A LES-LANGEVIN MODEL

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A LES-LANGEVIN MODEL

• B. Dubrulle
• Groupe Instabilite et Turbulence
• CEA Saclay

Colls R. Dolganov and J-P Laval N.
Kevlahan E.-J. Kim F. Hersant J. Mc
Williams S. Nazarenko P. Sullivan J. Werne
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IS IT SUFFICIENT TO KNOW BASIC EQUATIONS?
Giant convection cell
Dissipation scale
Solar spot
Granule
0.1 km
Waste of computational resources Time-scale
problem
Necessity of small scale parametrization
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Influence of decimated scales
Typical time at scale l
Decimated scales (small scales) vary very
rapidly We may replace them by a noise with short
time scale
Generalized Langevin equation
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Obukhov Model
Simplest case
No mean flow
Large isotropic friction
No spatial correlations
Gaussian velocities
Richardsons law
Kolmogorovs spectra
LES Langevin
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Influence of decimated scales transport
Stochastic computation
Turbulent viscosity
AKA effect
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Refined comparison
True turbulence
Additive noise
Gaussianity Weak intermittency
Non-Gaussianité Forte intermittence
Iso-vorticity
Spectrum
PDF of increments
LES Langevin
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LOCAL VS NON-LOCAL INTERACTIONS

• Navier-Stokes equations two types of triades

NON-LOCAL
LOCAL
L
L
l
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LOCAL VS NON-LOCAL TURBULENCE
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NON-LOCAL TURBULENCE
E
U
?
Analogy with MHD equations small scale grow via
 dynamo  effect
k
Conservation laws In inviscid case
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A PRIORI TESTS IN NUMERICAL SIMULATIONS
2D TURBULENCE
ltlt
Non-local
Local small/small scales
Local large/ large scales
3D TURBULENCE
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DYNAMICAL TESTS IN NUMERICAL SIMULATIONS
2D RDT
2D DNS
3D RDT
3D DNS
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THE RDT MODEL
Equation for large-scale velocity
Linear stochastic inhomogeneous equation (RDT)
Reynolds stresses
Equation for small scale velocity
Forcing (energy cascade)
Turbulent viscosity
Computed (numerics) or prescribed (analytics)
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THE FORCING
Iso-vorticity
Iso-force
PDF of increments
Correlations
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TURBULENT VISCOSITY
RDT
SES
DNS
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LANGEVIN EQUATION AND LAGRANGIAN SCHEME
Décomposition into wave packets
The wave packet moves with the fluid Its wave
number is changed by shear Its amplitude depends
on forces
coupling (cascade)
multiplicative noise
friction
additive noise
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COMPARISON DNS/SES
Fast numerical 2D simulation
Shear flow
Computational time 10 days 2 hours
phi_m obs
DNS
Lagrangian model
(Laval, Dubrulle, Nazarenko, 2000)
Hersant, Dubrulle, 2002
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SES SIMULATIONS
SES
Experiment
Hersant, 2003
DNS
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LANGEVIN MODEL derivation
Equation for small scale velocity
Turbulent viscosity
Forcing
Isoforce
PDF
LES Langevin
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Equation for Reynolds stress
with
Forcing due To cascade
Advection Distorsion By non-local interactions
Generalized Langevin equation
LES Langevin
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Performances
Comparaison DNS 384384384 et LES 212121
Intermittency
Spectrum
LES Langevin
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Performances (2)
s probability
Q vs R
LES Langevin
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THE MODEL IN SHEARED GEOMETRY
Basic equations
RDT equations for fluctuations with
stochastic forcing
Equation for mean profile
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ANALYTICAL PREDICTIONS
Mean flow dominates
Fluctuations dominates
Low Re
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TORQUE IN TAYLOR-COUETTE
No adjustable parameter
Dubrulle and Hersant, 2002