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Interaction of phase-locked modes: a new mechanism for the rapid growth of three-dimensional disturbances

Published online by Cambridge University Press:  26 April 2006

Xuesong Wu  and
Philip A. Stewart
Xuesong Wu
Affiliation:
Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK
Philip A. Stewart
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

In this paper, we have identified a new mechanism which can promote rapid growth of three-dimensional disturbances. The mechanism involves the interaction between a planar mode and an oblique mode, or a pair of oblique modes, which are phase-locked in the sense that they have the same phase speed. This allows a powerful nonlinear interaction to take place within the common critical layer(s). The disturbance is not required to form a subharmonic resonant triad, and hence the mechanism operates under much less restrictive conditions than does subharmonic resonance (although it is somewhat less powerful). We show that the quadratic interaction between the planar mode and the oblique modes drives an exceptionally large forced mode with the difference frequency, which in turn interacts with the planar mode to contribute the dominant nonlinear effect. This interaction can cause the oblique modes to grow super-exponentially provided that their magnitude is sufficiently small. As a result of the super-exponential growth, the oblique mode may soon become strong enough to produce a feedback effect on the planar mode, so that the interactions eventually become fully coupled. This subsequent stage takes slightly different forms depending on whether a single or a pair of oblique modes is present. Both cases are investigated. Particular attention is paid to symmetric plane shear layers, e.g. a planar wake or jet, for which subharmonic resonance of sinuous modes is inactive.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 316 , 10 June 1996 , pp. 335 - 372
Copyright
© 1996 Cambridge University Press

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