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 k -e Model)
· Stress Transport Models
· Material-Frame Indifference
· Continuum Approach
· Consistency of the k -e Model
· Rate-Dependent Model
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VISCOUS FLOW
The conservation laws for a continuous media are:
Mass
¶r + Ñ × (ru) = 0 ¶t
Momentum
du r = rf + Ñ × t dt
Angular Momentum
t T = t
Energy
re& = t : Ñu + Ñq + rh
Entropy Inequality
q rh r h& - Ñ ×( ) - > 0 T T
Constitutive Equation
Experimental evidence shows that for a viscous fluid, the stress is a function of velocity gradient. That is
tkl = -pd kl + G kl (u i, j )
The velocity gradient term may be decomposed as
ui, j = dij + wij
where d ij is the deformation rate tensor and wij is the spin tensor. These are given as
1 1 d = (u + u ), w = (u - u ) kl 2 k,l l,k kl 2 k,l l,k
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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
tkl = -pdkl + Fkl (dij )
For a Newtonian fluid, the constitutive equation is linear and is given as
tkl = (-p + lu i,i )dkl + 2md kl
The entropy inequality imposed the following restrictions on the coefficient of viscosity:
3l + 2m > 0 , m >0 ,
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
2 ¶u i ¶u i ¶p ¶ u i r( + u j ) = - + m , ¶t ¶x j ¶x i ¶x j¶x j
¶u i = 0 ¶xi
These form four equations for evaluating four unknowns u i , p.
TURBULENT FLOW
In turbulent flows the field properties become random functions of space and time. Thus
u i = U i + u i¢ ui = Ui , u¢i = 0
p = P + p¢ p = P , p¢ = 0
Substituting the decomposition into the Navier-Stokes equation and averaging leads to the Reynolds equation.
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Reynolds Equation
2 ¢ ¢ ¶U i ¶U i 1 ¶P ¶ U i ¶u i u j + U j = - + n - ¶t ¶x j r ¶x i ¶x j x j ¶x j
Here
T tij = -ru¢i u¢j =Turbulent Stress Tensor
First Order Modeling (Classical Phenomenology)
Boussinesq Eddy Viscosity:
dU tT = -ru¢v¢ = rn 21 T dy
T tij ¶U i ¶U j 1 = -u¢i u¢j = n T ( + ) - u¢k u¢k dij r ¶x j ¶x i 3
Prandtl Mixing Length
¶U ¶U tT = rl2 | | 21 m ¶y ¶y
2 ¶U n T ¶T n T = lm | |, - T¢v¢ = ¶y s T ¶y
Kolmogorov-Prandtl Expression
Eddy Viscosity
n T » cu l , u = velocity scale, l= length scale, c = const. Kinematic Viscosity
n µ cl , c = speed of sound, l = mean free path
Let ¶U ¶U u ~ l | |, l = l Þ n = l 2 | | m ¶y m T m ¶y
For free shear flows
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l m ~ cl 0 ( l 0 = half width)
Close to a wall
l m = ky (y = distance from the wall)
Local Equilibrium
For local equilibrium Mixing length Hypothesis Production = Dissipation ó
Shortcomings of the Mixing Length Model
¶U · When = 0 Þ n = 0 ¶y T · Lack of transport of scales of turbulence
· Estimating the mixing length, l m .
Reattachment Point
¶U At the reattachment point = 0 which leads to vanishing eddy diffusivity and ¶y thus negligible heat flux. Experiments, however, show that the heat flux is maximum at the reattachment point.
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One-Equation Models
Eddy Viscosity
1 n = c k1/ 2l , k = u¢u¢ = Turbulence Kinetic Energy T m 2 i i
Exact k-equation
d u¢u¢ ¶ u¢u¢ P¢ ¶U ¶u¢ ¶u¢ ¶ 2 u¢u¢ i i = - u¢ ( i i + ) - u¢u¢ i - n i i + n i i dt 2 ¶x k 2 r i j ¶x ¶x ¶x ¶x x 2 14243 144k 424443 14243j 142j 43j 144j2j 443 Convective TurbulenceDiffusion Transport Production Dissipation Viscous Diffusion where
d u¢i u¢i d ¶ ¶ = convective transport, = + U j dt 2 dt ¶t ¶x j
¶ u i¢u¢i P¢ u¢k ( + ) = turbulence diffusion ¶x k 2 r
¶U i u¢i u¢j = production ¶x j ¶u¢ ¶u¢ n i i = , = dissipation ¶x j ¶x j ¶ 2 u¢u¢ n i i = viscous diffusion ¶x jx j 2
Modeled k-equation
¶U 3/ 2 dk ¶ n T ¶k ¶Ui j ¶Ui k = ( ) + n T ( + ) - cD , dt ¶x j sk ¶x j ¶x j ¶x i ¶x j l where
¶ n T ¶ k ( ) = turbulence diffusion, sk »1(turbulence Prandlt number) ¶x j sk ¶x j
¶U ¶Ui j ¶Ui n T ( + ) = production, ¶x j ¶x i ¶x j
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k 3/ 2 c = e = dissipation. D l
Note that the turbulence length scale l is given by an algebraic equation.
Bradshaw’s Model
Modeled k-equation
dk ¶ t ¶U k 3/ 2 = (Bk Max ) + ak - c dt ¶y r ¶y D l where
¶U ak = production ¶y k 3 / 2 c = dissipation D l
- u¢v¢ = ak (shear stress µ kinetic energy)
tMax y y B = 2 g( ) , l = df ( ) rv 0 d d
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
dz ¶ n T ¶z n T ¶U 2 k = ( ) + z[c1 ( ) - c2 ] + Sz dt ¶y sz ¶y k ¶y n T where
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¶ n ¶z ( T ) = diffusion ¶y s z ¶y n ¶U c T ( )2 = production 1 k ¶y k c2 = destruction, n T
Sz = secondary source
Choices for z
Turbulence Time Scale = l 2 / k Turbulence frequency Scale = k / l2 Turbulence mean-square vorticity Scale = k / l2 ¶u¢ ¶u¢ Turbulence Dissipation = e = n i i ¶x j ¶x i z = kl
e -Equation (exact):
de ¶ ¶u¢ ¶u¢ ¶u¢ ¶ 2u¢ ¶ 2 u¢ = - (u¢e') - 2n i i k - 2n i i dt ¶x j ¶x ¶x ¶x ¶x ¶x ¶x ¶x 14j243 144k 24l 43l 144k42l 44k 43l Diffusion Generationby Viscousdestruction vortexstretching
Note that k 3/ 2 k 3/ 2 E(k) e ~ , l ~ Universal Equilibrium l e Thus , k 2 n ~ k l ~ , T e
Note also that e is also the amount of energy that paths through the Inertia entire spectrum of eddies of Subrange turbulence. k
Schematics of turbulence energy spectrum.
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Two-Equation Models
The k - e Model
c k 2 ¶U m ¶Ui j 2 n T = , - u¢i u¢j = n T ( + ) - kd ij e ¶x j ¶x i 3 k-equation
dk ¶ u ¶k ¶U ¶U ¶U = ( T ) + n ( i + j ) i - e dt ¶x s ¶x T ¶x ¶x ¶x 123 14j42k443j 144j424i443j dissipation Diffusion Pr oduction
e -equation
¶U 2 de ¶ u T ¶k e ¶U i j ¶U i e = ( ) + c e1n T ( + ) - ce2 , dt ¶x j se ¶x j k ¶x j ¶x i ¶x j 123k 1442443 14444244443 Distruction Diffusion Generation where
cm = 0.09 , ce1 = 1.45 , c e 2 = 1.9 , s k = 1, se = 1.3, (Jones and Launder, 1973)
Momentum
dU i 1 ¶P ¶ = - - u i¢u¢j dt r ¶x i ¶x j
Mass
¶U i = 0 ¶x i
Closure: Six Equations for six unknowns, v i , P, k , e .
Kolmogorov Model
dk 1 2 ¶ k ¶k = 2uTSijSij - k w + A¢¢ ( ) dt 14243 2 ¶x j w ¶x j Pr oduction 14243 Dissipation 1442443 Diffusion
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dw 7 ¶ k ¶k = - w2 + 2A¢ ( ) dt 10 ¶x j w ¶x j
¶U Ak 1 ¶Ui j n T = , Sij = ( + ) w 2 ¶x j ¶x i
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 k - e model captures the features of the flow more accurately when compared with the one-equation and mixing length model.
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Turbulent Recirculating Flow (Durst and Rastogi, 1979)
The k - e model predictions for turbulent flow in a channel with an obstructing block are compared with the experimental data.
b) Velocity profiles
Flow in a Square Cavity (Gosman and Young)
The k - e model predictions for a square cavity are shown in this section.
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Free-Stream Turbulence
The free stream turbulence affects the skin friction coefficient. The mixing length model can not predict such effects. The k - e 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 k - e model)
Closed Channel Flow
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Open Channel Flow
Jets Issuing in Co-flowing Streams (Rodi, 1982)
1 ¥ For a jet, q = ò U(U - U E )dy = Excess momentum thickness U E 0 Plane Jet Round Jet
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Shortcomings of the k - e Model
¶U ¶Ui j 2 - u¢i u¢j = n T ( + ) - kd ij ¶x j ¶x i 3
· 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: u¢i u¢j ~ k .
Stress Transport Models
Subtracting the Navier-Stokes equation form the Reynolds equation, we find an evolution equation for the turbulence fluctuation velocity. That is
2 ¶u¢i ¶u¢i 1 ¶p¢ ¶ u¢i ¶ ¶ ¶U i + U k = - + n + u¢i u¢k - (u¢i u¢k ) - u¢k (1) ¶t ¶x k r ¶x i ¶x k¶x k ¶x k ¶x k ¶x k
2 ¶u¢j ¶u¢j 1 ¶p' ¶ u¢j ¶ ¶ ¶U j + U k = - + n + u¢j u¢k - (u¢ju¢k ) - u¢k (2) ¶t ¶x k r ¶x j ¶x k ¶x k ¶x k ¶x k ¶x k
Multiplying Equation (1) by u¢j , and Equation (2) by u i¢, adding the resulting equations and averaging leads to the exact stress transport equation:
¶ ¶ ¶U ¶U ¶u¢ ¶u¢ p' ¶u¢ ¶u¢ ( + U )u¢u¢ = -[u¢u¢ j + u¢u¢ i ] - 2n i j + ( i + j ) ¶t k ¶x i j i k ¶x j k ¶x ¶x ¶x r ¶x ¶x 144424k443 14444k244443k 142k 43k 1442j 443i Convection Pr oduction Dissipation Pr essure-strain , ¶ p' ¶ - [u¢u¢u¢ + (u¢d + u¢d ) - n u¢u¢ ] ¶x i j k r i jk j ik ¶x i j 14k 444444424444444k 43 Diffusion where
¶U j ¶U i [u¢i u¢k + u¢ju¢k ] = production, ¶x k ¶x k
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¶u¢ ¶u¢ 2n i j = dissipation, ¶x k ¶x k
p' ¶u¢ ¶u¢ ( i + j ) = pressure strain r ¶x j ¶x i ¶ p' ¶ [u¢i u¢ju¢k + (u¢i d jk + u¢jdik ) - n u¢i u¢j ] = diffusion. ¶x k r ¶x k
Modeling Diffusion:
¢ ¢ k ¶u¢i u¢k ¶u¢k u¢i ¶u i u j - u¢i u¢ju¢k = c s (u¢i u¢l + u¢j u¢l + u¢k u¢l ) e ¶x l ¶x l ¶x l
Pressure diffusion » 0 Viscous diffusion » 0
Modeling Dissipation
¢ ¶u¢i ¶u j 2 2n = dije ¶x k ¶x k 3
Modeling Pressure-Strain
p' ¶u¢ ¶u¢ ¶u¢ ¶u¢ ¶u¢ ¶u¢ ( i + j ) = - dx G(x,x ) ( l m ) ( i + j ) ò 1 1 { 1 r ¶x j ¶x i ¶x m ¶x l ¶x j ¶x i x1 14444444244444443 (1) jij
¶U ¶u¢ ¶u¢ ¶u¢ + 2( l ) ( m ) ( i + j ) } ¶x 1 ¶x 1 ¶x ¶x 14m444l244j 443i (2) (2) jij +jji where e 2 j(1) = -c ( )(u¢u¢ - d k) (Return to isotropy) ij 1 k i j 3 ij
2 j(2) + j(2) = -g(P - Pd ) (Rapid term) ij ji ij 3 ij Here ¶U j ¶U i Pij = -(u¢i u¢k + u¢i u¢k ) ¶x k ¶x k
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Pressure-Strain Correlation
p' ¶u¢ ¶u¢ Modeling the pressure-strain correlation, ( i + j ) , is critical to the stress r ¶x j ¶x i transport equations.
Navier-Stokes Equation
2 ¶u i ¶u i 1 ¶p ¶ ui + u k = - + n (1) ¶t ¶x k r ¶x i ¶x j¶x j
Taking the divergence of (1), we find
¶ 2 u u 1 ¶ 2 p i k = - , (2) ¶x i ¶x k r ¶x i¶x i or
p ¶ 2u u Ñ 2 = - i k . (3) r ¶x i ¶x k
Averaging Equation (3), the result is:
p ¶ 2 U U ¶ 2 u¢u¢ Ñ 2 = - i k - i k . (4) r ¶x i ¶x k ¶x i ¶x k
Subtracting (4) from (3), we find
p' ¶U ¶u¢ ¶u¢ ¶u¢ ¶u¢ ¶u¢ Ñ 2 = -2 i k - i k + i k . (5) r ¶x k ¶x i ¶x k ¶x i ¶x k ¶x i
Introducing the Green function G(x,x1 ) for the Poisson equation. i.e.,
2 Ñ G(x,x1 ) = d(x - x1 ) , (6)
Equation (5) may be restated as
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¶U ¶u¢ ¶u¢ ¶u¢ ¶u¢ ¶u¢ p'= - G(x,x )[2 i k + i k - i k ]dx (7) ò 1 ¶x ¶x ¶x ¶x ¶x ¶x 1 x1 k i k i k i
The pressure-strain rate correlation then becomes p' ¶u¢ ¶u¢ ¶u¢ ¶u¢ ¶u¢ ¶u¢ ¶U ¶u¢ ¶u¢ ¶u¢ ( i + j ) = - G(x,x )[( l k ) ( i + j ) + 2( l ) ( k ) ( i + j )]dx r ¶x ¶x ò 1 ¶x ¶x 1 ¶x ¶x ¶x 1 ¶x 1 ¶x ¶x 1 j i x1 k l j i k l j i (8) or ¢ p' ¶u¢i ¶u j (1) (2) (2) ( + ) = jij + (jij + j ji ) (9) r ¶x j ¶x i Note that for unbounded regions
1 1 G(x,x1 ) = - . (10) 4p | x - x1 |
Modeled Stress Transport Equation
¶ ¶ ¶U ¶U 2 ( + U )u¢u¢ = -[u¢u¢ i + u¢u¢ j ] - d e ¶t k ¶x i j j k ¶x i k ¶x 3 ij 144424k 443 14444k 244443k 123 Convection Pr oduction Dissipation e 2 - c (u¢u¢ - d k) + (j(2) + j(2) ) + (j(w ) + j(w) ) 1 k i j 3 ij ij ji 14ij4244ji3 144444424444443 Wall effects Pr essure-strain ¶ k ¶u¢u¢ ¶u¢ u¢ ¶u¢u¢ + c { [u¢u¢ j k + u¢u¢ k i + u¢ u¢ i j ]} s ¶x e i l ¶x j l ¶x k l ¶x 144k 444444l 442444l 444444l 43 Diffusion where
¶U ¶U i j [u¢ju¢k + u¢i u¢k ] = production ¶x k ¶x k 2 d e = dissipation, 3 ij
e 2 c (u¢u¢ - d k) = pressure-strain 1 k i j 3 ij
¢ ¢ ¢ ¢ ¶ k ¶u ju k ¶u¢k u¢i ¶u i u j cs { [u¢i u¢l + u¢ju¢l + u¢k u¢l ]} = diffusion. ¶x k e ¶x l ¶x l ¶x l
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Dissipation Equation
¶ ¶ ¶ k ¶e e ¶U e 2 ( + U )e = c ( u¢ u¢ ) - c u¢u¢ i - c , ¶t k ¶x e ¶x e k i ¶x e1 k i k ¶x e2 k 144244k3 144k 424443i 1442443k 123 Convection Diffusion Generation Destruction where
¶ k ¶e ce ( u¢k u¢l ) = diffusion, ¶x k e ¶x k
e ¶U i ce1 u¢i u¢k = generation k ¶x k
e 2 c = destruction. e2 k
Reynolds Equation
¶ ¶ 1 ¶P ¶ ( + U j )U i = - - u¢i u¢j ¶t ¶x j r ¶x i ¶x j
Continuity Equation
¶U i = 0 ¶xi
Closure
There are eleven equations for eleven unknowns, U i , P , u¢i u¢j , e .
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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)
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Asymmetric Channel Flow (Launder, Reece and Rodi, 1975)
Mean Velocity and Turbulent Shear Stress
Turbulence Intensities
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Algebraic Stress Transport Model (Rodi, ZAMM 56, 1976)
A simplified stress transport model is given as:
d ¶ k ¶ ¶U ¶U u¢u¢ = c ( u¢ u¢ u¢ u¢ ) - u¢u¢ j - u¢u¢ i dt i j s ¶x e k m ¶x i j i k ¶x j k ¶x 14k 444244m 443 1444k 24443k Diffusion Production , (1) e 2 2 2 - c1 (u¢i u¢j - dij k) - g(Pij - dij P)- dije 14k 4444432444444331323 Pr essure-Strain Dissipation where
¶ k ¶ Dij = ( u¢k u¢l u¢i u¢j ) = diffusion, ¶x k e ¶x l ¶U j ¶U i Pij = u i¢u¢k - u¢j u¢k = production, ¶x k ¶x k e 2 2 c (u¢u¢ - d k) - g(P - d P) = pressure-strain, 1 k i j ij 3 ij ij 3
2 d e = dissipation. 3 ij
1 Here, P = P is the production rate of turbulent kinetic energy. Contracting 2 ii Equation (1), we find the transport equation for k:
dk ¶ k ¶k ¶U k = cs ( u¢k u¢m ) - u¢k u¢m -{e , (2) dt ¶x e ¶x ¶x 14k 4424443m 14243m Dissipation Diffusion Production where
¶ k ¶k D= ( u¢k u¢l ) =diffusion ¶x k e ¶x l
¶U k P = u¢k u¢l = production ¶x l
Rodi (1976) assumed that
d u¢u¢ dk u¢u¢ u¢u¢ - D = i j ( - D) = i j (P - e) , (3) dt i j ij k dt k
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Using (3) in (1) and rearranging, the result is:
é P 2 P ù ij - d ê2 1- g ij ú u¢u¢ = kê d + e 3 e ú . (4) i j 3 ij c 1 P ê 1 1+ ( -1)ú ê ú ë c1 e û
Equation (4) provides an algebraic expression for u¢i u¢j.
For simple shear flows, it may be shown that equation (4) reduces to the Kolmogorov-Prandtl hypothesis with
k 2 n = c (5) T m e and
1 P [1- (1- g )] 2 (1- g) c1 e cm = with g = 0.6 and c1 = 1.8 - 2.2 . (6) 3 c 1 P 2 1 [1+ ( -1)] c1 e
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 k - e 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.
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For example, the transport equations for u'2 , v'2 , w'2 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., 2 u¢2 v'2 - u¢v' > 0.
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Anisotropic Rate-Dependent Model
Averaged Balance Laws:
Mass
v i,i = 0
Linear Momentum
T rv& i = t ji,j + t ji ,j + rfi
Thermal Energy
T re& = qi,i + qi,i + tij v j,i + re + r
Fluctuation Energy
& T rk = t ij v j,i + K i,i - re
Clausius-Duhem Inequality
T T T rh& - (qi J),i - R i,i - rJ + rh& - Si,i > 0
Helmholtz free Energy Function
h hT y = e - , yT = k - J JT
Heat Flux-Coldness Correlation
T T R i = q i J
Fluctuation Energy Flux-Turbulence Coldness Correlation
T T Si = K i J - E i
Total Heat Flux
T Qi = qi + qi
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Clausius-Duhem Inequality
é æ hJ& ö 1 ù J - rçy& - ÷ - Q J + t v + re ê ç 2 ÷ i ,i ij j,i ú ë è J ø J û
ì é hTJ& T ù 1 1 ü T ï & T T T ï + J í- rêy - T 2 ú - T K i J,i + T + t ij v j,i - reý > 0 îï ë (J ) û J J E i,i þï
Constitutive Equations
Stress
t ij = -pdij + 2md ij
T ˆ T 2 ¶y Dd ij T ì 2 é1 ùü t ij = - rkdij + r + m í(2 + gt d kld kl )d ij + btê d kl d kldij - d ik d kj úý 3 ¶D Dt î ë3 ûþ
Jaumann Derivative
Dˆ d ij = d& + d w + d w Dt ij ik kj jk ki
Deformation Rate and Spin Tensors
1 1 1 d = (v + v ), w = (v - v ) , D = d d ij 2 i, j j,i ij 2 i, j j,i 2 ij ij
Heat Flux
æ m T ö ç ÷ Qi = ç k + C q ÷q,i è s ø
Fluctuation Energy flux
æ m T öé k ù ç ÷ K i = çm + k ÷êk ,i - t,i ú è s øë t û
Heat Capacity
¶ 2y C = -q ¶q 2
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Thermodynamic Constraints
mT > 0 , sq > 0 , g > 0 , b 2 < 48g
mT = C mrkt
Turbulence free Energy Function
é æ tao ö ù T ç ÷ m 2 y = kêlnç ÷ + aC t D + C 0 ú ë è k ø û
Turbulence Stress
ˆ T 2 T ì D 2 é1 ùü t ij = - rkdij + m í2dij + at dij + gt d lk dkl dij + btê d lkd kldij - d ik dkj úý 3 î Dt ë3 ûþ
Basic Equations
v i,i = 0
ˆ 2 ïì Dd ij 1 ïü & é ù T T 2 rv i = -êp + rkú + í2(m + m )d ij + m [at + bt( d lk d kldij - dik d kj ) + gt dlk d kl dij ]ý + rfi ë 3 û ,i ï Dt 3 ï î þ, j
T & é m ù rCq = ê(k + C q )q ,i ú + 2md ijd ij + re + r ë s û ,i
T Dˆ d & é m k ù T ij rk = ê(m + k )(k ,i - t,i )ú + P + atm d ij - re ë s t û,i Dt
T 2 2 P = m [2dij dij - btd ikd kjd ij + gt (d ijd ji ) ]
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Scale Transport Equations
é mT ù t æ mT öæ t ö k k rt& = (m + )t + C t1 P + C t3 çm + ÷ [k - t ][k - t ] ê T ,i ú ç k ÷ç 2 ÷ ,i ,i ,i ,i ë s û k è s øè k ø t t ,i æ mT ö 2aC m + çm + ÷[ ](t2 D) t - rCt2 C D ç T ÷ m 2 ,i ,i è s ø a 0 + 2aC t D
1 1 1 t1 t2 t3 > C > 0 , C > , > C > 0 , a0 > 0 a 0m a0 a 0m
m 2 a0m max .(a0 + 2aC t D)
k e = C D t
T T é m ù e e æ m öæ e ö k k & e1 3 re = ê(m + )e,i ú + C P + C çm + ÷ç ÷[k ,i - e ,i ][k ,i - e,i ] e k ç k ÷ 2 e e ë s û ,i è s øè k ø T m 2 2 æ m ö 2aC k e + çm + ÷[ ](D ) e - rC e2C e2 ç se ÷ k 2 e 2 ,i ,i k è ø a + 2aC mD 0 e 2
k 2 k mT = rCm , t = e e
When g = 0 , a = 0.93 , b = 0.54
b 2 g > , g = 0.005 48
Cm = 0.09 , Ce1 =1.45 , Ce2 =1.92 , sk =1, se = 1.3
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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-e model predictions
Comparison of mean velocity profile with Comparison of axial turbulence the experimental data of Kreplin and intensity with the experimental data of Eckelmann for Reynolds numbers of 8200 Kreplin and Eckelmann, DNS of Kim et and 5600. al. and k-e model.
Comparison of vertical turbulence Comparison of lateral turbulence
intensity with the experimental data of intensity with the experimental data of
Kreplin and Eckelmann, DNS of Kim et Kreplin and Eckelmann, DNS of Kim et al. and k-e model. al. and k-e model.
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Comparison of turbulence shear stress with the experimental data of Kreplin and Eckelmann, and DNS of Kim et al.
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.
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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).
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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 31 G. Ahmadi
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 32 G. Ahmadi
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 33 G. Ahmadi
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))
ME637 34 G. Ahmadi