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The linear growth of localized disturbances in unstable compressible flows

Published online by Cambridge University Press:  14 November 2011

S. Candler  and
A. D. D. Craik
S. Candler
Affiliation:
Department of Applied Mathematics, University of St Andrews, St Andrews, Fife KY16 9SS
A. D. D. Craik
Affiliation:
Department of Applied Mathematics, University of St Andrews, St Andrews, Fife KY16 9SS
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Synopsis

Accurate computation of the evolution of initially localized disturbances in compressible parallel flows is a tedious task requiring superposition of a large number of Fourier modes with differing temporal growth rates. An alternative approximate method, similar to that developed by Craik (1981, 1982) for viscous incompressible flows, is presented here. This involves asymptotic evaluation, by the saddle point method, of a double Fourier integral representation of the disturbance, with the actual dispersion relation replaced by a simpler analytic expression containing several parameters which may be adjusted to approximate the flow under investigation. Limiting cases yield informative results in simple closed form: these exemplify the possible shapes into which the disturbance may evolve. In particular, ‘splitting’ of the disturbance into two dominant regions is demonstrated.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 3-4 , 1984 , pp. 249 - 265
Copyright
Copyright © Royal Society of Edinburgh 1984

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