Analytical calculation of the translational velocity of linked vortices

Abstract

This paper determines analytically the velocity field induced by a configuration of linked vortices, with the aim of finding the translational velocity of such a configuration. The linked-vortex configuration consists of two vortices; the vortices are thin tubes of circular cross section lying on the surface of an immaterial torus of small aspect ratio r1r0 (where r1 is the cross section radius of a torus and r0 is its centerline radius). The induced velocity field has been found based on the vector potential associated with the Biot-Savart law by using a multipolar expansion; the comparison of this field with the material condition on the surface of the vortices allows one to calculate the translational velocity. The solution obtained retains effects to first order at the multipolar expansion, which corresponds to the effects the vortex curvature has at the vorticity distribution on its cross section. This solution agrees with the numerical results in the range of 2%. The method presented is generalized to the case of n linked vortices.