A LES-LANGEVIN MODEL
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Transcript of A LES-LANGEVIN MODEL
A LES-LANGEVIN MODEL
B. Dubrulle
Groupe Instabilite et Turbulence
CEA Saclay
Colls: R. Dolganov and J-P LavalN. KevlahanE.-J. KimF. HersantJ. Mc WilliamsS. NazarenkoP. SullivanJ. Werne
IS IT SUFFICIENT TO KNOW BASIC EQUATIONS?
E(k)kGrandes EchellesPetites EchellesFlux d’Energie(Viscosité Turbulente)(Paramétrisées)Explosion90 % des ressources informatiques
Waste of computational resourcesTime-scale problem
Necessity of small scale parametrization
Giant convectioncell
Solarspot
GranuleDissipation scale
0.1 km 103km 3⋅104 km 2⋅105 km
Influence of decimated scales
Typical time at scale l: δt≈lu
∝ l2
3
Decimated scales (small scales) vary very rapidlyWe may replace them by a noise with short time scale
u=u +u'
Dtu' i =Aiju' j +ξj
ξi x,t( )ξj x',t'( ) =κ ij x,x'( )δ t−t'( )
Generalized Langevin equation
Obukhov ModelSimplest case
u =0
Aij =−γδij, γ >>δt
κ ij x,x'( ) ∝γδij
No mean flow
Large isotropic frictionNo spatial correlations
P(
r x ,
r u ,t) =
32πεt2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ exp−
3x2
εt3 −3r x •
r u
εt2 −u2
εt
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
u∝ εt
x∝ ε2/3t3/ 2
u∝ x1/ 3
Gaussian velocities
Richardson’s law
Kolmogorov’s spectraLES: Langevin
Influence of decimated scales: transport
r x •
=r u +
r u '
r Ω =
r ∇ ×
r u
r Ω •
=(r Ω •
r ∇ )
r u +(
r Ω •
r ∇ )
r u '
∂tΩi +u k∇kΩi =Ωk∇kui +∇ k βkl∇l Ωi[ ]+2αkil∇kΩ l
βkl = uk' ul
'
αijk = ui'∂kuj
'Stochastic computation
Turbulent viscosity AKA effect
Refined comparison
True turbulenceAdditive noise
GaussianityWeak intermittency
Non-GaussianitéForte intermittence
˙ u =−γu+η
PDF of increments
SpectrumIso-vorticity
LES: Langevin
LOCAL VS NON-LOCAL INTERACTIONS
• Navier-Stokes equations : two types of triades∂tu +u•∇ u=−∇ p+ν Δu+ f
Nl
L
L
l
LOCAL NON-LOCAL
LOCAL VS NON-LOCAL TURBULENCE
NON-LOCAL TURBULENCE
€
∂tU + (U • ∇)U = −∇p + u ×ω + νΔU
∂tω =∇ × U ×ω( ) + ηΔω
€
E = U 2 + u2( )∫ dx
Hm = u • ω dx∫Hc = U • ω dx∫
Analogy with MHD equations: small scale grow via « dynamo » effect
Conservation lawsIn inviscid case
E
k
U
A PRIORI TESTS IN NUMERICAL SIMULATIONS
2D TURBULENCE
3D TURBULENCE
U ∇ u
u∇ u
U ∇ U
u∇ U
Local large/ large scales
Local small/small scales
Non-local
<<
DYNAMICAL TESTS IN NUMERICAL SIMULATIONS
2DDNS
3DDNS
2DRDT
3DRDT
THE RDT MODEL
∂t Ui +U j ∇ j Ui =−∇iP +ν ΔUi −∇ j uiU j +ujUi +uiuj( )
€
∂t ui + U j ∇ j ui = −u j∇ jU i −∇ i p + ν t Δ ui + f i
Equation for large-scale velocity
Equation for small scale velocity
Reynolds stresses
Turbulent viscosity Forcing (energy cascade)
Computed (numerics) or prescribed (analytics)
Linear stochastic inhomogeneous equation(RDT)
THE FORCING
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.05 0.1 0.15 0.2
< F( t )F( t
0 ) >
t-t0
1
10
100
1000
10 4
10 5
10 6
10 7
-100 -50 0 50 100
P(x)
x
CorrelationsPDF of increments
Iso-force Iso-vorticity
TURBULENT VISCOSITY
DNS RDTSES
νt =Cv
25
q−2E(q)dqk
∞
∫
LANGEVIN EQUATION AND LAGRANGIAN SCHEME
∂t ui +U j ∇ j ui =−∇i p+νt Δui + fi
GT u( ) x,k[ ]= dx'∫ f x−x'( )eik(x−x')u x'( )
x
k
Décomposition into wave packets
Dtu=−νTk2u+u•∇ 2
kk2
U •k−U⎛
⎝ ⎜
⎞
⎠ ⎟ +f
Dt x=U
Dtk=−∇ U •k( ) The wave packet moves with the fluidIts wave number is changed by shear
Its amplitude depends on forces
friction “additive noise”
coupling (cascade)“multiplicative noise”
COMPARISON DNS/SES
Fast numerical 2D simulation
Computational time10 days 2 hours
DNS Lagrangian model
(Laval, Dubrulle, Nazarenko, 2000)
QuickTime™ and aGIF decompressor
are needed to see this picture.
Shear flow
QuickTime™ and aBMP decompressor
are needed to see this picture.
Hersant, Dubrulle, 2002
SES SIMULATIONS
Experiment
DNS
SES
Hersant, 2003
LANGEVIN MODEL: derivation
€
∂t ui + U j ∇ j ui = −u j∇ jU i −∇ i p + ν t Δ ui + f i
Equation for small scale velocity
Turbulent viscosity
Forcing
∇ u u −u u ( )
1
10
100
1000
10 4
10 5
10 6
10 7
-100 -50 0 50 100
P(x)
x
Isoforce
LES: Langevin
Equation for Reynolds stress
τij =u iu j −u iu j +u iu' j +u' i u j +u'i u' j
=u iu j −u iu j + Lij −2νT Sij
∇ jLij =l i
∂t
r l =−
r ω ×
r l +
r ∇ ×
r l [ ]×
r u ( )
⊥+νtΔ
r l +
r ξ
r ξ =−
r ω ×
r f +
r ∇ ×
r f [ ]×
r u ( )
with
Generalized Langevin equation
Forcing dueTo cascade
AdvectionDistorsionBy non-local interactions
LES: Langevin
Performances
LES: Langevin
Spectrum Intermittency
Comparaison DNS: 384*384*384 et LES: 21*21*21
Performances (2)
LES: Langevin
Q vs R
s probability
€
Q =1
2SijS ji
R =1
3SijS jkSki
s = −3 6αβγ
α 2β 2γ 2( )
THE MODEL IN SHEARED GEOMETRY
Basic equations
∂tUθ =−1r2
∂rr2 uruθ +νΔUθ
Dtur =2krkθ
k2Ω +S( )ur +2Ωuθ 1−
kr2
k2
⎛
⎝ ⎜
⎞
⎠ ⎟ −νTk
2uθ +Fr
Dtuθ =2kθ
2
k2Ω +S( )ur −
krkθ
k22Ωuθ − 2Ω +S( )ur −νTk
2uθ +Fθ
Dtuz =2kθkz
k2Ω+S( )ur −
krkz
k22Ωuθ −νTk
2uz +Fz
Equation for mean profile
RDT equations for fluctuationswith stochasticforcing
ANALYTICAL PREDICTIONS
Mean flow dominates Fluctuations dominates
Low Re
G =1.46η2
1−η( )7/ 4 Re3/2
G =0.5η2
1−η( )3/2
Re2
ln(Re2)3/ 2
TORQUE IN TAYLOR-COUETTE
10 5
10 6
10 7
10 8
10 9
10 10
10 11
100 1000 10 4 10 5 10 6
G
Re
10 4
10 5
10 6
10 7
10 8
10 9
10 10
100 1000 10 4 10 5
G
R
η = 0.68
η = 0.935
η = 0.85No adjustable parameter
Dubrulle and Hersant, 2002