Modeling the influence of interactions on different variables in a turbulent thermoacoustic system

TL;DR

This work addresses thermoacoustic instability in turbulent ducted flames by introducing a reduced-order framework that treats the global oscillatory variables $p'$ (acoustic pressure) and $\dot{q}'$ (global heat release rate) as cubic Stuart-Landau oscillators driven by inter-subsystem forcing. Acoustic fluctuations are modeled with a Wiener-process forcing, while heat-release-rate fluctuations are driven by a Markov-modulated Poisson process, capturing multi-timescale interactions and spike-like energy releases. The model reproduces key experimental features, including multifractality in chaotic states, its loss during the chaos-to-order transition, and the emergence of periodicity and bifurcations in $\dot{q}'$, linking stochastic forcing to observed scaling laws. This reduced-order approach provides a tractable, physically grounded framework for predicting thermoacoustic transitions and can be extended to other complex turbulent systems.

Abstract

Turbulent reacting flows confined to ducts are plagued by thermoacoustic instability, a state in which a positive feedback between flow, flame and acoustic perturbations leads to the emergence of catastrophically high-amplitude oscillatory dynamics in the sound and global heat release rate fluctuations. Modeling the interdependence between local interactions and the global emergence of order in such spatially extended complex systems is exacting. Here, we present a novel reduced-order model to capture the influence of the local interactions on the variables exhibiting global emergence of order in a turbulent reacting flow system. We represent each variable that exhibits global oscillatory instability as an oscillator with a cubic nonlinearity. The oscillator is driven by a forcing term that represents the holistic influence of the inter-subsystem interactions on the global behavior. The forcing term essentially couples the local interactions and the globally emergent dynamics in the model. Further, the influence of the inter-subsystem interactions on the behavior of each subsystem is different. Therefore, we use different forcing terms for each variable inspired by the physical interactions in the system. The nonlinear oscillators representing the acoustic and the heat release rate oscillations are hence forced using Wiener and Markov-modulated Poisson processes, respectively. Using this approach, we are able to reproduce (i) the multifractal characteristics of acoustic pressure fluctuations during chaotic dynamics, (ii) the loss of multifractality through the experimentally observed scaling law behavior during the transition from chaos to order and (iii) the emergence of periodicity and bifurcation in heat release rate dynamics.

Figures (11)

• Figure 1: Flame-flow-acoustic interactions in turbulent thermoacoustic systems result in emergent order in the acoustic pressure. The schematic shows the interactions between the various subsystems in a turbulent thermoacoustic system. The configuration of the system includes a backward-facing step that creates a flow recirculation zone due to the sudden expansion in the cross section area. The vortices that are shed from the backward-facing step convect downstream and impinge on the bluff body surface. The mixture of hot fuel and air contained in these vortices undergoes combustion, causing a sudden spike (‘kick’matveev2003model) in the heat release rate fluctuations. Acoustic waves are generated in the duct and interact with the local flame and flow fluctuations thus affecting the combustion and vortex shedding processes. Thus, all the subsystems interact in a closed feedback loop.
• Figure 2: (a) The schematic of the turbulent combustor. (b) The schematic of the bluff body used in the experiments to stabilize the flame.
• Figure 3: Comparison of the time series of acoustic pressure fluctuation ($p'$) obtained from the experiments, normalized with the maximum amplitude of oscillation in the limit cycle regime (column (I)), from the model as per Eq. (\ref{['eqn:noise pressure']}) (column (II)), and the corresponding power spectrum. For the experiments, the mass flow rate of air is increased and thus the equivalence ratio $\phi$ decreases from (a) $\phi=1$, to (b) $\phi=0.76$, to (c) $\phi=0.56$, respectively. In the model, the parameter $\mu$ is increased from (d) to (f) as $-0.2$, $-0.05$, and $0.3$, respectively, for a constant noise intensity of $\eta=0.01$. The time series of $p’$ obtained from the model replicate the characteristic features of the time series of $p’$ obtained from the experiments.
• Figure 4: The bifurcation diagram for the normalized acoustic pressure fluctuations ($p’$) with respect to (a) flow control parameter ($\phi$) in the experiments and (b) the control parameter ($\mu$) for the model. A smooth bifurcation occurs at $\phi' = 0.74$ in experiments and at $\mu = 0$ in the model. (c) The multifractal spectrum of $p'$ signals obtained from the model at $\mu=-0.2,~0.05,\text{and}~0.3$. (d) Variation of the spectral amplitude ($A$) normalized with the spectral amplitude at the onset of thermoacoustic instability ($A_1$) of the dominant frequency of $p'$ with the Hurst exponent ($H$) when $\mu \in [-0.25,0.05]$ in the model, exhibits a power law scaling with the exponent $-1.993\pm 0.04$ (95% confidence). The multifractal spectrum shrinks as periodicity emerges and the Hurst exponent exhibits the universal scaling law observed from experiments in turbulent thermo-fluid and fluid systems pavithran2020universality.
• Figure 5: The time series of ((a),(d) and (g)) $\dot{q}’$ obtained from the experiments as the turbulent thermoacoustic system transitions from (I) low-amplitude aperiodic fluctuations to (III) high amplitude periodic fluctuations through (II) intermittency. The zoomed-in time series ((b),(e), and (h)) for $\dot{q}’$ illustrate the calculation of average arrival rates using a threshold and $k=3$ in Eq. (\ref{['avg arrival rate']}), therefore yielding an average arrival rate of $r=3/\Delta \tau$. The average arrival rates ((c),(f), and (i)) binned using Lambda algorithmheyman2003modeling plotted against time.