A new family of uniform vortices related to vortex configurations before merging

Abstract

Stimulated by experimental observations of vortex merging, we compute a new family of uniform-vorticity steady solutions of the Euler equations in two dimensions. In experiments with two co-rotating vortices, one finds that, prior to the convective merging phase, and the formation of vortex filaments, the initial pair diffuses into a single structure (with two vorticity peaks) in the form of a symmetric ‘dumb-bell’. In the present computations, our exploration of the existence of vortex solutions has been guided by the streamline patterns of the co-rotating reference frame, and by the simple concept that the vortex boundary must be one of these streamlines. By varying the parameters which define the vortex patches, we find a family of vorticity structures which pass from the limiting case of point vortices, through the case of two equal co-rotating uniform vortices (as previously computed by Saffman & Szeto 1980; Overman & Zabusky 1982; Dritschel 1985), to the regime where the vortices touch in the form of a dumb-bell. Further exploration of this family of solutions leads to an elliptic vortex, which joins precisely to the local transcritical bifurcation from elliptic vortices with $n\,{=}\,4$ perturbation symmetry that was found by Kamm (1987) and Saffman (1988). Finally, one reaches a limiting ‘cat's-eye’ vortex patch of two-fold symmetry ($m\,{=}\,2$), which constitutes an extension to the limiting shapes of $m$-fold symmetry ($m \,{>}\, 2$) found by Wu, Overman & Zabusky (1984).

Growing vortex patches

“…By procedures such as ͑i͒ the desingularization of point vortices to uniform vortex patches ͑or vice versa͒, ͑ii͒ the growing of new point vortices at corotation points of existing equilibria as done recently by Aref and Vainchtein, 6 ͑iii͒ the growing of new vortex patches at corotation points of existing equilibria as done here, and ͑iv͒ the smooth continuation of touching vortex patches to a merged equilibrium as done here and recently by Cerretelli and Williamson, 17 it appears that even basic equilibria with simple vorticity distributions can be continuously continued to more complicated ones with more elaborate vortical topology.…”

The merging of two co-rotating vortices: a numerical study

“…More recently, the same approach has been extended in an inviscid context to various nonuniform vorticity profiles [26,4] or else to uniform vorticity region which is not formed of two disconnected domains i.e. a dumb-bell shape patch [7]. This latter approach recovers the limiting V-states obtained in [27].…”

“…1(a) with a vortex ring surrounding the two vortices. It is also reminiscent of the model solution devised in [7]. Implementing the above ansatz in the vorticity equation, it is readily seen that in the co-rotating frame, the term u · ∇ω scales in Reynolds number like the viscous term ω/Re.…”

“…Filaments do play a role but region II is as important. Actually the relevance of region II implicitly appears in [7] in which the family of nonlinear solutions is extended to dumbbell forms. Finally, the other vortex does not contribute significantly except near t * p .…”

“…For truly three-dimensional flows, such processes are within the heart of three-dimensional turbulence or vortex reconnection [1]. In two-dimensional flows, vortex merging [2][3][4][5][6][7][8] is the principal ingredient of the inverse cascade [9][10][11] although three vortex interactions may be also significant. It also explains the thickening of the mixing layer width [12] and is considered in the context of aircraft wakes [13,14].…”