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Amplitude equations for wave packets in slightly inhomogeneous unstable flows

Published online by Cambridge University Press:  26 April 2006

T. F. Balsa
T. F. Balsa
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA
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Abstract

We derive several versions of the (complex) amplitude equation of an inviscid wave packet travelling on a slightly inhomogeneous (and possibly unsteady and viscous) unstable base flow. This is done with complete generality, without any reference to the dimensions of physical and propagation spaces, by using the usual high-frequency ansatz. The final results are extremely simple: volume integrals of a complex wave action density are conserved subject to an appropriate flux and a source term. The latter is expressible in a remarkably concise way in terms of the gradient of the base flow acceleration and vanishes when the base flow is inviscid. The simplicity of our results hinges on a transformation of the dependent variables and on a suitable decomposition of these in cross- and propagation spaces. Our results are also discussed with the help of three different Lagrangian densities and their associated kinematic wave theories which are based on a basic identity due to Hayes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 204 , July 1989 , pp. 433 - 455
Copyright
© 1989 Cambridge University Press

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