Turbulence Modelling in STREAM 1. List of Turbulence Models
Model Code Low-Re?
One-equation models
Wolfshtein (1969) WO
Norris and Reynolds (1975) NR
Linear k-ε models
Launder and Spalding (1974) – Standard k-ε HR
Yakhot et al. (1992) – RNG k-ε RG
Shih, Liou et al. – realisable model (1995) RE
Launder and Sharma (1974) LS
Lam and Bremhorst (1981) LB
Chien (1982) CH
Lien and Leschziner (1993) LL
Linear k-ω models
Wilcox (1988) WX
Wilcox (1994) W2
Menter (1994) – Baseline model BL
Menter (1994) – SST model FM
Non-linear k-ε models
Speziale (1987) SP
Rubinstein and Barton (1992) RB
Shih, Zhu and Lumley (1995) – quadratic realisable model SH
Gatski and Speziale (1993) GS
Lien, Chien and Leschziner (1996) CU
Craft, Launder and Suga (1996) KS
Apsley and Leschziner (1998) DA
Differential stress models
Gibson and Launder (1978) GL
Craft and Launder (1992) CL
Speziale, Sarkar and Gatski (1991) – SSG model SG
Shima (1998) NS
Hanjalić and Jakirlić (1995) HJ
Wilcox (1988b) – multiscale k-ω model WM
Most models have been tested only in incompressible flow. In compressible flow the Favre
(density-weighted) average is assumed to replace the Reynolds average in the specifications
that follow.
2. Eddy-Viscosity Models (Linear and Non-Linear) 2.1 Constitutive (Stress-Strain) Relationship
For linear eddy-viscosity models the stress-strain relationship is
ij
k
k
i
j
j
i
tijjix
U
x
U
x
Ukuu δ
3
2νδ
3
2
where
)modelsequationone(
)modelsω(ω
*α
)modelsε(ε
ρ
μν
μ
2/14/1
μ
2
μμ
lkC
kk
kk
fC
t
t
In the one-equation models in STREAM a transport equation is solved for k and the
lengthscale lμ is specified algebraically.
Between k-ε and k-ω models there is a rough correspondence:
kCμ
εω , μ*α f , μμ*β fC
For nonlinear eddy-viscosity models up to cubic order the stress-strain relationship in
incompressible flow may be written in the following form:
)(γ)}{}{(γ}){γ}{γ(
}){(β)(β}){(β
2
22
432222
3
2
2
2
1
2
312
32
2
312
1
μμ
wswsIwswswswswsws
wwswwsss
sa
Cf
where the following general notation is used for second-rank tensors:
)( ijTT , kkTtrace )(}{ TT , )( n
n traceT T , )δ( ijI
The dimensional mean strain and mean vorticity tensors are denoted in upper case by
i
j
j
i
ijx
U
x
US
2
1,
i
j
j
i
ijx
U
x
UW
2
1
whilst dimensionless quantities – anisotropy a, mean strain s and mean vorticity w – are
written in lower case and defined by
ij
ji
ijk
uua δ
3
2 , ijij Ss τ , ijij Ww τ
The turbulent timescale τ is given by
)modelsequationone(
)modelsω(ω*β
1
)modelsε(ε
τ
2/14/3
μ
ε
kC
l
k
kk
For compressible flows, Sij is replaced in constitutive relations by its deviatoric form
ijkkijij SSS δ*31
and for system rotation Ω, Wij is replaced by
kijkijij WW Ωε*
The following dimensionless shear parameters may be defined:
222 ssss ijij , )(22 2wwww ijij
(Both reduce to )/)(ε/( yUk in simple shear flow.)
The rate of production of turbulent kinetic energy (per unit mass) by mean shear is
j
i
ji
k
x
UuuP
)(
2.2 Transport Equations For Turbulence Variables
k equation
)ε(ρ)σ
μμ()ρ()ρ( )(
)(DP
x
k
xkU
xk
t
k
j
k
t
j
j
j
D is only non-zero for models which distinguish homogeneous and inhomogeneous
dissipation rates. In many original references (but not here) ε is often written as ε~ .
For other models ε is determined in the k equation by
modelsequationone
modelsωω*β
ε
ε
2/34/3
μ
l
kC
kk
ε equation
)(ρε
)ε(ρε
)σ
μμ()ερ()ρε( ε22ε
)(
11ε)ε(SS
kfCPfC
xxU
xtl
k
j
t
j
j
j
The additional source terms Sl and Sε are used to control the growth of the turbulent length
scale and correct near-wall viscous sublayer behaviour, respectively. Sl can be also used to
incorporate any non-conventional model for ε production, such as in the realisable model of
Shih, Liou et al. (1995); in that model Cε1 is set to 0 and all production of ε put in Sl.
ω equation
ω
2)(
)ω(ρ)βω
ν
α(ρ
ω)
σ
μμ()ωρ()ρω( SP
xxU
xt
k
tj
t
j
j
j
2.3 Viscosity-Dependent Parameters
Relevant non-dimensional lengths are defined by
ν
τ nyuy
ν
*2/1
nyky
where yn denotes the distance to the nearest wall, ρ/ττ wu and τw is wall stress.
Turbulent Reynolds numbers are
νε
2kRt ,
νω
kRw
2.4 Length Scales
These are used directly in one-equation models and indirectly in some two-equation models.
They are also used in a two-layer treatment of wall boundaries.
Wolfshtein (1969):
)e1(κ *016.0
μ
yyl , )e1(κ *263.0
ε
yyl
Norris and Reynolds (1975):
)e1(κ *0198.0
μ
yyl , 4/3
μ
ε/κ2*
*κ
Cy
yyl
The following values are assumed:
Cμ = 0.09, κ = 0.41
whilst in the k equation:
σ(k)
= 1.0
2.5 Coefficients in Linear k-ε Models
Stress-strain relationship and transport equations
Model Code Cμ Cε1 Cε2 σ(k)
σ(ε)
Sl
Launder and Spalding (1974) HR 0.09 1.44 1.92 1.0 1.3 0
Yakhot el al. (1992) RG 0.085 1.42 1.68 0.719 0.719 0
Shih, Liou et al. (1995) RE **
0 σ
1
sAA 0
νε9.1
k
k 1.0 1.2 ε)5
,43.0max( Ss
s
Launder and Sharma (1974) LS 0.09 1.44 1.92 1.0 1.3 0
Lam and Bremhorst (1981) LB 0.09 1.44 1.92 1.0 1.3 0
Chien (1982) CH 0.09 1.35 1.80 1.0 1.3 0
Lien and Leschziner (1993) LL 0.09 1.44 1.92 1.0 1.3 0
Viscous terms
Model Code fμ D f1 f2 Sε
Launder and
Sharma
(1974)
LS ))50/1(
4.3exp(
2
tR
22/1
)(ν2ix
k
1 2
e03.01 tR
2
νν2
kj
i
txx
U
Lam and
Bremhorst
(1981)
LB )
5.201(
)e1( 2*0165.0
t
y
R
0 3
μ
)05.0
(1f
2
e1 tR 0
Chien (1982) CH y0115.0e1 2
ν2
ny
k 1
2)6/(e22.01 tR
2/
2e
νε2 y
ny
Lien and
Leschziner
(1993)
LL )1(
ε
)1(
μ
l
l 0 1
2
e03.01 tR
2*0022.0)1(
22ε eεε y
kfC
Notes
The Lien and Leschziner (1993) model uses the one-equation model of Wolfshtein
(1969) to supply lengthscales )1(
μl and )1(
εl . The original model actually modifies f1
rather than Sε, but the two formulations are equivalent.
In the Shih, Liou et al. (1995) realisable linear model, 0.40 A , the constant *sA ,
which is derived as the positive root of a cubic equation, is given by
cos6*sA , )6(cos3
12/3
2
31
s
s
and the shear parameter σ* is
2/1)(*σ ijijijij wwss
The unfamiliar ε production term has been transferred to Sl.
Also, although this is not strictly a low-Re model, the ε removal term in that model
contains
νε
ε 2
k
rather than the more common ε2/k. This is implemented in the common form by
setting Cε2 as given in the table.
2.6 Coefficients in Linear k-ω Models
Model Code α* β* α β σ(k)
σ(ω)
Sω
Wilcox
(1988a)
WX 1
100
9
9
5
40
3
2 2 0
Wilcox
(1994a)
W2
61
640
1
ω
ω
R
R
4ω
4ω
)8
(1
)8
(18
5
100
9
R
R
7.21
7.210
1
9
5
ω
ω
R
R
40
3
2 2 0
Menter
(1994)
– Baseline
BL 1 0.09
440.0
553.0B
083.0
075.0B
0.1
5.0
1
B
856.0
5.0
1
B
ωω
71.10
kB
Menter
(1994)
– SST
FM )ω
ω031.0,1min(
2F
0.09
440.0
553.0B
083.0
075.0B
0.1
5.0
1
B
856.0
5.0
1
B
ωω
71.10
kB
Notes
Menter’s models are constructed as a blend of k-ω and k-ε models, phrased in k-ω form:
bFaFb
aB )1( 11
where:
ny
kr
ω*β
2/1
ω , ω
ν2ν
nyr ,
)0.0,ωmax(
ω2
ky
kr
n
g
)tanh( 4
11 argF , )2),500,max(min( νω1 grrrarg
)tanh( 2
22 argF , )500,2max( νω2 rrarg
2.7 Coefficients in Non-Linear k-ω Models
Non-linear stress-strain relationship
Model Code Cμ (β1, β2, β3) (γ1, γ2, γ3, γ4)
Speziale (1987) SP 0.09 (0.054, 0.054, 0) (0, 0, 0, 0)
Rubinstein and
Barton (1992)
RB 0.085 (0.230, 0.047, 0.189) (0, 0, 0, 0)
Shih, Zhu and
Lumley (1995)
SH **
0 σ
1
sAA )0,2,0(
)]([61
)91(
2/1
22
2/1
2
2
μ
ws
sC
(0, 0, 0, 0)
Gatski and Speziale
(1993)
GS *
1121 α C )0,α,α( *
22
*
33 CC , (0, 0, 0, 0)
Lien, Chen and
Leschziner (1996)
CU
ws 9.025.1
3/2
)19,15,3(
1000 3
μ
s
f
)80,0,16,16(μ
3
μ fC
Craft, Launder and
Suga (1996)
KS 3/2
η75.0
η0.351
)e36.0exp(1(3.0
)04.1,4.0,4.0(μμ fC )80,0,40,40(μ
3
μ fC
Apsley and
Leschziner (1998)
DA *22
3
1
*
12
σγβ1
)( Pfa
)0,),(6()σ
( *
22
*
11
*
22
*
11
2
*aaaa
f P )γβ,γ,γ,β()
σ(4
232
2322
312
*μPf
C
Turbulence transport equations
Model Code Cε1 Cε2 σ(k)
σ(ε)
Sl
Speziale (1987) SP 1.44 1.92 1.0 1.3 0
Rubinstein and
Barton (1992)
RB 1.42 1.68 0.719 0.719
ks
ssC
2
3
3
μ
ε
012.01
)138.4/(
Shih, Zhu and
Lumley (1995)
SH 1.44 1.92 1.0 1.3 0
Gatski and Speziale
(1993)
GS 1.44 1.83 1.0 1.3 0
Lien, Chen and
Leschziner (1996)
CU 1.44 1.92 1.0 1.3 0
Craft, Launder and
Suga (1996)
KS 1.44 1.92 1.0 1.3 YAP
Apsley and
Leschziner (1998)
DA 1.44 1.83 22 γ3/β1
0.1
22 γ3/β1
37.1
0
Viscous Terms
Model Code fμ D f1 f2 Sε
Lien, Chen and
Leschziner (1996)
CU )1(
)ε(
)1(
μ / ll 0 1 2
e3.01 tR 2*00375.0
)1(
22ε eεε y
kfC
Craft, Launder and
Suga (1996)
KS ]400
)90
(exp[1 2 tt RR 2
2/1
)(ν2ix
k
1 2
e3.01 tR )250()(
ε
ν0022.0 2
22
t
kj
it Rxx
Uks
Apsley and
Leschziner (1998)
DA 1 0 1 1 2*0038.0)1(
2ε eεε y
kC
Notes
The original Speziale (1987) model included terms involving DSij/Dt. These have
been found to provoke numerical instability and have, therefore, been omitted from
the stress-strain relationship.
In the Shih, Zhu and Lumley (1995) quadratic model, 5.60 A , the constant *sA ,
which is derived as the positive root of a cubic equation, is given by
cos6*sA , )6(cos3
12/3
2
31
s
s
and the shear parameter σ* is
2/1)(*σ ijijijij wwss
In the Gatski and Speziale (1993) model, *iC are shear-dependent terms based on the
regularisation of the 2-d solution as given by Speziale and Xu (1996):
)η6ηζ21)(ζ21(
η)η61)(ζ21(*
6
1
222
2
3552
1b
C
,
)ηζ21)(ζ21(
η)η1)(ζ21(*
6
3,2
22
2
3242
3,2b
C
where
2/1
2
1
3 )(α
α
2
1η s ,
2/1
2
1
2 )(α
αζ w , )0.4,3.6,0.7(),,( 321 bbb
gC )(α 234
1 , gC )2(αα 4121
2 , gC )2(αα 313
1)ε/(
1)(
121
eqm
kPCg ,
1
1
ε 1ε
2ε
)(
C
CP
eqm
k
The constants, which come from the SSG pressure-strain model, are
C1 = 6.8, C2 = 0.36, C3 = 1.25, C4 = 0.40
Note also that, for system rotation in this model only, wij is alternatively defined by
kijkijij
C
CWw Ωε
2
4τ
4
4
In the Craft et al. (1996) model,
),max(η ws
The Yap correction (Yap, 1987) in the dissipation equation is
)0,ε
γ)1γ(83.0max(2
2
kYAP ,
ε
)κ/(γ
2/34/3
0μ nykC , Cμ0 = 0.09
The low-Re terms in the Lien, Chen and Leschziner (1996) model are based on the
one-equation model of Norris and Reynolds (1975) for the mixing and dissipation
lengths near the wall. The original model actually modifies f1 rather than Sε, but the
two formulations are equivalent.
In the Apsley and Leschziner (1998) model, fμ is incorporated naturally into Cμ. *
αβa
and σ* are curve fits to the three independent anisotropy components and shear
parameter, respectively, from DNS data for plane channel flow:
3
2)040.0296.0exp(42.01 *2/1**
11 yya
3
2])000147.0001.0exp(1[404.0 2***
22 yya
])0189.000443.0exp(1[3.0 *2/1**
12 yya
])088.0exp(277.01[])45.0exp(1[33.3σ *2/3*** yyy
The constants β and γ are based on the values of the anisotropy components and
shear parameter in the log-law region and are given by
222.0β , 623.0γ
Modifications to Cμ, σ(k)
and σ(ε)
arise because, unlike the other non-linear models, the
first two cubic terms do not cancel out in simple shear flow. The non-equilibrium
parameter fp which accounts for departures of the local shear parameter 2/1
22
2 )()/()ε/(σ wsxUk ji from the calibration value σ* is given by
2*
00
0
)σ/σ)(1(411
2
ff
ff P , )0.1,σ09.0max(25.11 2*
0 f
The additional term in the dissipation equation is based upon a curve fit to DNS data
for dissipation length:
])279/exp(1[)28.1
1(179.0 2*
*
)1(
ε yy
yl n , )1(
ε
2/34/3
0μ)1(εl
kC
Additional Note
Some researchers (including Dr Apsley) have noted that numerical implementations
of the model can become numerically unstable for extremely fine grids. The root
cause appears to be as follows.
In simple shear, the model gives
a
axxa
k
uv
1
3*
12 , where 2
312 βγ a ,
*σ
σpfx
Unfortunately,
0
k
uv
x at
ax
3
1
With standard values of β and γ , this occurs at x = 0.947 and is uncomfortably close
to the “preferred” equilibrium value x = 1. The proximity to a stationary point means
that kuv / doesn’t change much even when shear U/y does, resulting in large
fluctuations in U(y) even for a shear stress fixed by the pressure gradient.
This can be eliminated by changing the values of β and γ – but that would damage
the agreement with experimental results for boundary-layer anisotropy. A better
solution (yet to be found!) would be a more stable numerical implementation.
3. Differential Stress Models
3.1 Transport Equations
)εΦ(ρ)ρ()ρ( ijijijijijk
k
jik
k
ji GPdx
uuUx
uut
)(ρε
)ε(ρ)
)ερ()ρε( ε22ε
)(
11ε
ε(
SSk
fCPfCx
dU
xtl
k
j
kj
j
where
)(k
i
kj
k
j
kiijx
Uuu
x
UuuP
, ii
k PP21)(
ijjiij ufufG
Modelled terms are given below. Note that:
)(i
k
kj
j
k
kiijx
Uuu
x
UuuD
ij
ji
ijk
uua δ
3
2 (anisotropy)
)(8
91 32 aaA (flatness)
ij
ij
ije δ3
2
ε
ε
)(8
91 32 eeE
3.2 Diffusion
)()ε
ρμδ( ji
l
lksklijk uu
x
uukCd
l
lkklk
x
uukCd
ε)
ε
ρμδ( ε
)ε(
3.3 Pressure-Strain Correlation
)()2()1( ΦΦΦΦ w
ijijijij
Slow Pressure-Strain
)δ3
2(
εΦ 1
)1(
ijjiij kuuk
C
or, for a non-linear extension:
)}{3
1(
ε
22
11
)1(
IaaaΦ
CC
Fast Pressure-Strain
)δ3
1()δ
3
1()δ
3
1(Φ 432
)2(
ijkkijijkkijijkkijij SSkCDDCPPC
or
)()}{3
2()}{
3
1(
ε121101
)2(
awwaIasassaIssΦ
CCC
The two expressions may be interconverted by:
43201 )(3
4CCCC , 3211 CCC , 3212 CCC
or
)(2
112112 CCC , )(
2
112113 CCC , 01114
3
4CCC
Wall Reflection Terms
)()( )Φ~
2
3Φ~
2
3δΦ
~(Φ w
kijkkjikijlkkl
w
ij fnnnnnn
Here,
)2()(
2
)(
1 Φε
Φ~
ij
w
ji
w
ij Cuuk
C
except, in the Craft-Launder (1992) model, where:
ijlk
l
kk
kjjk
k
i
ji
w
ij nnx
UkPkuu
x
Uuu
kC δ)
3
4.008.0()δ
3
2(1.0
εΦ~ )()(
1
In all cases, the wall-distance parameter is
n
w
y
kCf
κ
ε/2/34/3
μ)( , Cμ = 0.09
3.4 Dissipation
Except in the Hanjalić and Jakirlić model, dissipation is modelled as isotropic:
ijij εδε32
3.5 Individual Models
Gibson and Launder (1978)
Diffusion:
Cs = 0.22, Cε = 0.18
Pressure-strain:
C1 = 1.8, 01 C
C2 = 0.6, C3 = 0, C4 = 0
5.0)(
1 wC , 3.0)(
2 wC , 5.2lC
Dissipation equation:
Cε1 = 1.44, Cε2 = 1.92, Sl = 0
f1 = 1, f2 = 1, Sε = 0
Speziale, Sarkar and Gatski (1991) – SSG Model
Diffusion: not specified in the original paper; taken as
Cs = 0.22, Cε = 0.18
Pressure-strain:
ε
9.07.1)(
1
kPC , 05.11 C
2/1
201 65.08.0 aC , 625.011 C , 2.012 C
No separate wall-reflection
Dissipation equation: not specified in the original paper; taken as
Cε1 = 1.44, Cε2 = 1.83, Sl = 0
f1 = 1, f2 = 1, Sε = 0
Shima (1998)
Diffusion:
Cs = 0.22, Cε = 0.15
Pressure-strain:
)e1)(e1(45.2122 )60/()7(4/34/1
21tRAAaC
, 01 C
AC 7.02 , 2/1
3 3.0 AC , 4/1
2214 3.1)123.0(65.0 aCCAC
No separate wall-reflection
Dissipation equation:
Cε1 = 1.44 + β1 + β2, Cε2 = 1.92, Sl = 0
)0,1ε
min(4.1)0,15.2
λmin(25.0β
)(
1 kP
AA
)0,15.2
λmax(λ0.1β 2
2 A
)4,ε
min(λ2/3k
f1 = 1, ε
ε~
2 f , Sε = 0
)0,)(ν2εmax(ε~ 22/1
ix
k
Hanjalić and Jakirlić (1995)
Diffusion:
Cs = 0.22, Cε = 0.18
Pressure-strain:
22/1
1 EACC , 01 C
2/1
2 8.0 AC , C3 = 0, C4 = 0
)3.0,7.01max()(
1 CC w , )3.0,min()(
2 AC w , 5.2lC
tRa fAfC 4/1
25.2 , )6.0,min( 22
afa , ]1,)150
min[( 2/3tR
Rf
t
Here, )4.1,ε/
min(2/3
)(
nl
w
yC
kf
Dissipation (anisotropic):
*
ε32
ε ε)1(εδε ijijij ff
d
n
djinkikjkjkiji
ij
fk
u
fnnunnuunnuuuu
k 2
2
*
2
31
)(εε
22/1
ε EAf , t
dR
f1.01
1
Dissipation equation:
Cε1 = 1.44, Cε2 = 1.92, Sl = 0
f1 = 1, ε
ε~e)
1(1
2)6/(
2ε
2ε
2
tR
C
Cf ,
lj
k
li
kji
xx
U
xx
Uuu
kS
22
εε
ν25.0
)0,)(ν2εmax(ε~ 22/1
ix
k
3.6 Wilcox (1988b) – Multiscale Model
Wilcox’s multiscale model is built upon the premise that one can partition the energy
spectrum into large-scale, energy-bearing eddies and small-scale, isotropic, dissipative
eddies. The formulation is rather different from that of “traditional” Reynolds-stress transport
models and consists of transport equations for k, ω and the upper-partition stress tensor
)δ3
2( ijjiij euuT
where e is the energy of the lower-partition eddies, plus a tensor describing the exchange of
energy between upper and lower partitions. The most practical way of incorporating this into
the STREAM code is to rewrite Wilcox’s equations in terms of equations for jiuu , ω and kU
(the upper-partition turbulence energy).
)εΦ(ρ)ρ()ρ( ijijijijijk
k
jik
k
ji GPdx
uuUx
uut
ω
2)(
)ω(ρ)βω
ν
α(ρ
ω)
σ
μμ()ωρ()ρω( SP
xxU
xt
k
tj
t
j
j
j
ε)()1(ρ)ρ()ρ( 2/3)(
32k
kPCCkU
xk
t
UkU
j
j
U
where
iiuuk2
1 , kω*βε ,
ωρ
μν
ktt
Stress-Transport Equation
Diffusion:
ij
k
k
t
ijkx
kd δ)
σ
μμ(
)(
, σ
(k) = 2
Pressure-strain:
2/3)(
1 )(41k
kC
U
, 55
422 C ,
55
63 C ,
4
14 C
No wall-reflection terms
Dissipation:
ijij εδ
3
2ε
ω-equation
σ(ω)
= 2, 9
5α ,
40
3β
wS βωω
4. Wall Boundary Conditions
There are three approaches.
(i) Low-Re treatment:
assume that behaviour is adequately resolved right to the boundary;
apply viscous modifications to turbulence equations and constitutive relations;
these viscous modifications are applied throughout the flow.
(ii) Wall-function treatment:
if necessary, model what happens between the near-wall node and the boundary;
depending on variable, may set wall flux, cell-averaged source or fixed value at the
near-wall node;
viscous effects are included only for the near-wall cell and boundary.
(iii) Two-layer treatment (eddy-viscosity models only):
blend a high- or low-Re 2-equation model into a 1-equation model near a wall.
4.1 Wall-Boundary Effects on Individual Transport Variables
Variable Common Low-y+ Wall function (near-wall cell only)
Ui Wall value = 0 Viscous modifications to νt,
affecting fluxes.
Wall flux via effective viscosity μw
k Wall value = 0
Wall flux = 0
Viscous terms in transport
equation.
Cell-averaged production and
dissipation.
jiuu Wall value = 0
Wall flux = 0
Viscous terms in transport
equation
Cell-averaged production and
dissipation.
ε (if D = 0) Wall flux = 0 2
ν2εε
P
PPw
y
k
Viscous terms in transport
equation.
εw = εP = wall-function value at P
ε (if D 0) Wall flux = 0 εw= 0
Viscous terms in transport
equation.
N/A
ω Wall flux = 0 2
μ
ν2ωωω
P
PfacPw
yC
k
Viscous terms in transport
equation
ωw = ωP = wall-function value at P
Notes
Subscript P refers to the near-wall node, subscript w to the value on the boundary.
Using the treatment suggested by Menter (1994), 8002
ωμ
C
fac .
4.2 Wall-Function Approach
4.2.1 Basic Profiles
The wall-function formulae are deduced from the following basic assumptions which make a
smooth transition from laminar viscous sublayer to fully-turbulent log layer. Here, y denotes
the distance from the boundary, subscript P denotes the value of a variable at the near-wall
node (centre of the near-wall cell) and Δ denotes the thickness of the near-wall cell.
Total stress constant and an effective total viscosity:
y
Ueffw
ρνττ , 0)0( U
)()(κν
)(νν
νν0
ν
yyyyu
yyeff
where
2/14/1
μ PkCu
The dissipation rate is given by:
)(
)(κ
)(ε
εε
3
0
ε
yyyy
u
yy
d
w
with εw determined so as to make ε continuous at yε.
Note:
Profile points yν, yε and yd are defined below.
The definition of u0 is such that it would equal uτ in the log layer.
The implied equivalent one-equation model ( μ
2/14/1
μν lkCt ) would have length scale
)1(κ νμ
y
yyl ( νyy )
4.2.2 Derived Quantities
Assumed Mean-Velocity Profile
By integration:
)~()~(κ1lnκ
1
)~(~
ρ
τ
ννν
ν
2
00yyyyy
yyy
uu
U w
where
ν
~ 0 yuy
Important: y~ here is based on u0 rather than uτ.
Wall Stress and Effective Wall Viscosity
P
Pww
y
Uρντ
where
)~(
)~(κ1lnκ
1
~)~(1
ννν
νν
ν
yy
yyy
y
yy
P
P
P
P
w
Cell-Averaged Production and Dissipation
)Δ~
()Δ
~(κ1
)Δ~
(κ)Δ
~(κ1ln
Δκ
)ρ/τ(
)Δ~
(0
ν
ν
ν
ν
0
2
ν
)(
yy
yy
u
y
Pw
k
av
)Δ(
Δln
Δκ
)Δ(ε
εν
ε
ε
ε
3
0
ν
yyy
y
yy
yu
y
dd
d
w
av
Near-Wall Dissipation
εP is given directly from the assumed ε profile at y = yP; i.e.
)(
)(κ
)(ε
εε
3
0
ε
yyyy
u
yy
P
dP
Pw
P
4.2.3 Matching Depths
For smooth walls:
37.7ν y
4.27ε y , 9.4
dy
For arbitrarily-rough walls (Apsley, 2007) the viscous sublayer cutoff is given by:
)κlnκ
1(ν Bfy ,
)0()1(κ
1
)0(
)( κ xe
xx
xf x
and
)ln(κ
1CkBB srough ,
)(κ smoothrough BBeC
Here, Bsmooth and Brough are constants in the fully-smooth and fully-rough logarithmic wall
profiles, respectively:
smoothByU lnκ
1 and rough
s
Bk
yU ln
κ
1
STREAM assumes values κ = 0.41, Bsmooth = 5.2 and Brough = 8, whence C = 3.152.
Similarly, the dissipation-related constants are given by:
κ
1ν yyd
d
d
dyy
sysyy
ε
1
1ε
)/11(exp
κ
1, )0,max(κ1 ν1
ys
The last has to be determined iteratively, but converges quickly.
4.2.4 Other Variables
Omega
Where wall functions are used with the k-ω model, ε is deduced as above and the near-wall
value of ω then determined by
P
PP
kCμ
εω
Reynolds stresses
Where wall functions are used with differential-stress models, cell-averaged production is
first expressed in a local coordinate system with tangential (t) and normal (n) velocities
(relative to any wall velocity). Then:
)()( 2 av
k
av
tt PP , )(
μ
)( 248.0 av
k
av
tn PC
P , 0)( av
abP otherwise
Components of production are subsequently rotated to the global Cartesian system and the
equations treated in similar fashion to the k equation (including cell-averaged dissipation).
Note:
Earlier versions of STREAM set individual stresses at near-wall nodes rather than
cell-averaged production. This, however, gave problems when the positive tangential
direction could not be identified; for example, at impingement or separation points.
This does not affect the production terms as they vanish here anyway.
Because only cell-averaged Pij is specified in this way (not Dij or Sij) all differential-
stress models using wall functions in STREAM revert to the standard return-to-
isotropy form for the fast pressure strain in the near-wall cell. This is not ideal!
4.3 Two-Layer Approach (Two-Equation Eddy-Viscosity Models Only)
A blending function fb is used to blend the eddy viscosity μt and dissipation rate ε between
any two-equation eddy-viscosity model (high-Re or low-Re) and the one-equation model of
Wolfshtein (1969) defined earlier. The blending function is here taken as
)
352.4
60*tanh(1
2
1 yfb
To blend the eddy viscosity:
)1()2( μ)1(μμ eqn
tb
eqn
tbt ff
To blend the dissipation rate within its discretised equation ( pFFPP baa εε ), write
)1(ε)1()
ε(ε eqn
Pb
P
FFP
bp fa
abf
This rearranges as:
)1(ε
)1(εε eqn
PP
b
b
PFFp
b
P af
fba
f
a
so that any under-relaxation step for ε is preceded by a modification of coefficients:
b
p
pPf
aaa , )1(ε)1( eqn
PPbPP afbb
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