Acoustically induced instabilities in pipes by the Nyquist criterion

In process pipes, orifices and pressure relief valves are sometimes prone to self-sustained oscillations in the presence of acoustic reverberating conditions. Former studies have shown that this kind of instability can be adequately described by a linear approach inspired by control theory, where the pressure drop device behaves like an acoustic amplifier and the surrounding pipe as an acoustic feedback. In order to better understand the instability conditions, a simplified model of a device and its surroundings is elaborated: in harmonic regime, the device is assumed to amplify propagating plane pressure waves in a range of Strouhal numbers close to 0.2, whereas acoustic waves are reverberated by acoustic obstacles located several diameters upstream and downstream. The application of the Bode-Nyquist criterion brings out several features of actual acoustically-induced instabilities: first, the unstable frequencies are close to a series of discrete values equal to the acoustic natural frequencies of the pipe. Second, the unstable frequencies depend slightly on the flow velocity, because of the phase shift generated by the pressure drop device. Third, the instability is enhanced by the amplitude of the acoustic reflection coefficients and the proximity of the device to a pressure node. In dimension-less representation, the whistling frequencies exhibit successive branches, depending on the flow velocity. Comparisons with actual cases support this simplified representation. Former acoustic data obtained with a single hole orifice are revisited, and plotted in the dimensionless representation; the successive branches appear quite similar to the ones of the ideal model. Fluctuating pressures obtained with a safety relief valve experiencing chatter are processed, and bring out two branches in the same dimensionless representation. The results suggest that acoustically-induced instabilities are dominated by the acoustic response of the surrounding pipe, and that a gain/feedback representation is adequate for predicting the flow velocities associated with these instabilities.