Dynamical States and Bifurcations in Coupled Thermoacoustic Oscillators

Abstract

The emergence of rich dynamical phenomena in coupled self-sustained oscillators, primarily synchronization and amplitude death, has attracted considerable interest in several fields of science and engineering. Here, we present a comprehensive theoretical study on the manifestation of these exquisite phenomena in a reduced-order model of two coupled Rijke tube oscillators, which are prototypical thermoacoustic oscillators. We characterize the dynamical behaviors of two such identical and non-identical oscillators by varying both system parameters (such as the uncoupled amplitudes and the natural frequencies of the oscillators) and coupling parameters (such as coupling strength and coupling delay). The present model captures all of the dynamical phenomena -namely synchronization, phase-flip bifurcation, amplitude death, and partial amplitude death -observed previously in experiments on coupled Rijke tubes. By performing numerical simulations and deriving approximate analytical solutions, we systematically decipher the conditions and the nature of bifurcations underlying the aforementioned phenomena. This study provides insights into how the interplay between system and coupling parameters determines the dynamical behavior of coupled nonlinear oscillators in natural and engineering systems.

Rijke tube: A nonlinear oscillator

“…This type of coupling is introduced in an experimental system by connecting the two oscillators using a coupling tube, whose diameter is smaller than the diameter of the Rijke tube 272,280 . The increase in the length (l c ) and the diameter (d) of the coupling tube in experiments have direct correspondence with an increase in the time delay (τ) and the strength of coupling (K τ ) between the oscillators in the model 262,272 . On the other hand, dissipative coupling accounts for the dissipation of energy during the transfer of information from one oscillator to another as a consequence of mutual interaction.…”

“…The introduction of coupling between two such oscillators significantly changes their dynamical properties. In this case, Srikanth et al 262,269 found a forward shift in the occurrence of Hopf bifurcation along with a reduction in the amplitude of limit cycle oscillations when compared to these properties for an isolated Rijke tube oscillator. Recently, there has been an increased interest in studying the behavior of two coupled Rijke tube oscillators 262,[270][271][272][273][274] .…”

“…In this case, Srikanth et al 262,269 found a forward shift in the occurrence of Hopf bifurcation along with a reduction in the amplitude of limit cycle oscillations when compared to these properties for an isolated Rijke tube oscillator. Recently, there has been an increased interest in studying the behavior of two coupled Rijke tube oscillators 262,[270][271][272][273][274] . These studies have potential applications in understanding the interaction between multiple combustion systems of canannular type combustors used in gas turbine engines [275][276][277][278][279] .…”

“…The sources of dissipation of energy could arise due to the direct flow transfer from one system to another through the coupling tube or from the loss of acoustic energy due to the introduction of the coupling tube 272,280 . In addition to the variation of coupling parameters, the effect of change in system parameters, such as the amplitude and natural frequency of each oscillator in the uncoupled state, has been shown to play an important role in the coupled dynamics of Rijke tube oscillators 184,262,272 .…”

Mitigation of limit cycle oscillations in a turbulent thermoacoustic system via delayed acoustic self-feedback

“…At appropriate coupling parameter values, coupled systems approach the same state of oscillation quenching, known as amplitude death [8]. Through experimental and numerical analysis, amplitude death and partial amplitude death have been discovered in coupled laminar thermoacoustic systems [9][10][11][12]. In addition, a few studies have investigated the dynamics of mutually coupled turbulent combustors [13][14][15][16], and have also reported the presence of amplitude death in them [14].…”