Analysis of spurious sound due to vortical flow through permeable surfaces

Abstract

Aeroacoustic wave equations proposed by Lighthill and developed by Ffowcs-Williams and Hawkings (FW-H) have been widely used to analyze sound generated from turbulence and its interaction with solid surfaces, such as jet noise and airfoil noise. In engineering applications, the wave equation is usually solved by integral formulations derived by Farassat, where quadrupole volume sources outside permeable integral surfaces are usually ignored in subsonic flow. However, several existing studies have shown that the acoustic prediction result will be greatly contaminated by spurious sources due to vortical components crossing permeable integral surfaces. This paper analytically studies spurious source terms on permeable integral surfaces, and concludes that the convective FW-H equation is always exact to predict the aerodynamic noise because the material derivative and divergence operators in the monopole and dipole sources enable to automatically filter spurious sources located on the permeable integral surfaces. However, the formulations of Farassat fail to filter the spurious sound owing to replacing the spatial derivative by the temporal derivative. A three-dimensional convective frequency-domain version of the Kirchhoff integral formulation is developed to resolve this issue, because it does not use velocity fluctuations as the input variable. Numerical test cases are performed to validate the analytical result and the developed formulation.