On the formation of small-time curvature singularities in vortex sheets

Abstract

The Kelvin–Helmholtz model for the evolution of an infinitesimally thin vortex sheet in an inviscid fluid is mathematically ill-posed for general classes of initial conditions. However, if the initial data, say imposed at t = 0, are in a certain class of analytic functions then the problem is well-posed for a finite time until a singularity forms, say at t = ts, on the vortex-sheet interface, e.g. as illustrated by Moore (1979, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. Roy. Soc. Lond. A, 365, 105–119). However, if the problem is analytically continued into the complex plane, then the singularity, or singularities, exist for t < ts away from the physical real axis. More specifically, Cowley et al. (1999, On the formation of Moore curvature singularities in vortex sheets. J. Fluid Mech., 378, 233–267) found that for a class of analytic initial conditions, singularities can form in the complex plane at t = 0+. They posed asymptotic expansions in the neighbourhood of these singularities for 0 < t ≪ 1 and found numerical solutions to the governing similarity differential equations. In this paper we obtain new exact solutions to these equations, show that the singularities always correspond to local ${\textstyle \frac {3}{2}}$-power singularities and determine both the number and precise locations of all branch points. Further, our analytical approach can be extended to a more general class of initial conditions. These new exact solutions can assist in resolving the small-time behaviour for the numerical solution of the Birkhoff–Rott equations.