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 explorations for 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 and 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).
- Publication:
-
APS Division of Fluid Dynamics Meeting Abstracts
- Pub Date:
- November 2003
- Bibcode:
- 2003APS..DFD.FH001C