Global time evolution of viscous vortex rings

Abstract

This article gives an overview of growing knowledge of translation speed of an axisymmetric vortex ring, with focus on the influence of viscosity. Helmholtz–Lamb’s method provides a shortcut to manipulate the translation speed at both small and large Reynolds number, for a vortex ring starting from an infinitely thin core. The resulting asymptotics significantly improve Saffman’s formula (1970) and give closer lower and upper bounds on translation speed in an early stage. At large Reynolds numbers, Kelvin–Benjamin’s kinematic variational principle achieves a further simplification. At small Reynolds numbers, the whole life of a vortex ring is available from the vorticity obeying the Stokes equations, which is closely fitted, over a long time, by Saffman’s second formula.