Forced synchronization of periodic and aperiodic thermoacoustic oscillations: lock-in, bifurcations and open-loop control

Abstract

Synchronization is a universal concept in nonlinear science but has received little attention in thermoacoustics. In this numerical study, we take a dynamical systems approach to investigating the influence of harmonic acoustic forcing on three different types of self-excited thermoacoustic oscillations: periodic, quasi-periodic and chaotic. When the periodic system is forced, we find that: (i) at low forcing amplitudes, it responds at both the forcing frequency and the natural (self-excited) frequency, as well as at their linear combinations, indicating quasi-periodicity; (ii) above a critical forcing amplitude, the system locks in to the forcing; (iii) the bifurcations leading up to lock-in and the critical forcing amplitude required for lock-in depend on the proximity of the forcing frequency to the natural frequency; (iv) the response amplitude at lock-in may be larger or smaller than that of the unforced system and the system can exhibit hysteresis and the jump phenomenon owing to a cusp catastrophe; and (v) at forcing amplitudes above lock-in, the oscillations can become unstable and transition to chaos, or switch between different stable attractors depending on the forcing amplitude. When the quasi-periodic system is forced at a frequency equal to one of the two characteristic frequencies of the torus attractor, we find that lock-in occurs via a saddle-node bifurcation with frequency pulling. When the chaotic system is forced at a frequency close to the dominant frequency of its strange attractor, we find that it is possible to destroy chaos and establish stable periodic oscillations. These results show that the open-loop application of harmonic acoustic forcing can be an effective strategy for controlling periodic or aperiodic thermoacoustic oscillations. In some cases, we find that such forcing can reduce the response amplitude by up to 90 %, making it a viable way to weaken thermoacoustic oscillations.

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

“…Such a type of suppression phenomenon was experimentally evidenced in the control of ion-sound instability (Keen & Fletcher 1969), in self-excited ionization waves (Ohe & Takeda 1974) and even in the control of thermoacoustic instability (Guan et al 2018). Further, the reduction in response amplitude of thermoacoustic oscillation forced at non-resonant frequencies has been reported in previous studies (Lubarsky et al 2003;Bellows et al 2008;Guan et al 2018;Kashinath et al 2018). However, they did not focus on explaining the mechanism behind such a quenching of natural oscillations due to forcing.…”

“…Further, characterizing the dynamical states and mechanisms of amplitude reduction in the context of dynamical systems theory is essential for a better understanding of the phenomenon. In a recent study by Kashinath, Li & Juniper (2018), forced synchronization leading to periodic, quasiperiodic and chaotic states is investigated quite elaborately using a simple model of a ducted laminar flame; however, experimental realization of this study is still limited.…”

Forced synchronization of quasiperiodic oscillations in a thermoacoustic system

“…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).…”

“…Quasiperiodicity in self-excited thermoacoustic systems Both passive and active methods are available to control thermoacoustic oscillations (Candel 2002;Lieuwen & Yang 2005), but most assume a priori that such oscillations are periodic with a single characteristic frequency and a time-independent amplitude, i.e. they assume period-1 limit cycles (Kashinath, Li & Juniper 2018). This assumption, however, is not always valid (Juniper & Sujith 2018).…”