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Fully coupled resonant-triad interactions in a free shear layer

Published online by Cambridge University Press:  26 April 2006

R. Mallier  and
S. A. Maslowe
R. Mallier
Affiliation:
Department of Mathematics, McGill University, Montreal, P. Q., H3A 2K6, Canada
S. A. Maslowe
Affiliation:
Department of Mathematics, McGill University, Montreal, P. Q., H3A 2K6, Canada
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Abstract

We report the results of an investigation of the weakly nonlinear evolution of a triad of waves, each slightly amplified on a linear basis, that are superimposed on a tanh y mixing layer. The triad consists of a plane wave and a pair of oblique modes that act as a subharmonic of order 1/2. The oblique modes are inclined at approximately ±60°. to the mean flow direction and because the resonance conditions are satisfied exactly the analysis is entirely self-consistent as an asymptotic theory. The nonlinearity first occurs within the critical layer and the initial interaction is of the parametric resonance type. This produces faster than exponential growth of the oblique waves, behaviour observed recently in the experiments of Corke & Kusek (1993). The critical-layer dynamics lead subsequently to coupled integro-differential equations governing the amplitude evolution and, as first shown in related work by Goldstein & Lee (1992) on boundary layers in an adverse pressure gradient, these equations develop singularities in a finite time.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 278 , 10 November 1994 , pp. 101 - 121
Copyright
© 1994 Cambridge University Press

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