The Ffowcs Williams–Hawkings equation is widely used in computational aeroacoustics to post-process unsteady simulations and provide the sound at distances beyond the accurate range of the grid. It distinguishes the monopole/dipole contributions from solid surfaces Σ and that from quadrupoles present in the volume of the fluid. Curle showed that at low Mach numbers, the solid-surface terms are stronger than the volume term. Very few studies have included the volume term itself, but many have used a permeable Ffowcs Williams–Hawkings surface Σ, which in principle surrounds the quadrupoles, giving valid results independent of Mach number; in reality for external flows, the surface cannot surround all the quadrupoles. However, in spite of doubts over the solid-surface approach even at low Mach numbers, it is widely used because of its simplicity and the difficulties associated with turbulence crossing the permeable surface. We consider Mach numbers M up to 0.25, which challenges approximations based on the property that “M ≪1.” An additional attraction of the solid-surface approach is the idea of identifying the “true” source of the sound by computing separately the Ffowcs Williams–Hawkings integrals for different components. We wish to determine whether this “self-evident” argument gives an effective approach, and in general to assess Curle’s approximations, using a sphere-dipole problem and three model problems related to landing-gear noise, namely an isolated rectangular body, a fuselage with a cavity, and one with a bluff body under it. One key test is the shielding of sound toward various directions; any approach that misses this shielding is suspect. The overall conclusion is, again, that Curle’s approximations succeed in some cases but are quite inaccurate in the audible range for an airliner in approach and for high-speed trains, and also that separating the components’ contributions is often misleading. In contrast we verify that, for our model problems and with adequate grid resolution and surface placement, the permeable-surface results are as accurate as the simulation itself is, irrespective of distance, direction, or Mach number.

中文翻译：

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基于固体和渗透表面 Ffowcs Williams-Hawkings 积分解的噪声预测差异

Ffowcs Williams-Hawkings 方程广泛用于计算气动声学，以对非定常模拟进行后期处理，并在超出网格准确范围的距离提供声音。它区分了固体表面 Σ 的单极/偶极贡献和流体体积中存在的四极贡献。Curle 表明，在低马赫数下，固体表面项强于体积项。很少有研究包括体积项本身，但许多研究使用了可渗透的 Ffowcs Williams-Hawkings 表面 Σ，它原则上围绕四极杆，给出独立于马赫数的有效结果；实际上，对于外部流动，表面不能包围所有四极杆。然而，尽管即使在低马赫数下也对固体表面方法存有疑问，它被广泛使用，因为它的简单性和与穿过可渗透表面的湍流相关的困难。我们考虑马赫数 M 高达 0.25，这挑战了基于“M ≪1”属性的近似值。固体表面方法的另一个吸引力是通过分别计算不同分量的 Ffowcs Williams-Hawkings 积分来识别声音的“真实”来源的想法。我们希望确定这种“不言自明”的论点是否提供了一种有效的方法，并且一般来说，使用球偶极子问题和与起落架噪声相关的三个模型问题，即孤立的矩形体、机身一个空腔，一个在它下面有一个钝体。一项关键测试是对各个方向的声音的屏蔽；任何错过这种屏蔽的方法都是可疑的。总体结论再次表明，Curl 的近似值在某些情况下是成功的，但在接近的客机和高速列车的可听范围内非常不准确，而且分离组件的贡献通常会产生误导。相比之下，我们验证了，对于我们的模型问题，并且具有足够的网格分辨率和表面放置，渗透表面结果与模拟本身一样准确，而与距离、方向或马赫数无关。