An optimality condition on the minimum energy threshold in subcritical instabilities

Carlo Cossu

doi:10.1016/j.crme.2005.02.002

An optimality condition on the minimum energy threshold in subcritical instabilities

Presented by: Patrick Huerre

Carlo Cossu ¹

¹ Laboratoire d'hydrodynamique (LadHyX), École polytechnique – CNRS, 91128 Palaiseau cedex, France

Comptes Rendus. Mécanique, Volume 333 (2005) no. 4, pp. 331-336.

Abstract
Résumé

For flows subject to subcritical instabilities the stability of the basic flow can be guaranteed only for perturbations of energy lower than a critical threshold δ. The computation of this threshold for the Navier–Stokes equations is still out of reach. More surprisingly, this computation has not been attempted for low dimensional models of subcritical transition. In this Note guidelines are provided for the computation of the minimum energy threshold δ and of the corresponding nonlinear optimal perturbations. In particular it is demonstrated that nonlinear optimal perturbations are constrained by the requirement that they must satisfy a local minimum condition. These results are applied to the analysis of four-dimensional models proposed in F. Waleffe, Phys. Fluids 7 (1995) and Phys. Fluids 9 (1997).

Dans le cas d'instabilité sous-critique, la stabilité de l'écoulement de base peut être garantie seulement pour des perturbations d'énergie inférieure au seuil critique δ. Le calcul direct de ce seuil est inaccessible pour les équations de Navier–Stokes. Plus surprenant est le fait que ce calcul n'a pas été tenté pour des modèles de basse dimension de transition sous-critique. Dans cette Note des indications générales sont fournies pour le calcul de δ et des perturbations non linéaires optimales associées. Notamment, nous démontrons que les perturbations non linéaires optimales doivent satisfaire une condition de minimum local. Ces résultats sont appliqués à l'analyse de systèmes à quatre dimensions proposés in F. Waleffe, Phys. Fluids 7 (1995) et Phys. Fluids 9 (1997).

Received: 2004-12-01
Accepted: 2005-01-18
Published online: 2005-02-26

DOI: 10.1016/j.crme.2005.02.002

Keywords: Fluid mechanics, Instability, Dynamical systems, Optimal perturbations
Mots-clés : Mécanique des fluides, Instabilité, Systèmes dynamiques, Perturbations optimales

Author's affiliations:

Carlo Cossu ¹

¹ Laboratoire d'hydrodynamique (LadHyX), École polytechnique – CNRS, 91128 Palaiseau cedex, France

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     title = {An optimality condition on the minimum energy threshold in subcritical instabilities},
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     pages = {331--336},
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Carlo Cossu. An optimality condition on the minimum energy threshold in subcritical instabilities. Comptes Rendus. Mécanique, Volume 333 (2005) no. 4, pp. 331-336. doi : 10.1016/j.crme.2005.02.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2005.02.002/

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