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A vortex sheet modelling of boundary-layer noise

Published online by Cambridge University Press:  20 April 2006

J. E. Ffowcs Williams  and
M. Purshouse
J. E. Ffowcs Williams
Affiliation:
Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ
M. Purshouse
Affiliation:
Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ
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Abstract

In this paper we describe a simple way of modelling boundary-layer effects in analytical flow noise studies. We develop an exact analogy between the real flow and one in which there is a step velocity profile. This profile is intended to model a boundary layer in an idealized way and we recognize it in our Green's function for the problem. We insist that the Green's function is bounded, a step that makes it non-causal and similar to those used in recent jet-noise analogies. We derive an expression for the induced pressure which consists of surface and volume terms, just as in Lighthill's theory, but, because both contain elements to be evaluated in future time, we argue that the turbulence must be able to respond to linear surface stimulus and avoid the otherwise inevitable violation of causality. This novel feature distinguishes our analysis from applications of Lighthill's theory to boundary-layer-induced noise. The response of the turbulence may be large when the surface is driven at low boundary-layer Strouhal number. But it is negligible at high Strouhal number, and in that limit the surface terms are found to depend only on the instantaneous boundary geometry and its rate of change. This leads to a simple expression for ‘boundary-layer fluid loading’, in which the finite boundary-layer scale emerges explicitly. To illustrate the physical consequences of this result, we use it to estimate the impedance of a baffled piston vibrating beneath a boundary layer. Potential theory predicts that flow should destabilize the piston motion while experiments usually indicate the reverse. We find that the boundary layer is responsible for the discrepancy and that experimentally observed behaviour is predicted quite reasonably by our model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 113 , December 1981 , pp. 187 - 220
Copyright
© 1981 Cambridge University Press

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