Microsoft PowerPoint - ch10.ppt~ attempts to predict turbulence
DNS (Direct Numerical Simulation):
~ finest grid Δx ~ Kolmogorov’s dissipation length scale
~ time increment Δt ~ Kolmogorov’s dissipation time scale
~ simulation time period = several L/q’s
Resolution 1:
Resolution 2:
Resolution 3:
Resolution 4:
10. DNS, LES, and RANS 10. DNS, LES, and RANS
Estimation of the number of grids required for a DNS with ReL = 104:
Kolmogorov’s dissipation time scale ~ ( ) 2121 Re−η =εν=τ Lq L
43Re −=η LLKolmogorov’s dissipation length scale ~
( ) 9493 10~Re~~# LL η
Estimation of the number of time steps required for a DNS with ReL = 104:
100~Re~~# 21 L
10. DNS
This animation demonstrates the evolution of near-wall coherent structures as they convect downstream in a fully-developed turbulent channel flow. The gray structures indicate where the fluid is "spinning" fastest: such regions are identified by solids (also known as "isosurfaces") which enclose all regions of the flow in which the discriminant of the velocity gradient tensor is greater than some positive threshold, indicating fluid motion which is focus in nature. Such regions also roughly (but not exactly) indicate where the pressure of the fluid is lowest, as do the visible cores of tornados; thus, the shaded regions above roughly correspond to tiny unstable "tornado-like" structures in the turbulent flow. Only one eighth of the computational domain used for this simulation, which used approximately 8.5 million grid points, is shown. The animation is shown in a reference frame which convects at approximately 2/3 of the bulk velocity to slow the apparent movement of these structures so that their evolution may be more easily observed. Simulation by Thomas Bewley (UCSD). Visualization by Ned Hammond (Stanford).
Color contours of the vorticity magnitude in Direct Numerical Simulations of supersonic turbulent jet with near acoustic field. The simulations were performed by Jonathan Freund, Parviz Moin and Sanjiva K. Lele.
10. DNS
Color surfaces (red is positive, blue is negative) of the instantaneous streamwise vorticity in Direct Numerical Simulations of flow over a cylinder at Re=300. The simulations were performed by Arthur Kravchenko.
10. DNS
10. DNS
Resolution 3: LES
Resolution 4: DNS
1. select a space filter );( σ− xyG
( ) )2exp( 2 1; 22 σ−− σπ
=σ− xyxyG
( ) ( ) xy xy
xyG −π
0= ∂
0= ∂
1 traceless
10. LES
( )ijkkij sub ij δη−η−≡τ 3
1
jiijjiij uuuuuu ′′++′+′≡η
~ effects of small eddies on large eddies through interactions in between
~ effects of small eddies generated by large eddies on large eddies
~ theoretically computable (no need in modeling)
~ usually modeled together with subgrid stresses because its magnitude is usually on the order of truncation errors. In that case,
( ) ( )ijkkijijkkij sub ij LL δ−−δη−η−≡τ 3
1 3 1
1
ρ 3 1mod
~ solve for the large eddy averaged velocity and the modified pressure
~ subgrid stresses need modeling
j
sub subi i i i i i i ij i i ij
i j j j j j
uu u t x
∂∂ + ∂ ∂
Energy exchange rate between resolved and subgrid eddies
sub i ij
∂τ ∂
(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99) ~ simplest, commonly used
( ) ( ) 212 2 / ijijSt SSσCν =
~ always positive (no backward cascade)
~ isotropic
(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99)
•Determination of the proportional constant:
Energy exchange rate between resolved and subgrid eddies:
( ) ><σ>= ∂ ∂
∫ σπ
232 223
3 2
(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99)
• Determination of the proportional constant:
At large Reynolds number: >ε>≈<ε< g
213
43243
43
K
structure function: ( ) >+−≡< 2 2 ),(,),( trxutxurxF
For homogeneous isotropic turbulence:
>+<−><== )()(2)()(2)(),( 22 rxuxuxuxurFrxF iiii
∫

10. LES 10. LES
( ) ( )( ) ( ) expi i iiu x u x r k ik r dk∫∫∫< + >= Φ ⋅
( ) ( )max 2 2
k
∫ ∫ ∫= Φ θ θ φ θ
For isotropic homogeneous turbulence:
k
∫ ∫= π Φ θ θ θ
( ) ( ) 0
ikr ikr −
kr kr
0 ( , ) ( )
k
i i iiu u k t dk E k dk∫∫∫ ∫= Φ =
∫ ∫ π π
φθθΦ= 2
0 0
( ) ( )max 2
kr ∫< + >= π Φ
Recall
( ) ( )max
0
kr ∫< + >=
At large Reynolds number:
( ) 21
211 )( )(~~
tg dkkEk 0
2 2/3 5 3
2/3 4/332 4t K cC k= ν < ε> ⋅
1/3 4/331 2 4t K cC k−= ν < ε> ⋅
( ) 3/ 21 22/3 1/ 2
5 1
k C
23




( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )












iiKK F t SxCxFxC ωω+⋅Δ⋅⋅=Δ⋅Δ⋅⋅=ν −− 222321
2 23 043.0)(105.0
10. LES
(c) dynamic model (1992, JFM 238,352-336; 1991, Phys. Fluids A 3, 1760-1765)
idea: use two filters, one with width Δx and the other with αΔx (α>1)
ydxxyGtyutxu ii ∫ Δ−≡ );(),(),(
ydxxyGtyutxu ii ∫ Δα−≡ );(),(),(~
ydxxyGtyutxu ii ∫ Δα−≡ );(),(),(~
jijiij uuuuT −≡~ ~
If Smagorinsky’s model is applied, then
ijijkkij SSxCTT 2 3 1 2 Δ=δ−
= effect of eddies of size < Δx on those of size > Δx
( ) ijijkkij SSxC ~~
2 2 3 1 Δα=δℑ−ℑ



∂
∂ +
∂ ∂
~~ 2~~ 222
3 1
10. LES
mnmn
ijij
α−Δ=δ−
• Another possibility: choose C to be the one that minimizes E(C)
( )( )ijijkkijijijkkij CMLLCMLLCE 2~~2~~)( 3 1
3 1 −δ−−δ−≡
( )1 3
( ) 0 2 2ij ij kk ij ij E C M L L CM
C ∂ = =− − δ − ∂
10. LES
10. LES ( )1
30 2 2ij ij kk ij ijM L L CM=− − δ −
1 30 2ij ij ij kk ij ij ijM L M L CM M= − δ −
1 3 2ij ij jj kk ij ijM L M L CM M− =
( )2 2 ij ij ijM x S S S S≡Δ −α
10. LES
=
( )2 2 ij ij ijM x S S S S≡Δ −α
jijiij uuuuL ~~~ −≡
ijijkkij SSxCTT 2 3 1 2 Δ=δ− (subgrid stresses)
Other subgrid models:
(1994) Center for Turbulence, Proc. Of the Summer Program 271-299
(1995) JFM 286, 229-255
• A negative C is possible (backward cascade).
• mathematically inconsistent (dynamic localization methods)
• Too strong negativeness causes instability.
• The real C is found to be intermittent.
10. LES ( )1 , 2
M M = =
1/ 21 3 2ij kk ij ijT T C x k S− δ = Δ ⋅ ⋅
10. LES
)(2)()( 2 kEkkT t kE
ν−= ∂
∂ + ν = ∂ 0 ( ; , )
∫ ∫∫ =
+ + = ∫∫≡
< = >
∫∫ ∫∫= +
10. LES
p q k p q k q k
< >
+ + = <
k E k T k p q dpdq t
< + + = ∫∫
+ + = ∫∫ν =
+ + = ∫∫ν =
( ) ( ) ( ) ( )( ) 3
2 2, , kpq xy zT k p q E q k E p p E k
q += θ ⋅ ⋅ ⋅ −
x y
z p
q k
( )
( )
0 ˆ ˆ ˆ i ij m j m
p q k k u i k u p u q dpdq
t + − = ∫∫
k
<
ν+ν
Kraichnan,1976, Eddy viscosity in two and three dimensions, J. Atmos. Sci. 33 1521–36.
10. LES
T k k dk
′ ′ = the fraction of the total energy
transfer across kc which comes from wavenumbers between k and kc
Color shades of the instantaneous passive scalar in Large Eddy Simulations of turbulent flow in coaxial jet combustor with swirl. The simulations were performed by Charles Pierce and Parviz Moin.
10. LES
Color shades of the instantaneous streamwise velocity in Large Eddy Simulations of turbulent flow in plane diffuser. The simulations were performed by Massimiliano Fatica, Rajat Mittal, Hans Kaltenbach and Parviz Moin.
10. LES
Color contours of the instantaneous vorticity magnitude in Large Eddy Simulations of flow over a cylinder at Re=3900. The simulations were performed by Arthur Kravchenko.
10. LES 10. DNS, LES, and RANS
RANS (Reynolds Averaged Navier-Stokes Equations)
0= ∂



∂
∂ +
∂ ∂
t ij Kuu
Zero equation One equation Model : model K-equation Two equation Model : equation equation −ε+−K Reynolds stress models: model t
ijτ -equations
10. RANS
§ Zero equation
2yl ≈Sublayer:
flow plate-flatfor 26 ; exp1 ≈
• Eddy Viscosity Concept: free)-e(divergenc 2 ijt
t ij Sν= ρ
ε= μ 2 KC
• Eddy Viscosity Concept: free)-e(divergenc 2 ijt
t ij Sν= ρ
ijij j
i ji
j jiij

§ One-Equation Model
10. RANS
§ Two-Equation Model
i
j
production
destruction

ε− ∂ ∂
0= ∂
Model equations directly.t ijτ
~ viscous diffusion/dissipation effect
~ turbulent advection
be expressed in terms of ji j
i uu x u ′′− ∂ ∂
i=j :
′′′′′′′′′ , , , , , 133221



10. RANS
10. RANS
10. RANS 10. DNS, LES, and RANS
Comparisons among DNS, LES and RANS:
DNS
• simulate motion of all scales (limited only by numerical errors)
• able to capture all details of the turbulent flow fields
• tremendously large amount of computations and huge data size
LES
• need modeling effects of subgrid eddies on large eddies
• able to see fluctuations over length scale >= Δx
RANS
• compute only ensemble averaged quantities (least computations)