Perturbation response and pinch-off of vortex rings and dipoles

Abstract

The nonlinear perturbation response of two families of vortices, the Norbury family of axisymmetric vortex rings and the Pierrehumbert family of two-dimensional vortex pairs, is considered. Members of both families are subjected to prolate shape perturbations similar to those previously introduced to Hill's spherical vortex, and their response is computed using contour dynamics algorithms. The response of the entire Norbury family to this class of perturbations is considered, in order to bridge the gap between past observations of the behaviour of thin-cored members of the family and that of Hill's spherical vortex. The behaviour of the Norbury family is contrasted with the response of the analogous two-dimensional family of Pierrehumbert vortex pairs. It is found that the Norbury family exhibits a change in perturbation response as members of the family with progressively thicker cores are considered. Thin-cored vortices are found to undergo quasi-periodic deformations of the core shape, but detrain no circulation into their wake. In contrast, thicker-cored Norbury vortices are found to detrain excess rotational fluid into a trailing vortex tail. This behaviour is found to be in agreement with previous results for Hill's spherical vortex, as well as with observations of pinch-off of experimentally generated vortex rings at long formation times. In contrast, the detrainment of circulation that is characteristic of pinch-off is not observed for Pierrehumbert vortex pairs of any core size. These observations are in agreement with recent studies that contrast the formation of vortices in two and three dimensions. We hypothesize that transitions in vortex formation, such as those occurring between wake shedding modes and in vortex pinch-off more generally, might be understood and possibly predicted based on the observed perturbation responses of forming vortex rings or dipoles.

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