Dissipation by flows Festival de thorie, Aix-en-Provence, 07.2011 - - PowerPoint PPT Presentation

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Dissipation by flows Festival de thorie, Aix-en-Provence, 07.2011 - - PowerPoint PPT Presentation

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Dissipation by flows Festival de thorie, Aix-en-Provence, 07.2011 Romain Nguyen van yen 1,3 , Marie Farge 1 , Kai Schneider 2 1 LMD-CNRS, ENS Paris, France 2 M2P2-CNRS et CMI, Universit dAix-Marseille, France 3 FB Mathematik &

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Dissipation by flows

Festival de théorie, Aix-en-Provence, 07.2011

Romain Nguyen van yen1,3, Marie Farge1, Kai Schneider2

1 LMD-CNRS, ENS Paris, France

2 M2P2-CNRS et CMI, Université d’Aix-Marseille, France

3 FB Mathematik & Informatik, FU Berlin, Allemagne

Supported by the Humboldt Foundation, Euratom-CEA and the French Federation for Fusion Studies, the PEPS program of CNRS and IDRIS-CNRS

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Outline

  • 1. Introduction: dissipative singularities,

macroscopic randomization, and wavelets.

  • 2. 2D periodic Navier-Stokes turbulence

– Methodology – Molecular dissipation – Wavelet-based macroscopic dissipation

  • 3. 2D wall-bounded Navier-Stokes (1 slide)
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black slide

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Graphical examples

Mast tokamak (CCFE, UK) Earth (Apollo 17) Ultraviolet sun (TRACE, NASA) Biker in a wind tunnel

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A first attempt: molecular dissipation

  • Flows = collective motions of many particles described

macroscopically.

  • Prediction usually impossible from principles of Lagrangian

mechanics.

  • But statistical assumptions are possible.
  • The main statistical assumption is the closeness to a certain

statistical equilibrium (local thermodynamic equilibrium, molecular chaos…),

  • Global relaxation can usually be proved (growth of entropy),
  • This phenomenon is called molecular dissipation.
  • R. Balian, From microphyics to macrophysics, Springer (2006)
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Not the end of the story

  • For example in incompressible Navier-Stokes this leads to

the equation:

  • In practice relaxation often occurs on time-scales that are

independent on microscopic coupling coefficients. This coefficient may be very small !

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Ways out of this paradox

  • 1. Singular behavior counteracts the smallness of the coupling

coefficient and opens access to microscopic dissipation

  • 2. Microscopic dissipation is not the relevant relaxation

mechanism anymore, another relaxation takes over (possibly associated to a macroscopic statistical equilibrium).

  • 3. A combination of the above two.

Example : Burgers equation flow entropy Example : T.D. Lee equilibria in Galerkin truncated systems. Landau damping (isentropic relaxation)

  • Fig. from Sreenisvasan (1984)

Molecular dissipation becomes Re- independent

energy dissipation rate measured experimentally in flows behind grids

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Dissipation as randomization

time trajectory of reduced model

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Dissipation as randomization

time trajectories of a more complete model

Dissipation can be seen as voluntary forgetfulness. The goal is to make predictions from incomplete knowledge.

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Flow dissipation seen in Fourier space

FINE SCALE COARSE SCALE

L.F. Richardson, Diffusion regarded as a compensation for smoothing (1930) R.H. Kraichnan, On Kolmogorov’s inertial range theories (1974)

position Critique of Kraichnan Nonlinear interaction at a given scale and Nonlinear transfer between scales

  • ccur on the same timescale !

In other words, there is no good reason to think that the flow is better equilibrated at fine scales than at coarse scales.

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Flow dissipation seen in wavelet space

FINE SCALE COARSE SCALE COHERENT INCOHERENT

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Orthogonal wavelet bases

scaling function wavelet energy spectra

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Orthogonal wavelet bases

dilated / translated wavelets corresponding energy spectra

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Orthogonal wavelet bases

2d scaling function and wavelets

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Wavelets and spatial localization

Two-step function Brownian motion

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Wavelet nonlinear thresholding

  • Orthogonal wavelet decomposition:
  • Idea: split wavelet coefficients between two sets, “large

coefficients” and “small coefficients”: Where large and small are defined with respect to a certain threshold:

  • r

, where

  • D. Donoho & I. Johnstone (1992), M. Farge, N. Kevlahan, K. Schneider (1999)
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Navier-Stokes initial-boundary value problem

Equation (no body forces  decaying flow) Boundary conditions : periodic Initial conditions In 2D this is a well-posed problem

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Molecular dissipation in 2D Navier-Stokes

energy enstrophy vorticity

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Numerical study

initial condition Re ≈ 17 000 Re ≈ 66 000 Re ≈ 266 000 Re ≈ 1062 000 Re ≈ 4 248 000 time evolution time evolution time evolution max min

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Numerical study

max min

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Energy dissipation at vanishing viscosity?

increasing Re

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Enstrophy dissipation at vanishing viscosity?

increasing Re

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Enstrophy dissipation at vanishing viscosity?

Tran & Dritschel, JFM 559 (2006): “There is no Re-independent measure of dissipation”

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Scale-wise statistics

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Wavelet denoising

= +

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Scale-wise coherent vorticity extraction

Total vorticity Coherent vorticity Incoherent vorticity

= +

PDF of wavelet coefficients at scale j=8

The threshold is defined at each scale by: standard deviation constant parameter

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Statistical properties of the split

  • The global PDF of the incoherent part is close to a Gaussian
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Statistical properties of the split

  • 3
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Dissipation of coherent enstrophy

MOLECULAR DISSIPATION DISSIPATION OF COHERENT ENSTROPHY INCOHERENT ENSTROPHY

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Inter-scale and intra-scale transfers

Ex: j = 9 negative dissipation intra-scale transfer

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Retroaction of the dissipated flow

  • Model the incoherent

wavelet coefficients by random variables,

  • Maximum entropy

distribution, with constraints: (1) (2) (3) It doesn’t fit, but we proceed anyway.

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Retroaction of the dissipated flow

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Summary

  • We have introduced a “dissipation mechanism” for 2D

turbulence based on a split of the flow between explicit and dissipated components.

  • The associated enstrophy dissipation rate does not vanish at

vanishing viscosity.

  • The dissipation rate can be directly related to the nonlinear

transfers and studied quantitatively scale-wise.

  • Negative dissipation is allowed.
  • We have shown that the retroaction of the dissipated flow

has a diffusive effect on the explicit flow at short times.

R.H. Kraichnan & S. Chen Is there a statistical mechanics of turbulence? (1989) R.H. Kraichnan, Reduced descriptions of hydrodynamic turbulence (1988) RNVY, M. Farge, K. Schneider, doi: 10.1016/j.physd.2011.05.022

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A result on dissipative singularities

  • There are no dissipative singularities in 2D Navier-Stokes in

the absence of walls and for bounded vorticity fields.

  • However this happens in the presence of boundaries.

RNVY, M. Farge, K. Schneider, PRL 106, 184502

time

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Acknowledgements

  • Thanks to Claude Bardos, Dmitry Kolomenskiy, Xavier

Garbet, Greg Hammett, any many others.

  • The Kicksey-Winsey code can be downloaded at :

http://justpmf.com/romain

  • Papers are available on :

http://wavelets.ens.fr/ Thank you!

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Le temps efface tout comme effacent les vagues Les travaux des enfants sur le sable aplani Nous oublierons ces mots si précis et si vagues Derrière qui chacun nous sentions l'infini. Marcel Proust