The rolling up of a semi-infinite vortex sheet

Abstract

The rolling up of a semi-infinite initially straight vortex sheet is studied analytically. In its initial state the circulation in the sheet increases as the square root of the distance from its edge. Previous investigations have asserted that the asymptotic form for the equation of the rolled up portion given by Kaden could be improved on by finding higher terms in a locally determined asymptotic expansion. This assertion is contested and it is suggested that the correction to Kaden can not be found unless the shape of the whole vortex sheet is known. The correction proposed renders the turns of the spiral slightly elliptical, the precise magnitude involving an integral over the entire vortex sheet. While a useful analytical solution cannot be found this way, it is suggested that the result would be useful in a numerical study.