Aerodynamic sound and the low-wavenumber wall-pressure spectrum of nearly incompressible boundary-layer turbulence

Abstract

The low-wavenumber structure of fixed-frequency cross-spectral density of turbulent pressure fluctuations at a rigid plane boundary is determined for flows with small but finite Mach number. The method of matched asymptotic expansions is applied in the coordinate normal to the boundary. The boundary layer is an inner region, and prescribes an “effective” boundary velocity distribution for the outer region, which is governed by the acoustic wave equation. Those components of effective velocity with supersonic phase speeds account for the radiation of sound. For the inviscid infinite plate model, the wall-pressure spectrum has a nonintegrable singularity at the acoustic critical wavenumber. Because of undamped contributions to point pressure from distant acoustic sources, in fact, the infinite model fails in an inviscid medium with any degree of compressibility. A large but finite model is considered, and the nonintegrable singularity at the critical wavenumber is removed. The spectrum coincides otherwise with the infinite plate result. The finite extent of the model can represent either a real geometrical limitation or the effect of damping over long distances. The intensity in the radiated field is shown to vary with the eighth power of velocity, but with a coefficient proportional to the logarithm of the characteristic in-plane dimension.