Forced synchronization of quasiperiodic oscillations in a thermoacoustic system

In self-excited combustion systems, the application of open-loop forcing is known to be an effective strategy for controlling periodic thermoacoustic oscillations, but it is not known whether and under what conditions such a strategy would work on thermoacoustic oscillations that are not simply periodic. In this study, we experimentally examine the effect of periodic acoustic forcing on a prototypical thermoacoustic system consisting of a ducted laminar premixed flame oscillating quasiperiodically on an ergodic T² torus at two incommensurate natural frequencies, f₁ and f₂ . Compared with that of a classical period-1 system, complete synchronization of this T1,2² system is found to occur via a more intricate route involving three sequential steps: as the forcing amplitude, [STIX]x1D716f , increases at a fixed forcing frequency, ff , the system transitions first (i) to ergodic T1,2,f³ quasiperiodicity; then (ii) to resonant T1,f² quasiperiodicity as the weaker of the two natural modes, f₂ , synchronizes first, leading to partial synchronization; and finally (iii) to a P1f limit cycle as the remaining natural mode, f₁ , also synchronizes, leading to complete synchronization. The minimum [STIX]x1D716f required for partial and complete synchronization decreases as ff approaches either f₁ or f₂ , resulting in two primary Arnold tongues. However, when forced at an amplitude above that required for complete synchronization, the system can transition out of P1f and into T1,2,f³ or T2,f² . The optimal control strategy is to apply off-resonance forcing at a frequency around the weaker natural mode ( f₂ ) and at an amplitude just sufficient to cause P1f , because this produces the largest reduction in thermoacoustic amplitude via asynchronous quenching. Analysis of the Rayleigh index shows that this reduction is physically caused by a disruption of the positive coupling between the unsteady heat release rate of the flame and the f₁ and f₂ acoustic modes. If the forcing is applied near the stronger natural mode ( f₁ ), however, resonant amplification can occur. We then phenomenologically model this T1,2² thermoacoustic system as two reactively coupled van der Pol oscillators subjected to external sinusoidal forcing, and find that many of its synchronization features – such as the three-step route to P1f , the double Arnold tongues, asynchronous quenching and resonant amplification – can be qualitatively reproduced. This shows that these features are not limited to our particular system, but are universal features of forced self-excited oscillators. This study extends the applicability of open-loop control from classical period-1 systems with just a single time scale to ergodic T² quasiperiodic systems with two incommensurate time scales.