Dynamical states associated with the shift in whistling frequency in aeroacoustic system

Self-sustained aeroacoustic oscillations arising from the interactions between the hydrodynamic and acoustic fields are perceived as a whistle. Such whistling can lead to large amplitude acoustic oscillations that have disastrous consequences for engineering systems such as large segmented solid rocket motors and large gas pipelines. The whistling corresponds to the state of limit cycle oscillations (LCO) in dynamical systems theory. An aeroacoustic system exhibits different dynamical states when the bulk flow velocity is varied as a control parameter. Understanding the dynamical states and the transitions between them, as the control parameter is varied, is crucial in designing control strategies for such aeroacoustic oscillations. Previous studies have shown that as the control parameter varies, in an aeroacoustic system that has a flow through orifices, the whistling frequency shifts. We show that such a change in frequency occurs via three different scenarios— (1) direct transition between the two LCOs as an abrupt transition, (2) via a state of intermittency, and (3) via a state of aperiodicity. In the current aeroacoustic system, the abrupt transition between the LCOs is manifested as a bursting behaviour where the amplitude of the acoustic pressure fluctuations abruptly switches between the high and low-amplitude LCOs. Further, we show that the dynamical state and the transition between them during the frequency shift have a correlation with the magnitude of the frequency shift. Using recurrence theory we show that there is a change in the dynamical state of the system during the frequency shift. Further, we use synchronisation theory to investigate the coupled behaviour of the velocity ($u^{\prime }$) and the acoustic pressure ($p^{\prime }$) fluctuations during the different dynamical states. Our findings imply that $u^{\prime }$ and $p^{\prime }$ exhibit phase synchronisation (PS) during the state of LCO, corresponding to whistling. In contrast, $u^{\prime }$ and $p^{\prime }$ are desynchronised during the state of aperiodicity, corresponding to stable operation. Furthermore, the bursts of periodic oscillations during intermittency correspond to the phase-synchronised epochs of periodic $u^{\prime }$ and $p^{\prime }$, and the aperiodic epochs correspond to the desynchronised aperiodic $u^{\prime }$ and $p^{\prime }$.