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Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

Published online by Cambridge University Press:  11 February 2011

SJOERD W. RIENSTRA  and
MIRELA DARAU
SJOERD W. RIENSTRA*
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
MIRELA DARAU
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
*
†Email address for correspondence: s.w.rienstra@tue.nl
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Abstract

The Ingard–Myers condition, modelling the effect of an impedance wall under a mean flow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a two-dimensional incompressible limit, we show that the flow is absolutely unstable for h smaller than a critical hc and convectively unstable or stable otherwise. This critical hc is by nature independent of wavelength or frequency and is a property of liner and mean flow only. An analytical approximation of hc is given, which is complemented by a contour plot covering all parameter values. For an aeronautically relevant example, hc is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard–Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem.

JFM classification

Instability: Absolute/convective instability Acoustics: Aeroacoustics Boundary Layers: Boundary layer stability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 671 , 25 March 2011 , pp. 559 - 573
Copyright
Copyright © Cambridge University Press 2011

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