Steady Vortex Ring with Isochronous Flow in the Vortex Core

Steady-state solutions to the problem of a thin vortex ring in an inviscid incompressible fluid in infinite space are investigated. The Fraenkel procedure is used to construct the steady-state solutions. In this procedure a given vorticity distribution in plane flow with circular streamlines is transformed into a steady vortex ring using an expansion in the ring thinness parameter. For example, a two-dimensional vortex of constant vorticity is transformed into a steady vortex ring with the uniform distribution in which the absolute value of vorticity is proportional to the distance from the axis of symmetry. The principal aim of our study is to construct the algorithm of finding the flow for an isochronous vortex ring in which the periods of revolution are the same for all the liquid particles in the vortex core. The problem is that the two-dimensional distribution which goes over in the isochronous ring in accordance with the Fraenkel procedure is unknown in advance. In particular, the ring with the uniform distribution is not isochronous despite the isochronism of the initial two-dimensional flow. In this connection the Fraenkel procedure is significantly modified so that the initial two-dimensional vorticity distribution is determined in each of the steps of the iteration procedure. The solution for the vortex ring with the uniform distribution obtained in the present study is significantly used to construct the isochronous solution. The necessary corrections to the former solution are calculated in each step. Obtaining of the isochronous flow is the key step for the investigation of stability of three-dimensional oscillations of the vortex ring since the oscillation spectrum of this flow is discrete.