Quadrupole source term corrections based on correlation functions for Ffowcs Williams and Hawkings integrals

The quadrupole source term in the Ffowcs Williams and Hawkings (FW-H) integrals usually causes spurious contributions to the far field sound. The spurious contributions are contributed by the convection of the Lighthill stress tensor crossing the FW-H integral surface when the integral domain cannot include all the quadrupole sources. We proposed a quadrupole source term correction model within the frequency domain to correct the errors associated with the convection of Lighthill stress tensor. Based on Taylor's hypothesis, the proposed model uses the correlation functions of the Lighthill stress tensor to compute the convection velocity. Compared with the using of uniform upstream velocity as the convection velocity, the proposed model has the advantages of taking account for the non-uniform distribution of the convection velocity. We validate the proposed model by computing the far-field sound generated by the two-dimensional convecting vortex and the flows around a circular cylinder. The proposed model significantly improves the directivity of the far field sound generated by the flows around the circular cylinder.