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TURBULENCE MODELS AND
THEIR APPLICATIONS
Presented by:
T.S.D.Karthik
Department of Mechanical EngineeringIIT Madras
Guide:Prof. Franz Durst
10thIndo German Winter Academy 2011
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Outline3
Turbulence models introduction
Boussinesq hypothesis
Eddy viscosity concept
Zero equation model
One equation model
Two equation models
Algebraic stress model
Reyolds stress model
Comparison
Applications
Developments
Conclusion
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Turbulence models
A turbulence model is a procedure to close the system of meanflow equations.
For most engineering applications it is unnecessary to resolvethe details of the turbulent fluctuations.
Turbulence models allow the calculation of the mean flowwithout first calculating the full time-dependent flow field.
We only need to know how turbulence affected the mean flow.
In particular we need expressions for the Reynolds stresses.
For a turbulence model to be useful it: must have wide applicability,
be accurate,
simple,
and economical to run.
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Common turbulence models
Classical models. Based on Reynolds Averaged Navier-Stokes (RANS)equations (time averaged): Zero equation model: mixing length model.
One equation model
Two equation models: k- style models (standard, RNG, realizable), k-model, and ASM.
Seven equation model: Reynolds stress model.
The number of equations denotes the number of additional PDEs thatare being solved.
Large eddy simulation. Based on space-filtered equations. Timedependent calculations are performed. Large eddies are explicitlycalculated. For small eddies, their effect on the flow pattern is takeninto account with a sub-grid model of which many styles areavailable.
DNS
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Classification6
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Prediction Methods7
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Boussinesq hypothesis
Many turbulence models are based upon the Boussinesqhypothesis. It was experimentally observed that turbulence decays unless there is
shear in isothermal incompressible flows.
Turbulence was found to increase as the mean rate of deformationincreases.
Boussinesq proposed in 1877 that the Reynolds stresses could be linkedto the mean rate of deformation.
Using the suffix notation where i, j, and k denote the x-, y-, andz-directions respectively, viscous stresses are given by:
Similarly, link Reynolds stresses to the mean rate ofdeformation
i
j
j
iij
x
u
x
u
8
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Eddy Viscosity Concept
One of the most widely used concept
Reynoldsstress tensor
A new quantity appears: the turbulent viscosity or eddy viscosity
(t).
The second term is added to make it applicable to normal
turbulent stress.
The turbulent viscosity depends on the flow, i.e. the state of
turbulence.
The turbulent viscosity is not homogeneous, i.e. it varies in space.
i
j
j
i
tji
T
ijx
U
x
Uuu ''
9
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Eddy Viscosity Concept
It is, however, assumed to be isotropic. It is the same in
all directions. This assumption is valid for many flows,
but not for all (e.g. flows with strong separation or swirl).
The turbulent viscosity may be expressed as
This concept assumes that Reynolds stress tensor can be
characterized by a single length and time scales.
cccct tlorlu /)( 2
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Major Drawbacks
Interaction among eddies is not elastic as in the case for molecularinteractions in kinetic theory of gases.
For many turbulent flows, the length and time scale of characteristic
eddies is not small compared with the flow domain (boundarydominated flows).
The eddy viscosity is a scalarquantity which may not be true for simple
turbulent shear flows. It also fails to distinguish between plane shear,plane strain and rotating plane shear flows.
Successful 2D shear flows. Erroneous results for simple shear flows
such as wall jets and channel flows with varying wall roughness.
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Zero Equation Model - Mixing
Length Model
On dimensional grounds one can express the kinematic turbulentviscosity as the product of a velocity scale and a length scale:
If we then assume that the velocity scale is proportional to the
length scale and the gradients in the velocity (shear rate, which hasdimension 1/s):
we can derive Prandtls(1925) mixing length model:
Algebraic expressions exist for the mixing length for simple 2-Dflows, such as pipe and channel flow.
)()/()/( 2 msmsmt
y
U
yU
mt
2
12
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Equations for mixing length
Wall boundary layers
= boundary layer thickness
y = distance from the wall
= 0.09
K = Von-Karman Constant
Developed pipe flows
R = radius of the pipe or the half width of the duct
)//(
)//(
Kyl
KyKyl
m
m
42 )1(06.0)1(08.014.0R
y
R
y
R
lm
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Mixing Length Model Discussion
Advantages: Easy to implement.
Fast calculation times.
Good predictions for simple flows where experimental correlations forthe mixing length exist.
Used in higher models Disadvantages:
Completely incapable of describing flows where the turbulent lengthscale varies: anything with separation or circulation.
Only calculates mean flow properties and turbulent shear stress.
Cannot switch from one type of region to another History effects of turbulence are not considered.
Use:
Sometimes used for simple external aero flows.
Pretty much completely ignored in commercial CFD programs today.
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One Equation Model
Different transport equation for k is solved.
L is defined algebraically, in a similar manner as mixing length.
PDE for turbulent KE : Diffusion, Production and Dissipation terms
kl
ku
mt
2/1
c
ix
ju'
ix
ju'
ix
jU
'j
'ui
uP
)'i
'uj
'uju
ix
'ju'ju
'ip'u(i
D
where
Pi
xi
D
ixk
iU
2
12
15
Turbulence Models and Their Applications
c
D
i
j
j
i
i
j
t
ik
t
ii
il
kC
x
U
x
U
x
U
x
k
xx
kU
2
3
Modeled Equation
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One Equation Model Discussion
Economical and accurate for:
Attached wall-bounded flows.
Flows with mild separation and recirculation.
Developed for use in unstructured codes in the aerospace industry.
Popular in aeronautics for computing the flow around aero plane
wings, etc.
Weak for:
Massively separated flows.
Free shear flows.
Decaying turbulence.
Complex internal flows.
Characteristic length still experimental.
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Two Equation Models
The k-model
Equations for k and , together with the eddy-viscosity stress-
strain relationship constitute the k-turbulence model.
is the dissipation rate of k.
If kand are known, we can model the turbulent viscosity as:
22/32/1 kkk
t
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The k-model
K equation:
Model (simplified) equation for k after using Boussinesq
assumption by which the fluctuation terms can be linked to the
mean flow is as follows:
2
2
09.0 kwith
xx
k
x
k
xx
U
x
U
x
U
x
kU
t
k
t
iiik
t
ii
j
j
i
i
j
t
i
i
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Turbulent Dissipation
We can define the rate of dissipation per unit mass as:
Equation for
''.2ijij
ee
ndestructioViscousY
stretchingby vortexProductionP
citymean vortiofgradientbyProductionP
fieldflowmeanofndeformatiobyProductionP
fieldflowmeanofndeformatiobyProductionP
nfluctuatiopressurebytransportDiffusiveD
nfluctuatiobytransportDiffusiveD
transportviscousDiffusiveD
4
3
2
1
p
f
v
4321
where
YPPPPDDDDtD pfv
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Turbulent Dissipation
ix
u
ix
p
kxk
u
kx
kx
kx
k
DtD ''2
'
2
'2
2'''
2
2'
'2''''
2
lx
kx
iu
lx
ku
lx
iu
kx
ku
lx
kx
iU
lx
iu
ku
kx
iU
kx
klu
ix
lu
lx
ku
lx
iu
vDfD
pD
3
P2
P1
P 4
P Y
20
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Model Equation for
A model equation for is derived by multiplying the kequation by (/k) and introducing model constants.
Closure coefficients found emperically
K- model leads to all normal stresses being equal,which is usually inaccurate.
i
j
i
j
j
it
zz
jk
t
jj
j
t
x
U
x
U
x
UP
KC
KPC
xxxU
kC
2
21
2
1k 92.12zC 44.11zC 09.0C 33.1z
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Applications22
Flow on a backward facing step using k-epsilon model
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Single and multiple jet flows using k-epsilon models
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k- model discussion
Advantages:
Relatively simple to implement.
Leads to stable calculations that converge relatively easily.
Reasonable predictions for many flows.
Disadvantages:
Poor predictions for:
swirling and rotating flows,
flows with strong separation,
axis symmetric jets,
certain unconfined flows, and
fully developed flows in non-circular ducts.
Valid only for fully turbulent flows.
Requires wall function implementation.
Modifications for flows with highly curved stream lines.
Production of turbulence in highly strained flows is over predicted.
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More two equation models
The k-model was developed in the early 1970s. Its strengths
as well as its shortcomings are well documented.
Many attempts have been made to develop two-equationmodels that improve on the standard k-model.
We will discuss some here:
k- RNG model. k- realizable model.
k- model.
Algebraic stress model.
Non-linear models.
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RNG k-
k- equations are derived from the application of a rigorous statistical technique(Renormalization Group Method) to the instantaneous Navier-Stokes equations.
Similar in form to the standard k-equations but includes:
Additional term in equation for interaction between turbulence dissipation and mean
shear.
The effect of swirl on turbulence.
Analytical formula for turbulent Prandtl number.
Differential formula for effective viscosity.
Improved predictions for:
High streamline curvature and strain rate. Transitional flows, separated flows.
Wall heat and mass transfer.
Also, time dependent flows with large scale motions (vortex shedding)
But still does not predict the spreading of a round jet correctly.
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Realizable k- model
Shares the same turbulent kinetic energy equation as the
standard k-model.
Improved equation for .
Variable Cinstead of constant.
Improved performance for flows involving:
Planar and round jets (predicts round jet spreading correctly).
Boundary layers under strong adverse pressure gradients or
separation. Rotation, recirculation.
Strong streamline curvature.
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Realizable k- Cequations
Eddy viscosity computed from.
kU
AA
Ck
C
s
*
0
2
t
1,
ijijijijSSU *
WAAs 6cos3
1
,cos6,04.4
1
0
ijij
kijiijSSS
S
SSSW
~,~
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Realizable k- positivity of
normal stresses
Boussinesq viscosity relation:
Normal component:
Normal stress will be negative if:
2
tijt ;3
2-
kCk
x
u
x
uuu
i
j
j
i
ji
23
2
22
x
UkCku
3.73
1
Cx
Uk
31
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k- model
This is another two equation model. In this model is an inversetime scale that is associated with the turbulence.
This model solves two additional PDEs:
A modified version of the kequation used in the k-model. A transport equation for .
The turbulent viscosity is then calculated as follows:
Its numerical behavior is similar to that of the k-models.
It suffers from some of the same drawbacks, such as theassumption that tis isotropic.
kt
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The SST Model33
The SST (Shear Stress Transport) model is an eddy-viscosity model which
includes two main novelties:
1. It is combination of a k-!model (in the inner boundary layer) and k-
model (in the outer region of and outside of the boundary layer);
2. A limitation of the shear stress in adverse pressure gradient regions isintroduced.
Actual Model
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Non-linear models
The standard k- model is extended by including second andsometimes third order terms in the equation for the Reynoldsstresses.
One example is the Speziale model:
Here f() is a complex function of the deformation tensor,
velocity field and gradients, and the rate of change of thedeformation tensor.
The standard k-model reduces to a special case of this modelfor flows with low rates of deformation.
These models are relatively new and not yet used very widely.
)/,,/,(*423
2''
2
32
2
xUtEEfk
CCEk
Ckuu Dijijjiij u
34
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Reynolds stress model
RSM closes the Reynolds-Averaged Navier-Stokes equations bysolving additional transport equations for the six independentReynolds stresses.
Transport equations derived by Reynolds averaging the product of themomentum equations with a fluctuating property.
Closure also requires one equation for turbulent dissipation. Isotropic eddy viscosity assumption is avoided.
Resulting equations contain terms that need to be modeled.
RSM is good for accurately predicting complex flows. Accounts for streamline curvature, swirl, rotation and high strain rates.
Cyclone flows, swirling combustor flows.
Rotating flow passages, secondary flows.
Flows involving separation.
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Turbulence Models and Their Applications
Reynolds stress transport
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Reynolds stress transport
equation
The exact equation for the transport of the Reynolds stress Rij:
This equation can be read as:
rate of change of plus
transport of Rijby convection, equals
rate of production Pij, plus
transport by diffusion Dij, minus
rate of dissipation ij
, plus
transport due to turbulent pressure-strain interactions ij, plus
transport due to rotation ij.
This equation describes six partial differential equations, one for the
transport of each of the six independent Reynolds stresses.
ijijijijD
ijP
Dt
ijDR
'' ji uuijR
36
Turbulence Models and Their Applications
R ld t t t
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Reynolds stress transport
equation
The various terms are modeled as follows: Production Pijis retained in its exact form.
Diffusive transport Dijis modeled using a gradient diffusion assumption.
The dissipation ij, is related to as calculated from the standard
equation, although more advanced models are available also.
Pressure strain interactions ij, are very important. These include pressurefluctuations due to eddies interacting with each other, and due tointeractions between eddies and regions of the flow with a different meanvelocity. The overall effect is to make the normal stresses more isotropic
and to decrease shear stresses. It does not change the total turbulentkinetic energy. This is a difficult to model term, and various models areavailable. Common is the Launder model. Improved, non-equilibriummodels are available also.
Transport due to rotation ijis retained in its exact form.
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RSM equations
m
ijm
m
j
imijx
UR
x
URP:exactProduction
)''('''j
uiki
ujk
pk
uj
ui
uijk
J
kxijk
J
ijD
:exacttransportDiffusive
waystandardtheincalculatedviscositykinematicturbulenttheis
:modeltransportDiffusive
t
ij
k
t
m
ij
k
t
m
ij Rgraddiv
x
R
xD
)(
38
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RSM equations continued
ijij
k
j
k
i
ijx
u
x
u
3
2''
2
:modelnDissipatio:exactnDissipatio
pressuretheis
:modelstrainPressure
:exactstrainPressure
P
PPCkRk
C
x
u
x
up
ijijijijij
i
j
j
i
ij
)()(
'''
3
2
23
2
1
vectorrotationtheis
indicestheondepending1or0,,is
:(exact)termRotational
k
1
)(2
ijk
jkmimikmjmkij
e
eReR
39
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Algebraic stress model
The same kand equations are solved as with the standard k-model.
However, the Boussinesq assumption is not used.
The full Reynolds stress equations are first derived, and then some
simplifying assumptions are made that allow the derivation of algebraic
equations for the Reynolds stresses.
Thus fewer PDEs have to be solved than with the full RSM and it is
much easier to implement.
The algebraic equations themselves are not very stable, however, and
computer time is significantly more than with the standard k-model. This model was used in the 1980s and early 1990s. Research continues
but this model is rarely used in industry anymore now that most
commercial CFD codes have full RSM implementations available
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Setting boundary conditions
Characterize turbulence at inlets and outlets (potential backflow). k-models require kand .
Reynolds stress model requires Rijand .
Other options:
Turbulence intensity and length scale.
Length scale is related to size of large eddies that contain most of energy.
For boundary layer flows, 0.4 times boundary layer thickness: l 0.4d99.
For flows downstream of grids /perforated plates: l opening size.
Turbulence intensity and hydraulic diameter. Ideally suited for duct and pipe flows.
Turbulence intensity and turbulent viscosity ratio.
For external flows:
10/1
t
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Some Applications42
Turbulent Annular Flow
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43
The figures show plots of the normalized tangential velocity, for each ofthe turbulence models, plotted with the experimental data.
RNG k - epsilon model produces the best results with the standard k -epsilon model giving the worst but this variation is small compared totheir deviations from the experimental data.
The k-L mixing length model does lead to an answer which predicts themovement of the maximal tangential velocity from the inner wall to thecentre of the annulus better than the other models. Where theimplementation of the model fails is in its prediction of the flow near thewalls.
Turbulence Models and Their Applications
Flocculation tank Analogy with flow over a back
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44
Flocculation tank Analogy with flow over a back
step
In the middle of the channel, the flow separate due to the small step size of height h. The
flow reattaches at about 7 times the step height further downstream - similar to the 180
degree bend in the flocculation tank where we have flow separation and reattachment
downstream
Analyzed using K-,K-SST, K-realizable, K-RNG, RSM turbulence models and compared
with experimental data.
Plotting the derivative du/dy, the change in direction of velocity in x direction with respect
to y at the wall, the reattachment point is easily identified. At the wall, separated flow will
give a negative du/dy, while reattaches flow has a positive du/dy value.Turbulence Models and Their Applications
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45
Turbulenc
e Model
K-e K-W SST K-e
realizable
RSM
Reattach
ment
Ratio
0.195/0.
038 =
5.13
0.242/0.0
38 = 6.37
0.235/0.0
38 = 6.18
0.2/0.038
= 5.26
Flocculation tank Analogy with flow over a back
step
The K- model under-predicts the
reattachment length. K- SST and K-
realizable gives the most accurate
representation of the back step flow with
reattachment length. However, from literature
reviews, K- realizable is more proven for a
variety of types of flows.
Below in Figure, the stream contours (of the
averaged velocity) of the Re=48,000 for the k-
realizable model case closely approximate the
experimental results.
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Flow over airfoil46
Contour plots of the predicted turbulentviscosity around an airfoil obtained with
four different steady-state turbulence
models: an algebraic model, a one-
equation model, and a duo of two-
equation models.
While the Spalart-Allmaras and Chien k-
epsilon models are in rough agreement
with each other, the SST and Baldwin-
Lomax models predict a very differentturbulent viscosity distribution.
If you looking solely at performance in the shear layer, you might want to
choose either the Spalart or Chien models.
Turbulence Models and Their Applications
Comparison of RANS turbulence
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Comparison of RANS turbulence
models
Model Strengths Weaknesses
Zero Equation
Model
Economical (1-eq.); good track
record for mildly complex B.L.
type of flows.
Not very widely tested yet; lack of sub-
models (e.g. combustion, buoyancy).
STD k-
Robust, economical,
reasonably accurate; long
accumulated performance data.
Mediocre results for complex flows with
severe pressure gradients, strong streamline
curvature, swirl and rotation. Predicts thatround jets spread 15% faster than planar jets
whereas in actuality they spread 15% slower.
RNG k-
Good for moderately complex
behavior like jet impingement,
separating flows, swirling flows,
and secondary flows.
Subjected to limitations due to isotropic
eddy viscosity assumption. Same problem
with round jets as standard k-.
Realizable
k-
Offers largely the same benefits
as RNG but also resolves the
round-jet anomaly.
Subjected to limitations due to isotropic
eddy viscosity assumption.
Reynolds Stress
Model
Physically most complete model
(history, transport, and
anisotropy of turbulent stresses
are all accounted for).
Requires more cpu effort (2-3x); tightly
coupled momentum and turbulence
equations.
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Recommendation
Start calculations by performing 100 iterations or so with standard k- modeland first order upwind differencing. For very simple flows (no swirl orseparation) converge with k- model.
If the flow involves jets, separation, or moderate swirl, converge solution withthe realizable k- model.
If the flow is dominated by swirl (e.g. a cyclone or un-baffled stirred vessel)converge solution deeply using RSM and a second order differencing scheme.If the solution will not converge, use first order differencing instead.
Ignore the existence of mixing length models and the algebraic stress model.
Only use the other models if you know from other sources that somehowthese are especially suitable for your particular problem (e.g. Spalart-Allmaras for certain external flows, k- RNG for certain transitional flows, ork-).
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Turbulence Models and Their Applications
Other Numerical Methods:
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Other Numerical Methods:
DNS (Direct Numerical Simulation)
Very accurate
High computing time
Works on small Reynolds number flows
Used to verify the turbulence model
Some arbitrary initial velocity field is set up and the Navier-Stokes
equations are used directly to describe the evolution of this field
over time.
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LES (Large Eddy Simulation)
More accurate than RANS
More computing time than RANS
Middle route between DNS and RANS
Can work on larger Reynolds number and more complex flows
Simulations for large scales and RANS for small scales
Both DNS and LES require to solve the instantaneous Navier-Stokesequations in time and three-dimensional space.
Hybrid ApproachRANS + LES
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Developments
EddyViscosity
Models
ReynoldsStressModels
ProbabilityDensity
Functions
Accuracy increasesComplexity increases
Computing time increases
Usability decreases
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Conclusion52
Different turbulence modelsstrengths and weaknesses
Usage of models specific to certain flows
ApplicationsWhere to use what?
Compromise between accuracy and computing power
Other developments
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References
Turbulent Flows Fundamentals, Experiments and Modeling,G.Biswas and V.Eswaran, Narosa Publishing House, 2002
Turbulence Modeling for CFDWilcox, D.C, 1993
Fluid Mechanics - An Introduction to the Theory of Fluid Flows,Franz Durst, Springer, 2008
Turbulence model validation - https://confluence.cornell.edu/
http://www.tfd.chalmers.se/doct/comp turb model
FluentModeling Turbulence
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