A Spherical Multipole Expansion of Acoustic Analogy for Propeller Noise

Abstract

This work develops a spherical-multipole expansion of Goldstein's acoustic analogy, 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 lift-like loading, drag-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. The results show good accuracy in their respective regimes of validity, together with substantial computational savings.

Figures (9)

• Figure 1: Reference geometry and observer coordinates for the rotating two-bladed propeller. The rotor spins about the $z$-axis with angular velocity $\Omega$ in the fixed Cartesian frame $(x,y,z)$. The observer is located at $\mathbf{x}_0$ at distance $r_0$ from the origin, with spherical elevation $(\theta_0)$ and azimuth angles $(\phi_0)$. The source position in the frame attached to the propeller is denoted by $\mathbf{x}'$.
• Figure 2: Decomposition of the harmonic directivity for the first non-zero harmonic $m=2$. The curves show the isolated contributions of the first two degrees, $\ell=2$ and $\ell=3$, their coherent sum, and the complete harmonic directivity obtained from the full multipole expansion.
• Figure 3: Colour map of $10\log_{10}|j_\ell(mM_t)|$ [dB] in the $(mM_t,\ell)$-plane. The region $\ell<m$ is not admissible. The dashed line is $\ell=10\,mM_t$.
• Figure 4: Lifting-surface geometry.The $x'$-axis is attached to the rotor, passing through the mid-chord of a reference blade section. The trailing-edge position of each section is then described by the azimuthal offset $\phi'_{TE}$ measured from this line, with counter-clockwise angles taken as positive (negative value in the present example).
• Figure 5: Lifting-line geometry.