Acoustic radiation of Tollmien–Schlichting waves as they undergo rapid distortion

Abstract

This paper is primarily concerned with acoustic radiation of instability waves as they undergo rapid distortion, which is one of the fundamental mechanisms by which instability modes generate sound in subsonic flows. To fix the idea, we consider the case where the abrupt distortion is associated with scattering of a Tollmien–Schlichting (T-S) wave by the mean flow induced by a localized surface roughness in a compressible subsonic boundary layer with an $O(1)$ free-stream Mach number. The sound field was calculated by extending the asymptotic approach based on the triple-deck formalism, developed previously. This approach allows us to identify and approximate the sound source systematically by seeking the solution for the near field hydrodynamics as an asymptotic series in ascending powers of $\epsilon=R^{-1/8}$, where $R$ is the Reynolds number at the roughness site. It is found that the first four terms in the expansion act as octupole, quadrupole, dipole and monopole, respectively, and they make equal order-of-magnitude contributions to the acoustic far field. Some rather delicate source cancellations are noted. As a by-product, the analysis also shows that a localized roughness also influences the energetics of the T-S wave, and that effect can be characterized by a transmission coefficient, defined as the ratio of the T-S wave amplitude after the scattering to that before the scattering.