1. 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

2. IS IT SUFFICIENT TO KNOW BASIC EQUATIONS? Waste of computational resources Time-scale problem Necessity of small scale parametrization Giant convection cell Solar spot Granule Dissipation scale 0.1 km

3. 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

4. Obukhov Model Simplest case No mean flow Large isotropic friction No spatial correlations Gaussian velocities Richardson’s law Kolmogorov’s spectra LES: Langevin

5. Influence of decimated scales: transport Stochastic computation Turbulent viscosityAKA effect

6. Refined comparison True turbulenceAdditive noise Gaussianity Weak intermittency Non-Gaussianité Forte intermittence PDF of increments Spectrum Iso-vorticity LES: Langevin

7. LOCAL VS NON-LOCAL INTERACTIONS Navier-Stokes equations : two types of triades L L l LOCAL NON-LOCAL

8. LOCAL VS NON-LOCAL TURBULENCE

9. NON-LOCAL TURBULENCE Analogy with MHD equations: small scale grow via « dynamo » effect Conservation laws In inviscid case E k U 

10. A PRIORI TESTS IN NUMERICAL SIMULATIONS 2D TURBULENCE 3D TURBULENCE Local large/ large scales Local small/small scales Non-local <<

11. DYNAMICAL TESTS IN NUMERICAL SIMULATIONS 2D DNS 3D DNS 2D RDT 3D RDT

12. THE RDT MODEL 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)

13. THE FORCING Correlations PDF of increments Iso-force Iso-vorticity

14. TURBULENT VISCOSITY DNS RDTSES

15. LANGEVIN EQUATION AND LAGRANGIAN SCHEME x k Décomposition into wave packets The wave packet moves with the fluid Its wave number is changed by shear Its amplitude depends on forces friction “additive noise” coupling (cascade) “multiplicative noise”

16. COMPARISON DNS/SES phi_m obs Fast numerical 2D simulation Computational time 10 days2 hours DNSLagrangian model (Laval, Dubrulle, Nazarenko, 2000) Shear flow Hersant, Dubrulle, 2002

17. SES SIMULATIONS Experiment DNS SES Hersant, 2003

18. LANGEVIN MODEL: derivation Equation for small scale velocity Turbulent viscosity Forcing Isoforce PDF LES: Langevin

19. Equation for Reynolds stress with Generalized Langevin equation Forcing due To cascade Advection Distorsion By non-local interactions LES: Langevin

20. Performances LES: Langevin Spectrum Intermittency Comparaison DNS: 384*384*384 et LES: 21*21*21

21. Performances (2) LES: Langevin Q vs R s probability

22. THE MODEL IN SHEARED GEOMETRY Basic equations Equation for mean profile RDT equations for fluctuations with stochastic forcing

23. ANALYTICAL PREDICTIONS Mean flow dominatesFluctuations dominates Low Re

24. TORQUE IN TAYLOR-COUETTE No adjustable parameter Dubrulle and Hersant, 2002