The importance of quadrupole sources in prediction of transonic tip speed propeller noise

A theoretical analysis is presented for the harmonic noise of high speed, open rotors. Far field acoustic radiation equations based on the Ffowcs-Williams/Hawkings theory are derived for a static rotor with thin blades and zero lift. Near the plane of rotation, the dominant sources are the volume displacement and the rho U(2) quadrupole, where u is the disturbance velocity component in the direction blade motion. These sources are compared in both the time domain and the frequency domain using two dimensional airfoil theories valid in the subsonic, transonic, and supersonic speed ranges. For nonlifting parabolic arc blades, the two sources are equally important at speeds between the section critical Mach number and a Mach number of one. However, for moderately subsonic or fully supersonic flow over thin blade sections, the quadrupole term is negligible. It is concluded for thin blades that significant quadrupole noise radiation is strictly a transonic phenomenon and that it can be suppressed with blade sweep. Noise calculations are presented for two rotors, one simulating a helicopter main rotor and the other a model propeller. For the latter, agreement with test data was substantially improved by including the quadrupole source term.