The dynamics of rolled vortex sheets tightly winded around slender vortex filaments in inviscid incompressible flow

Abstract

The purpose of this paper is to develop an asymptotic theory for the core of a tightly winded rolled vortex sheet, when winding arises around a line. The physical picture replaces the core by a region of distributed vorticity, the so-called vortex filament. The (pseudo) mathematical picture relies on an algorithm which allows to relate (in a quite convincing but non rigorous way) a flow with vorticity spread over a region to a corresponding irrotational flow with vorticity concentrated on a rolled sheet. Of course, the correspondence holds only in the asymptotic limit that the turns of the sheet are infinitely close to each other. The technique for that, which was developed by Guiraud & Zeytounian, 1977, relies on a multiple scale expansion using a closeness parameter which is the ratio of the distance from one turn of the sheet to the next one, to a length characteristic of the flow. The leading approximation is, then, a rotational flow. In the present paper, an application of the theory is given when this leading approximation has the structure of a slender vortex filament and is itself given by an inner and outer expansion technique. Inner and outer approximations to the dynamics of vortex filaments is reviewed, the results of Guiraud & Zeytounian are presented and, finally, the specific application is worked out.