Motion and oscillations of a circular perturbed vortex ring

The long bending distortions of the central line of a perturbed circular slender vortex ring with an axial velocity component are studied with the equation of motion of Callegari and Ting [1] rather than with the equation due to a cut-off method as in Widnall and Sullivan [2]. The link between the evolution of the perturbation and the inner structure of the vortex is then shown and the study of the perturbations of small amplitudes gives an analytical expression of the period of oscillation for a circular perturbed inviscid vortex. A numerical code that solves the full nonlinear filament evolution equation has been developed and validated by the results of the linear analysis. In the small amplitude limit, the numerical simulations and the analytical approach are in excellent agreement both for an inviscid and a viscous vortex. The evolution of a perturbation of finite amplitude is also given.