Sound Radiation from an Annular Duct with Jet Flow and a Lined Centerbody

Analytical solutions for the problem of radiation of sound from a duct with flow have been shown to serve as an important first model for acoustic engine-aircraft engineering applications, useful for understanding the physics and for validating and benchmarking numerical solutions. The probably most important steps were taken by Levine & Schwinger and Munt. He showed that application of the Kutta condition, in order to determine the amount of vorticity shed from the trailing edge, goes together with excitation of the KelvinHelmholtz instability wave of the jet. In the present work we will extend this work further by considering the effect of lining on the centerbody and on the afterbody, which is mathematically a much different problem. The lining is of impedance type, while the Ingard-Myers boundary condition is taken to include the effect of the mean flow. The flow is assumed to be subsonic. Differences in the (otherwise uniform) mean flow velocity, density and temperature are taken into account. A vortex sheet which separates the jet from the outer flow emanates from the edge. Due to this velocity discontinuity of the mean flow the jet is unstable. An analytical solution satisfying the full or partial Kutta condition at the trailing edge is found by means of the Wiener-Hopf technique. The jet instability wave is taken apart from the rest of the solution and the effect of applying the Kutta condition to the scattered field is shown, both in far field and near field. The influence of lining on the radiation is displayed graphically. The problem of a lined afterbody is of particular interest, because its Wiener-Hopf solution embodies a novel application of the so-called “weak factorization”. It should be noted that our primary goal here is to archive the mathematical solution, rather than to unravel all physical possibilities and explore the whole spectrum of problem parameters and their combinations.