Statistical approach of turbulence
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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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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


Statistical approach of turbulence

Out-of-equilibrium systems vs. Classical equilibrium systems

Degrees of freedom:


Statistical approach of turbulence

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’


2d and quasi 2d results

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


Approach principle

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


Statistical approach of turbulence

2D vs axisymmetric (1)

axisymmetric

2D

No vortex stretching

Vortex stretching

Angular momentum conservation

Vorticity conservation

2D experiment

Coherent

structures

Bracco et al. Torino


2d versus axisymmetric 2

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


2d versus axisymmetric 3

Vertical vorticity:

Azimuthal vorticity:

angular momentum:

poloidal velocity:

azimuthal vorticity:

2D versus axisymmetric (3)

Basic equations

2D:

AXI:

Variables

of interest:


Statistical approach of turbulence

(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


Statistical approach of turbulence

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


Statistical description 2

Statistical description (2)

Probability distribution to observe

at point r

Mixing Entropy:

Coarse-grained A. M.

Coarse-grained constraints:

Robust constraints

Fragile constraints


Statistical description 3bis

Statistical description (3bis)

Maximisation of S under conservation constraints

The Gibbs State

Equilibrium state

Equation for most

probable fields

Steady solutions of Euler equation


Steady states 1

F

T1

T2

Two thermostats T1>T2

Steady States (1)

  • What happens when the flow is mechanically stirred and viscous?


Statistical approach of turbulence

Steady States (2)

NS:

Working hypothesis(Leprovost et al. 05’):


Statistical approach of turbulence

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?


Statistical approach of turbulence

Von Kármán Flow - LDV measurement


Data processing 1

Data Processing (1)


Data processing 2

Data Processing (2)

fmpv

Time-averaged


Test beltrami flow with 60 noise

A steady solution of Euler equation:

Test: Beltrami Flow with 60% noise


Data processing 3

>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


Comparison to numerical study

Comparison to numerical study

Re=5000

viscous stirring

Re=3000

“inertial” stirring

Simulation: Piotr Boronski (Limsi, Orsay, France)


F function

Dependence on viscosity (1)

Legend

(+)

(-)

F function:


G function

Dependence on viscosity (2)

G function:

Legend

(+)

(-)


Dependence on forcing

92.5mm

(+)

50mm

Dependence on forcing

Re = 190 000

Re = 250 000

Re = 500 000


Conclusions

Conclusions


Perspectives

Perspectives


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