The Rolling up of a Semi-Infinite Vortex Sheet
Proceedings of the Royal Society A
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 th a t 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 sug gested th a t the correction to Kaden can
... tion 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 th a t the result would be useful in a numerical study. I ntroduction This paper concerns the solution of the following problems. A semi-infinite flat plate y = 0, x > 0 is supporting irrotational flow of an inviscid, incompressible fluid in which the complex potential rf> + i f r is -yzi, where y is a con where the velocity field is /<¥ \ d x 'd y ) ' The plate dissolves a t time t = 0 leaving behind a semi-infi circulation in the portion (0, x) of the sheet being 2yx%. This configuration cannot persist, because the self-induced velocity while vanishing for 0 is infinite a t x = 0. Thus deformation of the vortex sheet will start a t the tip x = O.f The problem is to determine the subsequent evolution of the vortex sheet. This unsteady two-dimensional flow is related in an approximate way to the steady three-dimensional flow past a wing of large aspect ratio placed a t small incidence in a steady stream of velocity W.