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

PDF

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