INTRODUCTION TO TURBULENCE MODELING - Web Space that the turbulence length scale l is given by an...

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ME637 G. Ahmadi 1 INTRODUCTION TO TURBULENCE MODELING Goodarz Ahmadi Department of Mechanical and Aeronautical Engineering Clarkson University Potsdam, NY 13699-5725 In this section, an introduction to the historical development in turbulence modeling is provided. Outline Viscous Fluid Turbulence Classical Phenomenological models (Mixing Length) One-Equation Models Two-Equation Models (The e - k Model) Stress Transport Models Material-Frame Indifference Continuum Approach Consistency of the e - k Model Rate-Dependent Model

Transcript of INTRODUCTION TO TURBULENCE MODELING - Web Space that the turbulence length scale l is given by an...

ME637 G. Ahmadi 1

INTRODUCTION TO TURBULENCE MODELING

Goodarz Ahmadi Department of Mechanical and Aeronautical Engineering

Clarkson University Potsdam, NY 13699-5725

In this section, an introduction to the historical development in turbulence modeling is provided. Outline • Viscous Fluid • Turbulence • Classical Phenomenological models (Mixing Length) • One-Equation Models • Two-Equation Models (The ε−k Model) • Stress Transport Models • Material-Frame Indifference • Continuum Approach • Consistency of the ε−k Model • Rate-Dependent Model

ME637 G. Ahmadi 2

VISCOUS FLOW The conservation laws for a continuous media are: Mass

0)(t

=ρ⋅∇+∂ρ∂

u

Momentum

tfu

⋅∇+ρ=ρdtd

Angular Momentum

tt =T Energy

he ρ+∇+∇=ρ qu:t& Entropy Inequality

0Th

)T

( >ρ

−⋅∇−ηρq&

Constitutive Equation

Experimental evidence shows that for a viscous fluid, the stress is a function of velocity gradient. That is

)u(Gp j,iklklkl +δ−=τ

The velocity gradient term may be decomposed as

ijijj,i du ω+= where ijd is the deformation rate tensor and ijω is the spin tensor. These are given as

),uu(21

d k,ll,kkl += )uu(21

k,ll,kkl −=ω

ME637 G. Ahmadi 3

The principle of Material Frame-Indifference of continuum mechanics implies that the stress is generated only by the deformation rate of media and the spin has no effect. This is because both stress and deformation rate tensors are frame-indifferent while spin is not. Thus, the general form of the constitutive equation is given as )d(Fp ijklklkl +δ−=τ For a Newtonian fluid, the constitutive equation is linear and is given as klkli,ikl d2)up( µ+δλ+−=τ The entropy inequality imposed the following restrictions on the coefficient of viscosity:

023 >µ+λ , 0>µ , Using the constitutive equation in the balance of momentum leads to the celebrated Navier-Stokes equation. For an incompressible fluid the Navier-Stokes and the continuity equations are given as

jj

i2

ij

ij

i

xxu

xp)

xu

ut

u(

∂∂∂

µ+∂∂−=

∂∂

+∂

∂ρ ,

0=∂∂

i

i

xu

These form four equations for evaluating four unknowns .p,u i

TURBULENT FLOW In turbulent flows the field properties become random functions of space and time. Thus iii uUu ′+= ii Uu = , 0u i =′ pPp ′+= Pp = , 0p =′ Substituting the decomposition into the Navier-Stokes equation and averaging leads to the Reynolds equation.

ME637 G. Ahmadi 4

Reynolds Equation

j

ji

jj

i2

ij

ij

i

x

uu

xxU

xP1

xU

Ut

U∂

′′∂−

∂∂

ν+∂∂

ρ−=

∂∂

+∂

Here ji

Tij uu ′′ρ−=τ =Turbulent Stress Tensor

First Order Modeling (Classical Phenomenology) Boussinesq Eddy Viscosity:

dydU

vu TT21 ρν=′′ρ−=τ

ijkki

j

j

iTji

Tij uu

31)

x

U

xU

(uu δ′′−∂

∂+

∂∂

ν=′′−=ρ

τ

Prandtl Mixing Length

yU

|yU

|l2m

T21 ∂

∂∂∂

ρ=τ

|yU

|l2mT ∂

∂=ν ,

yTvT

T

T

∂∂

σν

=′′−

Kolmogorov-Prandtl Expression Eddy Viscosity

lucT ≈ν , u = velocity scale, scalelength=l , c = const. Kinematic Viscosity λ∝ν c , c = speed of sound, path freemean =λ Let

|yU

|~ m ∂∂

lu , ⇒= mll |yU

|2mT ∂

∂=ν l

For free shear flows

ME637 G. Ahmadi 5

0m c~ ll ( 0l = half width) Close to a wall

ym κ=l (y = distance from the wall) Local Equilibrium Shortcomings of the Mixing Length Model

• When ⇒=∂∂

0yU

0T =ν

• Lack of transport of scales of turbulence • Estimating the mixing length, .ml

At the reattachment point 0yU

=∂∂

which leads to vanishing eddy diffusivity and

thus negligible heat flux. Experiments, however, show that the heat flux is maximum at the reattachment point.

Mixing length Hypothesis

For local equilibrium Production = Dissipation ó

Reattachment Point

ME637 G. Ahmadi 6

One-Equation Models Eddy Viscosity

l2/1T kcµ=ν , iiuu

21

k ′′= = Turbulence Kinetic Energy

Exact k-equation

44344214342143421444 3444 2143421

DiffusionViscous

ii

jj

2

nDissipatio

j

i

j

i

oductionPr

j

iji

DiffusionTurbulence

iik

k

TransportConvective

ii

2uu

xxxu

xu

xU

uu)P

2uu

(ux2

uudtd ′′

∂∂

ν+∂

′∂∂

′∂ν−

∂∂′′−

ρ′

+′′

′∂

∂−=

′′

where

2uu

dtd ii ′′

= convective transport, j

j xU

tdtd

∂∂

+∂∂

=

)P

2uu

(ux

iik

k ρ′

+′′

′∂

∂= turbulence diffusion

j

iji x

Uuu

∂∂′′ = production

j

i

j

i

xu

xu

∂′∂

∂′∂

ν = , = dissipation

2uu

xxii

jj

2 ′′∂

∂ν = viscous diffusion

Modeled k-equation

l

2/3

Dj

i

i

j

j

iT

jk

T

j

kcxU

)x

U

xU

()xk(

xdtdk −

∂∂

∂+

∂∂

ν+∂∂

σν

∂∂= ,

where

)xk(

x jk

T

j ∂∂

σν

∂∂ = turbulence diffusion, 1k ≈σ (turbulence Prandlt number)

j

i

i

j

j

iT x

U)

x

U

xU

(∂∂

∂+

∂∂

ν = production,

ME637 G. Ahmadi 7

ε=l

2/3

D

kc = dissipation.

Note that the turbulence length scale l is given by an algebraic equation.

Bradshaw’s Model Modeled k-equation

l

2/3

DMax k

cyU

ak)Bk(ydt

dk−

∂∂

τ∂∂

=

where

yU

ak∂∂

= production

lk

cD

2/3

= dissipation

akvu =′′− (shear stress ∝ kinetic energy)

)y(gv

B 20

Max

δρτ

= , )y

(fδ

δ=l

Shortcomings of One-Equation Models • Use of an algebraic equation for the length scale is too restrictive. • Transport of the length scale is not accounted for. Transport of Second Scale (Boundary Layer) The transport of a second turbulence scale, z, is given as z-Equation

zT

22T

1z

T S]kc)yU(

kc[z)

yz(

ydtdz +

ν−

∂∂ν

+∂∂

σν

∂∂=

where

ME637 G. Ahmadi 8

)yz(

y z

T

∂∂

σν

∂∂ = diffusion

2T1 )

yU

(k

c∂∂ν

= production

T2

kc

ν = destruction,

zS = secondary source Choices for z

Turbulence Time Scale = k/2l

Turbulence frequency Scale = 2/k l Turbulence mean-square vorticity Scale = 2/k l

Turbulence Dissipation = i

i

j

i

xu

xu

∂′∂

∂′∂

ν=ε

lkz = ε -Equation (exact):

444 3444 2144 344 2143421

ndestructioViscous

lk

i2

lk

i2

stretchingvortexbyGeneration

l

k

l

i

k

i

Diffusion

jj xx

uxx

u2

xu

xu

xu

2)'u(xdt

d∂∂

′∂∂∂

′∂ν−

∂′∂

∂′∂

∂′∂

ν−ε′∂∂−=ε

Note that

l

2/3k~ε ,

ε

2/3k~l

Thus

ε

ν2

T

k~k~ l

Note also that ε is also the amount of energy that paths through the entire spectrum of eddies of turbulence.

Schematics of turbulence energy spectrum.

k

E(k)

,

,

Universal Equilibrium

Inertia Subrange

ME637 G. Ahmadi 9

Two-Equation Models The ε−k Model

ε

=ν µ2

T

kc, ij

i

j

j

iTji k

32)

x

U

xU

(uu δ−∂

∂+

∂∂

ν=′′−

k-equation

321444 3444 2144 344 21 ndissipatio

oductionPr

j

i

i

j

j

iT

Diffusion

jk

T

j xU

)x

U

xU

()xk

(xdt

dkε−

∂∂

∂+

∂∂

ν+∂∂

συ

∂∂

=

ε -equation

3214444 34444 2144 344 21 nDistructio

2

2

Generation

j

i

i

j

j

iT1

Diffusion

j

T

j kc

xU

)x

U

xU

(k

c)xk

(xdt

d ε−

∂∂

∂+

∂∂ε

ν+∂∂

συ

∂∂

εεε

,

where 09.0c =µ , 45.1c 1 =ε , 9.1c 2 =ε , 1=kσ , 3.1=σε , (Jones and Launder, 1973) Momentum

jiji

i uuxx

P1dt

dU ′′∂∂−

∂∂

ρ−=

Mass

0xU

i

i =∂∂

Closure: Six Equations for six unknowns, iv , P, k , ε . Kolmogorov Model

44 344 214342143421

Diffusion

jjnDissipatio

2

oductionPr

ijijT )xkk

(x

Ak21

SS2dtdk

∂∂

ω∂∂′′+ω−υ=

ME637 G. Ahmadi 10

)xkk

(x

A2107

dtdw

jj

2

∂∂

ω∂∂′+ω−=

ω=ν

AkT , )

x

U

xU

(21S

i

j

j

iij ∂

∂+

∂∂

=

Comparison of Model Predictions

In this section comparisons of the predictions of the mixing length and one and

two-equation models with the experimental data for simple turbulent shear flows are presented.

Development of Plane Mixing Layer (Rodi, 1982)

It is seen that the ε−k model captures the features of the flow more accurately

when compared with the one-equation and mixing length model.

ME637 G. Ahmadi 11

Turbulent Recirculating Flow (Durst and Rastogi, 1979)

The ε−k model predictions for turbulent flow in a channel with an obstructing block are compared with the experimental data.

Flow in a Square Cavity (Gosman and Young)

The ε−k model predictions for a square cavity are shown in this section.

b) Velocity profiles

ME637 G. Ahmadi 12

Free-Stream Turbulence The free stream turbulence affects the skin friction coefficient. The mixing length model can not predict such effects. The ε−k model does a reasonable job in predicting the increase of skin friction coefficient with the free stream turbulence.

Turbulent Channel Flow (Rodi, 1980)

Distribution of mean velocity and turbulence quantities in fully developed two-dimensional channel flows was predicted by Rodi (1980) using an algebraic stress model (a modified ε−k model)

Closed Channel Flow

ME637 G. Ahmadi 13

Open Channel Flow

Jets Issuing in Co-flowing Streams (Rodi, 1982)

For a jet, ∫∞

=−=θ0

EE

thicknessmomentumExcessdy)UU(UU1

Plane Jet Round Jet

ME637 G. Ahmadi 14

Shortcomings of the ε−k Model

iji

j

j

iTji k

32)

x

U

xU

(uu δ−∂

∂+

∂∂

ν=′′−

• Limited to an eddy viscosity assumption. • Eddy viscosity and diffusivity are assumed to be isotropic. • Convection and diffusion of the shear stresses are neglected. • Normal turbulent stresses are not considered. • Main assumption is: k~uu ji ′′ .

Stress Transport Models

Subtracting the Navier-Stokes equation form the Reynolds equation, we find an evolution equation for the turbulence fluctuation velocity. That is

k

ikki

kki

kkk

i2

ik

ik

i

xU

u)uu(x

uuxxx

uxp1

xu

Ut

u∂∂′−′′

∂∂

−′′∂∂

+∂∂

′∂ν+

∂′∂

ρ−=

∂′∂

+∂

′∂ (1)

k

jkkj

kkj

kkk

j2

jk

jk

j

x

Uu)uu(

xuu

xxx

u

x'p1

x

uU

t

u

∂′−′′

∂∂−′′

∂∂+

∂∂

′∂ν+

∂∂

ρ−=

′∂+

′∂ (2)

Multiplying Equation (1) by ju′ , and Equation (2) by iu ′ , adding the resulting equations and averaging leads to the exact stress transport equation:

44444444 344444444 21

44 344 21434214444 34444 21444 3444 21

Diffusion

jik

ikjjkikjik

strainessurePr

i

j

j

i

nDissipatio

k

j

k

i

oductionPr

k

ikj

k

jki

Convection

jik

k

]uux

)uu('p

uuu[x

)x

u

xu

('p

x

u

xu

2]xU

uux

Uuu[uu)

xU

t(

′′∂∂

ν−δ′+δ′ρ

+′′′∂∂

′∂+

∂′∂

ρ+

′∂

∂′∂

ν−∂∂′′+

∂′′−=′′

∂∂

+∂∂

− ,

where

]xU

uux

Uuu[

k

ikj

k

jki ∂

∂′′+∂

∂′′ = production,

ME637 G. Ahmadi 15

k

j

k

i

x

u

xu

2∂

′∂

∂′∂

ν = dissipation,

)x

u

xu

('p

i

j

j

i

′∂+

∂′∂

ρ = pressure strain

]uux

)uu('p

uuu[x ji

kikjjkikji

k

′′∂∂

ν−δ′+δ′ρ

+′′′∂

∂ = diffusion.

Modeling Diffusion:

)x

uuuu

xuu

uuxuu

uu(kcuuul

jilk

l

iklj

l

kiliskji ∂

′′∂′′+

∂′′∂′′+

∂′′∂′′

ε=′′′−

Pressure diffusion ≈ 0 Viscous diffusion ≈ 0 Modeling Dissipation

εδ=∂

′∂

∂′∂

ν ijk

j

k

i

32

x

u

xu

2

Modeling Pressure-Strain

{

}4444 34444 21

4444444 34444444 21

)2(ji

)2(ij

1

)1(ij

)x

u

xu

()xu

()xU

(2

)x

u

xu

()xu

xu

()(Gd)x

u

xu

('p

i

j

j

i1

l

m1

m

l

i

j

j

i1

l

m

m

l11

i

j

j

i

ϕ+ϕ

ϕ

′∂+

∂′∂

∂′∂

∂∂

+

′∂+

∂′∂

∂′∂

∂′∂

−=∂

′∂+

∂′∂

ρ ∫x

xx,x

where

)k32

uu)(k

(c ijji1)1(

ij δ−′′ε−=ϕ (Return to isotropy)

)P32

P( ijij)2(

ji)2(

ij δ−γ−=ϕ+ϕ (Rapid term)

Here

)xU

uux

Uuu(P

k

iki

k

jkiij ∂

∂′′+∂

∂′′−=

ME637 G. Ahmadi 16

Pressure-Strain Correlation

Modeling the pressure-strain correlation, )x

u

xu

('p

i

j

j

i

′∂+

∂′∂

ρ, is critical to the stress

transport equations. Navier-Stokes Equation

jj

i2

ik

ik

i

xxu

xp1

xu

ut

u∂∂

∂ν+

∂∂

ρ−=

∂∂

+∂

∂ (1)

Taking the divergence of (1), we find

ii

2

ki

ki2

xxp1

xxuu

∂∂∂

ρ−=

∂∂∂

, (2)

or

ki

ki2

2

xxuup

∂∂∂

−=ρ

∇ . (3)

Averaging Equation (3), the result is:

ki

ki2

ki

ki2

2

xxuu

xxUUp

∂∂′′∂

−∂∂

∂−=

ρ∇ . (4)

Subtracting (4) from (3), we find

i

k

k

i

i

k

k

i

i

k

k

i2

xu

xu

xu

xu

xu

xU

2'p

∂′∂

∂′∂

+∂

′∂∂

′∂−

∂′∂

∂∂

−=ρ

∇ . (5)

Introducing the Green function )(G 1xx, for the Poisson equation. i.e.,

)()(G211 xxxx, −δ=∇ , (6)

Equation (5) may be restated as

ME637 G. Ahmadi 17

∫ ∂′∂

∂′∂

−∂

′∂∂

′∂+

∂′∂

∂∂

−=1x

11 xxx, d]xu

xu

xu

xu

xu

xU

2)[(G'pi

k

k

i

i

k

k

i

i

k

k

i (7)

The pressure-strain rate correlation then becomes

∫ ∂

′∂+

∂′∂

∂′∂

∂∂

+∂

′∂+

∂′∂

∂′∂

∂′∂

−=∂

′∂+

∂′∂

ρ1x

11 xxx, d)]x

u

xu

()xu

()xU

(2)x

u

xu

()xu

xu

()[(G)x

u

xu

('p

i

j

j

i1

l

k1

k

l

i

j

j

i1

l

k

k

l

i

j

j

i

(8) or

)()x

u

xu

('p )2(ji

)2(ij

)1(ij

i

j

j

i ϕ+ϕ+ϕ=∂

′∂+

∂′∂

ρ (9)

Note that for unbounded regions

||

141

)(G1

1 xxxx,

−π−= . (10)

Modeled Stress Transport Equation

4444444444 34444444444 21

4434421444444 3444444 21

3214444 34444 21444 3444 21

Diffusion

l

jilk

l

iklj

l

kjli

ks

effectsWall

)w(ji

)w(ij

strainessurePr

)2(ji

)2(ijijji1

nDissipatio

ij

oductionPr

k

jki

k

ikj

Convection

jik

k

]}x

uuuu

xuu

uux

uuuu[

k{

xc

)()()k32uu(

kc

32

]x

Uuu

xU

uu[uu)x

Ut

(

′′∂′′+

∂′′∂′′+

′′∂′′

ε∂∂

+

ϕ+ϕ+ϕ+ϕ+δ−′′ε−

εδ−∂

∂′′+

∂∂′′−=′′

∂∂

+∂∂

where

]x

Uuu

xU

uu[k

jki

k

ikj ∂

∂′′+

∂∂′′ = production

εδ ij32

= dissipation,

)k32

uu(k

c ijji1 δ−′′ε = pressure-strain

]}x

uuuu

xuu

uux

uuuu[k{

xc

l

jilk

l

iklj

l

kjli

ks ∂

′′∂′′+

∂′′∂′′+

′′∂′′

ε∂∂ = diffusion.

ME637 G. Ahmadi 18

Dissipation Equation

32144 344 21444 3444 2144 344 21

nDestructio

2

2

Generation

k

iki1

Diffusion

iik

k

Convection

kk k

cxU

uuk

c)x

uuk

(x

c)x

Ut

−∂∂′′ε

−∂

ε∂′′ε∂

∂=ε

∂∂

+∂∂

εεε ,

where

)x

uuk

(x

ck

lkk ∂

ε∂′′ε∂

∂ε = diffusion,

k

iki1 x

Uuu

kc

∂∂′′ε

ε = generation

k

c2

2

εε = destruction.

Reynolds Equation

jiji

ij

j uuxx

P1U)

xU

t( ′′

∂∂

−∂∂

ρ−=

∂∂

+∂∂

Continuity Equation

0=∂∂

i

i

xU

Closure

There are eleven equations for eleven unknowns, iU , P , jiuu ′′ , ε .

ME637 G. Ahmadi 19

Comparison of Model Predictions In this section comparisons of the predictions of the stress transport model with

the experimental data for simple turbulent shear flows are presented.

Curved Mixing Layer (Gibson and Rodi, 1981)

ME637 G. Ahmadi 20

Asymmetric Channel Flow (Launder, Reece and Rodi, 1975)

Mean Velocity and Turbulent Shear Stress

Turbulence Intensities

ME637 G. Ahmadi 21

Algebraic Stress Transport Model (Rodi, ZAMM 56, 1976)

A simplified stress transport model is given as:

321444444 3444444 21

444 3444 214444 34444 21

nDissipatio

ij

StrainessurePr

ijijijji1

oductionPr

k

ikj

k

jki

Diffusion

mmk

ksji

32)P

32P()k

32uu(

kc

xU

uux

Uuu)juiu

xuu

k(

xcuu

dtd

εδ−δ−γ−δ−′′ε−

∂∂′′−

∂′′−′′

∂∂′′

ε∂∂

=′′

, (1)

where

)uux

uuk

(x

D jil

lkk

ij ′′∂∂′′

ε∂∂

= = diffusion,

k

ikj

k

jkiij x

Uuu

x

UuuP

∂∂′′−

∂′′= = production,

)P32

P()k32

uu(k

c ijijijji1 δ−γ−δ−′′ε = pressure-strain,

εδ ij32

= dissipation.

Here, iiP21

P = is the production rate of turbulent kinetic energy. Contracting

Equation (1), we find the transport equation for k:

{nDissipatio

oductionPr

m

kmk

Diffusion

mmk

ks x

Uuu)

xkuuk(

xc

dtdk ε−

∂∂′′−

∂∂′′

ε∂∂=

43421444 3444 21, (2)

where

)xk

uuk

(x

Dl

lkk ∂

∂′′ε∂

∂= =diffusion

l

klk x

UuuP

∂∂′′= = production

Rodi (1976) assumed that

)P(k

uu)D

dtdk(

k

uuDuu

dtd jiji

ijji ε−′′

=−′′

=−′′ , (3)

ME637 G. Ahmadi 22

Using (3) in (1) and rearranging, the result is:

−ε

+

δε

−εγ−

+δ=′′)1P(

c11

P32P

c1

32

kuu

1

ijij

1ijji . (4)

Equation (4) provides an algebraic expression for .uu ji ′′

For simple shear flows, it may be shown that equation (4) reduces to the Kolmogorov-Prandtl hypothesis with

ε=ν µ

2

T

kc (5)

and

2

1

1

1 )]1P

(c1

1[

)]P1(c11[

c)1(

32

c−

ε+

εγ−−

γ−=µ with 6.0=γ and 2.28.11 −=c . (6)

Conclusions (Existing Models) • Available models can predict the mean flow properties with reasonable accuracy.

Small adjustments of parameters are sometimes necessary! • First-order modeling gives reasonable results only when a single length and velocity

scale characterizes turbulence. • The ε−k model gives relatively accurate results when a scalar eddy viscosity is

sufficient to characterize the flow. That is there is no preferred direction for example through the action of a body force.

• The stress transport models have the potential to most accurately represent the mean turbulent flow fields.

Deficiencies of Existing Models • Adjustments of coefficients are sometimes needed. • The derivation of the models are somewhat arbitrary. • There is no systematic method for improving a model when it loses its accuracy. • Models for complicated turbulent flows (such as multiphase flows) are not available. • Realizability and other fundamental principles are sometimes violated.

ME637 G. Ahmadi 23

For example, the transport equations for 222 'w,'v,'u must always lead to positive values of these quantities. In addition, the transport equations for the cross terms must also lead to cross correlations that satisfy Schwarts inequalities. i.e.,

.0'vu'vu222 >′−′

ME637 G. Ahmadi 24

Anisotropic Rate-Dependent Model Averaged Balance Laws: Mass 0v i,i = Linear Momentum i

Tj,jij,jii fttv ρ++=ρ &

Thermal Energy rvtqqe i,jij

Ti,ii,i +ρε+++=ρ&

Fluctuation Energy ρε−+=ρ i,ii,j

Tij Kvtk&

Clausius-Duhem Inequality 0SrR)q( T

i,iTT

i,ii,i >−ηρ+ϑ−−ϑ−ηρ && Helmholtz free Energy Function

ϑη

−=ψ e , T

TT k

ϑη

−=ψ

Heat Flux-Coldness Correlation ϑ= T

iTi qR

Fluctuation Energy Flux-Turbulence Coldness Correlation

iT

iTi EKS −ϑ=

Total Heat Flux T

iii qqQ +=

ME637 G. Ahmadi 25

Clausius-Duhem Inequality

0vtE1

K1

)(

vtQ1

i,jTij

i,iT

Ti,iT2T

TTTT

i,jiji,i2

>

ρε−+ϑ

+ϑϑ

ϑ

ϑη−ψρ−ϑ+

ρε++ϑ

ϑ−

ϑ

ϑη−ψρ−ϑ

&&

&&

Constitutive Equations Stress ijijij d2pt µ+δ−=

( )

−δβτ+γτ+µ+

∆∂ψ∂ρ+δρ−= kjikijklklijklkl

2TijT

ijTij dddd

31ddd2

Dt

dD̂k

32t

Jaumann Derivative

kijkkjikijij ddd

Dt

dD̂ω+ω+= &

Deformation Rate and Spin Tensors

( )i,jj,iij vv21

d += , )vv(21

i,jj,iij −=ω , ijijdd21

=∆

Heat Flux

i,

T

i CQ θ

σµ

+κ= θ

Fluctuation Energy flux

τ

τ−

σµ

+µ= i,i,k

T

ik

kK

Heat Capacity

2

2

Cθ∂ψ∂

θ−=

ME637 G. Ahmadi 26

Thermodynamic Constraints

0T >µ , 0>σθ , 0>γ , γβ 482 < τρ=µ µ kCT Turbulence free Energy Function

+∆τα+

τ=ψ µ

α

02T CC

klnk

o

Turbulence Stress

−δβτ+γτ+ατ+µ+δρ−= kjikijkllkijkllk

2ijij

Tij

Tij dddd

31

ddddDtD̂

d2k32

t

Basic Equations 0v i,i =

i

j,

ijkllk2

kjikijkllkijT

ijT

i,i f]ddd)dddd

31

(Dt

dD̂[d)(2k

32

pv ρ+

γτ+−δβτ+ατµ+µ+µ+

ρ+−=ρ&

rdd2)C(C ijij

i,

i,

T

+ρε+µ+

θ

σµ

+κ=θρ θ&

ρε−ατµ++

τ

τ−

σµ

+µ=ρ ijijT

i,

i,i,k

T

dDt

dD̂P)

kk)((k&

])dd(ddddd2[P 2

jiij2

ijkjikijijT γτ+βτ−µ=

ME637 G. Ahmadi 27

Scale Transport Equations

D

i,i,2

20

T

T

i,i,i,i,2k

T

i,

i,T

T

CC)](C2

C2[

]kk][kk[k

CPk

C)(

2

31

τµ

µ

ττ

ρ−τ∆τ∆τα+α

α

σµ

+µ+

ττ

−ττ

τ

σµ+µ+τ+

τ

σµ+µ=τρ&

0C1

1

m0

>>α

τ , 0

1C 2

α>τ , 0C

13

m0

>>α

τ , 00 >α

)C2.(max 2

0m0 ∆τα+αα µ

τ

=εk

CD

kCC)

k](

kC2

C2[

]k

k][k

k[k

CPk

C)(

22

i,i,2

2

2

2

0

T

i,i,i,i,2k

T

i,

i,

T

2

31

ερ−ε

ε∆

ε∆α+α

α

σµ

+µ+

εε

−εε

ε

σµ

+µ+ε

+

ε

σµ

+µ=ερ

εε

µ

µ

ε

εεε

&

ε

ρ=µ µ2

T kC ,

ε=τ

k

When 0=γ , 93.0=α , 54.0=β

48

2βγ > , 005.0=γ

09.0C =µ , 45.1C 1 =ε , 92.1C 2 =ε , 1k =σ , 3.1=σε

ME637 G. Ahmadi 28

Comparison with the Experimental Data for Duct Flow In this section the rate-dependent model predictions are compared with the experimental data of Kreplin and Eckelmann, DNS of Kim et al. and the k-ε model predictions

Comparison of mean velocity profile with the experimental data of Kreplin and

Eckelmann for Reynolds numbers of 8200 and 5600.

Comparison of axial turbulence intensity with the experimental data of

Kreplin and Eckelmann, DNS of Kim et al. and k-ε model.

Comparison of vertical turbulence intensity with the experimental data of

Kreplin and Eckelmann, DNS of Kim et al. and k-ε model.

Comparison of lateral turbulence intensity with the experimental data of

Kreplin and Eckelmann, DNS of Kim et al. and k-ε model.

ME637 G. Ahmadi 29

Comparison with Experimental the Data for Backward Facing Step Flows In the section the rate-dependent model predictions are compared with the experimental data of Kim et al. and the algebraic model of Srinivasan et al.

Schematics of the flow over a backward facing step.

Comparison of turbulence shear stress with the experimental data of Kreplin and Eckelmann, and DNS of Kim et al.

ME637 G. Ahmadi 30

Comparison of the mean velocity profiles with the data of Kim et al. (1978).

(Dashed lines are the model predictions of Srinivasan et al. (1983).

Comparison of the turbulence kinetic energy profiles with the data of Kim et al. (1978).

(Dashed lines are the model predictions of Srinivasan et al. (1983).

ME637 G. Ahmadi 31

Comparison of the turbulence dissipation profiles.

(Dashed lines are the model predictions of Srinivasan et al. (1983).

Comparison of the axial turbulence intensity profiles with the data of Kim et al. (1978).

(Dashed lines are the model predictions of Srinivasan et al. (1983).

ME637 G. Ahmadi 32

Comparison of the vertical turbulence intensity profiles with the data of Kim et al. (1978). (Dashed lines are the model predictions of Srinivasan et al. (1983).

Comparison of the turbulence shear stress profiles with the data of Kim et al. (1978). (Dashed lines are the model predictions of Srinivasan et al. (1983).

ME637 G. Ahmadi 33

Schematics of the flow in an axisymmetric pipe expansion.

Comparison of the mean velocity profiles with the data of Junjua et al. (1982) and

Chaturvedi (1963). (Dashed lines are the model predictions of Srinivasan et al. (1983))

Comparison of the axial turbulence intensity profiles with the data of Junjua et al. (1982)

and Chaturvedi (1963). (Dashed lines are the model predictions of Srinivasan et al. (1983))

ME637 G. Ahmadi 34

Comparison of the vertical turbulence intensity profiles with the data of Junjua et al. (1982) and Chaturvedi (1963). (Dashed lines are the model predictions of

Srinivasan et al. (1983))

Comparison of the turbulence shear stress profiles with the data of Junjua et al. (1982)

and Chaturvedi (1963). (Dashed lines are the model predictions of Srinivasan et al. (1983))