Progressive Deformation of a Curved Vortex Filament by its Own Induction

Abstract

Progressive three-dimensional deformation of an initially parabolic, plane curved vortex due to its own induction is numerically obtained in this paper by the use of the localized-induction concept instead of through the full Biot—Savart law. It is shown that, after the region near the vertex rises up from the plane of the initial orientation, a helical deformation takes place on the legs of the parabola. The helix, which rotates in the direction opposite to the circulatory rotation around the vortex filament, is found to travel away toward the far ends of the parabola as the vertex keeps rising up, while increasing the amplitude and extent of the helical deformation. Similar computations also carried out with a hyperbolic initial shape resulted in essentially the same conclusions as the parabola but with much better numerical stability, as well as with an exponential initial configuration, which grew into somewhat more complicated patterns, yet maintained the essential feature of the helical deformation.