A frequency-domain formulation is proposed to compute the far-field noise generated by flows with periodically oscillating or rotating boundaries. The proposed formulation significantly enhances the efficiency of the frequency-domain method in handling the multi-frequency sources with nonrectilinear motion. The novelty of the proposed method is that the frequency- and time-dependent components of the Ffowcs Williams and Hawkings (FW-H) integral are separated by using the far-field asymptotic Green’s function. The separation of the frequency- and time-dependent components avoids the need for an expensive time integration in computing the multi-frequency noise generated by flows with periodically moving boundaries. They proposed only one Fourier transform computation in obtaining the noise at different frequencies. The efficiency of the proposed formulation is investigated by analyzing the required number of floating-point operations. Its validity is examined by computing the noise from rotating or oscillating permeable boundaries around composite monopoles and a flapping wing. The proposed formulation is applicable to the FW-H integral with periodically oscillating or rotating boundaries when the maximum velocity on the moving boundary is subsonic.

提出了一种频域公式来计算由具有周期性振荡或旋转边界的流产生的远场噪声。所提出的公式显着提高了频域方法在处理具有非线性运动的多频率源时的效率。所提出方法的新颖之处在于，Ffowcs Williams 和 Hawkings (FW-H) 积分的频率和时间相关分量通过使用远场渐近格林函数分离。频率和时间相关分量的分离避免了在计算由具有周期性移动边界的流产生的多频噪声时进行昂贵的时间积分的需要。他们只提出了一次傅立叶变换计算来获得不同频率的噪声。通过分析所需的浮点运算次数来研究所提出公式的效率。通过计算复合单极子和扑翼周围的旋转或振荡可渗透边界的噪声来检查其有效性。当移动边界上的最大速度为亚音速时，所提出的公式适用于具有周期性振荡或旋转边界的 FW-H 积分。