On a Generalized Multipole Expansion with Application to Propeller Design Synthesis

Abstract

The radiation of sound by surfaces in motion in a non-uniform flow, including the effects of reflections from obstacles on noise, is specified by an extension of the Kirchhoff integral that leads to a generalized multipole expansion that extends the classical series of spherical harmonics to account for the effects of ( i) the moving medium and ( ii) the rotating sound sources. The corresponding radiation integrals are evaluated analytically for an arbitrary source distribution along the blades of a propeller at an angular inflow. The effects of the incident flow and propeller rotation on the amplitude and phase of sound, e.g. through the retarded time, lead to an extension of the generating function for Legendre polynomials; this provides the representation of the acoustic field as the generalized multipolar series. Each generalized multipole sound field is shown to consist of a fundamental blade passng frequency (BPF) plus all harmonics, with directivities specified by integrals of Bessel functions, that are evaluated analytically. The theory is validated by comparison with noise measurements using two different model propellers in two distinct aeroacoustic wind tunnels. The experimental results are used to illustrate the principle of propeller noise synthesis, relating the radiated sound field to the sound source distribution.