The breakdown of the linearized theory and the role of quadrupole sources in transonic rotor acoustics

Abstract

The retarded Green's function for the linearized version of the equation of the mixed type governing the potential flow around a rotating helicopter blade or a propeller (with no forward motion) is derived and is shown to constitute the unifying feature of the various existing approaches to rotor acoustics. This Green's function is then used to pinpoint the singularity predicted by the linearized theory of rotor acoustics which signals its experimentally confirmed breakdown in the transonic regime: the gradient of the near-field sound amplitude, associated with a linear flow which is steady in the blade-fixed rotating frame, diverges on the sonic cylinder at the dividing boundary between the subsonic and supersonic regions of the flow. Prom the point of view of the equivalent Cauchy problem for the homogeneous wave equation, this singularity is caused by the imposition of entirely non-characteristic initial data on a space—time hypersurface which, at its points of intersection with the sonic cylinder, is locally characteristic. It also emerges from the analysis presented that the acoustic discontinuities detected in the far zone are generated by the quadrupole source term in the Ffowcs Williams-Hawkings equation and that the impulsive noise resulting from these discontinuities would be removed if the flow in the transonic region were to be rendered unsteady (as viewed from the blade-fixed rotating frame).