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Curvature instability of a curved Batchelor vortex

Published online by Cambridge University Press:  06 February 2017

Francisco J. Blanco-Rodríguez  and
Stéphane Le Dizès Open the ORCID record for Stéphane Le Dizès [Opens in a new window]
Francisco J. Blanco-Rodríguez
Affiliation:
Aix Marseille Univ., CNRS, Centrale Marseille, IRPHE, 13384 Marseille, France
Stéphane Le Dizès*
Affiliation:
Aix Marseille Univ., CNRS, Centrale Marseille, IRPHE, 13384 Marseille, France
*
†Email address for correspondence: ledizes@irphe.univ-mrs.fr
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Abstract

In this paper, we analyse the curvature instability of a curved Batchelor vortex. We consider this short-wavelength instability when the radius of curvature of the vortex centreline is large compared with the vortex core size. In this limit, the curvature instability can be interpreted as a resonant phenomenon. It results from the resonant coupling of two Kelvin modes of the underlying Batchelor vortex with the dipolar correction induced by curvature. The condition of resonance of the two modes is analysed in detail as a function of the axial jet strength of the Batchelor vortex. In contrast to the Rankine vortex, only a few configurations involving $m=0$ and $m=1$ modes are found to become the most unstable. The growth rate of the resonant configurations is systematically computed and used to determine the characteristics of the most unstable mode as a function of the curvature ratio, the Reynolds number and the axial flow parameter. The competition of the curvature instability with another short-wavelength instability, which was considered in a companion paper (Blanco-Rodríguez & Le Dizès, J. Fluid Mech., vol. 804, 2016, pp. 224–247), is analysed for a vortex ring. A numerical error found in this paper, which affects the relative strength of the elliptic instability, is also corrected. We show that the curvature instability becomes the dominant instability in large rings as soon as axial flow is present (vortex ring with swirl).

JFM classification

Vortex Flows: Vortex Flows Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 814 , 10 March 2017 , pp. 397 - 415
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© 2017 Cambridge University Press 

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