On the stability of multiple helical vortices

Abstract

The classical problem of linear stability of a regular $N$-gon of point vortices to infinitesimal space displacements from an equilibrium of the vortex configuration is generalized to the one for $N$ helical vortices (couple, triplet, etc., $N\,{>}\, 1$) for the first time. As a consequence of this consideration, the analytical form for the stability boundaries has been obtained. This solution allows an efficient analysis to be made of the existence of stable helical vortex arrays, which were repeatedly observed in practice.

Such a stability problem was earlier considered in theory, but only for the case of a plane polygonal array of $N$ point vortices. As for helical vortices, owing to their complexity, intensive study has been mainly on the self-induced motion of the vortex.

The algebraic representation for the velocity of motion of the $N$ helical vortex array was originally obtained as an additional intermediate result. The new formula allows accurate calculations to be made within the whole range of helical pitch variations and has a simpler form than the known asymptotic expressions.

Solution of these two classical problems of vortex dynamics has significance both for theoretic and applied mechanics, as well as for many other areas of natural science, where the rotor (vortex) concept is the basic one.