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Turbulent boundary layer on a circular cylinder: the low-wavenumber surface pressure spectrum due to a low-Mach-number flow

Published online by Cambridge University Press:  21 April 2006

M. R. Dhanak
M. R. Dhanak
Affiliation:
Topexpress Ltd, 13/14 Round Church Street, Cambridge CB5 8AD, UK
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Abstract

The effect of surface curvature on the low-wavenumber-frequency spectrum of turbulence-induced surface pressure fluctuations is considered and an estimate for the associated flow noise is obtained. The form of the low-wavenumber cross-spectral density of pressure on the surface of an infinitely long, rigid circular cylinder of radius a due to a statistically stationary turbulent boundary-layer flow at a low Mach number is determined. Viscous effects are ignored. It is shown that, in contrast to the case of an infinite plane surface, the pressure spectrum is finite everywhere in the wavenumber plane (k, n/a) except for a logarithmic, integrable, singularity at the acoustic wavenumber corresponding to the axisymmetric mode (n/a = 0); k and n/a being, respectively, the downstream and circumferential wavenumber. For non-axisymmetric modes (|n/a| > 0), the spectrum has two finite peaks in the radiative domain |k| < ω/c; ω being the frequency and c being the sound speed. For ωa/c large, the peaks occur in the vicinity of the total acoustic surface wavenumbers κ = ± ω/c (κ = (k2 + n2 / a2)½) and the principal contribution which determines the peak characteristics can be identified as being due to creeping rays emanating from turbulent sources on the cylinder out of the line of sight of the associated receiver point. For large value of ωa/c, the point pressure spectrum and the associated radiated sound vary logarithmically with $(\omega a/c)^{\frac{2}{3}}$; corresponding estimates for cylinders of moderate and small radius are also obtained. For an almost plane cylinder, it is shown that the effect of curvature may be included by a suitable simple modification of the form of the pressure spectrum for an infinite plane surface.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 191 , June 1988 , pp. 443 - 464
Copyright
© 1988 Cambridge University Press

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References

Bergeron, R. F. 1973 Aerodynamic sound and the low-wavenumber wall-pressure spectrum of nearly incompressible boundary layer turbulence. J. Acoust. Soc. Am. 54, 123.Google Scholar
Chase, D. M. 1980 Modelling the wave vector-frequency spectrum of turbulent boundary layer wall pressure. J. Sound Vib. 70, 29.Google Scholar
Chase, D. M. & Noiseux, C. F. 1982 Turbulent wall pressure at low wavenumbers: Relation to nonlinear sources in planar and cylindrical flow. J. Acoust. Soc. Am. 72, 975.Google Scholar
Crow, S. C. 1967 Visco-elastic character of fine-grained isotropic turbulence. Phys. Fluids 10, 1587.Google Scholar
Crow, S. C. 1970 Aerodynamic sound emission as a singular perturbation problem. Stud. Appl. Maths 49, 21.Google Scholar
Dowling, A. P. 1983 The low wavenumber wall pressure spectrum on a flexible surface. J. Sound Vib. 88, 11.Google Scholar
Dowling, A. P., Williams, J. E. Ffowcs & Goldstein, M. E. 1978 Sound production in a moving stream. Phil. Trans. R. Soc. Lond. A 288, 321.Google Scholar
Williams, J. E. Ffowcs 1965 Surface-pressure fluctuations induced by boundary-layer flow at finite Mach numbers. J. Fluid Mech. 22, 507.Google Scholar
Williams, J. E. Ffowcs 1982 Boundary-layer pressures and the Corcos model: a development to incorporate low-wavenumber constraints. J. Fluid Mech. 125, 9.Google Scholar
Howe, M. S. 1979 The role of surface shear stress fluctuations in the generation of boundary layer noise. J. Sound Vib. 65, 159.Google Scholar
Howe, M. S. 1987 The singularity at the acoustic wavenumber of the turbulent boundary layer wall pressure spectrum. Proc. R. Soc. Lond. A 412, 389.Google Scholar
Jones, D. S. 1979 Methods in Electromagnetic Wave Propagation. Clarendon.
Lighthill, M. J. 1952 On sound generated aerodynamically. I. General Theory. Proc. R. Soc. Lond. A 211, 564.Google Scholar