Transport Equations

k

ijk

ijijij

ij

x

JP

Dt

DR

++=

Generation

+

k

ikj

k

j

kiijx

Uuu

x

UuuP

+ i

j

j

iij

xu

xup

k

j

k

iij

x

u

x

u

2

Pressure-StrainRedistribution

Dissipation

Turbulent

Diffusion

(modeled)

(related to )

(modeled)

(computed)

(incompressible flow w/o body

forces)

Reynolds Stress

Transport Eqns.

434214342144 344 21

)( jik

kjiikjjkiijk uux

uuuupupJ

++

Pressure/velocityfluctuations

Turbulenttransport

Moleculartransport


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ijij

3

2=

Dissipation Modeling

Dissipation rate is predominantly associated with small scale eddy

motions

Large scale eddies affected by mean shear

Vortex stretching process breaks eddies down into continually smaller

scales

The directional bias imprinted on turbulence by mean flow is graduallylost

Small scale eddies assumed to be locally isotropic

is calculated with its own (or related) transport equation Compressibility and near-wall anisotropy effects can be accounted for


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Turbulent Diffusion

Most closure models combine the pressure diffusion with the

triple products and use a simple gradient diffusion hypothesis

Overall performance of models for these terms is generally

inconsistent based on isolated comparisons to measured tripleproducts

DNS data indicate that abovep terms are negligible

( ) ( ) ( )

=

++

l

ji

lks

k

ji

k

jikikjkji

k x

uuuu

kC

xuu

xuu

puuu

x

'

=k

ji

k x

uukC

x

2

Or even a simpler model


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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

iij

x

u

x

up

i

j

j

i

i

j

i

ii x

u

x

u

x

uu

xx

p

+

=

21

+

+= ij

m

lml

l

ilj

l

j

liijijx

Uuu

x

Uuu

x

Uuucbc

3

221

i

j

j

i

ii x

U

x

u

xx

p

=

2

1 2

Rapid PartSlow Part

ijji kuukijb

3

2

where

wijijijij ,2,1, ++=

meanmean

gradientgradient

Pressure-Strain Modeling


fl


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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 Model

Expands the basic LRR model to include non-linear (quadratic) terms

Superior performance demonstrated for some basic shear flows

plane strain, rotating plane shear, axisymmetric expansion/contraction


fl t


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www.fluentusers.comp

Characteristics of RSM

Effects of curvature, swirl, and rotation are directly accounted for in

the transport equations for the Reynolds stresses.

When anisotropy of turbulence significantly affects the mean flow,

consider RSM

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

50-60% more cpu time per iteration and 15-20% additional memory Strong coupling between Reynolds stresses and the mean flow

number of iterations required for convergence may increase


fl t


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iu

Heat Transfer

The Reynolds averaging process produces an additional term in the

energy equation:

Analogous to the Reynolds stresses, this is termed the turbulent heat flux

It is possible to model a transport equation for the heat flux, but this is not

common practice

Instead, a turbulent thermal diffusivity is defined proportional to the

turbulent viscosity

The constant of proportionality is called the turbulent Prandtl number

Generally assumed thatPrt~ 0.85-0.9

Applicable to other scalar transport equations

Documents

Adv Turb v62 01 Overview