statistical approach of turbulence r. monchaux n. leprovost, f. ravelet, p-h. chavanis*, b....
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![Page 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](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649d4c5503460f94a2a0e4/html5/thumbnails/1.jpg)
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
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Out-of-equilibrium systems vs. Classical equilibrium systems
Degrees of freedom: N L
3
Re9
4
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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’
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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
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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
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2D vs axisymmetric (1)2D axisymmetric
Vorticity conservation Angular momentum conservation
No vortex stretching Vortex stretching
2D experiment
Coherentstructures
Bracco et al. Torino
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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
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2D versus axisymmetric (3)Basic equations
Vertical vorticity:2D:
Azimuthal vorticity:AXI:
azimuthal vorticity:
angular momentum:
poloidal velocity:
Variables of interest:
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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)
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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
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Statistical description (2)Probability distribution to observe
at point r
Mixing Entropy:
Coarse-grained A. M.
Coarse-grained constraints:
Robust constraints
Fragile constraints
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Statistical description (3bis)Maximisation of S under conservation constraints
Equilibrium state
Equation for mostprobable fields
The Gibbs State
Steady solutions of Euler equation
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Steady States (1)
What happens when the flow is mechanically stirred and viscous?
T1 T2
Two thermostats T1>T2
F
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Working hypothesis (Leprovost et al. 05’):
NS:
Steady States (2)
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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)
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Von Kármán Flow - LDV measurement
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Data Processing (1)
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Data Processing (2)
Time-averaged
fmpv
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Test: Beltrami Flow with 60% noise
A steady solution of Euler equation:
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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
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Comparison to numerical study
Simulation: Piotr Boronski (Limsi, Orsay, France)
Re=3000“inertial” stirring
Re=5000viscous stirring
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Dependence on viscosity (1)
(+)(-)
F function:
Legend
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Dependence on viscosity (2)
(+) (-)
G function:
Legend
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(+)
92.5mm
Re = 190 000Re = 250 000Re = 500 000
50mm
Dependence on forcing
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Conclusions
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Perspectives