Published online by Cambridge University Press: 21 April 2006
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.
To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.
To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.