Forced synchronization of quasiperiodic oscillations in a thermoacoustic system

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$\mathbb{T}^{2}$torus at two incommensurate natural frequencies,$f_{1}$and $f_{2}$. Compared with that of a classical period-1 system, complete synchronization of this$\mathbb{T}_{1,2}^{2}$system is found to occur via a more intricate route involving three sequential steps: as the forcing amplitude,$\unicode[STIX]{x1D716}_{f}$, increases at a fixed forcing frequency,$f_{f}$, the system transitions first (i) to ergodic$\mathbb{T}_{1,2,f}^{3}$quasiperiodicity; then (ii) to resonant$\mathbb{T}_{1,f}^{2}$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$\unicode[STIX]{x1D716}_{f}$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$\mathbb{T}_{1,2,f}^{3}$or$\mathbb{T}_{2,f}^{2}$. 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$\mathbb{T}_{1,2}^{2}$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$\mathbb{T}^{2}$quasiperiodic systems with two incommensurate time scales.