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Nonlinear wave interactions in shear flows. Part 2. Third-order theory

Published online by Cambridge University Press:  29 March 2006

J. R. Usher ,
A. D. D. Craik  and
F. Hendriks
J. R. Usher
Affiliation:
Department of Mathematics and Physics, Glasgow College of Technology, Glasgow, Scotland
A. D. D. Craik
Affiliation:
Department of Applied Mathematics University of St Andrews, Fife, Scotland
F. Hendriks
Affiliation:
IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598
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Abstract

The temporal evolution of a resonant triad of wave components in a parallel shear flow has been investigated at second order in the wave amplitudes by Craik (1971) and Usher & Craik (1974). The present work extends these analyses to include terms of third order and thereby develops the resonance theory to the same order of approximation as the non-resonant third-order theory of Stuart (1960, 1962).

Asymptotic analysis for large Reynolds numbers reveals that the magnitudes of the third-order interaction coefficients, like certain of those at second order, are remarkably large. The implications of this are discussed with particular reference to the roles of resonance and of three-dimensionality in nonlinear instability and to the range of validity of the perturbation analysis.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 70 , Issue 3 , 12 August 1975 , pp. 437 - 461
Copyright
© 1975 Cambridge University Press

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