The Sadovskii vortex in strain

Abstract

The point vortex is the simplest model of a two-dimensional vortex with non-zero circulation. The limitations introduced by its lack of core structure have been remedied by using desingularizations such as vortex patches and vortex sheets. We investigate steady states of the Sadovskii vortex in strain, a canonical model for a vortex in a general flow. The Sadovskii vortex is a uniform patch of vorticity surrounded by a vortex sheet. We recover previously known limiting cases of the vortex patch and hollow vortex, and examine the bifurcations away from these families. The result is a solution manifold spanned by two parameters. The addition of the vortex sheet to the vortex patch solutions immediately leads to splits in the solution manifold at certain bifurcation points. The more circular elliptical family remains attached to the family with a single pinch-off, and this family extends all the way to the simpler solution branch for the pure vortex sheet solutions. More elongated families below this one also split at bifurcation points, but these families do not exist in the vortex sheet limit.

The corotating hollow vortex pair: steady merger and break-up via a topological singularity

“…The solutions here sit within a broader class of vortex structures of ‘Sadovskii type’ (Sadovskii 1971; Saffman 1992); these are vortex patches with vortex sheets on their boundaries (rather than just vortex jumps). A recent study of such structures in ambient flow fields has been carried out by Freilich & Llewellyn Smith (2017) where a parameter is introduced that governs the amount of the total vorticity of the structure that is held in the patch compared to the boundary sheet. In the solutions considered here this parameter has been set implicitly by insisting that the fluid inside the hollow vortices is in pure solid body rotation which slaves the choice of this patch vorticity to the angular velocity of the combined structure.…”

“…In a natural extension of the analytical solution for a steady hollow vortex in a linear strain found by Llewellyn Smith & Crowdy (2012), Zannetti, Ferlauto & Llewellyn Smith (2016) recently calculated equilibrium hollow vortices embedded in a shear flow (analytical solutions do not appear to be available in this case). Other recent related work on steady vortex structures involves vortices of Sadovskii type comprising a vortex patch with a vortex sheet on its boundary rather than the usual vortex jump (Freilich & Llewellyn Smith 2017).…”

‘H-states’: exact solutions for a rotating hollow vortex

“…In the corotating frame the H-states have a combination of uniform vorticity and concentrated boundary vorticity. They share these features with Sadovskii vortices (Saffman 1992) which have also received renewed attention (Freilich & Llewellyn Smith 2017). Zannetti et al.…”

Motion of a Buoyant Vortex Patch

“…The resulting models, involving vortex patches bounded by vortex sheets, have been referred to in the literature as Prandtl-Batchelor flows or sheet patches [30]; we use the term bouyant vortex patch (BVP) in this paper. Sheet patch studies have mostly involved computation of equilibria in a variety of background flows ( [12], [17], [38]). For example, in the important paper of Kao & Caflisch [17] invariant translating vortex patch shapes as a function of density ratio and circulation were computed.…”