The acoustic field of high-speed turbulent jets is dominated by a small number of low-wavenumber azimuthal Fourier modes. Accordingly, it is of interest to directly obtain individual azimuthal modes of the acoustic field from simulation data or models of the jet near-field. To this end, we manipulate the Ffowcs Williams and Hawkings (FW–H) equation to obtain a new formulation that lives entirely in the azimuthal Fourier domain—it delivers individual azimuthal modes of the acoustic field as a function of the same azimuthal modes of the near-field FW–H source term or, upon linearization of the source terms, as a function of the same azimuthal modes of the near-field flow variables. As an added benefit, all surface integrals are converted into line integrals in the streamwise–radial plane. After verifying and validating our formulation using a monopole problem with an exact solution and large-eddy simulation data, respectively, we show how our method can be used to efficiently and naturally compute the acoustic field associated with resolvent modes of a Mach 1.5 jet, thus avoiding the need to compute the modes on a large computational domain to capture their acoustic radiation.

中文翻译：

---

Ffowcs Williams 和 Hawkings 方程的方位傅里叶域公式

高速湍流射流的声场由少数低波数方位傅立叶模式主导。因此，从射流近场的模拟数据或模型中直接获得声场的各个方位模式是有意义的。为此，我们操纵 Ffowcs Williams 和 Hawkings (FW-H) 方程以获得一个完全存在于方位傅立叶域中的新公式——它提供声场的各个方位模式作为相同方位模式的函数近场 FW-H 源项，或在源项线性化后，作为近场流变量的相同方位模式的函数。作为一个额外的好处，所有表面积分都转换为流向 - 径向平面中的线积分。