Asymptotic formula for velocity of a vortex ring: utility of topological ideas
Yasuhide Fukomoto
Kyushu University, Japan
Abstract:
A general formula is established for translation speed of an axisymmetric vortex ring whose core is not necessarily thin. We rely on Helmholtz-Lamb's method of calculating the total kinetic energy of fluid in two ways. Combined with the Navier-Stokes equations, we can skip the detailed solution for the flow field to extend Saffman's velocity formula of a viscous vortex ring to third order in the ratio of the core radius to the ring radius, a small parameter, for the entire range of the Reynolds number. At small Reynolds numbers, a solution that describes the whole life of a vortex ring is available.
We show how topological ideas, or Lagrangian approach, help to calculate motion of an inviscid vortex ring; a further simplification is achieved by resorting to the variation, under the topological constraints, of the kinetic energy with respect to the hydrodynamic impulse. A remark is given on utility of topological ideas for study of three-dimensional instability of a vortex tube.