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.

Interaction of a vortex ring with a single bubble: bubble and vorticity dynamics

“…Apart from these single core instabilities, the influence of the neighbouring oppositely signed core can also be important, as it overlays a strain field on the main vortex, as discussed for example by Leweke & Williamson (1998) in the case of the 'cooperative' instability of vortex pairs. The stability analysis of O'Farrell & Dabiri (2012) further suggests that thicker core vortex rings are inherently more prone to instability than thinner core rings, as also observed in our study. In summary, given all the features of the present modified vortex core at bubble capture, it is difficult to pinpoint the instability mechanism responsible its fragmentation.…”

“…In this section, we shall present some results for the case of a thicker vortex ring, with = 0.75, interacting with a bubble. We shall see that thicker rings, which are inherently more unstable (O'Farrell & Dabiri 2012), show even more dramatic effects during interactions with a bubble, leading to clear fragmentation of the vortex core.…”

“…The vortex rings formed can have thin or thick cores, this being indicated by a non-dimensional core radius ( = a/R, where a = equivalent core radius and R = vortex ring radius) (Norbury 1973). Laminar rings typically become unstable due to the azimuthal short-wave instability of the vortex core resulting in the formation of Kelvin waves (Widnall, Bliss & Tsai 1974), with thicker core rings being more prone to instability, as shown by O'Farrell & Dabiri (2012).…”

“…These results clearly show that the effects of the interaction of these thicker rings with a bubble are more dramatic than for the thin core rings. This may be attributed to the inherently more unstable nature of the thicker core rings, as discussed by O'Farrell & Dabiri (2012).…”

Transient force augmentation due to counter-rotating vortex ring pairs

“…The circulation of NeVRs created in this way can be controlled by setting the flow parameters (Weathers et al 2010;Liu et al 2012), but the overall configuration is fixed by the experimental set-up. In contrast to this, numerical simulations can use artificial set-ups to gain deeper insight into vortex dynamics (Pradeep & Hussain 2004;Bergdorf, Koumoutsakos & Leonard 2007;O'Farrell & Dabiri 2012).…”