Coherence resonance and stochastic bifurcation behaviors of simplified standing-wave thermoacoustic systems

Abstract

In this work, noise-induced motions (i.e., external fluctuations) in two modelled standing-wave thermoacoustic systems are studied when these systems are close to the deterministic stability boundary. These systems include (1) open-open (i.e., Rijke-type) and ( 2) closed-open boundary conditions. It is found from the smooth transitions of the stationary probability density function that the thermoacoustic system is destabilized via stochastic P bifurcation, as the external noise intensity is continuously increased. In addition, the increased noise intensity can shift the hysteresis region, which makes the system more prone to quasi-periodic oscillations, but also reduces the hysteresis area. The noise-induced coherence motions are observed numerically in the open-open system, which is denoted by the occurrence of a bell-shaped signal to noise ratio (SNR). The SNR is shown to be applicable as a precursor. It becomes larger and the optimal noise intensity is decreased as the modelled thermoacoustic system approaches the critical bifurcation point. In addition, coherence resonance is observed in the closed-open system. To validate the findings, experimental studies are conducted on an open-open Rijke tube. Good qualitative agreements are obtained. The present study shed lights on the stochastic and coherence behaviors of the standing-wave thermoacoustic systems with different boundary conditions. V

Rijke tube: A nonlinear oscillator

“…These dynamical changes include modification in the stability margins, occurrence of coherence and stochastic resonance, and excitation of new dynamical states [209][210][211][212] . In this section, we discuss various noise-induced dynamics in the sub-threshold and bistable regimes of Rijke tube oscillators 107,161,[213][214][215][216][217][218][219][220] . The regime corresponding to a single stable fixed-point solution (stable focus), observed prior to the Hopf point in supercritical Hopf bifurcation (Fig.…”

“…Coherence resonance has been studied in various Rijke tube systems both experimentally 215 and theoretically 217,220 . Figure 16 shows the occurrence of coherence resonance in a laminar premixed flame Rijke tube burner for increasing values of the noise intensity D 215 .…”

“…The existence of coherence resonance has an important application in thermoacoustic systems. We can use the increase in the coherent nature of pressure signals in the steady state regime of the system prior to the Hopf point as a precursor to an impending thermoacoustic instability 215,217,220 . Furthermore, the existence of coherence resonance has been examined for subcritical Hopf bifurcation 215,216 and supercritical Hopf bifurcation 218 individually as well as collectively 217 .…”

“…Saurab et al 216 experimentally investigated the effects of noise in a Rijke tube with laminar premixed flame and observed the presence of a P-bifurcation along with coherence resonance. Li et al 220 analytically studied the stability of the stochastic one-dimensional self-excited nonlinear standing wave thermoacoustic system 238 . Moreover, Li et al 239 identified the presence of two different types of P-bifurcations in this system, one with a crater-like PDF and the other had two peaks and one trough.…”

System identification near a hopf bifurcation via the noise-induced dynamics in the fixed-point regime

“…Gopalakrishnan and Sujith 2015) and how it triggers limit-cycle oscillations in the bistable regime (Burnley and Culick 2000;Jegadeesan and Sujith 2013); and (ii) to gather information about the system from its noise-induced dynamics. Such information can then be used to predict the onset of instability Kabiraj et al 2015;Gopalakrishnan et al 2016a;Murayama et al 2018;Hashimoto et al 2019;Kobayashi et al 2019;Li et al 2019), to distinguish between supercritical and subcritical bifurcations (Gupta et al 2017), and to extract deterministic quantities (Noiray and Schuermans 2013a;Boujo and Noiray 2017;Noiray 2017;Noiray and Denisov 2017).…”