The spontaneous appearance of a singularity in the shape of an evolving vortex sheet

Abstract

The evolution of a small amplitude initial disturbance to a straight uniform vortex sheet is described by the Fourier coefficients of the disturbance. An approximation to the exact evolution equation for these coefficients is proposed and it is shown, by an asymptotic analysis valid at large times, that the solution of the approximate equations develops a singularity at a critical time. The critical time is proportional to ln $(\epsilon ^{-1})$, where $\epsilon $ is the initial amplitude of the disturbance and the singularity itself is such that the nth Fourier coefficient decays like $n^{-2.5}$ instead of exponentially. Evidence-not conclusive, however-is present to show that the approximation used is adequate. It is concluded that the class of vortex layer motions correctly modelled by replacing the vortex layer by a vortex sheet is very restricted; the vortex sheet is an inadequate approximation unless it is everywhere undergoing rapid stretching.