Triadic instability of a non-resonant precessing fluid cylinder

Romain Lagrange; Patrice Meunier; Christophe Eloy

doi:10.1016/j.crme.2015.12.002

Triadic instability of a non-resonant precessing fluid cylinder

Romain Lagrange ¹ ; Patrice Meunier ²; Christophe Eloy ²

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
² Aix-Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384 Marseille, France

Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 418-433.

Abstract
Résumé

Flows forced by a precessional motion can exhibit instabilities of crucial importance, whether they concern the fuel of a flying object or the liquid core of a telluric planet. So far, stability analyses of these flows have focused on the special case of a resonant forcing. Here, we address the instability of the flow inside a precessing cylinder in the general case. We first show that the base flow forced by the cylinder precession is a superposition of a vertical or horizontal shear flow and of an infinite sum of forced modes. We then perform a linear stability analysis of this base flow by considering its triadic resonance with two free Kelvin modes. Finally, we derive the amplitude equations of the free Kelvin modes and obtain an expression of the instability threshold and growth rate.

Les écoulements de précession peuvent subir des instabilités dont la compréhension est cruciale, que ce soit pour prédire le mouvement du carburant liquide d'un objet volant ou celui des noyaux liquides des planètes telluriques. Jusqu'à présent, les analyses de stabilité de ces écoulements se sont focalisées sur le cas particulier d'un forçage à une fréquence de résonance. Ici, nous étudions l'instabilité d'un fluide dans un cylindre en précession, pour une fréquence de forçage quelconque. Premièrement, nous montrons que l'écoulement de base d'un fluide dans un cylindre en précession est une superposition d'un cisaillement vertical ou horizontal et d'une somme de modes forcés. Ensuite, nous analysons la stabilité de l'écoulement de base en considérant sa résonance triadique avec deux modes de Kelvin libres. Finalement, nous dérivons les équations d'amplitude des modes de Kelvin libres et obtenons une expression du seuil d'instabilité et du taux de croissance.

Received: 2015-03-05
Accepted: 2015-12-02
Published online: 2016-02-11

DOI: 10.1016/j.crme.2015.12.002

Keywords: Precession, Kelvin modes, Triadic resonance, Stability
Mots-clés : Précession, Modes de Kelvin, Résonance triadique, Stabilité

Author's affiliations:

Romain Lagrange ¹; Patrice Meunier ²; Christophe Eloy ²

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
² Aix-Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384 Marseille, France

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     author = {Romain Lagrange and Patrice Meunier and Christophe Eloy},
     title = {Triadic instability of a non-resonant precessing fluid cylinder},
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     language = {en},
}

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Romain Lagrange; Patrice Meunier; Christophe Eloy. Triadic instability of a non-resonant precessing fluid cylinder. Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 418-433. doi : 10.1016/j.crme.2015.12.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.12.002/

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