Quadrupole Corrections for the Permeable-Surface Ffowcs Williams–Hawkings Equation

This paper focuses on the spurious noise of the Ffowcs Williams and Hawkings equation with a permeable integral surface, which is now commonly used in far-field noise prediction for computational aeroacoustics methodologies. Spurious noise arises in the surface integral terms as an adverse effect of truncating a computational domain resolved for the quadrupole sources so that hydrodynamic fluctuations pass through the integral surface with mean flow. First, the quadrupole contribution is reformulated in the form of surface integrals for additive correction by introducing a frozen turbulence assumption for uniform mean flow. The math formulation is verified for simple convecting vortices. The derived formulation is then extended to nonuniform convection for locally subsonic flow. By adding the correction to the conventional Ffowcs Williams and Hawkings surface integrals, spurious noise is suppressed in two-dimensional wake flow. In a three-dimensional turbulent jet-noise simulation, the proposed correction, extended to nonuniform convection, is demonstrated to reduce the dependence of the prediction on the streamwise end-cap location of the integral surface. The present method requires no tunable parameters, with only a minimal additional cost compared to the conventional permeable-surface integrals in the far-field noise computation.