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Reappraisal of the Kelvin–Helmholtz problem. Part 2. Interaction of the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instabilities

Published online by Cambridge University Press:  25 February 1997

T. BROOKE BENJAMIN
Affiliation:
Mathematical Institute, Oxford University, 24-29 St. Giles, Oxford OX1 3LB, UK
THOMAS J. BRIDGES
Affiliation:
Department of Mathematical and Computing Sciences, University of Surrey, Guildford, Surrey GU2 5XH, UK

Abstract

Several new results on the bifurcation and instability of nonlinear periodic travelling waves, at the interface between two fluids in relative motion, in a parametric neighbourhood of a Kelvin–Helmholtz unstable equilibrium are presented. The organizing centre for the analysis is a canonical Hamiltonian formulation of the Kelvin–Helmholtz problem presented in Part 1. When the density ratio of the upper and lower fluid layers exceeds a critical value, and surface tension is present, a pervasive superharmonic instability is found, and as uu0, where u is the velocity difference between the two layers and u0 is the Kelvin–Helmholtz threshold, the amplitude at which the superharmonic instability occurs scales like (u0u)1/2 with u < u0. Other results presented herein include (a) new results on the structure of the superharmonic instability, (b) the discovery of isolated branches and intersecting branches of travelling waves near a critical density ratio, (c) the appearance of Benjamin–Feir instability along branches of waves near the Kelvin–Helmholtz instability threshold and (d) the interaction between the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instability at low amplitude.

Type
Research Article
Copyright
© 1997 Cambridge University Press

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