Post on 20-Oct-2021
10. DNS, LES, and RANS
~ attempts to predict turbulence
0=∂
∂=⋅∇
j
j
xu
u
( )i
jj
i
ij
jii fxx
uxp
xuu
tu
+∂∂
∂ν+
∂∂
ρ−=
∂
∂+
∂∂ 1
~ four equations for four unknowns uj and p
DNS (Direct Numerical Simulation):
~ solve Navier-Stokes equations directly without modeling
~ 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:
10. DNS, LES, and RANS
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 LqL
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~~# 21L
qL
ητ
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, ParvizMoin 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
a spatially evolving free shear layer Bert Debusschere
10. DNS
10. DNS, LES, and RANS
Resolution 3: LES
Resolution 4: DNS
10. LES
LES (Large Eddy Simulation) ~ spatial average
1. select a space filter );( σ− xyG
( ) )2exp(21; 22 σ−−σπ
=σ− xyxyG
( ) ( )xyxy
xyG−π
σ−π=σ−
sin2;
( )⎩⎨⎧ σ≤−σ
=σ−otherwise. 0
;2 if 1;
xyxyG
x2σ
y
ydxyGtyutxu ii ∫ σ−≡ );(),(),(
2. define large-eddy quantity:
),(),(),( txutxutxu iii ′+=
jjjj xtxuydxyGtyu
xydxyG
ytyu
xtxu
∂∂
=σ−∂∂
=σ−∂
∂=
∂∂
∫∫),();(),();(),(),(
ttxu
ttxu
∂∂
=∂
∂ ),(),(
0
i
ii
u
uu
′ ≠≠
jijijiijjijiji uuuuuuuuuuuuuu −+′′+′+′+=
subgrid stresses, ηij
Leonard stresses, Lij
10. LES
3. solve large-eddy motion :iu
0=∂
∂=⋅∇
j
j
xu
u
( )i
jj
i
ij
jii fxx
uxp
xuu
tu
+∂∂
∂ν+
∂∂
ρ−=
∂
∂+
∂∂ 21
~ four equations for four unknowns uj and p
0=∂
∂
j
j
xu
( ) ( )ijsubij
ji
jj
ikk
ij
jii Lx
fxx
upxx
uut
u−τ
∂∂
++∂∂
∂ν+⎟⎟
⎠
⎞⎜⎜⎝
⎛η+
ρ∂∂
−=∂
∂+
∂∂ 2
31
( )ijkkijsubij δη−η−≡τ 3
1 traceless
10. LES
4. Subgrid-stress models ~ modeling small-scale turbulence
( )ijkkijsubij δη−η−≡τ 3
1
jijiij uuuuL −≡
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,
( ) ( )ijkkijijkkijsubij LL δ−−δη−η−≡τ 3
131
10. LES
0=∂
∂
j
j
xu
( )j
subij
ijj
i
ij
jiix
fxx
upxx
uut
u∂
τ∂++
∂∂∂
ν+⎟⎟⎠
⎞⎜⎜⎝
⎛ρ∂
∂−=
∂
∂+
∂∂ 2
mod
( ) ( ) ijkkkkijijsubij LL δ+η−+η−≡τ 3
1
( )kkkk Lpp+η+
ρ≡
ρ 31mod
~ solve for the large eddy averaged velocity and the modified pressure
~ subgrid stresses need modeling
10. LES
( )j
subij
ijj
i
ij
jiix
fxx
upxx
uut
u∂
τ∂++
∂∂∂
ν+⎟⎟⎠
⎞⎜⎜⎝
⎛ρ∂
∂−=
∂
∂+
∂∂ 2
mod
10. LES
iu
⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭
( )212122
mod
iij
j
sub subi i i i ii i ij i i ij
i j j j j j
uu ut x
u p u u u uu u u fx x x x x x
∂∂ +∂ ∂
⎛ ⎞⎛ ⎞ ∂ ∂ ∂ ∂∂ ∂ ⎟⎜⎟⎜ ⎟=− + ν + τ −ν + −τ⎜⎟⎜ ⎟⎟ ⎜⎟⎜ ⎟∂ ρ ∂ ∂ ∂ ∂ ∂⎜⎝ ⎠ ⎝ ⎠
Energy exchange rate between resolved and subgrid eddies
sub iij
j
ux
∂τ∂
(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99)~ simplest, commonly used
( ) ( ) 212 2 /ijijSt SSσCν =
• eddy viscosity:
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
=
−=
i
j
j
iij
ijtsubij
xu
xu
S
Sντ
21
2
• mixing length assumption:
( ) 212 , ~~ ijijt SSSSuxν ≡σ⋅σΔ⋅Δ
~ always positive (no backward cascade)
~ isotropic
~ incorrect near walls
10. LES
(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99)
•Determination of the proportional constant:
Energy exchange rate between resolved and subgrid eddies:
( ) ><σ>=∂∂
τ>=<ε<32 SC
xu
Sj
isubijg
Recall for homogeneous isotropic turbulence:
∫σπ
≈>ωω=<0
2 )( 2 dkkEkS ii
At large Reynolds number:
( )∫∫σπ
−σπ
>ε<≈≈>ωω>=<<0
35322
0
22 2)( 2 dkkCkdkkEkS Kii
232223
32
><⎟⎠⎞
⎜⎝⎛πσ
⎟⎟⎠
⎞⎜⎜⎝
⎛>=ε< S
CK
10. LES
(a) Smagorinsky’s model (1963, Mon. Weather Rev. 91, 99)
• Determination of the proportional constant:
At large Reynolds number: >ε>≈<ε< g
213
43243
321
><
><⎟⎟⎠
⎞⎜⎜⎝
⎛π
=S
S
CC
KS
( ) 212 ijij SSS ≡
( ) ( ) 212 2 /ijijSt SSσCν =
ijtsubij Sντ 2−=
43
321
⎟⎟⎠
⎞⎜⎜⎝
⎛π
≈KC
10. LES
( )3 2 2
3 22 3 223S
K
C S SC
⎛ ⎞ ⎛ ⎞σ⎟⎜ ⎟⎜σ < >≈ < >⎟⎜ ⎟⎜⎟ ⎟⎜⎜ ⎟ ⎝ ⎠π⎝ ⎠
(b) structure-function model (1992, JFM 239,157-194)
structure function: ( ) >+−≡< 22 ),(,),( trxutxurxF
For homogeneous isotropic turbulence:
>+<−><== )()(2)()(2)(),( 22 rxuxuxuxurFrxF iiii
∫∞
⎟⎠⎞
⎜⎝⎛ −=
0
sin1)(4 dkkr
krkE
xkdk
krkrkErF c
kc
Δπ
=⎟⎠⎞
⎜⎝⎛ −≈ ∫ , sin1)(4)(
02
( )( ) >+−+−≡< )()()()()(2 rxuxurxuxurF iiii
structure function of resolved eddies:
∫ ⎟⎠⎞
⎜⎝⎛
ΔΔ
−≈Δck
dkxk
xkkExF0
2sin1)(4)(
10. LES 10. LES
( ) ( )( ) ( ) expi i iiu x u x r k ik r dk∫∫∫< + >= Φ ⋅
( ) ( )max 2 2
0 0 0exp cos sin
k
ii k ikr k d d dkπ π
∫ ∫ ∫= Φ θ θ φ θ
For isotropic homogeneous turbulence:
( ) ( )( )max 2
0 02 exp cos sin
k
iik k ikr d dkπ
∫ ∫= π Φ θ θ θ
( ) ( )0
0exp cos s
exp cosinikr d
ikrikr
ππ
∫θ
=−θ θ θ( ) ( )exp expikr ikr
ikr ikr−
= −
( ) ( )2 si inn 2si krikr
krkr
==
max1 12 2
0( , ) ( )
k
i i iiu u k t dk E k dk∫∫∫ ∫= Φ =
∫ ∫π π
φθθΦ=2
0 0
221 sin),()( ddktkkE ii
22 ( , )iik k t= π Φ
( ) ( )max 2
0
sin( ) ( ) 4
k
i i ii
kru x u x r k k dk
kr∫< + >= π Φ
10. LES For isotropic homogeneous turbulence:
Recall
( ) ( )max
0
sin( ) ( ) 2
k
i i
kru x u x r E k dk
kr∫< + >=
(b) structure-function model (1992, JFM 239,157-194)
At large Reynolds number:
∫ ⎟⎠⎞
⎜⎝⎛
ΔΔ
−⎟⎟⎠
⎞⎜⎜⎝
⎛≈Δ
−ck
cc dk
xkxk
kkkExF
0
35
2sin1)( 4)(
c
ckkE
xA )(4
2
38
Δ
π=
476738.0sin1 0
35 ≈ξ⎟⎟⎠
⎞⎜⎜⎝
⎛ξξ
−ξ≡ ∫π
− dA
On the other hand, expect from mixing length assumption:
( )21
211 )()(~~ ⎟⎟
⎠
⎞⎜⎜⎝
⎛=⋅Δ⋅Δν −
c
cccct k
kEkEkkux
21
238
221
)(4
)(⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
Δπ
Δ=⎟⎟
⎠
⎞⎜⎜⎝
⎛≡ν xF
AxC
kkE
C Fc
cFt
10. LES
(b) structure-function model (1992, JFM 239,157-194)• Determination of proportional constant (CF):
At large Reynolds number: ∫ν>=ε>≈<ε<ck
tg dkkEk0
2 )(2
23
23 −= KF CC
10. LES
2
02 ( )
ck
t k E k dk∫< ε>= ν
2 2/3 5 3
02
ck
t Kk C k dk−∫= ν ⋅ < ε>
2/3 4/3324t K cC k= ν < ε> ⋅
1/3 4/331 24t K cC k−= ν < ε> ⋅
( ) 3/ 21 22/3 1/2
51
/3 231 (2
)K K
cc
cF cC kEC kk
kC
−−⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⋅ ⋅ < ε> ⋅ ⋅⎝ ⎠
3/ 2312 F KC C= ⋅
(b) structure-function model (1992, JFM 239,157-194)
23
23 −= KF CC
)(105.0)(43
2 212
2321
238
223 xFxCxF
AxC KKt Δ⋅Δ⋅⋅≈
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
Δπ
Δ=ν −−
• computation of structure function
( )( ) >Δ+−Δ+−≡<Δ )()()()(),(2 xxuxuxxuxuxxF iiii
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ⎪
⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
Δ−−+Δ+−+
Δ−−+Δ+−+
Δ−−+Δ+−+
≈2
3332
333
2222
2222
2111
2111
61
xxuxuxxuxu
xxuxuxxuxu
xxuxuxxuxu
iiii
iiii
iiii
10. LES
21
238
221
)(4
)(⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
Δπ
Δ=⎟⎟
⎠
⎞⎜⎜⎝
⎛≡ν xF
AxC
kkE
C Fc
cFt
•Comparison between Smagorinsky’s model and structure function model:
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛Δ
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛Δ
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛Δ
∂∂
≈Δ2
3
2
2
2
12 222
61),( x
xux
xux
xuxxF iii
xxi Δ=Δ
2
1
1211111 )()( x
xu
xxuxu Δ∂∂
≈Δ+−
2
1
2211212 )()( x
xuxxuxu Δ∂∂
≈Δ+−
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ⎪
⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
Δ−−+Δ+−+
Δ−−+Δ+−+
Δ−−+Δ+−+
≈2
3332
333
2222
2222
2111
2111
2 61
xxuxuxxuxu
xxuxuxxuxu
xxuxuxxuxu
F
iiii
iiii
iiii
10. LES
iiKKFt SxCxFxC ωω+⋅Δ⋅⋅=Δ⋅Δ⋅⋅=ν −− 222321
223 043.0)(105.0
( ) 21243
23
21 /ijij
K
St SSσ
Cν ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛π
= SxCK ⋅Δ⋅⋅= − 223055.0
( )j
i
j
i
xu
xuxxxF
∂∂
∂∂Δ
=Δ3
,2
2 ( )ijijijij SSxΩΩ+
Δ=
3
2
( )iiijij SSxωω+
Δ= 2
6
2
•Comparison between Smagorinsky’s model and structure function model:
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−≡ );(),(),(~
0=∂
∂
j
j
xu
( )j
iji
jj
i
ij
jiixT
fxx
upxx
uut
u∂
∂++
∂∂∂
ν+⎟⎟⎠
⎞⎜⎜⎝
⎛ρ∂
∂−=
∂
∂+
∂∂
jijiij uuuuT −≡
10. LES
( )j
iji
jj
i
ij
jiix
fxx
upxx
uut
u∂
∂ℑ++
∂∂∂
ν+⎟⎟⎠
⎞⎜⎜⎝
⎛ρ∂
∂−=
∂
∂+
∂∂ ~~~~~~
jijiijij uuuuT −+≡ℑ ~~~ ~ jijiij uuuuL ~~~ −≡ ~~
jijiij uuuuT −≡~ ~
η Δx αΔx L
ijij LT ~~ −=
( )j
iji
jj
i
ij
jiixT
fxx
upxx
uut
u∂
∂++
∂∂∂
ν+⎟⎟⎠
⎞⎜⎜⎝
⎛ρ∂
∂−=
∂
∂+
∂∂
ijT
ijℑ
10. LES
ijijij TL ℑ−= ~~Germano identity:
If Smagorinsky’s model is applied, then
ijijkkij SSxCTT 231 2 Δ=δ−
= effect of eddies of size < Δx on those of size > Δx
( ) ijijkkij SSxC~~
2 231 Δα=δℑ−ℑ
= effect of eddies of size < αΔx on those of size > αΔx
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
=i
j
j
iij x
uxu
S~~~
21
⎟⎠⎞⎜
⎝⎛ Δα−Δ=δ− ijijijkkij SSxCSSxCLL
~~ 2~~ 222
31
10. LES
Assume C = constant:
• One possibility:
( ) ijijijkkijij SCMLLS 2~~31 =δ−
mnmn
ijij
SMSL
C~
21
=
ijijijijkkij CMSSSSxCLL 2~~
2~~ 2231 ≡⎟
⎠⎞⎜
⎝⎛ α−Δ=δ−
• Another possibility: choose C to be the one that minimizes E(C)
( )( )ijijkkijijijkkij CMLLCMLLCE 2~~2~~)( 31
31 −δ−−δ−≡
( )13
( ) 0 2 2ij ij kk ij ijE C M L L CM
C∂ = =− − δ −∂
10. LES
mnmn
ijij
MMML
C~
21
=
10. LES( )1
30 2 2ij ij kk ij ijM L L CM=− − δ −
130 2ij ij ij kk ij ij ijM L M L CM M= − δ −
13 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
mnmn
ijij
MMML
C~
21
=
( )2 2 ij ij ijM x S S S S≡Δ −α
jijiij uuuuL ~~~ −≡
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
=i
j
j
iij x
uxu
S~~~
21
( )j
iji
jj
i
ij
jiixT
fxx
upxx
uut
u∂
∂++
∂∂∂
ν+⎟⎟⎠
⎞⎜⎜⎝
⎛ρ∂
∂−=
∂
∂+
∂∂
0i
i
ux
∂ =∂
ijijkkij SSxCTT 231 2 Δ=δ− (subgrid stresses)
Other subgrid models:
• variations of dynamic model
(1992) Phys. Fluids A 4, 633-635
(1994) Center for Turbulence, Proc. Of the Summer Program 271-299
(1995) JFM 286, 229-255
• k-equation dynamic model
dynamic model:
• C = 0 in laminar flows and at walls.
• 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
ij ij
mn mn
L MC C x t
M M= =
1/ 213 2ij kk ij ijT T C x k S− δ = Δ ⋅ ⋅
10. LES
(d) spectral models (models in the wave space)
)(2)()( 2 kEkkTtkE
ν−=∂
∂
22 ( ) ( )k E k T kt
⎛ ⎞∂ ⎟⎜ + ν =⎟⎜ ⎟⎜⎝ ⎠∂ 0( ; , )
p q kk k
S k p q dpdq dk+ + ==
∫ ∫∫⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠
0( ; , )
p q kT k p q dpdq
+ + =∫∫≡
and or 0 0
( ; , ) ( ; , )c cp q
p q k p q kk p q k
T k p q dpdq T k p q dpdq+ + = + +
<=>
∫∫ ∫∫= +
: resolved eddies: subgrid eddies
c
c
k kk k
⎧ <⎪⎪⎨ >⎪⎪⎩
cascade due to subgrid stresses, to be modeled
10. LES
2
0 0 and or
2 ( ) ( ; , ) ( ; , )c cp q k p
p q k p q kq k
k E k T k p q dpdq T k p q dpdqt + + = + + =
< >
∫∫ ∫∫⎛ ⎞∂ ⎟⎜ + ν = +⎟⎜ ⎟⎜⎝ ⎠∂
and0
2( ; , ) 2 ( )c
tpp q k
q kT k p q dpdq k E k
+ + =<
∫∫= + ν
( )
2
nd0
a
2 ( ) ( ; , )c
tp q kp q k
k E k T k p q dpdqt
<+ + =∫∫
⎛ ⎞∂ ⎟⎜ + ν+ν =⎟⎜ ⎟⎜⎝ ⎠∂
0or
2
2 ( ) ( ; , )c
tp q
kk
p q
k E k T k p q dpdq>
+ + =∫∫ν =
10. LES0
or
2
2 ( ) ( ; , )c
tp q
kk
p q
k E k T k p q dpdq>
+ + =∫∫ν =
Isotropic homogeneous turbulence theory:
EDQNM (Eddy-damped quasi-normal Markovian by Orszag)
DIA (direct interaction approximation by Kraichnan)
( ) ( ) ( ) ( )( )3
2 2, , kpqxy zT k p q E q k E p p E k
q+= θ ⋅ ⋅ ⋅ −
xy
zp
qk
x,y,z : cosine value of the corresponding angles
( )
( )
2 / 3 5 / 3
1
1/ 3 2 / 3
K
kpq k p q
k K
E k C k
C k
−
−
= ε
θ = η +η +η
η = μ ε
10. LES
( ) ( )2
0ˆ ˆ ˆ i ij m j m
p q kk u i k u p u q dpdq
t + − =∫∫
⎛ ⎞∂ ⎟⎜ +ν =−Δ⎟⎜ ⎟⎜⎝ ⎠∂
for ck k<
( ) ( ) ( )2
,
ˆ ˆ ˆ c
i ij m j mt
p qp q k
k
k u i k u p u q dpdqt + =
<
∫∫⎛ ⎞∂ ⎟⎜ + =−Δ⎟⎜ ⎟⎜⎝ ⎠∂
ν+ν
Kraichnan,1976, Eddy viscosity in two and three dimensions, J. Atmos. Sci. 33 1521–36.
10. LES
( )( )( )
0
c
c
k
ck
c k
c
T k k dkk k
T k k dk
∫
∫
′ ′Π ≡−
′ ′= 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, RajatMittal, 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=∂
∂
j
j
xu
( )tijij
jii
j
ij
iix
Kpx
gxu
ut
uDtuD
τ+τ∂∂
ρ+⎟⎟
⎠
⎞⎜⎜⎝
⎛+
ρ∂∂
−=∂∂
+∂∂
≡1
32
(Here overbar represents an ensemble average.)
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
μ=τi
j
j
iij x
uxu
mean viscous
stress tensor
tensorstress )(turbulent Reynolds 32 =δρ+′′ρ−=τ ijji
tij Kuu
Zero equation One equation Model : model K-equationTwo equation Model : equation equation −ε+−KReynolds stress models: model t
ijτ -equations
10. RANS
§ Zero equation
Slt2≈ν• Mixing length models:
2yl ≈Sublayer:
yl κ≈Overlap layer:
constant≈lOuterlayer:
¶ Prandtl and Karman: ¶ van Driest Model
flow plate-flatfor 26 ; exp1 ≈⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−κ≈
+A
Ayyl
damping factor
A varies with flow conditions (pressure gradient, wall roughness, blowing/suction etc)
• Eddy Viscosity Concept: free)-e(divergenc 2 ijt
tij Sν=ρ
τ
: eddy viscosity (isotropic herein)tν
§ One-Equation Model
• dimensional analysis: ),( ε=ν Kft
sm2 22 sm 32 sm
ε= μ2 KC
[ ]L
KL
2/3
3
2~
velocityareavelocity~
massvelocitydrag
ρ
×××ρ×=ε
LKC
2/3
ε=ε∝L
2/1 ∝K eddy velocity
effective eddy size
• turbulent kinetic energy dissipation rate:
• Eddy Viscosity Concept: free)-e(divergenc 2 ijt
tij Sν=ρ
τ
10. RANS
turbulent diffusion term:
1 2
21
jKjiij x
Kt
Cuuuup∂∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡≅′′′−′′
ρ− :
2
jK x
KKC∂∂
ε≡
turbulent production term:
( )j
itij
j
iij
tij
j
iji x
uτ
xu
K τxu
uu∂∂
=∂∂
δρ−=∂∂
′′ρ− 32
• turbulent kinetic energy per unit mass K:
ijijj
iji
jjiij
j
jj
SSxu
uuxKuuuup
x
xKu
tK
DtDK
′′μ−∂∂′′ρ−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂∂
μ+′′′ρ−′′−∂∂
=
∂∂
ρ+∂∂
ρ=ρ
2 21
ε=
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
ε=ν=
ρ
τμ
i
j
j
iijt
tij
xu
xuKCS
222
10. RANS
ε−∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
ε+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂∂
ν+∂∂
ε∂∂
=∂∂
+∂∂
μ 22
j
i
i
j
j
i
jjK
jjj x
uxu
xuKC
xK
xKKC
xxKu
tK
0=∂
∂
j
j
xu
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
ε+
∂∂
ν∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
ρ∂∂
−=∂∂
+∂∂
μi
j
j
i
j
i
jii
j
ij
ixu
xuKC
xu
xKp
xg
xu
ut
u 2
32 2
ρτtij
LKC
2/3
ε=ε
6 equations for 6 unknowns ⎟⎟⎠
⎞⎜⎜⎝
⎛ε+
ρ , , , 3
2 KKpui
with 3 empirical constants ( )LCCCK εμ , ,
§ One-Equation Model
10. RANS
§ Two-Equation Model
• turbulent kinetic energy dissipation rate (per unit mass):j
i
j
ixu
xu
∂′∂
∂′∂
ν≡ε
mm
i
jj
i
m
j
m
i
j
i
j
m
j
i
m
j
i
j
m
i
mj
i
m
ij
jmm
j
m
i
m
ij
j
xxu
xxu
xu
xu
xu
xu
xu
xu
xu
xu
xxu
xu
u
xxp
xu
xu
xu
uxDt
D
∂∂′∂
∂∂′∂
ν−∂
′∂
∂′∂
∂′∂
ν−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂′∂
∂′∂
+∂
′∂
∂
′∂
∂∂
ν−∂∂
∂∂
′∂′ν−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂ε∂
ν+∂
′∂∂
′∂
ρν
−∂
′∂∂
′∂′ν−
∂∂
=ε
22
2
22
22
2
exact equation:turbulent diffusion molecular diffusion
production
destruction
10. RANS
ε⋅ε
−≡ ε KC 2:
mm
j
m
i
m
ij x
pxu
xu
xu
u∂
′∂∂
′∂
ρν
−∂
′∂∂
′∂′ν−2
turbulent diffusion terms:
jxKC
∂ε∂
ε≡ ε
2:
production terms:
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂′∂
∂′∂
+∂
′∂
∂
′∂
∂∂
ν−∂∂
∂∂
′∂′ν−j
m
j
i
m
j
i
j
m
i
mj
i
m
ij x
uxu
xu
xu
xu
xxu
xu
u 22 2
j
iji x
uuu
KC
∂∂′′ε
−≡ ε1:j
itij x
uskg
m∂∂
τ⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
⋅≅
3
jxsm
∂ε∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡≅
2
mm
i
jj
i
m
j
m
i
j
ixx
uxx
uxu
xu
xu
∂∂′∂
∂∂′∂
ν−∂
′∂
∂′∂
∂′∂
ν−22
22
destruction terms:
ε⎥⎦⎤
⎢⎣⎡≅
sec1
j
i
i
j
j
ixu
xu
xu
KCC∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
≡ εμ 12:
10. RANS
6 equations for 6 unknowns ⎟⎟⎠
⎞⎜⎜⎝
⎛ε+
ρ , , , 3
2 KKpui
with 5 empirical constants ( )21,, , , εεεμ CCCCCK
ε−∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
ε+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂∂
ν+∂∂
ε∂∂
=∂∂
+∂∂
μ 22
j
i
i
j
j
i
jjK
jjj x
uxu
xuKC
xK
xKKC
xxKu
tK
0=∂
∂
j
j
xu
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
ε+
∂∂
ν∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
ρ∂∂
−=∂∂
+∂∂
μi
j
j
i
j
i
jii
j
ij
ixu
xuKC
xu
xKp
xg
xu
ut
u 2
32 2
§ Two-Equation Model
KC
xu
xu
xu
KCCxx
KCxDt
D
j
i
i
j
j
i
lll
2
21
22 ε
−∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
+⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂ε∂
ν+∂ε∂
ε∂∂
=ε
εεμε
10. RANS
§ Reynolds-stress Model
RANS (Reynolds Averaged Navier-Stokes Equations)
0=∂
∂
j
j
xu
( )tijij
jii
j
ij
iixx
pgxu
ut
uDtuD
τ+τ∂∂
ρ+
∂∂
ρ−=
∂∂
+∂∂
≡11
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
μ=τi
j
j
iij x
uxu
mean viscous
stress tensor
tensorstress )(turbulent Reynolds =′′ρ−=τ jitij uu
Model equations directly.tijτ
10. RANS
m
mji
m
imj
m
jmi
m
j
m
i
mm
ji
ji
ij
m
jim
jiji
xuuu
xu
uuxu
uu
xu
xu
xxuu
xpu
xpu
xuu
utuu
DtuuD
∂
′′′∂−
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂′′+
∂
∂′′−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
′∂
∂′∂
−∂∂
′′∂ν+
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂′∂′+
∂′∂′
ρ−=
∂
′′∂+
∂
′′∂=
′′
2
1
2
Reynolds stress tensor equations
~ mean motion Lagrangian
~ pressure effects (nonlocal, linear, and nonlinear)
~ viscous diffusion/dissipation effect
~ production and reorientation by the mean motion
~ turbulent advection
nonlocal
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
∂′∂
+∂
′∂
ρ′
+⎭⎬⎫
⎩⎨⎧
ρ′
δ′+δ′∂∂
−=j
i
i
jjmiimj
m xu
xuppuu
x
10. RANS
turbulent diffusion terms:
( ) mjijmiimj uuupuu ′′′−ρ′
δ′+δ′−m
ji
xuum
∂
′′∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡≅
sec
2
m
jiK x
uuKC∂
′′∂⋅
ε≡ :
2
pressure-strain terms:
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂′∂
+∂
′∂
ρ′
j
i
i
j
xu
xup
⎭⎬⎫
⎩⎨⎧
∂∂′′δ−
∂∂′′+
∂
∂′′≡
m
nmnij
m
imj
m
jmi x
uuu
xu
uuxu
uuC32: 2
traceless, expected to be able to
be expressed in terms of jij
i uuxu ′′−∂∂
and
[ ]⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
′′−≅j
iji x
uuu
dissipation terms:
m
j
m
ixu
xu
∂
′∂
∂′∂
ν− 2 ⎟⎠⎞
⎜⎝⎛ δ−′′
ε−εδ−≡ Kuu
KC ijjiij 3
232 : 1
isotropic part
non-isotropic part
10. RANS
⎟⎠⎞
⎜⎝⎛ δ−′′ε
−εδ−
⎭⎬⎫
⎩⎨⎧
∂∂
′′δ−∂∂
′′+∂
∂′′+
⎭⎬⎫
⎩⎨⎧
∂∂′′+
∂
∂′′−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂
′′∂⎟⎟⎠
⎞⎜⎜⎝
⎛ν+
ε∂∂
=′′
KuuK
C
xu
uuxu
uuxu
uuC
xu
uuxu
uux
uuKCxDt
uuD
ijjiij
m
nmnij
m
imj
m
jmi
m
imj
m
jmi
m
jiK
m
ji
32
32
32
1
2
2
modeled Reynolds stress tensor equations
i=j :
ε−⎭⎬⎫
⎩⎨⎧
∂∂′′+
∂∂′′−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛ν+
ε∂∂
= 212
m
imi
m
imi
mK
m xu
uuxu
uuxKKC
xDtDK
6 equations for 6 new unknowns ⎟⎠⎞⎜
⎝⎛ ′′′′′′′′′ , , , , , 133221
23
22
21 uuuuuuuuu
10. RANS
KC
xu
uuK
Cxx
KCxDt
D
j
iji
lll
2
21
2 ε−
∂∂
′′ε−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂ε∂
ν+∂ε∂
ε∂∂
=ε
εεε
0=∂
∂
j
j
xu
⎟⎟⎠
⎞⎜⎜⎝
⎛′′−
∂∂
ν∂∂
+∂∂
ρ−=
∂∂
+∂∂
jij
i
jii
j
ij
i uuxu
xxpg
xu
ut
u 1
⎟⎠⎞
⎜⎝⎛ δ−′′ε
−εδ−
⎭⎬⎫
⎩⎨⎧
∂∂
′′δ−∂∂
′′+∂
∂′′+
⎭⎬⎫
⎩⎨⎧
∂∂′′+
∂
∂′′−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂
′′∂⎟⎟⎠
⎞⎜⎜⎝
⎛ν+
ε∂∂
=′′
KuuK
C
xu
uuxu
uuxu
uuC
xu
uuxu
uux
uuKCxDt
uuD
ijjiij
m
nmnij
m
imj
m
jmi
m
imj
m
jmi
m
jiK
m
ji
32
32
32
1
2
2
11 equations for 11 unknowns
⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′′′′′′′′ερ
, , , , , ,, , , , 1332212
32
22
1321 uuuuuuuuupuuu
§ Reynolds-stress Model
10. RANS
§ Algebraic Stress Model
Assume negligible turbulent convection and diffusion
⎟⎠⎞
⎜⎝⎛ δ−′′ε
−εδ−
⎭⎬⎫
⎩⎨⎧
∂∂
′′δ−∂∂
′′+∂
∂′′+
⎭⎬⎫
⎩⎨⎧
∂∂′′+
∂
∂′′−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂
′′∂⎟⎟⎠
⎞⎜⎜⎝
⎛ν+
ε∂∂
=′′
KuuK
C
xu
uuxu
uuxu
uuC
xu
uuxu
uux
uuKCxDt
uuD
ijjiij
m
nmnij
m
imj
m
jmi
m
imj
m
jmi
m
jiK
m
ji
32
32
32
1
2
2
⎭⎬⎫
⎩⎨⎧
∂∂
′′δ−∂∂
′′+∂
∂′′+
⎟⎠⎞
⎜⎝⎛ δ−′′ε
−εδ−⎭⎬⎫
⎩⎨⎧
∂∂′′+
∂
∂′′−=
m
nmnij
m
imj
m
jmi
ijjiijm
imj
m
jmi
xu
uuxu
uuxu
uuC
KuuK
Cxu
uuxu
uu
32
32
320
2
1
~ algebraic equations for the Reynolds stresses ~
10. RANS
Masud Behnia, Sacha Parneix& Paul Durbin
10. RANS
Masud Behnia, Sacha Parneix& Paul Durbin
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
• simulate motion of large eddies only (less computations)
• 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)
• need strong assumptions to model effects of fluctuations on the mean motion