turbulence models and their application

8/10/2019 Turbulence models and their application

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TURBULENCE MODELS AND

THEIR APPLICATIONS

Presented by:

T.S.D.Karthik

Department of Mechanical EngineeringIIT Madras

Guide:Prof. Franz Durst

10thIndo German Winter Academy 2011


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2


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Outline3

Turbulence models introduction

Boussinesq hypothesis

Eddy viscosity concept

Zero equation model

One equation model

Two equation models

Algebraic stress model

Reyolds stress model

Comparison

Applications

Developments

Conclusion

Turbulence Models and Their Applications


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Turbulence models

A turbulence model is a procedure to close the system of meanflow equations.

For most engineering applications it is unnecessary to resolvethe details of the turbulent fluctuations.

Turbulence models allow the calculation of the mean flowwithout first calculating the full time-dependent flow field.

We only need to know how turbulence affected the mean flow.

In particular we need expressions for the Reynolds stresses.

For a turbulence model to be useful it: must have wide applicability,

be accurate,

simple,

and economical to run.

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Common turbulence models

Classical models. Based on Reynolds Averaged Navier-Stokes (RANS)equations (time averaged): Zero equation model: mixing length model.

One equation model

Two equation models: k- style models (standard, RNG, realizable), k-model, and ASM.

Seven equation model: Reynolds stress model.

The number of equations denotes the number of additional PDEs thatare being solved.

Large eddy simulation. Based on space-filtered equations. Timedependent calculations are performed. Large eddies are explicitlycalculated. For small eddies, their effect on the flow pattern is takeninto account with a sub-grid model of which many styles areavailable.

DNS

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Classification6



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Prediction Methods7



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Boussinesq hypothesis

Many turbulence models are based upon the Boussinesqhypothesis. It was experimentally observed that turbulence decays unless there is

shear in isothermal incompressible flows.

Turbulence was found to increase as the mean rate of deformationincreases.

Boussinesq proposed in 1877 that the Reynolds stresses could be linkedto the mean rate of deformation.

Using the suffix notation where i, j, and k denote the x-, y-, andz-directions respectively, viscous stresses are given by:

Similarly, link Reynolds stresses to the mean rate ofdeformation

i

j

j

iij

x

u

x

u

8



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Eddy Viscosity Concept

One of the most widely used concept

Reynoldsstress tensor

A new quantity appears: the turbulent viscosity or eddy viscosity

(t).

The second term is added to make it applicable to normal

turbulent stress.

The turbulent viscosity depends on the flow, i.e. the state of

turbulence.

The turbulent viscosity is not homogeneous, i.e. it varies in space.

i

j

j

i

tji

T

ijx

U

x

Uuu ''

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Eddy Viscosity Concept

It is, however, assumed to be isotropic. It is the same in

all directions. This assumption is valid for many flows,

but not for all (e.g. flows with strong separation or swirl).

The turbulent viscosity may be expressed as

This concept assumes that Reynolds stress tensor can be

characterized by a single length and time scales.

cccct tlorlu /)( 2

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Major Drawbacks

Interaction among eddies is not elastic as in the case for molecularinteractions in kinetic theory of gases.

For many turbulent flows, the length and time scale of characteristic

eddies is not small compared with the flow domain (boundarydominated flows).

The eddy viscosity is a scalarquantity which may not be true for simple

turbulent shear flows. It also fails to distinguish between plane shear,plane strain and rotating plane shear flows.

Successful 2D shear flows. Erroneous results for simple shear flows

such as wall jets and channel flows with varying wall roughness.

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Zero Equation Model - Mixing

Length Model

On dimensional grounds one can express the kinematic turbulentviscosity as the product of a velocity scale and a length scale:

If we then assume that the velocity scale is proportional to the

length scale and the gradients in the velocity (shear rate, which hasdimension 1/s):

we can derive Prandtls(1925) mixing length model:

Algebraic expressions exist for the mixing length for simple 2-Dflows, such as pipe and channel flow.

)()/()/( 2 msmsmt

y

U

yU

mt

2

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Equations for mixing length

Wall boundary layers

= boundary layer thickness

y = distance from the wall

= 0.09

K = Von-Karman Constant

Developed pipe flows

R = radius of the pipe or the half width of the duct

)//(

)//(

Kyl

KyKyl

m

m

42 )1(06.0)1(08.014.0R

y

R

y

R

lm

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Mixing Length Model Discussion

Advantages: Easy to implement.

Fast calculation times.

Good predictions for simple flows where experimental correlations forthe mixing length exist.

Used in higher models Disadvantages:

Completely incapable of describing flows where the turbulent lengthscale varies: anything with separation or circulation.

Only calculates mean flow properties and turbulent shear stress.

Cannot switch from one type of region to another History effects of turbulence are not considered.

Use:

Sometimes used for simple external aero flows.

Pretty much completely ignored in commercial CFD programs today.

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One Equation Model

Different transport equation for k is solved.

L is defined algebraically, in a similar manner as mixing length.

PDE for turbulent KE : Diffusion, Production and Dissipation terms

kl

ku

mt

2/1

c

ix

ju'

ix

ju'

ix

jU

'j

'ui

uP

)'i

'uj

'uju

ix

'ju'ju

'ip'u(i

D

where

Pi

xi

D

ixk

iU

2

12

15


c

D

i

j

j

i

i

j

t

ik

t

ii

il

kC

x

U

x

U

x

U

x

k

xx

kU

2

3

Modeled Equation


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One Equation Model Discussion

Economical and accurate for:

Attached wall-bounded flows.

Flows with mild separation and recirculation.

Developed for use in unstructured codes in the aerospace industry.

Popular in aeronautics for computing the flow around aero plane

wings, etc.

Weak for:

Massively separated flows.

Free shear flows.

Decaying turbulence.

Complex internal flows.

Characteristic length still experimental.

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Two Equation Models

The k-model

Equations for k and , together with the eddy-viscosity stress-

strain relationship constitute the k-turbulence model.

is the dissipation rate of k.

If kand are known, we can model the turbulent viscosity as:

22/32/1 kkk

t

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The k-model

K equation:

Model (simplified) equation for k after using Boussinesq

assumption by which the fluctuation terms can be linked to the

mean flow is as follows:

2

2

09.0 kwith

xx

k

x

k

xx

U

x

U

x

U

x

kU

t

k

t

iiik

t

ii

j

j

i

i

j

t

i

i

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Turbulent Dissipation

We can define the rate of dissipation per unit mass as:

Equation for

''.2ijij

ee

ndestructioViscousY

stretchingby vortexProductionP

citymean vortiofgradientbyProductionP

fieldflowmeanofndeformatiobyProductionP

fieldflowmeanofndeformatiobyProductionP

nfluctuatiopressurebytransportDiffusiveD

nfluctuatiobytransportDiffusiveD

transportviscousDiffusiveD

4

3

2

1

p

f

v

4321

where

YPPPPDDDDtD pfv

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Turbulent Dissipation

ix

u

ix

p

kxk

u

kx

kx

kx

k

DtD ''2

'

2

'2

2'''

2

2'

'2''''

2

lx

kx

iu

lx

ku

lx

iu

kx

ku

lx

kx

iU

lx

iu

ku

kx

iU

kx

klu

ix

lu

lx

ku

lx

iu

vDfD

pD

3

P2

P1

P 4

P Y

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Model Equation for

A model equation for is derived by multiplying the kequation by (/k) and introducing model constants.

Closure coefficients found emperically

K- model leads to all normal stresses being equal,which is usually inaccurate.

i

j

i

j

j

it

zz

jk

t

jj

j

t

x

U

x

U

x

UP

KC

KPC

xxxU

kC

2

21

2

1k 92.12zC 44.11zC 09.0C 33.1z

21



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Applications22

Flow on a backward facing step using k-epsilon model



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Single and multiple jet flows using k-epsilon models

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k- model discussion

Advantages:

Relatively simple to implement.

Leads to stable calculations that converge relatively easily.

Reasonable predictions for many flows.

Disadvantages:

Poor predictions for:

swirling and rotating flows,

flows with strong separation,

axis symmetric jets,

certain unconfined flows, and

fully developed flows in non-circular ducts.

Valid only for fully turbulent flows.

Requires wall function implementation.

Modifications for flows with highly curved stream lines.

Production of turbulence in highly strained flows is over predicted.

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More two equation models

The k-model was developed in the early 1970s. Its strengths

as well as its shortcomings are well documented.

Many attempts have been made to develop two-equationmodels that improve on the standard k-model.

We will discuss some here:

k- RNG model. k- realizable model.

k- model.

Algebraic stress model.

Non-linear models.

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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 for interaction between turbulence dissipation and mean

shear.

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, separated flows.

Wall heat and mass transfer.

Also, time dependent flows with large scale motions (vortex shedding)

But still does not predict the spreading of a round jet correctly.

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Realizable k- model

Shares the same turbulent kinetic energy equation as the

standard k-model.

Improved equation for .

Variable Cinstead of constant.

Improved performance for flows involving:

Planar and round jets (predicts round jet spreading correctly).

Boundary layers under strong adverse pressure gradients or

separation. Rotation, recirculation.

Strong streamline curvature.

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Realizable k- Cequations

Eddy viscosity computed from.

kU

AA

Ck

C

s

*

0

2

t

1,

ijijijijSSU *

WAAs 6cos3

1

,cos6,04.4

1

0

ijij

kijiijSSS

S

SSSW

~,~

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Realizable k- positivity of

normal stresses

Boussinesq viscosity relation:

Normal component:

Normal stress will be negative if:

2

tijt ;3

2-

kCk

x

u

x

uuu

i

j

j

i

ji

23

2

22

x

UkCku

3.73

1

Cx

Uk

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k- model

This is another two equation model. In this model is an inversetime scale that is associated with the turbulence.

This model solves two additional PDEs:

A modified version of the kequation used in the k-model. A transport equation for .

The turbulent viscosity is then calculated as follows:

Its numerical behavior is similar to that of the k-models.

It suffers from some of the same drawbacks, such as theassumption that tis isotropic.

kt

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The SST Model33

The SST (Shear Stress Transport) model is an eddy-viscosity model which

includes two main novelties:

1. It is combination of a k-!model (in the inner boundary layer) and k-

model (in the outer region of and outside of the boundary layer);

2. A limitation of the shear stress in adverse pressure gradient regions isintroduced.

Actual Model


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Non-linear models

The standard k- model is extended by including second andsometimes third order terms in the equation for the Reynoldsstresses.

One example is the Speziale model:

Here f() is a complex function of the deformation tensor,

velocity field and gradients, and the rate of change of thedeformation tensor.

The standard k-model reduces to a special case of this modelfor flows with low rates of deformation.

These models are relatively new and not yet used very widely.

)/,,/,(*423

2''

2

32

2

xUtEEfk

CCEk

Ckuu Dijijjiij u

34



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Reynolds stress model

RSM closes the Reynolds-Averaged Navier-Stokes equations bysolving additional transport equations for the six independentReynolds stresses.

Transport equations derived by Reynolds averaging the product of themomentum 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 is good 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.

Flows involving separation.

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Reynolds stress transport


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equation

The exact equation for the transport of the Reynolds stress Rij:

This equation can be read as:

rate of change of plus

transport of Rijby convection, equals

rate of production Pij, plus

transport by diffusion Dij, minus

rate of dissipation ij

, plus

transport due to turbulent pressure-strain interactions ij, plus

transport due to rotation ij.

This equation describes six partial differential equations, one for the

transport of each of the six independent Reynolds stresses.

ijijijijD

ijP

Dt

ijDR

'' ji uuijR

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R ld t t t


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equation

The various terms are modeled as follows: Production Pijis retained in its exact form.

Diffusive transport Dijis modeled using a gradient diffusion assumption.

The dissipation ij, is related to as calculated from the standard

equation, although more advanced models are available also.

Pressure strain interactions ij, are very important. These include pressurefluctuations due to eddies interacting with each other, and due tointeractions between eddies and regions of the flow with a different meanvelocity. The overall effect is to make the normal stresses more isotropic

and to decrease shear stresses. It does not change the total turbulentkinetic energy. This is a difficult to model term, and various models areavailable. Common is the Launder model. Improved, non-equilibriummodels are available also.

Transport due to rotation ijis retained in its exact form.

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RSM equations

m

ijm

m

j

imijx

UR

x

URP:exactProduction

)''('''j

uiki

ujk

pk

uj

ui

uijk

J

kxijk

J

ijD

:exacttransportDiffusive

waystandardtheincalculatedviscositykinematicturbulenttheis

:modeltransportDiffusive

t

ij

k

t

m

ij

k

t

m

ij Rgraddiv

x

R

xD

)(

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RSM equations continued

ijij

k

j

k

i

ijx

u

x

u

3

2''

2

:modelnDissipatio:exactnDissipatio

pressuretheis

:modelstrainPressure

:exactstrainPressure

P

PPCkRk

C

x

u

x

up

ijijijijij

i

j

j

i

ij

)()(

'''

3

2

23

2

1

vectorrotationtheis

indicestheondepending1or0,,is

:(exact)termRotational

k

1

)(2

ijk

jkmimikmjmkij

e

eReR

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Algebraic stress model

The same kand equations are solved as with the standard k-model.

However, the Boussinesq assumption is not used.

The full Reynolds stress equations are first derived, and then some

simplifying assumptions are made that allow the derivation of algebraic

equations for the Reynolds stresses.

Thus fewer PDEs have to be solved than with the full RSM and it is

much easier to implement.

The algebraic equations themselves are not very stable, however, and

computer time is significantly more than with the standard k-model. This model was used in the 1980s and early 1990s. Research continues

but this model is rarely used in industry anymore now that most

commercial CFD codes have full RSM implementations available

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Setting boundary conditions

Characterize turbulence at inlets and outlets (potential backflow). k-models require kand .

Reynolds stress model requires Rijand .

Other options:

Turbulence intensity and length scale.

Length scale is related to size of large eddies that contain most of energy.

For boundary layer flows, 0.4 times boundary layer thickness: l 0.4d99.

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:

10/1

t

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Some Applications42

Turbulent Annular Flow



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43

The figures show plots of the normalized tangential velocity, for each ofthe turbulence models, plotted with the experimental data.

RNG k - epsilon model produces the best results with the standard k -epsilon model giving the worst but this variation is small compared totheir deviations from the experimental data.

The k-L mixing length model does lead to an answer which predicts themovement of the maximal tangential velocity from the inner wall to thecentre of the annulus better than the other models. Where theimplementation of the model fails is in its prediction of the flow near thewalls.


Flocculation tank Analogy with flow over a back


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step

In the middle of the channel, the flow separate due to the small step size of height h. The

flow reattaches at about 7 times the step height further downstream - similar to the 180

degree bend in the flocculation tank where we have flow separation and reattachment

downstream

Analyzed using K-,K-SST, K-realizable, K-RNG, RSM turbulence models and compared

with experimental data.

Plotting the derivative du/dy, the change in direction of velocity in x direction with respect

to y at the wall, the reattachment point is easily identified. At the wall, separated flow will

give a negative du/dy, while reattaches flow has a positive du/dy value.Turbulence Models and Their Applications


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Turbulenc

e Model

K-e K-W SST K-e

realizable

RSM

Reattach

ment

Ratio

0.195/0.

038 =

5.13

0.242/0.0

38 = 6.37

0.235/0.0

38 = 6.18

0.2/0.038

= 5.26


step

The K- model under-predicts the

reattachment length. K- SST and K-

realizable gives the most accurate

representation of the back step flow with

reattachment length. However, from literature

reviews, K- realizable is more proven for a

variety of types of flows.

Below in Figure, the stream contours (of the

averaged velocity) of the Re=48,000 for the k-

realizable model case closely approximate the

experimental results.



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Flow over airfoil46

Contour plots of the predicted turbulentviscosity around an airfoil obtained with

four different steady-state turbulence

models: an algebraic model, a one-

equation model, and a duo of two-

equation models.

While the Spalart-Allmaras and Chien k-

epsilon models are in rough agreement

with each other, the SST and Baldwin-

Lomax models predict a very differentturbulent viscosity distribution.

If you looking solely at performance in the shear layer, you might want to

choose either the Spalart or Chien models.


Comparison of RANS turbulence


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Comparison of RANS turbulence

models

Model Strengths Weaknesses

Zero Equation

Model

Economical (1-eq.); good track

record for mildly complex B.L.

type of flows.

Not very widely tested yet; lack of sub-

models (e.g. combustion, buoyancy).

STD k-

Robust, economical,

reasonably accurate; long

accumulated performance data.

Mediocre results for complex flows with

severe pressure gradients, strong streamline

curvature, swirl and rotation. Predicts thatround jets spread 15% faster than planar jets

whereas in actuality they spread 15% slower.

RNG k-

Good for moderately complex

behavior like jet impingement,

separating flows, swirling flows,

and secondary flows.

Subjected to limitations due to isotropic

eddy viscosity assumption. Same problem

with round jets as standard k-.

Realizable

k-

Offers largely the same benefits

as RNG but also resolves the

round-jet anomaly.

Subjected to limitations due to isotropic

eddy viscosity assumption.

Reynolds Stress

Model

Physically most complete model

(history, transport, and

anisotropy of turbulent stresses

are all accounted for).

Requires more cpu effort (2-3x); tightly

coupled momentum and turbulence

equations.

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Recommendation

Start calculations by performing 100 iterations or so with standard k- modeland first order upwind differencing. For very simple flows (no swirl orseparation) converge with k- model.

If the flow involves jets, separation, or moderate swirl, converge solution withthe realizable k- model.

If the flow is dominated by swirl (e.g. a cyclone or un-baffled stirred vessel)converge solution deeply using RSM and a second order differencing scheme.If the solution will not converge, use first order differencing instead.

Ignore the existence of mixing length models and the algebraic stress model.

Only use the other models if you know from other sources that somehowthese are especially suitable for your particular problem (e.g. Spalart-Allmaras for certain external flows, k- RNG for certain transitional flows, ork-).

48


Other Numerical Methods:


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Other Numerical Methods:

DNS (Direct Numerical Simulation)

Very accurate

High computing time

Works on small Reynolds number flows

Used to verify the turbulence model

Some arbitrary initial velocity field is set up and the Navier-Stokes

equations are used directly to describe the evolution of this field

over time.

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LES (Large Eddy Simulation)

More accurate than RANS

More computing time than RANS

Middle route between DNS and RANS

Can work on larger Reynolds number and more complex flows

Simulations for large scales and RANS for small scales

Both DNS and LES require to solve the instantaneous Navier-Stokesequations in time and three-dimensional space.

Hybrid ApproachRANS + LES

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Developments

EddyViscosity

Models

ReynoldsStressModels

ProbabilityDensity

Functions

Accuracy increasesComplexity increases

Computing time increases

Usability decreases

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Conclusion52

Different turbulence modelsstrengths and weaknesses

Usage of models specific to certain flows

ApplicationsWhere to use what?

Compromise between accuracy and computing power

Other developments



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References

Turbulent Flows Fundamentals, Experiments and Modeling,G.Biswas and V.Eswaran, Narosa Publishing House, 2002

Turbulence Modeling for CFDWilcox, D.C, 1993

Fluid Mechanics - An Introduction to the Theory of Fluid Flows,Franz Durst, Springer, 2008

Turbulence model validation - https://confluence.cornell.edu/

http://www.tfd.chalmers.se/doct/comp turb model

FluentModeling Turbulence

53

https://confluence.cornell.edu/https://confluence.cornell.edu/https://confluence.cornell.edu/https://confluence.cornell.edu/https://confluence.cornell.edu/


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