Whistling of deep cavities subject to turbulent grazing flow: intermittently unstable aeroacoustic feedback

Abstract

In this work, the classic problem of the aeroacoustic instability occurring in deep cavities subject to a low-Mach grazing flow is revisited experimentally and theoretically. This instability is caused by the constructive feedback between the acoustic modes of the cavity and the turbulent shear layer that forms at its opening. Systematic experiments are performed in order to construct a new theoretical model, which describes the aeroacoustic system as two linearly stable oscillators, with linear reactive coupling, nonlinear damping and nonlinear resistive coupling. This model constitutes the basis for a linear stability analysis, and for the prediction of limit cycle amplitudes by using a describing function approach and by searching the fixed points of amplitude equations. Moreover, it is shown that only supercritical Hopf bifurcations are found in this aeroacoustic system, and that, in contrast with many flow-induced vibration problems, frequency lock-in does not occur. In the last part of the paper, the intermittency observed in the vicinity of the supercritical Hopf bifurcations is successfully modelled by adding a coloured multiplicative noise to the grazing flow velocity in order to account for the effect of turbulence. The necessary conditions favouring intermittently stable or intermittently unstable intervals in such systems are identified using stochastic differential equations governing the aeroacoustic oscillations and Fokker-Planck equations ruling the probability density function of the acoustic envelope. This work is relevant for many musical and industrial configurations exhibiting this type of aeroacoustic instability, as well as for thermoacoustic instabilities in turbulent combustors for aeronautic and power generation applications.