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

Degrees of freedom: N L

3

Re9

4

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 resultsStatistical 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

• 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

2D vs axisymmetric (1)2D axisymmetric

Vorticity conservation Angular momentum conservation

No vortex stretching Vortex stretching

2D experiment

Coherentstructures

Bracco et al. Torino

2D versus axisymmetric (2)

Von KarmanTaylor-Couette

610Re Re 105

Presentation of Laboratory experiments

2D turbulence in a Ferro Magnetic fluid

Re 103

Jullien et al., LPS, ENS Paris

Daviaud et al. GIT, Saclay, France

2D versus axisymmetric (3)Basic equations

Vertical vorticity:2D:

Azimuthal vorticity:AXI:

azimuthal vorticity:

angular momentum:

poloidal velocity:

Variables of interest:

2D versus axisymmetric (4)

Inviscid stationary states

Inviscid Conservation laws

(Casimirs)

F and G are arbitraryfunctions in infinite number

infinite number of steady states

Casimirs (F)

Generalized helicity(G)

Statistical description (1)

• Mixing occurs at smaller and smaller scalesMore and more degrees of freedom

• Meta-equilibrium at a coarse-grained scaleUse of coarse-grained fields

• Coarse-graining affects some constraintsCasimirs are fragile invariant

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)Maximisation of S under conservation constraints

Equilibrium state

Equation for mostprobable fields

The Gibbs State

Steady solutions of Euler equation

Steady States (1)

What happens when the flow is mechanically stirred and viscous?

T1 T2

Two thermostats T1>T2

F

Working hypothesis (Leprovost et al. 05’):

NS:

Steady States (2)

Steady states of turbulent axisymmetric flow

F and G are arbitraryfunctions in infinite number

infinite number of steady states

- How are F and G selected?

- Role of dissipation and forcing in this selection?

Steady States (3)

Von Kármán Flow - LDV measurement

Data Processing (1)

Data Processing (2)

Time-averaged

fmpv

Test: Beltrami Flow with 60% noise

A steady solution of Euler equation:

Data Processing (3)

• F is fitted from the windowed plot• F is used to fit G

Whole flow 50% of the flow

Distance to center

<0.7

>0.85

intermediate

Flow Bulk

Comparison to numerical study

Simulation: Piotr Boronski (Limsi, Orsay, France)

Re=3000“inertial” stirring

Re=5000viscous stirring

Dependence on viscosity (1)

(+)(-)

F function:

Legend

Dependence on viscosity (2)

(+) (-)

G function:

Legend

(+)

92.5mm

Re = 190 000Re = 250 000Re = 500 000

50mm

Dependence on forcing

Conclusions

Perspectives