Low-order modeling of the mutual synchronization between two turbulent thermoacoustic oscillators

Fokker-Planck modeling of the stochastic dynamics of a Rijke tube

We derive and numerically validate a low-order oscillator model to capture the stochastic dynamics of a prototypical thermoacoustic system (a Rijke tube) undergoing a subcritical Hopf bifurcation in the presence of additive noise. We find that on the fixed-point branch before the bifurcation, the system is dominated by the first duct mode, and the Fokker-Planck solution for the first Galerkin mode can adequately predict the stochastic dynamics of the overall system. We also find that this analytical framework predicts well the dominant mode on the limit-cycle branch, but underperforms in the hysteretic bistable zone where the role of nonlinearities is more pronounced. Besides offering new insights into stochastic thermoacoustic behavior, this study shows that an analytical framework based on the Fokker-Planck equation can facilitate the early detection of thermoacoustic instabilities in a Rijke-tube model subjected to noise.

Robust design against frequency variation for amplitude death in delay-coupled oscillators

Amplitude death has the potential to suppress unwanted oscillations in various engineering applications. However, in some engineering applications, such as dc microgrids, airfoil systems, and thermoacoustic systems, oscillation frequency is highly susceptible to external influences, leading to considerable variations. To maintain amplitude death amidst these frequency variations, we propose a design procedure that is robust against frequency variation for inducing amplitude death in delay-coupled oscillators. We first analytically derive the oscillator frequency band in which amplitude death can occur. The frequency bandwidth is maximized when the coupling strength is inversely proportional to the connection delay. Furthermore, our analysis reveals that the oscillator frequency band is influenced by the minimum eigenvalue of the normalized adjacency matrix (i.e., network topology) and that bipartite networks exhibit limited robustness to frequency variations. Our design procedure maintains the stability of amplitude death even under substantial frequency variations and is applicable to various network topologies. Numerical simulations confirm the validity of the proposed design.

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

We report the occurrence of amplitude death (AD) of limit cycle oscillations in a bluff body stabilized turbulent combustor through delayed acoustic self-feedback. Such feedback control is achieved by coupling the acoustic field of the combustor to itself through a single coupling tube attached near the anti-node position of the acoustic standing wave. We observe that the amplitude and dominant frequency of the limit cycle oscillations gradually decrease as the length of the coupling tube is increased. Complete suppression (AD) of these oscillations is observed when the length of the coupling tube is nearly 3 / 8 times the wavelength of the fundamental acoustic mode of the combustor. Meanwhile, as we approach this state of amplitude death, the dynamical behavior of acoustic pressure changes from the state of limit cycle oscillations to low-amplitude chaotic oscillations via intermittency. We also study the change in the nature of the coupling between the unsteady flame dynamics and the acoustic field as the length of the coupling tube is increased. We find that the temporal synchrony between these oscillations changes from the state of synchronized periodicity to desynchronized aperiodicity through intermittent synchronization. Furthermore, we reveal that the application of delayed acoustic self-feedback with optimum feedback parameters completely disrupts the positive feedback loop between hydrodynamic, acoustic, and heat release rate fluctuations present in the combustor during thermoacoustic instability, thus mitigating instability. We anticipate this method to be a viable and cost-effective option to mitigate thermoacoustic oscillations in turbulent combustion systems used in practical propulsion and power systems.

Dynamical states and bifurcations in coupled thermoacoustic oscillators

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 the coupling strength and the coupling delay). The present model captures all 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 bifurcations underlying the aforementioned phenomena. The insights provided by this study can be used to understand the interactions between multiple cans in gas turbine combustors and develop control strategies to avert undesirable thermoacoustic oscillations in them.

Quenching and amplification of thermoacoustic oscillations in two nonidentical Rijke tubes interacting via time-delay and dissipative coupling

We numerically explore the quenching and amplification of self-excited thermoacoustic oscillations in two nonidentical Rijke tubes interacting via time-delay and dissipative coupling. On applying either type of coupling separately, we find that the presence of nonidentical heater powers can shrink the regions of amplitude death in both oscillators, while producing new regions of amplitude amplification in the weaker oscillator. We find that the magnitude of amplitude amplification grows with the heater power mismatch and with the total power input. These effects are also present when both types of coupling are applied simultaneously. This study highlights the critical role that nonidentical thermal loads can play in determining the amplitude response of coupled thermoacoustic systems, facilitating the design of control strategies for coupled oscillatorlike devices such as gas turbines.

Steady-state statistics, emergent patterns and intermittent energy transfer in a ring of oscillators

Networks of coupled nonlinear oscillators model a broad class of physical, chemical and biological systems. Understanding emergent patterns in such networks is an ongoing effort with profound implications for different fields. In this work, we analytically and numerically study a symmetric ring of N coupled self-oscillators of van der Pol type under external stochastic forcing. The system is proposed as a model of the thermo- and aeroacoustic interactions of sound fields in rigid enclosures with compact source regions in a can-annular combustor. The oscillators are connected via linear resistive coupling with nonlinear saturation. After transforming the system to amplitude-phase coordinates, deterministic and stochastic averaging is performed to eliminate the fast oscillating terms. By projecting the potential of the slow-flow dynamics onto the phase-locked quasi-limit cycle solutions, we obtain a compact, low-order description of the (de-)synchronization transition for an arbitrary number of oscillators. The stationary probability density function of the state variables is derived from the Fokker-Planck equation, studied for varying parameter values and compared to time series simulations. We leverage our analysis to offer explanations for the intermittent energy transfer between Bloch waves observed in acoustic pressure spectrograms observed of real-world gas turbines.