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Statistical approach of Turbulence

Statistical approach of Turbulence. R. Monchaux N. Leprovost, F. Ravelet, P-H. Chavanis*, B. Dubrulle, F. Daviaud and A. Chiffaudel. GIT-SPEC, Gif sur Yvette France *Laboratoire de Physique Théorique, Toulouse France. Out-of-equilibrium systems vs. Classical equilibrium systems.

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Statistical approach of Turbulence

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  1. Statistical approach of Turbulence R. Monchaux N. Leprovost, F. Ravelet, P-H. Chavanis*, B. Dubrulle, F. Daviaud and A. Chiffaudel GIT-SPEC, Gif sur Yvette France *Laboratoire de Physique Théorique, Toulouse France

  2. Out-of-equilibrium systems vs. Classical equilibrium systems Degrees of freedom:

  3. Statistical approach of turbulence: Steady states, equation of state, distributions • 2D: Robert and Sommeria 91’, Chavanis 03’ • Quasi-2D: shallow water, β-plane Bouchet 02’’ • 3D: still unanswered question (vortex stretching) Axisymmetric flows: intermediate situation • 2D and vortex stretching • Theoretical developments by Leprovost, Dubrulle and Chavanis 05’

  4. 2D and quasi-2D results • Statistical equilibrium state of 2D Euler equation (Chavanis): • Classification of isolated vortices: monopoles and dipoles • Stability diagram of these structures: dependence on a single control parameter • Quasi 2D statistical mechanics (Bouchet): • Intense jets • Great Red Spot

  5. Approach Principle • Basic equation: Euler equation • Forcing is neglected • Viscosity is neglected • Variable of interest: Probability to observe the conserved quantity at • Maximization of a mixing entropy at conserved quantities constraints

  6. 2D vs axisymmetric (1) axisymmetric 2D No vortex stretching Vortex stretching Angular momentum conservation Vorticity conservation 2D experiment Coherent structures Bracco et al. Torino

  7. 2D turbulence in a Ferro Magnetic fluid 2D versus axisymmetric (2) Taylor-Couette Von Karman Jullien et al., LPS, ENS Paris Daviaud et al. GIT, Saclay, France Presentation of Laboratory experiments

  8. Vertical vorticity: Azimuthal vorticity: angular momentum: poloidal velocity: azimuthal vorticity: 2D versus axisymmetric (3) Basic equations 2D: AXI: Variables of interest:

  9. (Casimirs) F and G are arbitrary functions in infinite number infinite number of steady states 2D versus axisymmetric (4) Inviscid Conservation laws Casimirs (F) Generalized helicity (G) Inviscid stationary states

  10. Statistical description (1) • Mixing occurs at smaller and smaller scales More and more degrees of freedom • Meta-equilibrium at a coarse-grained scale Use of coarse-grained fields • Coarse-graining affects some constraints Casimirs are fragile invariant

  11. Statistical description (2) Probability distribution to observe at point r Mixing Entropy: Coarse-grained A. M. Coarse-grained constraints: Robust constraints Fragile constraints

  12. Statistical description (3bis) Maximisation of S under conservation constraints The Gibbs State Equilibrium state Equation for most probable fields Steady solutions of Euler equation

  13. F T1 T2 Two thermostats T1>T2 Steady States (1) • What happens when the flow is mechanically stirred and viscous?

  14. Steady States (2) NS: Working hypothesis(Leprovost et al. 05’):

  15. F and G are arbitrary functions in infinite number infinite number of steady states Steady States (3) Steady states of turbulent axisymmetric flow - How are F and G selected? - Role of dissipation and forcing in this selection?

  16. Von Kármán Flow - LDV measurement

  17. Data Processing (1)

  18. Data Processing (2) fmpv Time-averaged

  19. A steady solution of Euler equation: Test: Beltrami Flow with 60% noise

  20. >0.85 intermediate <0.7 Flow Bulk Data Processing (3) Whole flow 50% of the flow Distance to center • F is fitted from the windowed plot • F is used to fit G

  21. Comparison to numerical study Re=5000 viscous stirring Re=3000 “inertial” stirring Simulation: Piotr Boronski (Limsi, Orsay, France)

  22. Dependence on viscosity (1) Legend (+) (-) F function:

  23. Dependence on viscosity (2) G function: Legend (+) (-)

  24. 92.5mm (+) 50mm Dependence on forcing Re = 190 000 Re = 250 000 Re = 500 000

  25. Conclusions

  26. Perspectives

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