Forced synchronization and asynchronous quenching of periodic oscillations in a thermoacoustic system

Abstract

We perform an experimental and theoretical study to investigate the interaction between an external harmonic excitation and a self-excited oscillatory mode ($f_{n0}$) of a prototypical thermoacoustic system, a horizontal Rijke tube. Such an interaction can lead to forced synchronization through the routes of phase locking or suppression. We characterize the transition in the synchronization behaviour of the forcing and the response signals of the acoustic pressure while the forcing parameters, i.e. amplitude ($A_{f}$) and frequency ($f_{f}$) of forcing are independently varied. Further, suppression is categorized into synchronous quenching and asynchronous quenching depending upon the value of frequency detuning ($|\,f_{n0}-f_{f}|$). When the applied forcing frequency is close to the natural frequency of the system, the suppression in the amplitude of the self-excited oscillation is known as synchronous quenching. However, this suppression is associated with resonant amplification of the forcing signal, leading to an overall increase in the response amplitude of oscillations. On the other hand, an almost 80 % reduction in the root mean square value of the response oscillation is observed when the system is forced for a sufficiently large value of the frequency detuning (only for $f_{f}<f_{n0}$). Such a reduction in amplitude occurs due to asynchronous quenching where resonant amplification of the forcing signal does not occur, as the frequency detuning is significantly high. Further, the results from a reduced-order model developed for a horizontal Rijke tube show a qualitative agreement with the dynamics observed in experiments. The relative phase between the acoustic pressure ($p^{\prime }$) and the heat release rate ($\dot{q}^{\prime }$) oscillations in the model explains the occurrence of maximum reduction in the pressure amplitude due to asynchronous quenching. Such a reduction occurs when the positive coupling between $p^{\prime }$ and $\dot{q}^{\prime }$ is disrupted and their interaction results in overall acoustic damping, although both of them oscillate at the forcing frequency. Our study on the phenomenon of asynchronous quenching thus presents new possibilities to suppress self-sustained oscillations in fluid systems in general.

Forced synchronization of quasiperiodic oscillations in a thermoacoustic system

“…According to Keen & Fletcher (1970), asynchronous quenching can be interpreted as an asymptotic loss of stability of the natural self-excited oscillations, arising from disturbances imposed by the external forcing, resulting in a decrease in the total oscillation amplitude of the forced self-excited system. For periodic thermoacoustic oscillations, asynchronous quenching has recently been shown to be caused by a reduced Rayleigh index (Guan et al 2019a;Mondal et al 2019). For quasiperiodic thermoacoustic oscillations, we find a similar reduction in the Rayleigh index (figure 7c: green region), as evidenced by the steep decreases in both RI * 1 and RI * t en route to P1 f .…”

“…In thermoacoustics, open-loop forcing has been shown to be able to weaken periodic self-excited oscillations in various combustion systems, ranging from the simple Rijke tube (Reynolds numbers of Re ∼ 10 3 with natural frequencies of f 1 ∼ 10 2 Hz; Guan, Murugesan & Li 2018;Kashinath et al 2018;Guan et al 2019a;Mondal, Pawar & Sujith 2019) to turbulent premixed combustors (Re ∼ 10 4 , f 1 ∼ 10 2 Hz; Bellows, Hreiz & Lieuwen 2008;Balusamy et al 2015). A recurring theme of these studies has been the application of periodic acoustic forcing to periodic thermoacoustic oscillations, accompanied by an examination of the nonlinear dynamics en route to and beyond the synchronization boundaries.…”

“…Forced synchronization of quasiperiodic thermoacoustic oscillations 393 From a control perspective, numerous studies have shown that, near the onset of synchronization, the thermoacoustic amplitude can be drastically reduced -typically to less than 20 % of that of the unforced state (Bellows et al 2008;Kashinath et al 2018;Guan et al 2019a,b;Mondal et al 2019). This reduction occurs via asynchronous quenching, a nonlinear process in which the amplitude of a self-excited oscillator is reduced by the external application of periodic forcing at a frequency far enough from the natural frequency to prevent resonant amplification of the forcing signal (Minorsky 1967).…”

“…Asynchronous quenching can be modelled with a forced VDP oscillator (Dewan 1972) and has been exploited for open-loop control in various fields, ranging from plasma physics (Keen & Fletcher 1970) to hydrodynamics (Staubli 1987). In thermoacoustics, asynchronous quenching has recently been shown to coincide with an inverse Neimark-Sacker bifurcation to synchronization and a reduced Rayleigh index (Guan et al 2019a;Mondal et al 2019). However, it is not known how effective this control strategy would be if it were applied to thermoacoustic oscillations that are quasiperiodic rather than periodic.…”

“…Specifically, it is not known how the strategy would have to be modified to cope with the presence of an additional (second) natural mode. For example, in the periodic case, the thermoacoustic amplitude can be reduced by applying forcing at a frequency sufficiently far from the natural frequency for asynchronous quenching to occur, as demonstrated by Guan et al (2019a) and Mondal et al (2019). However, if a second natural mode is present, its frequency could coincide with the forcing frequency, resulting in resonant amplification (Minorsky 1967).…”