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Stability of finite-amplitude interfacial waves. Part 2. Numerical results

Published online by Cambridge University Press:  20 April 2006

D. I. Pullin  and
R. H. J. Grimshaw
D. I. Pullin
Affiliation:
Department of Mechanical Engineering, University of Queensland, St Lucia, Qld 4067, Australia
R. H. J. Grimshaw
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Vie 3052, Australia
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Abstract

In the preceding paper (Grimshaw & Pullin 1985) we discussed the long-wavelength modulational instability of interfacial progressive waves in a two-layer fluid. In this paper we complement our analytical results by numerical results for the linearized stability of finite-amplitude waves. We restrict attention to the case when the lower layer is infinitely deep, and use the Boussinesq approximation. For this case the basic wave profile has been calculated by Pullin & Grimshaw (1983a, b). The linearized stability problem for perturbations to the basic wave is solved numerically by seeking solutions in the form of truncated Fourier series, and solving the resulting eigenvalue problem for the growth rate as a function of the perturbation wavenumber. For small or moderate basic wave amplitudes we show that the instabilities are determined by a set of low-order resonances. The lowest resonance, which contains the modulational instability, is found to be dominant for all cases considered. For higher wave amplitudes, the resonance instabilities are swamped by a local wave-induced Kelvin–Helmholtz instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 160 , November 1985 , pp. 317 - 336
Copyright
© 1985 Cambridge University Press

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