Frequency-domain quadrupole correction for the permeable-surface Ffowcs Williams and Hawkings integration

The permeable-surface Ffowcs Williams and Hawkings (FW–H) integration for computing the far-field sound has the advantage of encapsulating the sources and nonlinear propagation inside the integral surface. However, it suffers from spurious sound when the volume integral for quadrupole term outside the permeable surface is conventionally ignored. The spurious sound is often suppressed by using two distinct approaches, which modifies the FW–H integration and acoustic variables/sources, respectively. This work clarifies the connection between the two approaches by analyzing the integral of the quadrupole sources. We show that the modification of the acoustic sources can be reformulated as a modification of the FW–H integration, which means that the two distinct approaches are interconvertible. A new quadrupole correction model for the FW–H integration is proposed by delicately modifying the acoustic sources. The modified acoustic sources consist of the filtered Lighthill stress tensor, where a convection operator is used to filter out the acoustically inefficient components. The proposed quadrupole correction model is consistent with the previous work on the modification of the FW–H integration under special conditions with the uniform convection velocity. The proposed model is validated by computing the sound pressure generated by laminar and turbulent flows over bluff bodies. It is found that the sensitivity of the acoustic pressure to the FW–H surface's position is suppressed and the accuracy of the predicted sound is improved. The results suggest that the modification of acoustic variables/sources can be a powerful method to construct new quadrupole correction models for the permeable FW–H integration.