[论文解读] A Spherical Multipole Expansion of Acoustic Analogy for Propeller Noise

Felice Fruncillo, Paolo Luchini+1 more|arXiv (Cornell University)|Mar 19, 2026

Aerodynamics and Acoustics in Jet Flows被引用 0

一句话总结

论文发展了 Goldstein 声学类比的球面多极展开，以预测音调螺桨噪声，解耦源几何与观测位置，并展示前两个多极的快速收敛。

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.

研究动机与目标

• 为跨观测位置的旋转螺桨的准确音调噪声预测提供动机。
• Develop a spherical-multipole representation that decouples source terms from observer coordinates.
• 评估悬停亚音速螺桨的前导多极的收敛性与物理含义。
• 提供简化的 LS 和 LL 近似，以解释主导多极并降低计算量。

提出的方法

• 从 Goldstein 的声学类比出发，考虑表面力和法向速度源项。
• 通过球面多极展开表示格林函数，以分离源与观测依赖。
• 将复数多极系数 A_{ℓm} 表示为与 g_{ℓm} 相关的 f 与 V_n 的表面积分（Eq. 11）。
• 将谐波压力表示为 p̂_m(x0)=∑_{ℓ≥|m|} A_{ℓm} f_{ℓm}(x0)（Eq. 12）。
• 从 Π_m = (1/(2ρ0 a_s k_m^2)) ∑_{ℓ≥|m|} |A_{ℓm}|^2（Eq. 13）计算声学功率。
• 给出远场简化并讨论主导多极（ℓ=m, ℓ=m+1）及其物理含义。

实验结果

研究问题

• RQ1 球面多极展开的结构和充分性在捕捉螺桨音调辐射方面如何？
• RQ2 悬停在亚音速下，多极系数随 ℓ 的衰减有多快？
• RQ3 哪些物理载荷贡献（升力、阻力、厚度）映射到 Goldstein 类比中的主导多极？
• RQ4 LS 与 LL 近似相较于精确表面积分在预测指向性与功率分布方面如何？
• RQ5 该框架是否能给出直观的辐射功率表达式，而无需远场积分？

主要发现

• 悬停亚音速条件下，前两个非零多极（ℓ=m 和 ℓ=m+1）主导音调场。
• 更高阶多极（ℓ≥m+2）数量级上要小若干数量级，常常可以忽略。
• 主导多极内容和指向性取决于刀片几何形状（gamma）和端部马赫数 M_t，随着 M_t 增大，高阶多极贡献更强。
• 两种简化描述（提升面与提升线）在其有效区间再现主导多极行为，且提供显著的计算节省。
• 球面多极方法给出直接的辐射功率表达式，无需单独的远场压力积分。
• 指向性分析表明前两多极能够捕捉本征的角分布和谐波辐射的模态干涉形状。