Far-field approximations to the derivatives of Green\u2019s function for the Ffowcs Williams and Hawkings equation

Abstract

The surface correction to the quadrupole source term of the Ffowcs Williams and Hawkings integral in the frequency domain suffers from the computation of high-order derivatives of Green’s function. The far-field approximations to the derivatives of Green’s function have been used without derivation and verification in previous work. In this work, we provide the detailed derivations of the far-field approximations to the derivatives of Green’s function. The binomial expansions for the derivatives of Green’s function and the far-field condition are employed during the derivations to circumvent the difficulties in computing the high-order derivatives. The approximations to the derivatives of Green’s function are systemically verified by using the benchmarks two -dimensional convecting vortex and the co-rotating vortex pair. In addition, we provide the derivations of the approximations to the multiple integrals of Green’s function by using the far-field approximations to the derivatives.