Acoustic and hydrodynamic power of wave scattering by an infinite cascade of plates in mean flow

Abstract

We study the balance of outgoing and incoming power for acoustic wave scattering by a cascade of flat blades in uniform subsonic mean flow. In nonzero mean flow the Kutta condition at the trailing edge of the cascade plates leads to the production of vorticity in the form of unsteady vortex sheets attached at the trailing edges, which results in a hydrodynamic power flux and the exchange of energy between the unsteady vorticity and the acoustic field. Although the scattering problem itself has been subject to much previous research, a comprehensive study of the composition of the outgoing energy flux for cascade scattering has received little previous attention. As an application of our detailed study we show how the time-averaged energy balance can be used to explain certain symmetries of the field in the angle of incidence with respect to the cascade face when there is no mean flow and the effect of zero acoustic reflection at certain angles of incidence with mean flow. We also show that, depending on the angle of incidence of the acoustic wave and the velocity of the mean flow, the vortex sheets can either absorb or generate acoustic energy. This can, for certain parameter values, result in a net increase in outgoing sound power over the incident sound power, as we demonstrate on a number of numerical examples that are computed using the semi-analytical solution described in recent work by Maierhofer and Peake (J. Sound Vib., vol. 481 (2020), 115418).