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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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References

Browning, K. A. 1971 Structure of the atmosphere in the vicinity of large-amplitude Kelvin–Helmholtz billows. Q. J. R. Met. Soc. 97, 283–299.Google Scholar
Chen, B. & Saffman, P. G. 1980 Numerical evidence for the existence of new types of gravity waves of permanent form on deep water. Stud. Appl. Maths 62, 1–21.Google Scholar
Garbow, B. S. 1978 Algorithm 535: the QZ algorithm to solve the generalized eigenvalue problem for complex matrices. In Collected Algorithms of ACM.
Grimshaw, R. H. J. & Pullin, D. I. 1985 Stability of finite-amplitude interfacial waves. Part 1. Modulational instability for small-amplitude waves. J. Fluid Mech. 000, 000–000.Google Scholar
Harrison, W. J. 1908 The influence of viscosity on the oscillation of superposed fluids. Proc. Lond. Math. Soc. 6, 396–405.Google Scholar
Kaufman, L. 1975 Algorithm 496: the LZ algorithm to solve the generalized eigenvalue problem for complex matrices. In Collected Algorithms of ACM.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Mclean, J. W. 1982a Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
Mclean, J. W. 1982b Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.Google Scholar
Mclean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. & Yuen, H. C. 1981 Three-dimensional instability of finite-amplitude water waves. Phys. Rev. Lett. 46, 817–820.Google Scholar
Meiron, D. I. & Saffman, P. G. 1983 Overhanging interfacial gravity waves of large amplitude. J. Fluid Mech. 129, 213–218.Google Scholar
Meiron, D. I., Saffman, P. G. & Yuen, H. C. 1982 Calculation of steady three-dimensional deep-water waves. J. Fluid Mech. 124, 109–121.Google Scholar
Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1983a Nonlinear interfacial progressive waves near a boundary in a Boussinesq fluid. Phys. Fluids 26, 897–905.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1983b Interfacial progressive waves in a two-layer shear flow. Phys. Fluids 26, 1731–1739.Google Scholar
Saffman, P. G. & Yuen, H. C. 1980 A new type of three-dimensional deep-water wave of permanent form. J. Fluid Mech. 101, 797–808.Google Scholar
Su, M. Y. 1982 Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric wave patterns. J. Fluid Mech. 124, 73–108.Google Scholar
Su, M. Y., Bergin, M., Marler, P. & Myrick, R. 1982 Experiments on nonlinear instabilities and evolution of steep gravity-wave trains. J. Fluid Mech. 124, 45–72.Google Scholar
Thorpe, S. A., Hall, A. J., Taylor, C. & Allen, J. 1977 Billows in Loch Ness. Deep-Sea Res. 24, 371–379.Google Scholar
Woods, J. D. 1968 Wave-induced shear instability in the summer thermocline. J. Fluid Mech. 32, 791–800.Google Scholar
Yuen, H. C. 1983 Instability of finite-amplitude interfacial waves. In Waves on Fluid Interfaces (ed. R. E. Meyer), pp. 17–40. Academic.
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar