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Computational Fluid Dynamics
ClassicalTurbulence
Modeling
Grtar TryggvasonFall 2011
Computational Fluid Dynamics
A jet in a cross flow
cross section of a jet
Most engineering problems involveturbulent flows. Such flows involveare highly unsteady and contain a
large range of scales. However, in
most cases the mean or averagemotion is well defined.
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Flow over a sphere
The drag depends on the separation point
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A modest Reynolds number the separated boundary layer remains initiallylaminar (left), before becoming turbulent. If the boundary layer is tripped (right)
it becomes turbulent, so that it separates farther rearward. Theoverall drag is thereby dramatically reduced, in a way that occurs
naturally on a smooth sphere only at a Reynolds numbers ten timesas great. ONERA photograph, Werle 1980.From "An Album of Fluid Motion," by Van Dyke, Parabolic Press.
Instantaneous flow past asphere at R = 15,000.
Instantaneous flow past a sphereat R = 30,000 with a trip wire
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Examples of Reynolds numbers:
Flow around a 3 m long car at
100 km/hr:
Flow around a 100 m long submarineat 10 km/hr:
Re =LU
v=
3!27.78
1.5!10"5=5.5 !10
6
Kinematic viscosity(~20 C)
Water "= 10-6m2/s
Air "= 1.5 !10-5m2/s
1km/hr = 0.27778 m/s
Re =LU
v=
100!2.78
10"6
=2.78!108
Water flowing though a 0.01 m diameter pipe with a velocityof 1 m/s
Re =LU
v=
0.01!1
10"6
=104
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Reynolds Averaged Navier-Stokes (RANS): Onlythe averaged motion is computed. The effect of
fluctuations is modeled
Large Eddy Simulations (LES): Large scalemotion is fully resolved but small scale motion ismodeled
Direct Numerical Simulations (DNS): Every lengthand time scale is fully resolved
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Computational Fluid Dynamics
ReynoldsAveraged
Navier-StokesEquations
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To solve for the mean motion, we derive equations for the
mean motion by averaging the Navier-Stokes equations.The velocities and other quantities are decomposed into the
average and the fluctuation part
a = A+ a'
< a > = A
< a' > = 0
< a + b > = A + B
= cA
=!A
Defining an averagingprocedure that satisfies
the following rules:
This will hold forspatial averaging,
temporal averaging,and ensamble
averaging
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There are several ways to define the proper averages
For homogeneous turbulence we can use the space average
For steady turbulence flow we can use the time average
For the general case we use the ensemble average
< a > =
1
Ladx
0
!
< a > =1
Tadt
0
T
!
< a > = ar(x,t)ensambles! a = A+ a'
Computational Fluid Dynamics
!
!tu +" # uu = $
1
%"p+&"2u
u = U+ u'
p = P + p'
< a > = A
< a' > = 0
=cA
=!A
Start with the Navier-Stokes equations
Decompose the pressure and velocityinto mean and fluctuations:
a = A+ a'
Or, in general, for anydependant variable:
Computational Fluid Dynamics
!
!t
U+ " #UU= $1
%
"P+&"2U$ "# < u'u' >
Applying the averaging to the Navier-Stokesequations results in:
< u'u'>=
< u 'u'> < u 'v'> < u 'w'>
< u 'v'> < v'v'> < v'w'>
< u 'w'> < v'w'> < w 'w'>
"
### %
&&&
Reynolds stress tensor
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Physical interpretation
< uv >
Fast moving fluid particle
Slow moving fluid particle
Net momentum transferdue to velocity fluctuations
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Computational Fluid Dynamics
Closure:
Since we only have an equation for the mean flow,the Reynolds stresses must be related to the mean
flow.
No rigorous process exists for doing this!
THE TURBULENCE PROBLEM
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Zero and Oneequation models
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Introduce the turbulent eddy viscosity
!T=
l0
2
t0
ij=!"
T
#Ui
#xj
+
#Uj
#xi
$
%&
'
()
where
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Zero equation models
!T =l02 dU
dy
Prandtlmixing length
l0=!y
Smagorinsky model
Baldvin-Lomaz model
!T= l
0
22SijSij( )
1/ 2
!T =l02
"i"i( )1/ 2
Sij=1
2
!Ui
!xj+
!Uj
!xi
#$$
&''
!i=
"Ui
"x j #
"Uj
"xi
%&
& ()
)
Computational Fluid Dynamics
One equation models
!T= k
1/ 2t0
Where kis obtained by an equation describing its
temporal-spatial evolution
However, the problem with zero and one equationmodels is that t
0and l
0are not universal. Generally, it is
found that a two equation model is the minimum needed
for a proper description
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Two equation
models
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Computational Fluid Dynamics
To characterize the turbulence it seems reasonable tostart with a measure of the magnitude of the velocityfluctuations. If the turbulence is isotropic, the turbulent
kinetic energy can be used:
k=1
2< u'u'> + < v 'v'> + < w 'w'>( )
The turbulent kinetic energy does, however, notdistinguish between large and small eddies.
Computational Fluid Dynamics
To distinguish between large and small eddies we need tointroduce a new quantity that describe
!" # $u'i$u'i
$x j$x j
Usually, the turbulent dissipation rate is used
Smaller eddiesdissipate faster
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!T=C
k2
"
!
!tU+" #UU= $
1
%"P + &+&
T( )"2U
Solve for the average velocity
Where the turbulent kinematic eddy viscosity isgiven by
Computational Fluid Dynamics
!k
!t+Uj
!k
!x j="ij
!Ui!x j
#$+!
!x j%!k
!x j#1
2ui
'ui
'uj
' #1
&p'uj
'
()
+,
The exact k-equation is:
where !ij = " ui'uj
'
The exact epsilon-equation is considerably more complexand we will not write it down here.
Both equations contain transport, dissipation and
production terms that must be modeled
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!k
!t+U"#k=# " D
k#k+ production $dissipation
!"
!t+U#$"=$ # D
"$"+ production %dissipation
The general for for the equations for k and epsilon is:
These terms must be modeled
Closure involves proposing a form for the missing termsand optimizing free coefficients to fit experimental data
Computational Fluid Dynamics
Here
The k-epsilon model
!T=C
k2
"
!ij ==
2
3k"ij#$T
%Ui
%x j+
%Uj
%xi
'((
*++and
C1 =0.09; C2 =1.0; C3 =0.769; C4 =1.44; C5 =1.92
Dk
Dt= +!" (#+C
2#T)!k- $ij
%Ui
%x j&'
D!
Dt
=" # ($+C3$T)"!+C4!
k
%ij&Ui
&x j'C5
!2
kProduction Dissipation
Turbulenttransport
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Computational Fluid Dynamics
Other two equation turbulence models:RNG k-epsilon
Nonlinear k-epsilonk-enstrophyk-lok-reciprocal timeetc
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Turbulent transport of energy and species concentrations is
modeled in similar ways.
For temperature we have:
!T
!t+" # uT =$"
2T
u = U+ u'
T = +T'
!< T>
!t+" # U = $"
2< T> %"# < UT>
Gradient Transport Hypothesis:
!"T# < T>
Computational Fluid Dynamics
ModelPredictions
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Spreading rates:
exp k-e CmottPlane jet 0.10 - 0.11 0.108 0.102
Round jet 0.085-0.095 0.116 0.095Mixing layer 0.13 - 0.17 0.152 0.154
Computational Fluid Dynamics
From: C.G. Speziale: Analytical Methods for theDevelopment of Reynolds-stress closure in Turbulence.
Ann Rev. Fluid Mech. 1991. 23: 107-157
Computational Fluid Dynamics
From: C.G. Speziale: Analytical Methods for theDevelopment of Reynolds-stress closure in Turbulence.
Ann Rev. Fluid Mech. 1991. 23: 107-157
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Computational Fluid Dynamics
Results
From: C.G. Speziale: Analytical Methods for theDevelopment of Reynolds-stress closure in Turbulence.
Ann Rev. Fluid Mech. 1991. 23: 107-157
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Wall boundedturbulence
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Wall bounded turbulence
Fundamental assumption: determined by localvariables only
Mean flow
Only the meanshear rate andthe properties of
the fluid are
important
!w =dU
dy, ", #
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Define a shear velocity:
v*=
!w
"
!w = du
dy, ", #
kg /ms2[ ], kg /m3[ ], m2 /s[ ]
Normalize the length andvelocity near the wall
u+
=u
v*
y+
=y v
*
v
Called wall variables
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Thus, the velocity near the wall is
Velocity versusdistance from wall
u+
y+
u+
= y+
u+
=1
!
lny+
+C
10
!=0.4
C=5.5
v* =!w
"
!w = du
dy
u
+
=u
v*
y+ =y v*
v
Bufferlayer
Outerlayer
Viscoussub-layer
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For a practicalengineering problem
L = 1m; U = 1m/s; "= 10-6(water)
The Reynolds number is therefore:
For a flat plate, the average drag coefficient is
Re =LU
v=10
6
CD
=
FD
1
2!U
2LW
CD
=0.592Re!1/ 5
where
CD
=0.0037
!w
=
FD
LW= C
D
1
2"U
2=3.74
Thusand
And we findv
*=0.06
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Computational Fluid Dynamics
y =10!
!*=
10"10#6
0.06=1.667"10
#4m =0.1667 mm
Thickness of the viscous sub-layer
Find the thickness of the boundary layer
!
L=0.37Re
"1/ 5
!
L=0.0233m =23.3mm
To resolve the viscous sublayer at the same time as theturbulent boundary layer would require a large number of
grid points
The average thickness of the viscous sub-layer is 10 in units of y+:
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To deal with this problem it is common to use wallfunctionswhere the mean velocity is matched with
an analytical approximation to the viscosus sublayer.
For a reference, see: Patel, Rodi, and Scheuerer,
Turbulence Models for Near-Wall and Low ReynoldsNumber Flows: A Review. AIAA Journal, 23 (1985),1308-1319
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Second order closure
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The k-epsilon and other two equation modelshave several serious limitations, including the
inability to predict anisotropic Reynolds stress
tensors, relaxation effects, and nonlocaleffects due to turbulent diffusion.
For these problems it is necessary to modelthe evolution of the full Reynolds stress
tensor
Computational Fluid Dynamics
Derive equations for the Reynolds stresses:
!ui
!t+"u
iu
j = #
1
$"p +%"2u
i
The Navier-Stokes equations in component form:
ui
!ui!t
+"uiuj = - 1#"p+$"2u
i
&'
)*
Multiply the equation by the velocity
and averaging leds to equations for
!
!tu
iu
j
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The new equations contain terms like
which are not known. These terms are therefore modeled
uiu
iu
j
The Reynolds stress model introduces 6 new equations(instead of 2 for the k-e model. Although the models
have considerably more physics build in and allow, for
example, anisotrophy in the Reynolds stress tensor,these model have yet to be optimized to the point that
they consistently give superior results.
For practical problems, the k-e model or more recentimprovements such as RNG are therefore most commonly used!
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Computational Fluid Dynamics
DirectNumerical
Simulations
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In direct numerical simulations the full unsteadyNavier-Stokes equations are solved on asufficiently fine grid so that all length and time
scales are fully resolved. The sizeof the
problem is therefore very limited. The goal ofsuch simulations is to provide both insight andquantitative data for turbulence modeling
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Channel Flow
Streamwise velocity
Flow direction
Periodicstreamwise
andspanwise
boundaries
Wall
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Streamwise vorticity
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Streamwise vorticity
Turbulentshear stress
Turbulent eddies generate anearly uniform velocity profile
Channel Flow
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Turbulence are intrinsically linked to vorticity, yetlaminar flows can also be vortical so looking at the
vorticity is not sufficient to understand what is
going on in a turbulent flows. Several attemptshave been made to define properties of the
turbulent flows that identifies vortices (as opposedto simply vortical flows.
One of the most successful method is the lambda-2method of Hussain.
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Computational Fluid Dynamics
Visualizing turbulence
!u =
"u
"x
"u
"y
"u
"z
"v
"x
"v
"y
"v
"z
"w
"x
"w
"y
"w
"z
$
%%%%%%
'
((((((
S =1
2!u + !Tu( ) =
1
2
2"u
"x
"u
"y+
"v
"x
"u
"z+
"w
"x
"v
"x+
"u
"y2"v
"y
"v
"z+
"w
"y
"w
"x+
"u
"z
"w
"y+
"v
"z2"w
"z
$
%%%%%%
'
((((((
!=1
2"u - "Tu( ) =
1
2
0 #u
#y$#v
#x
#u
#z$#w
#x
#v
#x$#u
#y0
#v
#z$#w
#y
#w
#x$#u
#z
#w
#y$#v
#z0
&
''''''
)
******
Computational Fluid Dynamics
It can be shown that the second eigenvalue of
S2
+!2
define vortex structures
Referece: J. Jeong and F. Hussain, "On the identification ofa vortex," Journal of Fluid Mechanics, Vol. 285, 69-94,1995.
Other quantities have also been used, such as thesecond invariant of the velocity gradient:
Q =!ui
!x j
!uj
!x i
!2
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!2= "0.3
!2= "0.2
Computational Fluid Dynamics
Turbulence models are used to allow us tosimulate only the averaged motion, not theunsteady small scale motion.
Turbulence modeling rest on the assumptionthat the small scale motion is universaland
can be described in terms of the large scalemotion.
Although considerable progress has beenmade, much is still not known and results from
calculations using such models have to beinterpreted by care!
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For more information:
D. C. Wilcox, Turbulence Modeling for
CFD (2nded. 1998; 3rded. 2006).
The author is one of the inventors ofthe k-#model and the book promotes ituse. The discussion is, however,
general and very accessible, as well as
focused on the use of turbulence
modeling for practical applications inCFD
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