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Transverse instability of surface solitary waves. Part 2. Numerical linear stability analysis

Published online by Cambridge University Press:  17 June 2010

TAKESHI KATAOKA
TAKESHI KATAOKA*
Affiliation:
Department of Mechanical Engineering, Graduate School of Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan
*
†Email address for correspondence: [email protected]
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Abstract

In a previous work, Kataoka & Tsutahara (J. Fluid Mech., vol. 512, 2004a, p. 211) proved the existence of longitudinally stable but transversely unstable surface solitary waves by asymptotic analysis for disturbances of small transverse wavenumber. In the present paper, the same transverse instability is examined numerically for the whole range of solitary-wave amplitudes and transverse wavenumbers of disturbances. Numerical results show that eigenvalues and eigenfunctions of growing disturbance modes agree well with those obtained by the asymptotic analysis if the transverse wavenumber of the disturbance is small. As the transverse wavenumber increases, however, the growth rate of the disturbance, which is an increasing function for small wavenumbers, reaches a maximum and finally falls to zero at some finite wavenumber. Thus, there is a high-wavenumber cutoff to the transverse instability. For higher amplitude, solitary waves become longitudinally unstable, and the dependence of the eigenvalues on the transverse wavenumber exhibits various complicated patterns. We found that such eigenvalues versus transverse wavenumber can be simply grouped into three basic classes.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 657 , 25 August 2010 , pp. 126 - 170
Copyright
Copyright © Cambridge University Press 2010

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