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Instability and filamentation of finite-amplitude waves on vortex layers of finite thickness

Published online by Cambridge University Press:  26 April 2006

D. I. Pullin ,
P. A. Jacobs ,
R. H. J. Grimshaw  and
P. G. Saffman
D. I. Pullin
Affiliation:
Department of Mechanical Engineering, The University of Queensland, St Lucia, QLD 4067, Australia
P. A. Jacobs
Affiliation:
Department of Mechanical Engineering, The University of Queensland, St Lucia, QLD 4067, Australia
R. H. J. Grimshaw
Affiliation:
School of Mathematics, The University of New South Wales, PO Box 1, Kensington, NSW, Australia
P. G. Saffman
Affiliation:
Applied Mathematics, 217-50, Firestone Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
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Abstract

We study the instability of finite-amplitude waves on uniform vortex layers of finite thickness bounded by a plane rigid surface. A weakly nonlinear analysis of vorticity interface perturbations, and spectral stability calculations using the full equations of motion, together show that steady progressive waves are unstable to general subharmonic pertubations in the range 0.094 < d/λ < 1.7, where d is the mean layer thickness and λ is the primary wavelength. The relevance of this instability to ultimate interface filamentation is tested by performing several numerical contourdynamical simulations of the nonlinear interface evolution for initial disturbances consisting of the finite amplitude wave plus eigenfunctions obtained from the spectral calculations. The results indicate that within the band of unstable wavelengths, small perturbations to the steady non-uniform flow given by the finite amplitude wave motion (vortex equilibrium) are able to grow in magnitude, until at a time tf, the wave extremum encounters a hyperbolic critical point of the velocity field after which filamentation occurs. Arguments are put forward based on the unsteady simulations with the purpose of identifying the preferred frame of reference for viewing the kinematical events controlling the filamentation process. An estimate for tf is then made, and the mechanism of filamentation found is discussed in relation to the recently proposed nonlinear-cascade mechanism of Dritschel (1988a).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 209 , December 1989 , pp. 359 - 384
Copyright
© 1989 Cambridge University Press

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