Computationally efficient, frequency-domain quadrupole corrections for the Ffowcs Williams and Hawkings equation

In the present article, frequency-domain formulations of quadrupole corrections are derived in a computationally efficient form for the Ffowcs Williams and Hawkings (FW-H) equation with permeable control surfaces. Quadrupole corrections effectively reduce spurious noise associated with hydrodynamic fluctuations passing across integral surfaces, originally derived for Formulation 1A of Farassat in the time domain. When a corresponding frequency-domain formulation is sought, however, difficulty arises as its Green’s function is written in a convective form and different from that of Formulation 1A. First, the mathematical framework of the convective FW-H equation is shown to be equivalent to Formulation 1A by applying a simple Galilean transformation for rectilinear motion. Then, a frequency-domain formulation is derived via a Fourier transform applied directly to the time-domain quadrupole correction forms. The results of the derived formulation agree precisely with the time-domain solutions, in the verification study of vortex convection, as well as non-uniform entropy convection, in which spurious noise can be effectively removed by the present quadrupole correction integrals.