A numerical experiment to determine whether surface shear-stress fluctuations are a true sound source

The sound generated due to a localized flow over a large (compared to the acoustic wavelength) plane no-slip wall is considered. It has been known since 1960 that for inviscid flow the pressure, while appearing to be a source of dipole sound in a formal solution to the Lighthill equation, is, in fact, not a true dipole source, but rather represents the surface reflection of volume quadrupoles. The subject of the present work—namely, whether a similar surface shear stress term constitutes a true source of dipole sound—has been controversial. Some have boldly assumed it to be a true source and have used it to calculate the noise in boundary-layer flows. Others have argued that, like the surface pressure, the surface shear stress is not a valid source of sound but rather represents a propagation effect. Here, a numerical experiment is undertaken to investigate the issue. A portion of an otherwise static wall is oscillated tangentially in an acoustically compact region to create shear stress fluctuations. The resulting sound field, computed directly from the compressible Navier-Stokes equations, is nearly dipolar and its amplitude agrees with an acoustic analogy prediction that regards the surface shear as acoustically compact and as a true source of sound. However, there is a correction that becomes noticeable for observers near wall-grazing angles as the computational domain size Ld along the wall is increased. An estimate, validated by the simulations, shows that as Ld→∞ the correction to the sound in the Fraunhofer zone is proportional to δbl∕λ (the ratio of oscillatory boundary-layer thickness to acoustic wavelength) times a directivity factor that becomes large at angles close to grazing. For observers at such angles, Lighthill’s acoustic analogy does not apply.