A numerical simulation of the hole-tone feedback cycle based on an axisymmetric discrete vortex method and Curle's equation

An axisymmetric numerical simulation approach to the hole-tone self-sustained oscillation problem is developed, based on the discrete vortex method for the incompressible flow field, and a representation of flow noise sources on an acoustically compact impingement plate by Curle's equation. The shear layer of the jet is represented by 'free' discrete vortex rings, and the jet nozzle and the end plate by bound vortex rings. A vortex ring is released from the nozzle at each time step in the simulation. The newly released vortex rings are disturbed by acoustic feedback. It is found that the basic feedback cycle works hydrodynamically. The effect of the acoustic feedback is to suppress the broadband noise and reinforce the characteristic frequency and its higher harmonics. An experimental investigation is also described. A hot wire probe was used to measure velocity fluctuations in the shear layer, and a microphone to measure acoustic pressure fluctuations. Comparisons between simulated and experimental results show quantitative agreement with respect to both frequency and amplitude of the shear layer velocity fluctuations. As to acoustic pressure fluctuations, there is quantitative agreement w.r.t. frequencies, and reasonable qualitative agreement w.r.t. peaks of the characteristic frequency and its higher harmonics. Both simulated and measured frequencies f follow the criterion L/uc+L/c0=n/f where L is the gap length between nozzle exit and end plate, uc is the shear layer convection velocity, c0 is the speed of sound, and n is a mode number (n=12,1,32,…). The experimental results however display a complicated pattern of mode jumps, which the numerical method cannot capture.