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Acoustic implications of a thin viscous boundary layer over a compliant surface or permeable liner

Published online by Cambridge University Press:  26 April 2011

E. J. BRAMBLEY
E. J. BRAMBLEY*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: E.J.Brambley@damtp.cam.ac.uk
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Abstract

This paper considers the implications of a high-Reynolds-number thin parallel boundary layer on fluid–solid interaction. Two types of boundaries are considered: a compliant boundary which is flexible but impermeable, such as an elastic sheet or elastic solid, and a permeable boundary which is rigidly fixed, such as a perforated rigid sheet. The fluid flow consists of a steady flow along the boundary and a small time-dependent perturbation, with the boundary reacting to the perturbation. The fluid displacement due to the perturbation is assumed to be much smaller than the boundary-layer thickness. The analysis is equally valid for compressible and incompressible fluids. Numerical examples are given for compressible flow along a cylindrical duct, for both permeable and compliant cylinder walls. The difference between compliant and permeable walls is shown to be dramatic in some cases. High- and low-frequency asymptotics are derived, and shown to compare well to the numerics. When used with a mass–spring–damper boundary, this model is shown to lead to similar, but not identical, temporal instability with unbounded growth rate to that seen for slipping flow using the Myers boundary condition. It is therefore suggested that a regularization of the Myers boundary condition removing the unbounded growth rate may lead to, or at least inform, a regularization of the model presented here.

JFM classification

Acoustics: Aeroacoustics Compressible Flows: Compressible Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 678 , 10 July 2011 , pp. 348 - 378
Copyright
Copyright © Cambridge University Press 2011

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