Turbulent cascade of circulations

Gregory L. Eyink

doi:10.1016/j.crhy.2006.01.008

Turbulent cascade of circulations
[Cascade turbulente des circulations]

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

Comptes Rendus. Physique, Volume 7 (2006) no. 3-4, pp. 449-455.

Résumé
Abstract

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 : 2006-04-01

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

Affiliations des auteurs :

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

@article{CRPHYS_2006__7_3-4_449_0,
     author = {Gregory L. Eyink},
     title = {Turbulent cascade of circulations},
     journal = {Comptes Rendus. Physique},
     pages = {449--455},
     publisher = {Elsevier},
     volume = {7},
     number = {3-4},
     year = {2006},
     doi = {10.1016/j.crhy.2006.01.008},
     language = {en},
}

TY  - JOUR
AU  - Gregory L. Eyink
TI  - Turbulent cascade of circulations
JO  - Comptes Rendus. Physique
PY  - 2006
SP  - 449
EP  - 455
VL  - 7
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crhy.2006.01.008
LA  - en
ID  - CRPHYS_2006__7_3-4_449_0
ER  -

%0 Journal Article
%A Gregory L. Eyink
%T Turbulent cascade of circulations
%J Comptes Rendus. Physique
%D 2006
%P 449-455
%V 7
%N 3-4
%I Elsevier
%R 10.1016/j.crhy.2006.01.008
%G en
%F CRPHYS_2006__7_3-4_449_0

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/

Bibliographie
Cité par

[1] H. Helmholtz Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelles Journal, Volume 55 (1858), pp. 25-55

[2] L. Kelvin On vortex motion, Trans. Roy. Soc. Edin., Volume 25 (1869), pp. 217-260

[3] L. Onsager Statistical hydrodynamics, Nuovo Cimento, Volume 6 (1949), pp. 279-287

[4] G. Eyink Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer, Physica D, Volume 78 (1994), pp. 222-240

[5] P. Constantin; W. E; E.S. Titi Onsager's conjecture on the energy conservation for solutions of the Euler's equations, Commun. Math. Phys., Volume 165 (1994), pp. 207-209

[6] J. Duchon; R. Robert Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations, Nonlinearity, Volume 13 (2000), pp. 249-255

[7] G.I. Taylor Production and dissipation of vorticity in a turbulent fluid, Proc. R. Soc. London Ser. A, Volume 164 (1938), pp. 15-23

[8] P.G. Saffman Vortex Dynamics, Cambridge Univ. Press, Cambridge, UK, 1992

[9] R. Salmon Hamiltonian fluid mechanics, Annu. Rev. Fluid Mech., Volume 20 (1988), pp. 225-256

[10] D.D. Holm; J.E. Marsden; T. Ratiu Euler–Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., Volume 349 (1998), pp. 4173-4177

[11] C. Foias; D.D. Holm; E.S. Titi The Navier–Stokes–alpha model of fluid turbulence, Physica D, Volume 152–153 (2001), pp. 505-519

[12] J. Leray Sur les movements d'un fluide visqueux remplaissant l'espace, Acta Math., Volume 63 (1934), pp. 193-248

[13] P. Constantin Near identity transformations for the Navier–Stokes equations, Handbook of Mathematical Fluid Dynamics, vol. II, North-Holland, Amsterdam, 2003 (117–141)

[14] R.J. DiPerna; P.L. Lions Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989), pp. 511-547

[15] H. Triebel Theory of Function Spaces, Birkhauser, Basel, 1983

[16] G.L. Eyink Dissipation in turbulent solutions of 2D Euler equations, Nonlinearity, Volume 14 (2000), pp. 787-802

[17] K.R. Sreenivasan; C. Meneveau The fractal facets of turbulence, J. Fluid Mech., Volume 173 (1986), pp. 357-386

[18] L.C. Young An inequality of the Hölder type connected with Stieltjes integration, Acta Math., Volume 67 (1936), pp. 251-282

[19] M. Zähle Integration with respect to fractal functions and stochastic calculus. I, Prob. Theory Rel. Fields, Volume 111 (1998), pp. 333-374

[20] D. Bernard; K. Gawȩdzki; A. Kupiainen Slow modes in passive advection, J. Stat. Phys., Volume 90 (1998), pp. 519-569

[21] L.F. Richardson Atmospheric diffusion shown on a distance-neighbor graph, Proc. R. Soc. London Ser. A, Volume 110 (1926), pp. 709-737

[22] Y. Le Jan; O. Raimond Integration of Brownian vector fields, Ann. Prob., Volume 30 (2002), pp. 826-873

[23] Y. Le Jan; O. Raimond Flows, coalescence, and noise, Ann. Prob., Volume 32 (2004), pp. 1247-1315

[24] M. Chaves; K. Gawȩdzki; P. Horvai; A. Kupiainen; M. Vergassola Lagrangian dispersion in Gaussian self-similar velocity ensembles, J. Stat. Phys., Volume 113 (2003), pp. 643-692

[25] Y. Brenier The least action principle and the related concept of generalized flows for incompressible inviscid fluids, J. Amer. Math. Soc., Volume 2 (1989), pp. 225-255

Cité par Sources :

Commentaires - Politique