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The nonlinear instability of Hill's vortex

Published online by Cambridge University Press:  21 April 2006

C. Pozrikidis
C. Pozrikidis
Affiliation:
Department of Chemical Engineering, University of Illinois, 1209 W. California St., Box C-3, Urbana, IL 61801, USA Present address: Research Laboratories, Eastman Kodak Company, Rochester, NY 14650, USA.
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Abstract

The nonlinear instability of Hill's spherical vortex, subject to axisymmetric perturbations is considered. The problem is formulated as a nonlinear integrodifferential equation for the motion of the vortex boundary. This equation is solved employing a numerical procedure which involves a piecewise representation of the vortex contour with discrete elements. This formulation offers an efficient method for studying a variety of vortex flows in axisymmetric geometry.

Our results indicate that if Hill's vortex becomes a prolate spheroid, a certain amount of rotational fluid is detrained from the rear stagnation point of the vortex, leaving behind a reduced vortex of approximately spherical shape. The amount of detrained fluid is a function of the initial deformation. If the vortex becomes an oblate spheroid, irrotational fluid is entrained into the vortex from the rear stagnation point, reaches the front vortex boundary, and circulates along the vortex boundary in a spiral pattern. In this fashion, the vortex reduces to a nearly steady vortex ring whose asymptotic structure is a function of the initial deformation. The structure of the asymptotic rings arising from oblate vortices is similar to that of steady rings described by Norbury (1973). The vortex speed is shown to tend to a constant value for prolate perturbations, and to fluctuate around a mean value for oblate perturbations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 168 , July 1986 , pp. 337 - 367
Copyright
© 1986 Cambridge University Press

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