Shock-induced tipping in a thermoacoustic system

Abstract

Tipping refers to the transition of a system from one state to another. In this study, we focus on shock-induced tipping, which occurs due to a sudden and large disturbance in a control parameter, which is referred to as the shock. This shock drives the system from one dynamical state to another. We present the first experimental demonstration of shock-induced tipping using a prototypical thermoacoustic system, the horizontal Rijke tube. In a thermoacoustic system, unsteady heat release and sound waves interact through positive feedback, leading to self-sustained, high-amplitude oscillations known as limit cycles. The system transitions from a quiescent state to a state of self-sustained oscillations when a shock is introduced in the power supplied to the heat source (an electrically heated grid). This shock is created by abruptly increasing the voltage supplied to the grid, which takes the system into a bistable region. To explain the underlying mechanism linking the shock in the supplied power to the observed tipping behaviour, we model the system by modifying the governing equations of the Rijke tube to incorporate the heat transfer properties of the grid. We demonstrate that the shock in the supplied power manifests as a shock in the grid temperature, causing the system to fall into the basin of attraction of an alternate stable state. The tipping event depends on the magnitude of the shock and the temperature of the grid. Understanding the mechanisms underlying shock-induced tipping is crucial for developing systems with improved safety and reliability.

Figures (5)

• Figure 1: Schematic of the horizontal Rijke tube, comprising of a long square duct, decoupler, mass flow controller, connecting rods supplying current to an electrically heated grid, and a piezoelectric pressure transducer for measuring acoustic signals. (b) Bifurcation diagram constructed by plotting the RMS value of acoustic pressure oscillations against the the voltage supplied to the grid. The bistable region of the system is represented in yellow.
• Figure 2: (a) When the voltage is increased at a constant rate of 3 mV/s, within 720 s to $V_t = 2.16~\text{V}$, without a shock, the system remains in the quiescent state, reflected as the noise floor with a mean near zero. (b) The voltage is linearly increasing at a constant rate of 3 mV/s and at $\tau_t = 50$ s, the value of voltage is abruptly increased from $0.5~\text{V}$ to $2.16~\text{V}$ ($V_t$) as shown by the arrow, within $0.3$ ms and then maintained at this value until $750~\text{s}$. The shock induced drives the system from a quiescent state to high-amplitude periodic oscillations.
• Figure 3: (a) Root mean square (RMS) values of the nondimensional acoustic pressure fluctuation $(p')$, obtained from the Rijke tube model, plotted against $K$. The system jumps from quiescent state to the state of LCO, with a hysteresis region marked in yellow. The values of $K$ corresponding to the Hopf and fold points marked in the diagram as $\nu_h$ and $\nu_f$ are 2.05 and 1.45 respectively. (b) The variation of $K$ with $\tau$ which models the shock applied to the system. At a particular instant of $\tau$, denoted by $\tau_t$ = 0.15, the value of $K$ is increased within $10^{-3}$ units of nondimensional runtime from 0.015 to 2.04, giving a shock of magnitude, $dK/d\tau$ = 2025
• Figure 4: (a) The stability map plotted for different $\nu_t$ and $\tau_t$ of the system. The red regions indicate the presence of LCO, while the green regions correspond to the quiescent state. We observe that as $\tau_t$ increases, the system must be driven to a higher value of $\nu_t$, i.e., closer to $\nu_h$ for the shock to tip the system to LCO. (b) The variation of $T_g$ with $K$ is computed to construct the bifurcation plot with the grid temperature. The red markers denote the grid temperature at the exact point of tipping for the cases in Fig. \ref{['stabilitymap']} where the system tipped to LCO. This transition occurred when the temperature exceeded the critical threshold $T_{\text{rms}} = 1.29$ for the combinations of $\nu_t$ and $\tau_t$ that lead to tipping to LCO in Fig. \ref{['stabilitymap']}. In contrast, the green markers show the final grid temperature for cases that did not tip, as their temperatures remained below this critical threshold, preventing the onset of LCO.
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