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Multiple asymptotic solutions for axially travelling waves in porous channels

Published online by Cambridge University Press:  25 September 2009

JOSEPH MAJDALANI
JOSEPH MAJDALANI*
Affiliation:
Mechanical, Aerospace and Biomedical Engineering Department, University of Tennessee Space Institute, 411 B. H. Goethert Parkway, Tullahoma, TN 37388-9700, USA
*
†Email address for correspondence: [email protected]
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Abstract

Travelling waves in confined enclosures, such as porous channels, develop boundary layers that evolve over varying spatial scales. The present analysis employs a technique that circumvents guessing of the inner coordinate transformations at the forefront of a multiple-scales expansion. The work extends a former study in which a two-dimensional oscillatory solution was derived for the rotational travelling wave in a porous channel. This asymptotic solution was based on a free coordinate that could be evaluated using Prandtl's principle of matching with supplementary expansions. Its derivation required matching the dominant term in the multiple-scales expansion to an available Wentzel-Kramers-Brillouin (WKB) solution. Presently, the principle of least singular behaviour is used. This approach leads to a multiple-scales approximation that can be obtained independently of supplementary expansions. Furthermore, a procedure that yields different types of WKB solutions is described and extended to arbitrary order in the viscous perturbation parameter. Among those, the WKB expansion of type I is shown to exhibit an alternating singularity at odd orders in the perturbation parameter. This singularity is identified and suppressed using matched asymptotic tools. In contrast, the WKB expansion of type II is found to be uniformly valid at any order. Additionally, matched asymptotic, WKB and multiple-scales expansions are developed for several test cases. These enable us to characterize the essential vortico-acoustic features of the axially travelling waves in a porous channel. All solutions are numerically verified, compared and discussed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 636 , 10 October 2009 , pp. 59 - 89
Copyright
Copyright © Cambridge University Press 2009

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