Forced synchronization of quasiperiodic oscillations in a thermoacoustic system. (27th September 2019) - Guan, Yu; Gupta, Vikrant; Wan, Minping; Li, Larry K. B.
Abstract : 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 T^2 torus at two incommensurate natural frequencies, f_1 and f_2 . Compared with that of a classical period-1 system, complete synchronization of this T_1, 2^2 system is found to occur via a more intricate route involving three sequential steps: as the forcing amplitude, [STIX]x1D716_f, increases at a fixed forcing frequency, f_f, the system transitions first (i) to ergodic T_1, 2, f^3 quasiperiodicity; then (ii) to resonant T_1, f^2 quasiperiodicity as the weaker of the two natural modes, f_2, synchronizes first, leading to partial synchronization; and finally (iii) to a P1_f limit cycle as the remaining natural mode, f_1, also synchronizes, leading to complete synchronization. The minimum [STIX]x1D716_f required for partial and complete synchronization decreases as f_f approaches either f_1 or f_2, resulting in two primary Arnold tongues. However, when forced at an amplitude above that required for complete synchronization, the system can transition out of P1_f and into T_1, 2, f^3 or T_2, f^2 . The optimal control strategy is to apply off-resonance forcing at a frequency around the weaker natural mode ( f_2 ) and at an amplitude just sufficient to cause P1_f, 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 f_1 and f_2 acoustic modes. If the forcing is applied near the stronger natural mode ( f_1 ), however, resonant amplification can occur. We then phenomenologically model this T_1, 2^2 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 P1_f, 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 T^2 quasiperiodic systems with two incommensurate time scales.