Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces

TLDR

The study compares the Lighthill acoustic analogy (FW‑H equation) with the Kirchhoff formulation for moving surfaces. The authors introduce new Sobolev‑norm‑based metrics to compare input data for quadrupole and Kirchhoff noise predictions. These metrics provide a framework for evaluating input data in both quadrupole noise calculations and Kirchhoff noise predictions. The FW‑H equation outperforms the Kirchhoff formulation by incorporating nonlinear flow effects without requiring an impenetrable surface, remains valid in nonlinear regions, and this advantage is numerically confirmed, whereas the Kirchhoff approach can incur hard‑to‑detect errors when the integration surface lies outside the linear region.

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 primary advantage of the Kirchhoff formulation (namely, that 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 to be impenetrable. The FW-H equation is analytically superior for aeroacoustics because it is based on the conservation laws of fluid mechanics rather than on the wave equation. Thus, the FW-H equation is valid even if the integration surface is in the nonlinear region. This advantage is demonstrated numerically. With the Kirchhoff approach, substantial errors can result if the integration surface is not positioned in the linear region, and these errors may be hard to identify. Finally, new metrics, based on the Sobolev norm, are introduced that may be used to compare input data for both quadrupole noise calculations and Kirchhoff noise predictions.