Green ’ s function discretization of Pridmore-Brown wave operator

A numerical method is presented for sound transmission through non-uniform flows with no restrictions on the ratio between the flow variation scale and the acoustic wavelength. Because it is based on the solution of a third-order convective linear wave equation for the pressure perturbation in the frequency domain, the method described on the following pages can be applied in the vortex sheet limit without any treatment of the wake shed by a trailing edge under sound excitation. This is an idealized representation of an aero-engine exhaust where noise generated by the fan system is transmitted through the annular bypass duct, diffracted by the duct lip, refracted by the external bypass stream, and radiated to the far field. The wave equation is obtained by linearization of the acoustic analogy equation put forth by Lilley (1974) around a time-independent base flow. The main interest of this formulation is not in the proper separation between sound propagation and generation, which is indeed unambiguous only for a unidirectional transversely sheared base flow, but in the wave operator itself, originally derived by Pridmore-Brown (1958). The numerical scheme is based on the Green’s function discretization technique previously developed for a second-order equation for the acoustic potential, and herein extended to Pridmore-Brown wave operator in order to avoid any explicit treatment of the potential discontinuity across the vortex sheet in turbofan bypass noise predictions.