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Stability of finite-amplitude interfacial waves. Part 3. The effect of basic current shear for one-dimensional instabilities

Published online by Cambridge University Press:  21 April 2006

D. I. Pullin  and
R. H. J. Grimshaw
D. I. Pullin
Affiliation:
Department of Mechanical Engineering, University of Queensland, St Lucia, Queensland, 4067 Australia
R. H. J. Grimshaw
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria, 3052 Australia Present address: School of Mathematics, University of N.S.W., P.O. Box 1, Kensington, N.S.W. 2033, Australia.
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Abstract

We consider the linearized stability of interfacial progressive waves in a two-layer inviscid fluid, for the case when there is a basic current shear in either, or both, of the fluids. For this configuration the basic wave has been calculated by Pullin & Grimshaw (1983b). Our results here are mainly restricted to two-space-dimensional instabilities (i.e. one-dimensional in the propagation space), and are obtained both analytically and numerically. The analytical results are for the long-wavelength modulational instability of small-amplitude waves. The numerical results are restricted to the case when the lower fluid is infinitely deep, and for the Boussinesq approximation. They are obtained by solving the linearized stability problem with truncated Fourier series, and solving the resulting eigenvalue problem for the growth rate. For small values of the basic current shear, and for small or moderate basic wave amplitude, the instabilities are determined by a set of low-order resonances; for larger basic wave amplitude, these are dominated by the onset of a local wave-induced Kelvin–Helmholtz instability. For larger values of the basic current shear, this interpretation is modified owing to the appearance of a number of new effects.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 172 , November 1986 , pp. 277 - 306
Copyright
© 1991 Cambridge University Press

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