Basic Displacements in the Problem of Core Perturbations of a Thin Isochronous Vortex Ring

Abstract

Periodic perturbations in the core of a thin isochronous vortex ring in an inviscid incompressible fluid are investigated in the linear approximation. The aim of the study is to construct the system of basic displacements, namely, the complete system of solutions of the Helmholtz equation for vorticity perturbations inside the core of a vortex ring with a given frequency in the form of expansion in the ring thinness parameter μ. The structure of basic displacements depends substantially on the fact to what extent the frequency of the forcing action is close to the resonance frequencies of the system. If the difference between these frequencies is small, then, in addition to the ring thinness μ, the second small parameters arises in the problem. This leads to significant complication of the procedure of obtaining the solution and appearance of considerable corrections in the subsequent approximations of the expansion procedure. The case of isochronous vortex ring in which the periods of revolution of liquid particles are identical is considered. Obtaining the threedimensional oscillations in such flows turns out to be the simplest since there are no perturbations of the continuous spectrum for the isochronous ring. The system of basic displacements is the necessary element in deriving the dispersion relation for the eigen-oscillations of the vortex ring. The solutions obtained can also serve as an instrument to analyze the reaction of flows with curvilinear vortex lines or flows localized in toroidal regions to the external excitation.