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.

Lock-in phenomenon of vortex shedding in flows excited with two commensurate frequencies: a theoretical investigation pertaining to combustion instability

“…At this point, it is worth to mention the parallels with the experimental study by Guan et al. (2019 a ). They periodically forced a ducted laminar premixed flame, exhibiting quasi-periodic unforced (natural) oscillations.…”

“…In our investigations, two forcing frequencies are used, while in Guan et al. (2019 a ), two natural frequencies are present. In both studies, three modes are involved.…”

Analysis of unsteady mixed convection of Cu–water nanofluid in an oscillatory, lid-driven enclosure using lattice Boltzmann method

“…The physical problem and the corresponding numerical solution are therefore unsteady. This type of forcing has been already used to study nonlinear phenomena in many other fluid mechanical systems, such as low-density jets [22,23] and thermoacoustic systems [24,25].…”

On two-frequency quasi-periodic perturbations of systems close to two-dimensional Hamiltonian ones with a double limit cycle

“…In recent years there has been considerable interest in the study of multi-frequency quasiperiodic oscillations [1][2][3][4][5][6][7][8][9][10][11][12][13]. Such phenomena are widespread in nature and technology, in almost all areas of physics (radiophysics, electronics, laser physics, astrophysics, and many others), as well as in biology, chemistry, and medicine (see the above papers as well as monographs [14][15][16]).…”