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Statistical approach of TurbulencePowerPoint Presentation

Statistical approach of Turbulence

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Statistical approach of Turbulence

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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:

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’

- 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

- 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)

axisymmetric

2D

No vortex stretching

Vortex stretching

Angular momentum conservation

Vorticity conservation

2D experiment

Coherent

structures

Bracco et al. Torino

2D turbulence in a

Ferro Magnetic fluid

Taylor-Couette

Von Karman

Jullien et al., LPS, ENS Paris

Daviaud et al. GIT, Saclay, France

Presentation of Laboratory experiments

Vertical vorticity:

Azimuthal vorticity:

angular momentum:

poloidal velocity:

azimuthal vorticity:

Basic equations

2D:

AXI:

Variables

of interest:

(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 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

Probability distribution to observe

at point r

Mixing Entropy:

Coarse-grained A. M.

Coarse-grained constraints:

Robust constraints

Fragile constraints

Maximisation of S under conservation constraints

The Gibbs State

Equilibrium state

Equation for most

probable fields

Steady solutions of Euler equation

F

T1

T2

Two thermostats T1>T2

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

Steady States (2)

NS:

Working hypothesis(Leprovost et al. 05’):

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?

Von Kármán Flow - LDV measurement

fmpv

Time-averaged

A steady solution of Euler equation:

>0.85

intermediate

<0.7

Flow Bulk

Whole flow

50% of the flow

Distance to center

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

Re=5000

viscous stirring

Re=3000

“inertial” stirring

Simulation: Piotr Boronski (Limsi, Orsay, France)

Dependence on viscosity (1)

Legend

(+)

(-)

Dependence on viscosity (2)

Legend

(+)

(-)

92.5mm

(+)

50mm

Re = 190 000

Re = 250 000

Re = 500 000