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

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.