Extension of the FW-H equation for acoustic velocity prediction

Acoustic velocity vector offers unique advantages in characterizing the acoustic field and flow sound decomposition. However, despite its wide application in extrapolating the far-field acoustic pressure, the classical FW-H equation cannot be directly used to predict the far-field acoustic velocity. Based the acoustic radiation from a point source in uniform flows, we established an explicit physical correlation in the frequency domain between the far-field acoustic velocity and pressure, leveraging the convective Green's function and the linearized momentum equation. Consequently, a novel analytical formulation for acoustic velocity in the frequency domain, FV3A-M, is proposed in this paper. The validity of FV3A-M is demonstrated by its successful application to benchmark cases involving a monopole and a dipole radiating in a uniform flow. Furthermore, The noise generated by a low-Mach-number turbulent flow past the rod-airfoil configuration is numerically obtained by DDES and the acoustic pressure extrapolated with the FW-H equation agree well with the experimental data. In the observation plane, the directivity of the streamwise component of acoustic velocity at the primary frequency resembles the characteristics of typical quadrupole acoustic pressure fields, whereas the distributions of the remaining two components exhibit characteristics of dipole acoustic pressure fields. Notably, the prediction of far-field acoustic velocity through FV3A-M is only a by-product of the integration process of FW-H equation's acoustic pressure extrapolation, significantly broadening the application scenarios of the FW-H equation.