A superior Kirchhoff method for aeroacoustic noise prediction: The Ffowcs Williams–Hawkings equation

Abstract

The Lighthill acoustic analogy, as embodied in the Ffowcs Williams–Hawkings (FW–H) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchhoff advantage (namely, nonlinear flow effects are included in the surface integration) is also available to the FW–H method if the integration surface used in the FW–H equation is not assumed impenetrable. The FW–H equation is analytically superior for aeroacoustics because it is based upon the conservation laws of fluid mechanics rather than the wave equation. Hence, the FW–H equation is valid even if the integration surface is in the nonlinear region. This is demonstrated numerically for helicopter rotor applications in the paper. The Kirchhoff approach can lead to substantial errors if the integration surface is not positioned in the linear region (i.e., if the input data are not a solution to the wave equation). These errors may be hard to identify in some cases.