A Spherical Multipole Expansion of Acoustic Analogy for Propeller Noise
This work develops a spherical-multipole expansion of the surface terms of an acoustic-analogy formulation, for the prediction of tonal noise from rotating propellers. The acoustic field is expressed through spherical multipoles, which separate source integrals from the observer dependence. This decoupling leads to computational efficiency: once the multipole coefficients are computed from blade geometry and aerodynamics, the sound field at any observer location is obtained by a simple evaluation of spherical harmonics and radial propagation factors, avoiding repeated integrations for each observer point. Moreover, this enables a straightforward radiated power calculation, without resorting to far-field pressure integrals. For hovering subsonic propellers, the results show a rapid convergence of the expansion. For each harmonic, the dominant radiation is accurately captured by the first two non-zero multipoles, corresponding to the leading symmetric and antisymmetric contributions with respect to the plane of rotation. To interpret the physical content of these leading terms, two simplified descriptions of the source integral are developed. The first is a lifting-surface formulation, suited to blades at small incidence, in which the thin-airfoil approximation allows to separate thrust-like loading, torque-like loading, and thickness contributions. The second is a lifting-line formulation, suited to high-aspect-ratio blades, in which the surface integral is reduced to spanwise integrals of compact sectional moments. The validity of the two formulations is assessed through comparisons of directivity, power distribution over harmonics and time-domain waveforms. Finally, both models are validated against experimental datasets taken from the literature, showing excellent agreement for tonal directivity and blade-passing harmonics.