The self-induced motion of vortex sheets

Abstract

A method is presented for following the self-induced motion of vortex sheets. In this method, we use a piecewise analytic representation of the sheet consisting of circular arcs with trigonometric polynomials for the circulation. The procedure is used to study the evolution of the motion in two special cases: a circular vortex sheet with sinusoidal circulation distribution and an infinite plane vortex sheet subject to periodic disturbances. The first problem was studied by Baker (1980) as a test of the method of Fink & Soh (1978), while the second has been studied by a number of authors, notably Meiron, Baker & Orszag (1982). In each case, we follow the motion of the sheet up to the appearance of a singularity at a finite time. The singularity takes the form of an exponential spiral with the simultaneous development of singularities in the curvature and in the circulation distribution. In the final stages of the calculations, up to 155 marker points are used to specify the position of the sheet. If it were possible to execute a stable calculation with equally spaced point vortices, approximately 106 points would be required to achieve the same resolution. Problems with instabilities have been reduced, but not entirely eliminated, and prevent a rigorous verification of the results obtained.