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Overview of Turbulence Modeling
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Outline
Background
Characteristics of Turbulent Flow
Scales
Eliminating the small scales
Reynolds Averaging
Filtered Equations
Turbulence Modeling Theory RANS Turbulence Models in FLUENT
Turbulence Modeling Options in Fluent
Near wall modeling, Large Eddy Simulation (LES)
Turbulent Flow Examples Comparison with Experiments and DNS
Turbulence Models
Near Wall Treatments
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What is Turbulence?
Unsteady, irregular (aperiodic) motion in which transported quantities
(mass, momentum, scalar species) fluctuate in time and space Fluid properties exhibit random variations
statistical averaging results in accountable, turbulence related transport
mechanisms
Contains a wide range of eddy sizes (scales)
typical identifiable swirling patterns
large eddies carry small eddies
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Turbulent boundary layer on a flat plateTurbulent boundary layer on a flat plate
Homogeneous, decaying, gridHomogeneous, decaying, grid--generated turbulencegenerated turbulence
Two Examples of Turbulence
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Energy Cascade
Larger, higher-energy eddies, transfer energy to smaller eddies via
vortex stretching Larger eddies derive energy from mean flow
Large eddy size and velocity on order of mean flow
Smallest eddies convert kinetic energy into thermal energy via viscous
dissipation
Rate at which energy is dissipated is set by rate at which they receive
energy from the larger eddies at start of cascade
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Vortex Stretching
Existence of eddies implies vorticity
Vorticity is concentrated along vortex lines or bundles Vortex lines/bundles become distorted from the induced velocities of
the larger eddies
As end points of a vortex line randomly move apart
vortex line increases in length but decreases in diameter
vorticity increases because angular momentum is nearly conserved
Most of the vorticity is contained within the smallest eddies
Turbulence is a highly 3D phenomenon
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Smallest Scales of Turbulence
Smallest eddy (Kolmogorov) scales:
large eddy energy supply rate ~ small eddy energy dissipationrate = -dk/dt k (u2+v2+w2) is (specific) turbulent kinetic energy [l2/t2]
is dissipation rate ofk[l2/t3]
Motion at smallest scales dependent upon dissipation rate, , andkinematic viscosity, [l2/t]
From dimensional analysis:
= (3/)1/4; = (/)1/2; v = ()1/4
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Small scales vs. Large scales Largest eddy scales:
Assume lis characteristic of larger eddy size
Dimensional analysis is sufficient to estimate order of large eddy supplyrate ofkas k /turnover
turnoveris a time scale associated with the larger eddies the order ofturnovercan be estimated as l/ k1/2
Since ~ k /turnover, ~ k3/2/l orl~ k3/2/ Comparing lwith ,
where ReT = k1/2l / (turbulence Reynolds number)
4/34/3
4/12/3
4/13Re)/(
)/(T
lklll = 1>>l
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Implication of Scales
Consider a mesh fine enough to resolve smallest eddies and large
enough to capture mean flow features Example: 2D channel flow
Ncells~(4l /)3
or
Ncells ~ (3Re)9/4
where
Re = uH /2
ReH = 30,800 Re = 800 Ncells = 4x107 !
H4/13 )/(
ll
l
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Direct Numerical Simulation
DNS is the solution of the time-dependent Navier-Stokes
equations without recourse to modeling
Numerical time step size required, t ~ For 2D channel example
ReH = 30,800
Number of time steps ~ 48,000
DNS is not suitable for practical industrial CFD
DNS is feasible only for simple geometries and low turbulent
Reynolds numbers
DNS is a useful research tool
+
=
+
j
i
kik
ik
i
x
U
xx
p
x
UU
t
U
u
Ht ChannelD
Re
003.02
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Removing the Small Scales
Two methods can be used to eliminate need to resolve small scales:
Reynolds Averaging Transport equations for mean flow quantities are solved
All scales of turbulence are modeled
Transient solution t is set by global unsteadiness
Filtering (LES) Transport equations for resolvable scales
Resolves larger eddies; models smaller ones
Inherently unsteady, t set by small eddies
Both methods introduce additional terms that must be modeled for
closure
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l = l/ReT
3/4
Prediction Methods
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RANS Modeling - Velocity Decomposition
Consider a point in the given flow field:
( ) ( ) ( )txutxUtxu iii ,,,rrr
+=
u'i
Ui ui
time
u
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RANS Modeling - Ensemble Averaging
Ensemble (Phase) average:
Applicable to nonstationary flows such as periodic or quasi-periodic flows
involving deterministic structures
( ) ( )( )=
=
N
n
n
iN
i txuN
txU1
,1
lim,rr
U
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( )
+
+
+=
+++
+
j
ii
jik
iikk
ii
x
uU
xx
pp
x
uUuU
t
uU )()()(
)(
.,0;0;;0; etc=+=
Deriving RANS Equations
Substitute mean and fluctuating velocities in instantaneous Navier-
Stokes equations and average:
Some averaging rules:
Given = + and = +
Mass-weighted (Favre) averaging used for compressible flows
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RANS Equations
Reynolds AveragedNavier-Stokes equations:
New equations are identical to original except :
The transported variables, U, , etc., now represent the mean flow
quantities
Additional terms appear:
Rij
are called the Reynolds Stresses
Effectively a stress
These are the terms to be modeled
( )j
ji
j
i
jik
ik
i
xuu
xU
xxp
xUU
tU
+
+
=
+
jiij uuR =
ji
j
i
j
uux
U
x
(prime notation dropped)
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Turbulence Modeling Approaches
Boussinesq approach
isotropic relies on dimensional analysis
Reynolds stress transport models
no assumption of isotropy
contains more physics
most complex and computationally expensive
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The Boussinesq Approach
Relates the Reynolds stresses to the mean flow by a turbulent (eddy)
viscosity, t
Relation is drawn from analogy with molecular transport of momentum
Assumptions valid at molecular level, not necessarily valid at
macroscopic level
t is a scalar (Rij aligned with strain-rate tensor, Sij) Taylor series expansion valid iflmfp|d
2U/dy2|
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Modeling t Oh well, focus attention on modeling t anyways
Basic approach made through dimensional arguments Units oft = t/are [m2/s]
Typically one needs 2 out of the 3 scales:
velocity - length - time
Models classified in terms of number of transport equations solved,e.g.,
zero-equation
one-equation
two-equation
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Zero Equation Model
Prandtl mixing length
model:
Relation is drawn from same analogy with molecular transport of
momentum:
The mixing length model:
assumes that vmix is proportional to lmix& strain rate:
requires that lmixbe prescribed
lmix must be calibrated for each problem Very crude approach, but economical
Not suitable for general purpose CFD though can be useful where a
very crude estimate of turbulence is required
+
==i
j
j
iijijijmixt
x
U
x
USSSl
2
1;22
mfpthv2
1l= mixmixv
2
1lt =
ijijSSl 2v mixmix
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Other Zero Equation Models
Mixing length observed to behave differently in flows near solidboundaries than in free shear flows
Modifications made to the Prandtl mixing length model to account fornear wall flows
Van Driest- Reduce mixing length in viscous sublayer (inner boundarylayer) with damping factor to effect reduced mixing
Clauser- Define appropriate mixing length in velocity defect (outerboundary) layer
Klebanoff- Account for intermittency dependency
Cebeci-Smith and Baldwin-Lomax
Accounts for all of above adjustments in two layer models Mixing length models typically fail for separating flows
Large eddies persist in the mean flow and cannot be modeled from localproperties alone
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One-Equation Models
Traditionally, one-equation models were based on transport equation
fork(turbulent kinetic energy) to calculate velocity scale, v = k1/2
Circumvents assumed relationship between v and turbulence length scale
(mixing)
Use of transport equation allows history effects to be accounted for
Length scale still specified algebraically based on the mean flow very dependent on problem type
approach not suited to general purpose CFD
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+
=
+
jjii
jjj
i
ijj
j upuuux
k
xx
U
Rx
k
Ut
k
'2
1
unsteady &
convective
production
dissipation
molecular
diffusion
turbulent
transport
pressure
diffusion
k
i
k
i
x
u
x
u
=
Turbulence Kinetic Energy Equation
Exact kequation derived from sum of products of Navier-Stokes
equations with fluctuating velocities (Trace of the Reynolds Stress transport equations)
where (incompressible form)
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Modeled Equation fork The production, dissipation, turbulent transport, and pressure
diffusion terms must be modeled
Rij in production term is calculated from Boussinesq formula
Turbulent transport and pressure diffusion:
= CDk3/2/l from dimensional arguments
t = CDk2/ (recall t k1/2l)
CD, k, and lare model parameters to be specified
Necessity to specify llimits usefulness of this model
Advanced one equation models are complete
solves for eddy viscosity
jk
jjiix
kupuuu
=+
t'2
1 Using t/kassumes kcan be transported by
turbulence as can U
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Spalart-Allmaras Model Equations
~
,f,~ 31
3
3
v11t += cvvf
1v2222 1
1f,~~
vv
ff
dSS
+=+
( )22
6
2
6/1
6
3
6
6
3~
~,g,
1
dSrrrcr
ggf w
w
ww
cc
+=
+
+=
( )2
1
2
2~
1
~
~~~1~~~
+
+
+=
dfc
xc
xxSc
Dt
Dww
j
b
jj
b
0~:conditionboundaryWall =
modified turbulent viscositymodified turbulent viscosity
distance from walldistance from wall
damping functionsdamping functions
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=
i
j
j
i
x
U
x
US2
1;2 ijijij
)-Smin(0,C ijijprodij +S
Spalart-Allmaras Production Term
Default definition uses rotation rate tensor only:
Alternative formulation also uses strain rate tensor:
reduces turbulent viscosity for vortical flows
more correctly accounts for the effects of rotation
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Spalart-Allmaras Model
Spalart-Allmaras model developed for unstructured codes in aerospace
industry Increasingly popular for turbomachinery applications
Low-Re formulation by default
can be integrated through log layer and viscous sublayer to wall
Fluents implementation can also use law-of-the-wall Economical and accurate for:
wall-bounded flows
flows with mild separation and recirculation
Weak for:
massively separated flows
free shear flows
simple decaying turbulence
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Two-Equation Models
Two transport equations are solved, giving two independent scalesfor calculating t Virtually all use the transport equation for the turbulent kinetic
energy, k
Several transport variables have been proposed, based ondimensional arguments, and used for second equation
Kolmogorov, : t k /, l k1/2/, k / is specific dissipation rate defined in terms of large eddy scales that define supply rate ofk
Chou, : t k2/, l k3/2/ Rotta, l: t k1/2l, k3/2/l
Boussinesq relation still used for Reynolds Stresses
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Standard k- Model Equations
ijijtjkj
SSSSxk
xDtDk 2;2t =+
+=
( )
2
2t1
t CSCkxxDt
D
jj
+
+
=
kk--transport equationtransport equation
--transport equationtransport equationproductionproduction dissipationdissipation
2,,, CCik
coefficientscoefficients
turbulent viscosityturbulent viscosity
2
k
Ct =
inverse time scaleinverse time scale
Empirical constants determined from benchmark
experiments of simple flows using air and water.
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Simple flows render simpler model equations
Coefficients can be isolated and compared with experiment e.g.,
Uniform flow past grid
Standard k- equations reduce to just convection and dissipation terms
Homogeneous Shear Flow
Near-Wall (Log layer) Flow
kC
xU
x
kU
2
2d
d;
d
d ==
Closure Coefficients
l S i C Ad d l i i
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BuoyancyBuoyancyproductionproduction
DilatationDilatationDissipationDissipation
RT
k
xgS
x
k
xDt
Dk
it
tit
jkj
2
Pr
2t
+
+
=
Standard k- Model
High-Reynolds number model
(i.e., must be modified for the near-wall region) The term standard refers to the choice of coefficients
Sometimes additional terms are included
production due to buoyancy
unstable stratification (gT >0) supports kproduction dilatation dissipation due to compressibility
added dissipation term, prevents overprediction of spreading rate in
compressible flows
Fl U S i C Ad d Fl T i i
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Standard k- Model Pros & Cons
Strengths:
robust economical
reasonable accuracy for a wide range of flows
Weaknesses:
overly diffusive for many situations
flows involving strong streamline curvature, swirl, rotation, separating
flows, low-Re flows
cannot predict round jet spreading rate
Variants of the k- model have been developed to address itsdeficiencies
RNG and Realizable
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RNG k- Model Equations Derived using renormalization group theory
scale-elimination technique applied to Navier-Stokes equations
(sensitizes equations to specific flow regimes)
kequation is similar to standard k- model
Additional strain rate term in equation most significant difference between standard and RNG k- models
Analytical formula for turbulent Prandtl numbers
Differential-viscosity relation for low Reynolds numbers
Boussinesq model used by default
( ) ( ) whereCSCkxxDt
Dt
jj
*2
21eff +
=
t
kS
C
CC
+=
=
+
+=
eff
0
3
0
3
2
*
2
tscoefficienare,
1
1
--transport equationtransport equation
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RNG k- Model Pros &Cons
For large strain rates:
where > 0, is augmented, and therefore kand t are reduced
Option to modify turbulent viscosity to account for swirl
Buoyancy and compressibility terms can be included
Improved performance over std. k- model for rapidly strained flows
flows with streamline curvature
Still suffers from the inherent limitations of an isotropic eddy-
viscosity model
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Standard k- model could not ensure: Positivity of normal stresses
Schwarzs inequality of shear stresses
Modifications made to standard model
kequation is same; new formulation fort and C is variable
equation is based on a transport equation for the mean-square vorticityfluctuation
02 u
( ) uuuu 222
Realizable k- Model: Motivation
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How can normal stresses become negative?
Standard k- Boussinesq viscosity relation:
Normal component:
Normal stress will be negative if:
3
2- ij
2
kx
U
x
UkCuu
i
j
j
iji
+
=
23
2
22
x
UkCku
=
3.73
1 >
Cx
Uk
Realizable k- Model: Realizability
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Realizable k-Model: C
C is not a constant, but varies as a function of mean velocity field and
turbulence (0.09 in log-layerSk/= 3.3, 0.05 in shear layer ofSk/= 6)
C contours for 2D backward-facing step
C along
bottom-wall
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Realizable k- Model Equations
kU
AA
Ck
C
s
*
0
2
t
1,
+
==
where
ijijkijkij
ijijijij SSSS
SSSWSSU ==+ ~,~,*
( )WAA s 6cos3
1,cos6,04.4 10
===
0.1,/,5
,43.0max 21 ==
+= CSkC
++
+
= kCSCxxDt
D
jj
2
21
t
--transport equationtransport equation
turbulent viscosityturbulent viscosity
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Realizable k- Model Pros & Cons
Performance generally exceeds the standard k- model
Buoyancy and compressibilty terms can be included Good for complex flows with large strain rates
recirculation, rotation, separation, strong p
Resolves the round-jet/plane jet anomaly
predicts the speading rate for round and plane jets
Still suffers from the inherent limitations of an isotropic eddy-viscosity
model
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Standard and SST k- Models k-models are a popular alternative to k-
~ / k
t k /
Wilcoxs original model was found to be quite sensitive to
inlet and far-field boundary values of
Can be used in near-wall region without modification Latest version contains several refinements:
reduced sensitivity to boundary conditions
modifcation for the round-jet/plane-jet anomaly
compressibility effects
low-Re (transitional) effects
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Standard k-Model The most well-known Wilcox k-model until recently was his 1988
model (will be referred to as Wilcox original k-model)
Fluent v6 Standard k- model is Wilcox 1998 model
Wilcox original k-is a subset of the Wilcox 1998 model, and can be
recovered by deactivating some of the options and changing some of the
model constants
+
+
=
+
+
=
=
j
t
jj
i
ij
jk
t
jj
iij
t
xxf
x
U
kDt
D
xk
xkf
xU
DtDk
k
2
*
*
*
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Standard k-Turbulent Viscosity
Turbulent viscosity is computed from:
The dependency of* uponReTwas designed to recover the correct
asymptotic values in the limiting cases. In particular, note that:
kt *=
0.1,Re,6
125
9,
3,
Re1
Rewhere
*
*
0
*
0**
===
==
+
+=
kR
R
R
Tk
ii
kT
kT
turbulent)(fullyas TRe1*
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g
Turbulence Apr 2005
Standard k-Turbulent Kinetic Energy
Note the dependence uponRe , Mt , andk
Dilatation dissipation is accounted for via Mt term
The cross-diffusion parameter (k) is designed to improve free shear
flow predictions
444 3444 2143421
321
kofDiffusion
kofratenDissipatio
kofproduction
+
+
=jk
t
jj
iij
x
k
xkf
x
U
Dt
Dk
**
( )( )
( )09.0,5.1,0.2
8,Re1
Re154
1
**
4
4
**
***
===
=+
+=
+=
k
T
T
i
ti
RR
R
MF( )
44 344 21parameterdiffusion-cross
jj
k
k
k
k
k
tt
tttt
tt
t
xx
kf
RTaMa
kM
MMMM
MMMF
=
>++
=
===
>
=
3
2
2
022
0
2
0
2
0
1,
04001
6801
01
,4
1,
2
0
*
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Note the dependence upon Re , Mt , and
Vortex-stretching parameter () designed to remedy the plane/round-jetanomaly
Standard k-Specific Dissipation Equation
+
+
=j
t
jj
iij
xxf
x
U
kDt
D
2
( )
( )
=
+
=
=
=+
+=
+=
====+
+=
i
j
j
iij
i
j
j
iij
kijkij
t
i
ii
T
T
x
U
x
U
x
U
x
US
SfMF
RR
R
2
1,
2
1
5.1,,801
701,1
0.2,95.2,9
1,
25
13,
Re1
Re
*
3*
**
00
*
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Standard k-Model Sub-models & Options (I) Transitional flow option
Corresponds to all terms involving ReTterms in the model
equations
Deactivatedby default Can benefit low-Re flows where the extent of the
transitional flow region is large
Compressibility Effects option
Takes effects viaF(Mt) Accounts for dilatation dissipation
Available with ideal-gas option only and is turned onby
default
Improve high-Mach number free shear and boundary
layer flow predictions - reduces spreading rates
kk
kdds
x
u
x
u
=+=
3
4,
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Standard k-Model Sub-models & Options (II) Shear-Flow Corrections option
Controls both cross-diffusion and vortex-stretching
terms - Activatedby default
Cross-diffusion term (in k-equation)
Designed to improve the model performance for free
shear flows without affecting boundary layer flows
Vortex-stretching term
Designed to resolve the round/plane-jet anomaly
Takes effects for axisymmetric and 3-D flows but
vanishes for planar 2-D flows
( )3*,801701
=++
=kijkij S
f
44 344 21 parameterdiffusion-cross
jj
k
k
k
k
k
xx
kf
=
>
+
+
=
3
2
21
,0
4001
6801
01
*
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Menters SST k-ModelBackground
Many people, including Menter (1994), have noted that:
Wilcox original k-model is overly sensitive to the freestream value(BC) of, while k-model is not prone to such problem
k-model has many good attributes and perform much better than k-models forboundary layer flows
Most two-equation models, including k-models, over-predict turbulentstresses in the wake (velocity-defect) region, which leads to poor
performance of the models for boundary layers under adverse pressure
gradient and separated flows
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Menters SST k-Model Main Components The SST k-model consists of
Zonal (blended) k-/k-equations (to address item 1 and 2 in the previous
slide)
Clipping of turbulent viscosity so that turbulent stresses stay within what
is dictated by the structural similarity constant. (Bradshaw, 1967) -
addresses item 3 in the previous slide
Inner layer
(sublayer,
log-layer)Wilcox original k-model
l
23
k
=
Wall
Outer layer
(wake and
outward)
k-model transformed
from std. k-model
Modified Wilcox k-model
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Menters SST k-ModelInner Layer The k-model equations for the inner layerare taken from the Wilcox
original k- model with some constants modified
++=
+
+
=
j
t
jj
iij
t
jk
t
jj
iij
xxxU
DtD
x
k
xk
x
U
Dt
Dk
1
21
1
1
*
( )41.0,,09.0
0.2,176.1,075.0
1
*2*
11
*
111
===
===
k
( )
( )
==
=
22
2
22
21
1
500,
09.0
2maxarg,argtanh
,amax
yy
kF
F
kat
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Menters SST k-Model Outer Layer The k-model equations for the outer layerare obtained from by transforming
the standard k-equations via change-of-variable
Turbulent viscosity computed from:
jj
j
t
jj
i
ijt
jk
t
jj
iij
xx
k
xxx
U
Dt
D
x
k
xk
x
U
Dt
Dk
+
+
+
=
++
=
12 2
2
2
2
2
2
*
( ) 41.0,,09.0168.1,0.1,0828.0
2
*2*
22
*
222
===
===
k
kt =
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Menters SST k-ModelBlending the Equations The two sets of equations and the model constants are blended in such
a way that the resulting equation set transitions smoothly from one
equation to another.
( )
=
=
=
202
2
2
2*1
4
11
10,1
2max
4,
500,maxminarg
argtanh
jj
k
k
xx
kCD
yCD
k
yy
k
F
( )
( )
,,,where
1
1
2111
outer
1
inner
1
k
FF
Dt
DkF
Dt
DkF
=
+=
++
+
layeroutlerthein
layerinnerthein
0
1
1
1
=
F
F
Wilcox original k-model
l
23
k=
Wall
k-model transformed
from std. k-model
Modified Wilcox k-model
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Menters SST k- ModelBlended k-Equations The resulting blended equations are:
Wall
( )jj
j
t
jj
iij
t
jk
t
jj
i
ij
xxkF
xxx
U
Dt
D
x
k
xk
x
U
Dt
Dk
+
+
+
=
+
+
=
112 21
2
*
( ) ,,,,1 2111 kFF =+=
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Menters SST k-Model Turbulent Viscosity Honors the structural similarity constant for boundary layers
(Bradshaw, 1967)
Turbulent stress implied by turbulence models can be written as:
In many flow situations (e.g. adverse pressure gradient flows), productionof TKE can be much larger than dissipation (Pk>> ), which leads to
predicted turbulent stress larger than what is implied by the structural
similarity constant How can turbulent stress be limited? - A simple trick is to clip turbulent
viscosity such that:
PkayU ktt 1 ===
1967)(Bradshaw,11 ak
vukavu ==
kat 1
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Menters SST k-Model Clippingt
Turbulent viscosity for the inner layeris computed from:
Remarks
F2 is equal to 1 inside boundary layer and goes to zero far from the wall
and free shear layers The name SST (shear-stress transport) is a big word for this simple trick
Note that the vorticity magnitude is used (strain-rate magnitude could alsobe used)
( )
( )
magnitude)(vorticityijij
t
yy
kF
Fkak
Fka
==
=
=
2
500,
09.0
2maxarg,argtanh
,min,amax
22
2
22
2
1
21
1
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Menters SST k-Model Submodels & Options SST k-model comes with:
Transitional Flows option (Off by
default) Compressibility Effects option when
ideal-gas option is selected (On bydefault)
The original SST k-model in theliterature does not have any of theseoptions
These submodels are being borrowedfrom Wilcox 1998 model - should beused with caution
Do not activate any options to recover theoriginal SST model
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k-ModelsBoundary Conditions Wall boundary conditions
The enhanced wall treatment (EWT) is the sole near-wall option fork-
models. Neither the standard wall functions option nor the non-equilibriumwall functions option is available fork-models in FLUENT 6
The blended laws of the wall are used exclusively
values at wall adjacent cells are computed by blending the wall-limiting
value (y->0) and the value in the log-layer The k-models can be used with either a fine near-wall mesh or a coarse
near-wall mesh
For other BCs (e.g., inlet, free-stream), the following relationship is used
internally, whenever possible, to convert to and from different turbulencequantities:
09.0, ** == k
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Faults in the Boussinesq Assumption
Boussinesq:Rij = 2tSij Is simple linear relationship sufficient?
Rij is strongly dependent on flow conditions and history
Rij changes at rates not entirely related to mean flow processes
Rij is not strictly aligned with Sij for flows with:
sudden changes in mean strain rate
extra rates of strain (e.g., rapid dilatation, strong streamline
curvature)
rotating fluids
stress-induced secondary flows
Modifications to two-equation models cannot be generalized for
arbitrary flows
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0)()( =+ ijji uNSuuNSu
Reynolds Stress Models
Starting point is the exact transport equations for the
transport of Reynolds stresses,Rij
six transport equations in 3d
Equations are obtained by Reynolds-averaging the product
of the exact momentum equations and a fluctuating velocity.
The resulting equations contain several terms that must be
modeled
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Reynolds Stress Transport Equations
k
ijk
ijijij
ij
x
JP
Dt
DR
++=
Generation
+
k
ikj
k
j
kiijx
Uuu
x
UuuP
+ i
j
j
iij
xu
xup
k
j
k
iij
x
u
x
u
2
Pressure-StrainRedistribution
Dissipation
Turbulent
Diffusion
(modeled)
(related to )
(modeled)
(computed)
(incompressible flow w/o body
forces)
Reynolds Stress
Transport Eqns.
434214342144 344 21
)( jik
kjiikjjkiijk uux
uuuupupJ
++
Pressure/velocityfluctuations
Turbulenttransport
Moleculartransport
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ijij
3
2=
Dissipation Modeling
Dissipation rate is predominantly associated with small scale eddy
motions
Large scale eddies affected by mean shear
Vortex stretching process breaks eddies down into continually smaller
scales
The directional bias imprinted on turbulence by mean flow is graduallylost
Small scale eddies assumed to be locally isotropic
is calculated with its own (or related) transport equation Compressibility and near-wall anisotropy effects can be accounted for
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Turbulent Diffusion
Most closure models combine the pressure diffusion with the
triple products and use a simple gradient diffusion hypothesis
Overall performance of models for these terms is generally
inconsistent based on isolated comparisons to measured tripleproducts
DNS data indicate that abovep terms are negligible
( ) ( ) ( )
=
++
l
ji
lks
k
ji
k
jikikjkji
k x
uuuu
kC
xuu
xuu
puuu
x
'
=k
ji
k x
uukC
x
2
Or even a simpler model
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Pressure-strain term of same order as production
Pressure-strain term acts to drive turbulence towards
an isotropic state by redistributing the Reynolds stresses
Decomposed into parts
Model of Launder, Reece & Rodi (1978)
+
i
j
j
iij
x
u
x
up
i
j
j
i
i
j
i
ii x
u
x
u
x
uu
xx
p
+
=
21
+
+= ij
m
lml
l
ilj
l
j
liijijx
Uuu
x
Uuu
x
Uuucbc
3
221
i
j
j
i
ii x
U
x
u
xx
p
=
2
1 2
Rapid PartSlow Part
ijji kuukijb
3
2
where
wijijijij ,2,1, ++=
meanmean
gradientgradient
Pressure-Strain Modeling
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Pressure-Strain Modeling Options
Wall-reflection effect
contains explicit distance from wall damps the normal stresses perpendicular to wall
enhances stresses parallel to wall
SSG (Speziale, Sarkar and Gatski) Pressure Strain Model
Expands the basic LRR model to include non-linear (quadratic) terms
Superior performance demonstrated for some basic shear flows
plane strain, rotating plane shear, axisymmetric expansion/contraction
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Characteristics of RSM
Effects of curvature, swirl, and rotation are directly accounted for in
the transport equations for the Reynolds stresses.
When anisotropy of turbulence significantly affects the mean flow,
consider RSM
More cpu resources (vs. k- models) is needed
50-60% more cpu time per iteration and 15-20% additional memory Strong coupling between Reynolds stresses and the mean flow
number of iterations required for convergence may increase
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iu
Heat Transfer
The Reynolds averaging process produces an additional term in the
energy equation:
Analogous to the Reynolds stresses, this is termed the turbulent heat flux
It is possible to model a transport equation for the heat flux, but this is not
common practice
Instead, a turbulent thermal diffusivity is defined proportional to the
turbulent viscosity
The constant of proportionality is called the turbulent Prandtl number
Generally assumed thatPrt~ 0.85-0.9
Applicable to other scalar transport equations