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Structure and stability of hollow vortex equilibria

Published online by Cambridge University Press:  01 December 2011

Stefan G. Llewellyn Smith  and
Darren G. Crowdy
Stefan G. Llewellyn Smith
Affiliation:
Institut de Mécanique des Fluides de Toulouse, UMR CNRS/INPT/UPS 5502, Allée Camille Soula, 31400 Toulouse, France Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
Darren G. Crowdy*
Affiliation:
Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, UK
*
†Email address for correspondence: sgls@ucsd.edu
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Abstract

This paper considers the structure and linear stability of two-dimensional hollow vortex equilibria. Equilibrium solutions for a single hollow vortex in linear and nonlinear straining flows are derived in analytical form using free streamline theory. The linear stability properties of this solution class are then determined numerically and a new type of resonance-induced displacement instability is identified. It is found to be a consequence of the fact that one of the shape distortion modes of a circular hollow vortex has the same frequency as one of the modes corresponding to displacement of the vortex centroid. The instability is observed in the case of an isolated hollow vortex situated in straining flow of order three. We also revisit the hollow vortex row solution due to Baker, Saffman & Sheffield (J. Fluid Mech., vol. 74, 1976, p. 1469), and since it is currently lacking in the literature, we present a full linear stability analysis of this solution using Floquet analysis.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 691 , 25 January 2012 , pp. 178 - 200
Copyright
Copyright © Cambridge University Press 2011

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