Acoustic scattering in a duct with mean swirling flow

Abstract

In this paper we consider the diffraction of waves by a sharp edge in three-dimensional flow with non-zero mean vorticity. This is an extension of the famous Sommerfeld problem of the diffraction of waves by a sharp edge in quiescent conditions. The precise problem concerns an infinitely long annular circular cylinder, which contains a concentric semi-infinite circular cylinder which acts as a splitter. The mean flow has both axial and swirl components, and cases in which the splitter is arranged with either a leading edge or a trailing edge relative to the axial flow are considered. This is a model of a number of practical situations in the aeroengine context. We treat both sonic and nearly-convected incident disturbances, and two regimes are considered; one in which the azimuthal order, $m$, of the incident waves is $O(1)$, and a second in which $m\,{\gg}\,1$. A solution for $m\,{=}\,O(1)$ in the case of rigid-body swirl is found using the Wiener–Hopf technique, and special care is needed to handle the infinite accumulation of scattered nearly-convected modes which results from the presence of the mean vorticity. Simplification in the limit $m\,{\gg}\,1$ then allows us to consider more general swirl distributions. A number of effects arise due to the presence of mean vorticity. This includes the generation of sound at a trailing edge due to the scattering of a nearly-convected disturbance, which is to be contrasted with the way in which a convected gust passes a trailing edge silently in uniform mean flow.