model-A gravity. Buoyant bubbles in an inclined channel flow. 2. Computational
Fluid Dynamics. Why turbulence modeling. Reynolds Averaged Numerical ...
Computational Fluid Dynamics! http://www.nd.edu/~gtryggva/CFD-Course/!
“Classical”! Turbulence Modeling! Grétar Tryggvason! Spring 2011!
Computational Fluid Dynamics! Outline!
2!
Computational Fluid Dynamics! Multiscale Issues!
Some communities have defined two types of multiscale problems.!
Buoyant bubbles in an inclined channel flow! Average Velocity-B!
Type A Problems: Dealing with Isolated Defects! Type B Problems: Constitutive Modeling Based on the Microscopic Models ! Reference: W. E and B. Enquist, The heterogeneous multiscale methods, Comm. Math. Sci. 1 (2003), 87—133.!
gravity! Thin film model-A!
Computational Fluid Dynamics!
A jet in a cross flow! Why turbulence modeling! Reynolds Averaged Numerical Simulations! Zero and One equation models! Two equations models! Model predictions! Wall bounded turbulence! Second order closure! Direct Numerical Simulations! Large-eddy simulations! Summary! cross section of a jet!
Computational Fluid Dynamics!
Most engineering problems involve turbulent flows. Such flows involve are highly unsteady and contain a large range of scales. However, in most cases the mean or average motion is well defined. !
Computational Fluid Dynamics!
Flow over a sphere!
Instantaneous flow past a sphere at R = 15,000.!
The drag depends on the separation point!
Instantaneous flow past a sphere at R = 30,000 with a trip wire
A modest Reynolds number the separated boundary layer remains initially laminar (left), before becoming turbulent. If the boundary layer is tripped (right) it becomes turbulent, so that it separates farther rearward. The overall drag is thereby dramatically reduced, in a way that occurs naturally on a smooth sphere only at a Reynolds numbers ten times as great. ONERA photograph, Werle 1980.! From "An Album of Fluid Motion," by Van Dyke, Parabolic Press. !
Computational Fluid Dynamics! Examples of Reynolds numbers:!
Kinematic viscosity ! (~20 °C)!
Flow around a 3 m long car at 100 km/hr:!
Re =
LU 3 ! 27.78 = = 5.5 !10 6 v 1.5 !10"5
Flow around a 100 m long submarine at 10 km/hr:!
Water ν = 10-6 m2/s! Air ν = 1.5 ✕10-5 m2/s! 1km/hr = 0.27778 m/s!
LU 100 ! 2.78 Re = = = 2.78 !10 8 v 10"6
Water flowing though a 0.01 m diameter pipe with a velocity of 1 m/s!
Re =
LU 0.01!1 = = 10 4 v 10"6
Reynolds Averaged Navier-Stokes (RANS): Only the averaged motion is computed. The effect of fluctuations is modeled! Large Eddy Simulations (LES): Large scale motion is fully resolved but small scale motion is modeled! Direct Numerical Simulations (DNS): Every length and time scale is fully resolved!
Computational Fluid Dynamics!
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 !
This will hold for spatial averaging, temporal averaging, and ensamble averaging!
It can be shown that for turbulent flow the ratio of the size of the smallest eddy to the length scale of the problem! ! ! " O(Re#1/ 2 ) " O(Re#3 / 4 ) L L In 3D! In 2D! If about 10 grid points are needed for Re=10 (the driven cavity problem) ! Re ! 3d ! ! 2d! 103!~ 3003 ! ~ 1002! 4 3 10 ! ~ 2000 ! ~ 3002! 105!~ 100003 ! ~ 10002! Largest computations today use about 40003 points!
Computational Fluid Dynamics!
a = A + a'
Computational Fluid Dynamics!
Defining an averaging procedure that satisfies the following rules:!
= A < a' > = 0 < a+ b> = A+ B < ca > = cA < !a > = !A
Computational Fluid Dynamics!
Reynolds Averaged Navier-Stokes Equations!
Computational Fluid Dynamics!
There are several ways to define the proper averages ! For homogeneous turbulence we can use the space average! L
=
1 adx L !0
For steady turbulence flow we can use the time average!
=
1 T
T
! adt 0
For the general case we use the ensemble average!
=
! a (x,t)
r ensambles
a = A + a'
Computational Fluid Dynamics!
Computational Fluid Dynamics!
Start with the Navier-Stokes equations!
! 1 u + " # uu = $ "p + &" 2u !t %
Decompose the pressure and velocity into mean and fluctuations:!
u = U + u' p = P + p' Or, in general, for any dependant variable:!
Applying the averaging to the Navier-Stokes equations results in:!
< a >= A < a' > = 0 < ca > =cA < !a > = !A
a = A + a'
! U + " # UU = - $1 "P + %" 2U + "# < u'u'> !t ! < u' u'> < u'v'> < u' w'> $ # & < u'u'>= # < u'v'> < v'v'> < v' w'> & #"< u' w'> < v' w'> < w' w'>&% Reynold s stress tensor!
Computational Fluid Dynamics!
Computational Fluid Dynamics!
Physical interpretation! Closure:!
< uv > Fast moving fluid particle! Net momentum transfer due to velocity fluctuations! Slow moving fluid particle!
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!
Computational Fluid Dynamics!
Computational Fluid Dynamics!
Introduce the turbulent eddy viscosity !
Zero and One equation models!
$ #U #U ' i < u'u'> ij = !" T && i + )) % #x j #x j ( where!
!T =
l02 t0
Computational Fluid Dynamics!
Zero equation models!
Computational Fluid Dynamics!
One equation models!
! T = k1/ 2 t 0
Prandtl mixing length!
! T = l02
dU dy
l0 = !y
Smagorinsky model!
! T = l02 (2Sij Sij )
1/ 2
Baldvin-Lomaz model!
! T = l02 (" i" i )
1/ 2
1 " !U !U j % Sij = $$ i + ' 2 # !x j !x i '& $ "U " U ' j ! i = && i # )) % "x j " x i (
Where k is obtained by an equation describing its temporal-spatial evolution! However, the problem with zero and one equation models is that t0 and l0 are not universal. Generally, it is found that a two equation model is the minimum needed for a proper description !
Computational Fluid Dynamics!
Two equation models!
Computational Fluid Dynamics!
To characterize the turbulence it seems reasonable to start with a measure of the magnitude of the velocity fluctuations. If the turbulence is isotropic, the turbulent kinetic energy can be used:!
k=
1 (< u' u'> + < v'v'> + < w' w'>) 2
The turbulent kinetic energy does, however, not distinguish between large and small eddies.!
Computational Fluid Dynamics!
To distinguish between large and small eddies we need to introduce a new quantity that describe! Smaller eddies dissipate faster! Usually, the turbulent dissipation rate is used!
$u'i $u'i ! "# $x j $ x j
Computational Fluid Dynamics!
Solve for the average velocity!
! 1 U + " # UU = $ "P + (& + & T )" 2U !t % Where the turbulent kinematic eddy viscosity is given by!
! T = Cµ
k2 "
Computational Fluid Dynamics!
The exact k-equation is:!
The general for for the equations for k and epsilon is:!
* !k !k !U i ! ' !k 1 ' ' ' 1 +Uj = " ij #$ + # ui ui u j # p' u'j , )% !t !x j !x j !x j ( !x j 2 & + where! ! ij = " ui u j '
Computational Fluid Dynamics!
'
The exact epsilon-equation is considerably more complex and we will not write it down here.! Both equations contain transport, dissipation and production terms that must be modeled!
Computational Fluid Dynamics!
!k + U " #k = # " Dk #k + production $ dissipation !t
!" + U # $" = $ # D" $" + production % dissipation !t These terms must be modeled ! Closure involves proposing a form for the missing terms and optimizing free coefficients to fit experimental data!
Computational Fluid Dynamics!
The k-epsilon model!
Dk %U i = +! " (# + C2# T )!k - $ ij &' Dt %x j
D! ! &U i !2 = " # ($ + C3$ T )"! + C4 % ij ' C5 Dt k &x j k Turbulent! transport!
Here!
!T = C
k2 "
and!
Production! Dissipation!
& %U %U ) 2 j ! ij =< u'i u'j >= k"ij # $ T (( i + ++ 3 ' %x j %x i *
Two major numerical difficulties! The equations may be stiff in some regions of the flow requiring very small time step. This can be overcome by an implicit scheme.! In reality k goes to zero at the walls. In simulations this usually takes place so close to the wall that it is not resolved by the grid. To overcome this we usually use a wall function or a damping function!
C1 = 0.09; C2 = 1.0; C3 = 0.769; C4 = 1.44; C5 = 1.92
Computational Fluid Dynamics!
Other two equation turbulence models:! !RNG k-epsilon! !Nonlinear k-epsilon! !k-enstrophy! !k-lo! !k-reciprocal time! !etc!
Computational Fluid Dynamics!
Turbulent transport of energy and species concentrations is modeled in similar ways.! For temperature we have:!
!T + " # uT = $" 2T !t
u = U + u' T =< T > +T'
! + " # U < T >= $" 2 < T > %"# < UT > !t Gradient Transport Hypothesis:!
< UT >! " T # < T >
Computational Fluid Dynamics!
Computational Fluid Dynamics!
Spreading rates:!
Model Predictions!
Computational Fluid Dynamics!
From: C.G. Speziale: Analytical Methods for the Development of Reynolds-stress closure in Turbulence. Ann Rev. Fluid Mech. 1991. 23: 107-157!
Computational Fluid Dynamics!
! Plane jet Round jet Mixing layer
! exp !0.10 - 0.11 !0.085-0.095 !0.13 - 0.17
! k-e ! ! 0.108! ! 0.116! ! 0.152!
Computational Fluid Dynamics!
From: C.G. Speziale: Analytical Methods for the Development of Reynolds-stress closure in Turbulence. Ann Rev. Fluid Mech. 1991. 23: 107-157!
Computational Fluid Dynamics!
Results!
Wall bounded turbulence! From: C.G. Speziale: Analytical Methods for the Development of Reynolds-stress closure in Turbulence. Ann Rev. Fluid Mech. 1991. 23: 107-157!
!Cmott! !0.102! !0.095! !0.154!
Computational Fluid Dynamics!
Computational Fluid Dynamics!
Define a shear velocity: !
Wall bounded turbulence! Fundamental assumption: determined by local variables only!
!w = µ
!w "
v* =
du , ", # dy
[kg /ms ], [kg /m ], [m /s] 2
3
2
Mean flow! Normalize the length and velocity near the wall! Only the mean shear rate and the properties of the fluid are important!
!w =
u+ =
dU , ", # dy
u v*
y+ =
Called wall variables !
Computational Fluid Dynamics!
For parallel flow!
!
0!
d dp d du < u'v'>= " + µ dy dx dy dy
Integrate from the wall to y:!
∫
y
0
⎛ d d du ⎞ ⎜⎝ ρ dy < u ' v ' >= dy µ dy ⎟⎠ dy
Resulting in:!
! < u'v'>= µ
Computational Fluid Dynamics!
!w Near the wall v* = " the fluid knows nothing about du what drives it. ! w = µ dy Thus we ignore u + the pressure u = * gradient! v y+ =
y v* v
du " #w dy
! < u'v'>= µ
du " #w dy
Very close to the wall:! < u'v'>! 0
du = !w dy ! u(y) = w y µ
so approximately! µ Integrating!
* u ! w y " (v ) y v * y = = = * * v µ v µ v* # 2
u+ = y +
or:!
Further away from the wall:! µ du ! 0
dy
! < u'v'>= "# w
du Taking:! < u'v'>= ! T dy 2 du and! l0 = !y where! ! T = lo dy 2 du du " du % < u'v'>= l = $!y ' dy dy # dy &
yields!
Giving:!
2 o
!y
# du =" w dy $
du u' ! y dy
!w "
!w = µ
du dy
u v* y v* y+ = v
Very close to the wall!
Computational Fluid Dynamics!
v* =
!w "
We have!
!w = µ
du dy
Using the nondimensional values!
u v* y v* y+ = v
u+ = Or simply assume:!
v* =
u+ =
Using the nondimensional values!
Computational Fluid Dynamics!
so approximately!
y v* v
!y
!y +
# du =" w dy $
# v* du + = " w =1 + dy $
integrating!
" du
+
=
1 !
"
dy + y+
giving!
u+ =
1 ln y + + C !
v* =
!w "
!w = µ
du dy
u v* y v* y+ = v
u+ =
Computational Fluid Dynamics!
Thus, the velocity near the wall is! Velocity versus distance from wall!
u+
! = 0.4 C = 5.5
1 u = ln y + + C ! + + u =y +
Computational Fluid Dynamics!
v* =
!w "
!w = µ
du dy
u v* y v* y+ = v
Viscous Buffer sub-layer! layer!
L = 1m; U = 1m/s; ν = 10-6 (water)! The Reynolds number is therefore:!
u+ =
y+
10!
For a practical engineering problem!
Re =
For a flat plate, the average drag coefficient is!
CD = 0.592Re!1/ 5
where!
CD =
Outer layer!
Computational Fluid Dynamics! Or!
CD = 0.0037 FD = CD 12 "U 2 = 3.74 LW
And we find!
1 2
FD !U 2 LW
Computational Fluid Dynamics!
Thickness of the viscous sub-layer!
y=
The average shear stress is therefore!
!w =
LU = 10 6 v
10! 10 "10#6 = = 1.667 "10#4 m = 0.1667 mm !* 0.06
Find the thickness of the boundary layer!
! L
= 0.37Re"1/ 5
! L
= 0.0233m = 23.3mm
v * = 0.06 The average thickness of the viscous sub-layer is 10 in units of y+:!
Computational Fluid Dynamics!
To deal with this problem it is common to use wall functions where 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 Reynolds Number Flows: A Review. AIAA Journal, 23 (1985), 1308-1319!
To resolve the viscous sublayer at the same time as the turbulent boundary layer would require a large number of grid points!
Computational Fluid Dynamics!
Second order closure!
Computational Fluid Dynamics!
Computational Fluid Dynamics!
Derive equations for the Reynolds stresses:! The k-epsilon and other two equation models have several serious limitations, including the inability to predict anisotropic Reynolds stress tensors, relaxation effects, and nonlocal effects due to turbulent diffusion.! For these problems it is necessary to model the evolution of the full Reynolds stress tensor!
The Navier-Stokes equations in component form:!
∂ ui 1 + ∇uiu j = − ∇p + ν∇ 2ui ∂t ρ Multiply the equation by the velocity!
% !u ( ui ' i + "ui u j = - #1 "p + $" 2 ui * & !t ) and averaging leds to equations for !
! ui u j !t
Computational Fluid Dynamics!
Computational Fluid Dynamics!
The new equations contain terms like!
ui ui u j which are not known. These terms are therefore modeled! 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 recent improvements such as RNG are therefore most commonly used!!
Computational Fluid Dynamics!
For more information:! D. C. Wilcox, Turbulence Modeling for CFD (2nd ed. 1998; 3rd ed. 2006). ! The author is one of the inventors of the k-ω model and the book promotes it use. The discussion is, however, general and very accessible, as well as focused on the use of turbulence modeling for practical applications in CFD!
Turbulence models are used to allow us to simulate only the averaged motion, not the unsteady small scale motion.! Turbulence modeling rest on the assumption that the small scale motion is universal and can be described in terms of the large scale motion.! Although considerable progress has been made, much is still not known and results from calculations using such models have to be interpreted by care!!
