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Nonlinear wave interactions in shear flows. Part 1. A variational formulation

Published online by Cambridge University Press:  29 March 2006

J. R. Usher  and
A. D. D. Craik
J. R. Usher
Affiliation:
Department of Mathematics, Teesside Polytechnic, Middlesbrough, England Present address: Department of Mathematics, Glasgow College of Technology, Glasgow, Scotland.
A. D. D. Craik
Affiliation:
Department of Applied Mathematics, University of St Andrews, Fife, Scotland
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Abstract

A modified version of Bateman's variational formulation of the incompressible Navier-Stokes equations and boundary conditions (see Dryden, Murnaghan & Bateman 1956) is introduced. This is employed to examine a particular nonlinear problem of hydrodynamic stability which was treated previously, using a ‘direct’ approach, by Craik (1971). This problem concerns the resonant interaction at second order of a triad of wave modes in a parallel shear flow.

The present method is conceptually attractive; it also has the major advantage over the ‘direct’ method of a substantial reduction in algebraic complexity, which allows results to be derived far more readily. Also, some further improvements are made upon Craik's previous analysis. Such a variational approach may often be simpler than present conventional methods of tackling nonlinear viscous-flow problems. The present paper shows how other problems of nonlinear stability and wave interactions may be tackled in this way.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 66 , Issue 2 , 6 November 1974 , pp. 209 - 221
Copyright
© 1974 Cambridge University Press

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