Adjoint-accelerated Bayesian inference in thermoacoustics

Abstract

We demonstrate a new approach to thermoacoustic modelling, in which we use an efficient Bayesian inference framework to assimilate experimental data into thermoacoustic models. The framework provides four main tools: (i) parameter inference, (ii) uncertainty quantification, (iii) model comparison, and (iv) optimal experiment design. With Bayesian parameter inference, we use experimental observations to infer the most probable parameters of a thermoacoustic model. At the same time, we quantify the uncertainty in the parameters and the model. If there are several plausible models, we use Bayesian model comparison to quantitatively rank the models to select the best one. Parameter inference and model comparison can require a lot of data if the experiments are not designed well. With Bayesian optimal experiment design, we use the model to identify the most informative experiments, reducing the experimental cost and effort. Bayesian inference is often considered too computationally expensive to be applied to problems in fluid dynamics. This is because many Bayesian inference frameworks use sampling techniques to construct the posterior, which require thousands of model evaluations. This is not practical if the model evaluations are expensive, as is usually the case in fluid dynamics. In this thesis, we demonstrate an approximate Bayesian inference framework, which requires far fewer model evaluations than sampling methods. In this framework, parameter inference reduces to a quadratic optimization problem, which we solve using gradient-based optimization, with gradients calculated using adjoint methods. We then estimate the posterior uncertainty using Laplace's method, which we evaluate using first and second order adjoint methods. Under the assumptions of Laplace's method, Bayesian model comparison metrics are evaluated through simple algebraic expressions. Similarly, optimal experiment design becomes computationally cheap when Laplace's method and the adjoint method are combined. We apply this framework to three simple thermoacoustic systems: an electrically heated Rijke tube, a ducted laminar conical flame, and a ducted turbulent conical flame. With the electrically heated Rijke tube, we demonstrate the full set of tools in the Bayesian framework. Using these tools, we create a thermoacoustic model that (i) is quantitatively accurate over the entire operating range, (ii) has quantified uncertainty bounds, (iii) can extrapolate beyond the observed data, and (iv) can be trained on a few optimal experiments. This is a significant improvement on previous attempts to model the electrically heated Rijke tube in the literature. With both sets of ducted flames, we demonstrate a second use for Bayesian inference in thermoacoustics: obtaining flame transfer functions, and their uncertainties, from pressure observations. We do this with the flame in situ, without a need for optical access. If the flame's response is sensitive to its environment, which is usually the case, then this is preferable to measuring flame transfer functions ex situ using optical methods. This thesis provides a proof of concept by applying adjoint-accelerated Bayesian inference to a set of canonical problems in thermoacoustics. In doing so, we show that the Bayesian framework provides a powerful set of tools for combining the work of the experimentalist and the modeller in a mutually beneficial way. The resulting models are accurate, interpretable, and capable of extrapolation. The models can be trained with only a few experimental observations, making Bayesian inference feasible for applications where experiments are expensive. This framework is currently being implemented on the Rolls--Royce SCARLET industrial test rig, which will investigate how the tools handle more complex thermoacoustic systems. Work is also being done on applying the framework to assimilate video footage of flames into models of the flame dynamics. This will allow us to predict flame transfer functions at operating conditions and frequencies that have not been observed experimentally. This, combined with the acoustic network models presented in this thesis, would form a powerful design tool that could be used to inform more elegant and robust interventions to thermoacoustic instability with fewer prototyping iterations.