Comptes Rendus
no. 4
Patterns and dynamics: homage to Pierre Coullet / Formes et dynamique : hommage à Pierre Coullet
A case of strong nonlinearity: Intermittency in highly turbulent flows
Yves Pomeau1; Martine Le Berre2 ; Thierry Lehner3
1 Ladhyx, École polytechnique, 91128 Palaiseau, France
2 Ismo, Université Paris-Sud, 91405 Orsay cedex, France
3 Luth, Observatoire de Paris-Meudon, 92195 Meudon, France
Comptes Rendus. Mécanique, Patterns and dynamics: homage to Pierre Coullet / Formes et dynamique: hommage à Pierre Coullet, Volume 347 (2019) no. 4, pp. 342-356.

It has long been suspected that flows of incompressible fluids at large or infinite Reynolds number (namely at small or zero viscosity) may present finite time singularities. We review briefly the theoretical situation on this point. We discuss the effect of a small viscosity on the self-similar solution to the Euler equations for inviscid fluids. Then we show that single-point records of velocity fluctuations in the Modane wind tunnel display correlations between large velocities and large accelerations in full agreement with scaling laws derived from Leray's equations (1934) for self-similar singular solutions to the fluid equations. Conversely, those experimental velocity–acceleration correlations are contradictory to the Kolmogorov scaling laws.

On pense depuis longtemps que les écoulements fluides incompressibles à grand, sinon infini, nombre de Reynolds présentent des singularités localisées en temps et en espace. Nous étudions l'effet d'une petite viscosité sur les solutions auto-semblables des équations des fluides. Nous montrons ensuite que des enregistrements de fluctuations de vitesse dans la soufflerie de Modane présentent des corrélations entre grandes vitesses et grandes accélérations, en accord complet avec les lois d'échelle déduites des solutions auto-similaires des équations trouvées par Leray en 1934. En revanche, ces corrélations sont en contradiction avec les lois d'échelle déduites de la théorie de Kolmogorov.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2019.03.002
Keywords: Navier–Stokes, Euler, Turbulence, Singularities, Leray, Intermittency
Mots-clés : Navier–Stokes, Euler, Turbulence, Singularités, Leray, Intermittence

Yves Pomeau 1; Martine Le Berre 2; Thierry Lehner 3

1 Ladhyx, École polytechnique, 91128 Palaiseau, France
2 Ismo, Université Paris-Sud, 91405 Orsay cedex, France
3 Luth, Observatoire de Paris-Meudon, 92195 Meudon, France 
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Yves Pomeau; Martine Le Berre; Thierry Lehner. A case of strong nonlinearity: Intermittency in highly turbulent flows. Comptes Rendus. Mécanique, Patterns and dynamics: homage to Pierre Coullet / Formes et dynamique: hommage à Pierre Coullet, Volume 347 (2019) no. 4, pp. 342-356. doi : 10.1016/j.crme.2019.03.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2019.03.002/

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