Scattering of stable and unstable waves in a flow duct

Abstract

The transmission and reflection properties of sound at sharp edges are the key elements when calculating the performance of absorptive and reactive splitter silencers using the Building Block method. These acoustic properties are here determined for a two-dimensional duct, having an air flow in its lower part only. In one, half-infinite, section an acoustically hard wall separates the moving and still media, while in the other there is an infinitely shear layer, that is, a vortex sheet is present. Leading and trailing edges are dealt with, while scattering problems, expressed through Fourier methods as Wiener-Hopf equations, solved under causality and edge behaviour constraints, provide a unique solution. Explicit expressions for scattering matrices are constructed and numerical examples are presented and discussed. A vital part of the theory is an analysis of the modal structure in the part of the duct having the infinitely thin shear layer. To this end Green's function is determined, and by applying causality the unique solution is found as an inverse Fourier integral, which is expressed as an infinite sum of modes. Therefore the modal system is complete. These modes, the properties of which are essential in establishing causality, have been studied by analytical techniques and the results are reported. The existence of ordinary acoustic modes as well as hydrodynamic ones is verified and the latter propagate only downstream. One of the hydrodynamic modes is found to be unstable for all frequencies, a consequence of causality: its amplitude grows rather than decays with the downstream distance. An example is given of an unusual acoustic mode that gets cut-on at a certain frequency and stays cut-on for higher frequencies except for a frequency interval where it is cut-off.