On the use of a Prandtl-Glauert-Lorentz transformation for acoustic scattering by rigid bodies with a uniform flow

Abstract

It is well-known that the convective wave equation with a uniform mean flow can be transformed into a standard wave equation without flow by a Prandtl-Glauert-Lorentz type transformation. This paper examines the boundary condition to be used in the transformed coordinates for acoustic scattering by rigid bodies. Recently, based on the energy conservation equation for acoustic waves propagating in a uniform flow, a new solid wall boundary condition, dubbed the Zero Energy Flux (ZEF) solid surface boundary condition, has been proposed. Instead of the commonly used solid surface boundary condition that the normal acoustic velocity be zero, the ZEF condition requires that the acoustic energy flux be zero. In this paper, we point out that when formulated in the acoustic velocity potential and under the ZEF boundary condition, the normal derivative of the velocity potential remains to be zero when computed in the transformed coordinates. As such, numerical methods developed for solving the standard wave equation can be directly applied to the convective wave equation through a use of the Prandtl-Glauert-Lorentz transformation. Utilization of the transformation is particularly beneficial for numerical methods based on the boundary integral formulation because of the simplified kernel function in the transformed coordinates, for both the time domain and the frequency domain approaches. Useful relations for applying the transformation are derived and numerical examples that demonstrate the effectiveness of the transformation and ZEF boundary condition are presented.