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Turbulence: Does Energy Cascade Exist?

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Abstract

To answer the question whether a cascade of energy exists or not in turbulence, we propose a set of correlation functions able to test if there is an irreversible transfert of energy, step by step, from large to small structures. These tests are applied to real Eulerian data of a turbulent velocity flow, taken in the wind grid tunnel of Modane, and also to a prototype model equation for wave turbulence. First we demonstrate the irreversible character of the flow by using multi-time correlation function at a given point of space. Moreover the unexpected behavior of the test function leads us to connect irreversibility and finite time singularities (intermittency). Secondly we show that turbulent cascade exists, and is a dynamical process, by using a test function depending on time and frequency. The cascade shows up only in the inertial domain where the kinetic energy is transferred more rapidly (on average) from the wavenumber \(k_{1}\) to \(k_{2}\) than from \(k_{1}\) to \(k'_{2}\) larger than \(k_{2}\).

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Notes

  1. See also the note “Hydrodynamical turbulence as a problem in non-equilibrium statistical mechanics” published on the web (www.ihes.fr/ruelle/PUBLICATIONS/turbulenceX)

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Acknowledgments

We thank Dr. Yves Gagne for having provided us the Eulerian velocity data of Modane’s experiment, which were taken by him et his collaborators, and we thank ONERA for facility.

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Authors and Affiliations

  1. Institut Jean Le Rond d’Alembert, Sorbonne Universités, CNRS & UPMC Univ Paris 06, UMR 7190, 75005, Paris, France

    Christophe Josserand

  2. Institut des Sciences Moléculaires d’Orsay ISMO-CNRS, Université Paris-Sud Bat, 210 91405, Orsay Cedex, France

    Martine Le Berre

  3. Laboratoire LUTH, UMR CNRS n°8102, Observatoire de Paris-Meudon, 5 Place Jules Janssen, 92190, Meudon cedex, France

    Thierry Lehner

  4. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Yves Pomeau

Authors
  1. Christophe Josserand
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  2. Martine Le Berre
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  3. Thierry Lehner
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  4. Yves Pomeau
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Correspondence to Christophe Josserand.

Additional information

Personal note by C. J. : It is a great honor for my colleagues and myself to contribute to this Special Memorial Issue Dedicated to Leo Kadanoff. The work presented here trace back in fact to my stay in Chicago as a post-doc with Leo, where I was investigating how irreversible dynamics could emerge from reversible systems. I remember emotionally my discussions with Leo during my postdoc where his advices were always of great help, often concluding with his characteristic voice by “I would (not) encourage you to go in that direction”! My two years in Chicago have widely influenced my scientific activity thanks to Leo’s personal advices and to the outstanding atmosphere there whose Leo was at the heart.

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Josserand, C., Le Berre, M., Lehner, T. et al. Turbulence: Does Energy Cascade Exist?. J Stat Phys 167, 596–625 (2017). https://doi.org/10.1007/s10955-016-1642-5

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  • DOI: https://doi.org/10.1007/s10955-016-1642-5

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