Comptes Rendus
no. 3-4
Turbulent cascade of circulations
[Cascade turbulente des circulations]
Gregory L. Eyink1, 2
1 CNLS, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2 Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, MD 21210, USA
Comptes Rendus. Physique, Volume 7 (2006) no. 3-4, pp. 449-455.

La circulation autour d'une boucle fermée est un invariant de Lagrange pour des solutions classiques des équations d'Euler incompressibles dans un espace à dimension quelconque. Cependant, les solutions singulières qui s'appliquent à des écoulements turbulents ne conservent pas forcément les constantes du mouvement classiques. Dans cette contribution, nous généralisons le théorème de Kelvin sur la conservation des circulations aux solutions singulières d'Euler et donnons les conditions nécessaires pour la dissipation anormale des circulations. Nous discutons le rôle important du théorème de Kelvin dans la dynamique turbulente qui déforme les tourbillons, et proposons une version du théorème qui pourrait s'appliquer aux solutions singulières.

The circulation around any closed loop is a Lagrangian invariant for classical, smooth solutions of the incompressible Euler equations in any number of space dimensions. However, singular solutions relevant to turbulent flows need not preserve the classical integrals of motion. Here we generalize the Kelvin theorem on conservation of circulations to distributional solutions of Euler and give necessary conditions for the anomalous dissipation of circulations. We discuss the important role of Kelvin's theorem in turbulent vortex-stretching dynamics and conjecture a version of the theorem which may apply to suitable singular solutions.

Publié le :
DOI : 10.1016/j.crhy.2006.01.008
Keywords: Turbulence, Circulation, Euler equations, Kelvin theorem
Mot clés : Turbulence, Circulation, Équations d'Euler, Théorème de Kelvin
Gregory L. Eyink 1, 2

1 CNLS, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2 Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, MD 21210, USA 
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Gregory L. Eyink. Turbulent cascade of circulations. Comptes Rendus. Physique, Volume 7 (2006) no. 3-4, pp. 449-455. doi : 10.1016/j.crhy.2006.01.008. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2006.01.008/

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