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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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References

Anderson, J. S. 1977 The effect of an air flow on a single side branch Helmholtz resonator in a circular duct. J. Sound Vib. 52, 423.Google Scholar
Benjamin, T. B. 1964 Fluid flow with flexible boundaries. Proc. 11th Int. Cong. Appl. Mech., Munich, p. 109.
Crighton, D. G. 1968 Wave motion and vibration induced by turbulent flow. Ph.D. thesis, University of London.
Curle, N. 1955 The influence of solid boundaries upon aerodynamic sound. Proc. Roy. Soc. A 231, 412.Google Scholar
Dowling, A. P., Ffowcs Williams, J. E. & Goldstein, M. E. 1978 Sound production in a moving stream. Phil. Trans. Roy. Soc. A 288, 321.Google Scholar
Ffowcs Williams, J. E. 1965 Sound radiation from turbulent boundary layers formed on compliant surfaces. J. Fluid Mech. 22, 347.Google Scholar
Ffowcs Williams, J. E. 1966 The influence of simple supports on the radiation from turbulent flow near a plane compliant surface. J. Fluid Mech. 26, 641.Google Scholar
Ffowcs Williams, J. E. 1972 The acoustics of turbulence near sound absorbent liners. J. Fluid Mech. 51, 737.Google Scholar
Ffowcs Williams, J. E. 1974 Sound production at the edge of a steady flow. J. Fluid Mech. 66, 791.Google Scholar
Ffowcs Williams, J. E. & Lovely, D. J. 1975 Sound radiation into uniformly flowing fluid by compact surface vibration. J. Fluid Mech. 71, 689.Google Scholar
Goldstein, M. E. 1976 Aeroacoustics. McGraw-Hill.
Gradshteyn, I. S. & Ryzhik, I. M. 1965 Table of Integrals, Series and Products, 4th edn. Academic.
Jones, D. S. 1973 The effect of radiation due to a moving source on a vortex sheet. Proc. Roy. Soc. Edinb. 72 (14), 195.Google Scholar
Landahl, M. T. 1967 A wave guide model for turbulent shear flow. J. Fluid Mech. 29, 441.Google Scholar
Lighthill, M. J. 1952 On sound generated aerodynamically. I. General theory. Proc. Roy. Soc. A 211, 564.Google Scholar
Lilley, G. M. 1974 On the noise from jets. AGARD Conf. Proc. no. 131, paper 13.Google Scholar
Lovely, D. J. 1974 Novel methods of aerodynamic sound generation. Ph.D. thesis, University of London.
Mani, R. 1976 The influence of jet flow on jet noise. Part I. The noise of unheated jets. J. Fluid Mech. 73, 753.Google Scholar
Mechel, F. 1960 Schalldämpfung and Schallverstärkung in Luftströmungen durch absorbierend ausgekleidete Kanale. Acustica 10, 133.Google Scholar
Meyer, E., Mechel, F. & Kurtze, G. 1958 Experiments on the influence of flow on sound attenuation in absorbing ducts. J. Acoust. Soc. Am. 30, 165.Google Scholar
Moore, C. J. 1977 The role of shear layer instability waves in jet exhaust noise. J. Fluid Mech. 80, 321.Google Scholar
Morfey, C. L. & Tester, B. J. 1976 Developments in jet noise modelling. J. Sound Vib. 46, 79.Google Scholar
Morgan, J. D. 1975 The interaction of sound with a subsonic cylindrical vortex layer. Proc. Roy. Soc. A 344, 341.Google Scholar
Panton, R. L. & Miller, J. M. 1975 Excitation of a Helmholtz resonator by a turbulent boundary layer. J. Acoust. Soc. Am. 58, 800.Google Scholar
Powell, A. 1960 Aerodynamic noise and the plane boundary. J. Acoust. Soc. Am. 32, 982.Google Scholar
Schlichting, H. 1955 Boundary Layer Theory. Pergamon.
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.