Supersonic Quadrupole Noise Theory for High-Speed Helicopter Rotors

High-speed helicopter rotor impulsive noise prediction is an important problem of aeroacoustics. The deterministic quadrupoles have been shown to contribute significantly to high-speed impulsive (HSI) noise of rotors, particularly when the phenomenon of delocalization occurs. At high rotor-tip speeds, some of the quadrupole sources lie outside the sonic circle and move at supersonic speed. Brentner has given a formulation suitable for efficient prediction of quadrupole noise inside the sonic circle. In this paper, we give a simple formulation based on the acoustic analogy that is valid for both subsonic and supersonic quadrupole noise prediction. Like the formulation of Brentner, the model is exact for an observer in the far field and in the rotor plane and is approximate elsewhere. We give the full analytic derivation of this formulation in the paper. We present the method of implementation on a computer for supersonic quadrupoles using marching cubes for constructing the influence surface (Sigma surface) of an observer space- time variable (x; t). We then present several examples of noise prediction for both subsonic and supersonic quadrupoles. It is shown that in the case of transonic flow over rotor blades, the inclusion of the supersonic quadrupoles improves the prediction of the acoustic pressure signature. We show the equivalence of the new formulation to that of Brentner for subsonic quadrupoles. It is shown that the regions of high quadrupole source strength are primarily produced by the shock surface and the flow over the leading edge of the rotor. The primary role of the supersonic quadrupoles is to increase the width of a strong acoustic signal.