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On the formation of Moore curvature singularities in vortex sheets

Published online by Cambridge University Press:  10 January 1999

STEPHEN J. COWLEY
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
GREG R. BAKER
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA
SALEH TANVEER
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA

Abstract

Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane, we find a consistent asymptotic solution to the problem posed by Moore. Our solution includes the shape of the vortex sheet as the curvature singularity forms. Analytic results are confirmed by comparison with numerical solutions. Further, for a wide class of initial conditions (including perturbations of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t=0+ in the complex plane. We show that these singularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vortex-sheet shape at times close to singularity formation.

Type
Research Article
Copyright
© 1999 Cambridge University Press

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