INTRODUCTION TO TURBULENCE MODELING...Free-Stream Turbulence The free stream turbulence affects the...
Transcript of INTRODUCTION TO TURBULENCE MODELING...Free-Stream Turbulence The free stream turbulence affects the...
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−
∂∂
+ρ
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
∂∂
=ε
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∂
∂=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))