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Fully coupled resonant-triad interaction in an adverse-pressure-gradient boundary layer

Published online by Cambridge University Press:  26 April 2006

M. E. Goldstein  and
Sang Soo Lee
M. E. Goldstein
Affiliation:
Lewis Research Center, Cleveland, OH 44135, USA
Sang Soo Lee
Affiliation:
Sverdrup Technology, Inc., Lewis Research Center Group, Cleveland, OH 44135. USA
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Abstract

The nonlinear resonant-triad interaction, proposed by Raetz (1959). Craik (1971), and others for a Blasius boundary layer, is analysed here for an adverse-pressure-gradient boundary layer. We assume that the adverse pressure gradient is in some sense weak and, therefore, that the instability growth rate is small. This ensures that there is a well-defined critical layer located somewhere within the flow and that the nonlinear interaction is effectively confined to that layer. The initial interaction is of the parametric resonance type, even when the modal amplitudes are all of the same order. This means that the oblique instability waves exhibit faster than exponential growth and that the growth rate of the two-dimensional mode remains linear. However, the interaction and the resulting growth rates become fully coupled, once oblique-mode amplitudes become sufficiently large, but the coupling terms are now quartic, rather than quadratic as in the Craik (1971) analysis. More importantly, however, new nonlinear interactions, which were not present in the Craik-type analyses, now come into play. These interactions eventually have a dominant effect on the instability wave development.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 245 , December 1992 , pp. 523 - 551
Copyright
© 1992 Cambridge University Press

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