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The stability of plane Poiseuille flow between flexible walls

Published online by Cambridge University Press:  29 March 2006

C. H. Green  and
C. H. Ellen
C. H. Green
Affiliation:
Imperial College, London Present address: The Nuclear Power Group, Radbroke Hall, Knutsford, Cheshire.
C. H. Ellen
Affiliation:
Imperial College, London
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Abstract

This paper examines the linear stability of antisymmetric disturbances in incompressible plane Poiseuille flow between identical flexible walls which undergo transverse displacements. Using a variational approach, an approximate solution of the problem is formulated in a form suitable for computational evaluation of the (complex) wave speeds of the system. A feature of this formulation is that the varying boundary conditions (and the Orr-Sommerfeld equation) are satisfied only in the mean; this reduces the labour involved in determining the approximate solution for a variety of wall conditions without increasing the difficulty of obtaining solutions to a given accuracy. In this paper the symmetric stream function distribution across the channel is represented by a series of cosines whose coefficients are determined by the variational solution. Comparisons with previous work, both for the flexible-wall and rigid-wall problems, show that the method gives results as accurate as those obtained previously by other methods while new results, for flexible walls, indicate the presence of a higher wave-number stability boundary which joins the distorted Tollmien-Schlichting stability boundary at lower wave-numbers. In some cases this upper unstable region, which is characterized by large amplification rates, may determine the critical Reynolds number of the system.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 51 , Issue 2 , 25 January 1972 , pp. 403 - 416
Copyright
© 1972 Cambridge University Press

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