Real-time thermoacoustic data assimilation

Low-order thermoacoustic models are qualitatively correct, but they are typically quantitatively inaccurate. We propose a time-domain method to make qualitatively low-order models quantitatively (more) accurate. First, we develop a Bayesian data assimilation method for a low-order model to self-adapt and self-correct any time that reference data, for example from experiments, becomes available. Second, we apply the methodology to infer the thermoacoustic states, heat release parameters, and model errors on the fly without storing data (real-time). Third, we analyse the performance of the data assimilation with synthetic data and interpret the results physically. We apply the data assimilation algorithm to all nonlinear thermoacoustic regimes, from limit cycles to chaos, in which acoustic pressure measurements from microphones are assimilated. Fourth, we propose practical rules for thermoacoustic data assimilation based on physical observations on the dynamics. An increase, reject, inflate strategy is proposed to deal with the rich nonlinear behaviour, the bifurcations of which are sensitive to small perturbations to the parameters. We show that (i) the correct acoustic pressure and parameters can be accurately inferred; (ii) the learning is robust because it can tackle large uncertainties in the observations (up to 50% the mean values); (iii) the uncertainty of the prediction and parameters is naturally part of the output; and (iv) both the time-accurate solution and statistics can be successfully inferred. Physical time scales for assimilation are proposed in non-chaotic regimes (with the Nyquist-Shannon criterion) and in chaotic regimes (with the Lyapunov time). Data assimilation opens up new possibility for real--time prediction of thermoacoustics by synergistically combining physical knowledge and data.