Abstract In this paper, the sound generated by three-dimensional flexible flapping wings during hovering flight is numerically studied by using an immersed boundary method. The wing shape, prescribed motion of the wing leading edge, Reynolds number, wing-to-fluid mass ratio and wing flexibility are systematically examined. The numerical results show that these governing parameters have a significant influence on the aerodynamic performances and consequently the acoustic outputs. The comparisons among four wing shapes indicate that the wing shape does not influence the sound directivity. The wing with larger area close to the wing-tip generates more lift associated with a larger acoustic output. The systematic examinations of four pitching amplitudes suggest that the optimal one is π ∕ 2 , which provides the highest efficiency with the lowest acoustic output. The effects of the flexibility are examined at two wing-to-fluid mass ratios (i.e., m ∗ = 1 . 0 and 5.0) with the dimensionless flapping frequency ω ∗ ranging from 0 to 0.5. The results show that an appropriate flexibility enhances the aerodynamic performance and reduces the acoustic outputs. The optimal choice is to use a flexible wing whose inertia is comparable to its aerodynamics, i.e., m ∗ = 1 . 0 and ω ∗ = 0 . 3 , by which the highest efficiency is achieved with a relatively low acoustic outputs (30.5% lower than that of a rigid wing) and high lift generation. By collecting all the numerical data together, a linear relationship of the sound power and the root-mean-square of the power coefficient is observed. By using the fitted equation, the acoustic outputs of a flapping wing can be directly obtained from its aerodynamics, which can be used to simplify the engineering design and optimization process considering acoustic outputs.

中文翻译：

---

悬停飞行中三维柔性扑翼发声数值研究

摘要 本文采用浸入边界法对悬停飞行过程中三维柔性扑翼产生的声音进行了数值研究。系统地检查了机翼形状、机翼前缘的规定运动、雷诺数、机翼与流体质量比和机翼柔韧性。数值结果表明，这些控制参数对空气动力学性能和声学输出有显着影响。四种翼型之间的比较表明，翼型不影响声音的指向性。靠近翼尖的面积较大的机翼会产生更大的升力，并伴随着更大的声输出。四个俯仰幅度的系统检验表明最佳的一个是 π ∕ 2 ，以最低的声输出提供最高的效率。在两个机翼与流体质量比（即，m ∗ = 1 . 0 和 5.0）下检查灵活性的影响，无量纲扑动频率 ω ∗ 范围从 0 到 0.5。结果表明，适当的柔韧性提高了空气动力学性能并降低了声输出。最佳选择是使用惯性与其空气动力学相当的柔性机翼，即 m ∗ = 1 。0 和 ω ∗ = 0 。如图3所示，通过相对较低的声输出（比刚性机翼低30.5%）和高升力产生实现最高效率。通过将所有数值数据收集在一起，可以观察到声功率与功率系数的均方根的线性关系。通过使用拟合方程，