Kinematic variational principle for motion of vortex rings

Yasuhide Fukumoto, Graduate School of Mathematics, Kyushu University

Vortex rings are prominent coherent structures in a diversity of fluid flows. I will show how topological ideas work to derive a theoretical upper bound on translation speed of a vortex ring. According to Kelvin-Benjamin's principle, a steady distribution of vorticity, relative to a moving frame, is realized as the state that maximizes the total kinetic energy, under the constraint of constant hydrodynamic impulse, with respect to variations preserving the vorticity-field topology. Combined with an asymptotic solution of the Euler equations for a family of vortex rings, we can skip the detailed solution for the flow field to obtain the translation velocity of a vortex ring valid to third order in a small parameter, the ratio of the core radius to the ring radius. Including small viscosity, Saffman's velocity formula of a viscous vortex ring is extended to third order, which gives an improved upper bound on the translation speed. Similarity of this principle is found with Rasetti-Regge's theory for three-dimensional motion of a vortex filament.