a les-langevin model
DESCRIPTION
A LES-LANGEVIN MODEL. Colls: R. Dolganov and J-P Laval N. Kevlahan E.-J. Kim F. Hersant J. Mc Williams S. Nazarenko P. Sullivan J. Werne. B. Dubrulle Groupe Instabilite et Turbulence CEA Saclay. IS IT SUFFICIENT TO KNOW BASIC EQUATIONS?. Giant convection cell. Dissipation - PowerPoint PPT PresentationTRANSCRIPT
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