Thermo-acoustic instability is one of the most persistent and costly problems facing gas turbine and rocket manufacturers. It arises when heat release rate fluctuations from a flame lock into acoustic modes in a combustion chamber. This causes large amplitude oscillations even if the thermodynamic efficiency of the conversion from heat to work is small. Although the mechanism has been well understood for over a century and carefully studied through experiments and numerical simulations, engines that have been designed to be thermo-acoustically stable often turn out to be unstable when tested.
During the cold war, the USA and Russia spent billions to eliminate it from their designs. For example, NASA performed 2000 full-scale tests on the F1 engine of the Apollo Program in order to obtain a stable engine by inspired trial and error. The physical mechanism of the instability was well known and the scientists and engineers devoted to it were highly capable, so why was it so hard to eliminate?
The answer is in most books and papers on the subject since the 1950's: Thermoacoustic instability is pathologically sensitive to small design changes. This is demonstrated in this Annual Review paper on Sensitivity in Thermoacoustics:
This sensitivity arises because the time delay between acoustic perturbations at the fuel injector and subsequent heat release rate perturbations at the flame is the same order as the acoustic period. Any design change that alters the flame time delay or the acoustic period therefore strongly influences thermoacoustic stability. The problem is that most design changes will alter one or the other. It is therefore a fool's errand to try to model the physical mechanism of thermoacoustic instability with quantitative accuracy ab initio. The mechanism might be correct and the parameters nearly accurate, but the model will almost certainly not be predictive because of this extreme sensitivity to parameters.
On the positive side, this sensitivity causes the model parameters to be highly observable from experimental data. This makes thermo-acoustic instability an ideal application for Bayesian inference because (i) the physics is well-known, (ii) model parameters are observable from data, (iii) data is available from laboratory to industrial scale rigs, (iv) we know from experience that thermo-acoustic instability can always been eliminated with small changes. The challenge is to design a quantitatively-accurate model that can predict those changes.
We use Laplace’s method combined with adjoint methods to first and second order, which is technically more difficult to implement than methods such as Markov Chain Monte Carlo, but is thousands of times faster, meaning that we can compare dozens of candidate models.
The final assembled model is therefore as small as possible, quantitatively accurate, and physically interpretable. The model extrapolates successfully because it is physics-based. Much data is required to select the model but, once selected, little data is required to train it:
Full details can be found in the following paper and thesis; the paper contains Matlab code that creates each figure:
Thermoacoustics is a branch of fluid mechanics, and is as such governed by the conservation laws of mass, momentum, energy and species. While computational fluid dynamics (CFD) has entered the design process of many applications in fluid mechanics, its success in thermoacoustics is limited by the multi-scale, multi-physics nature of the subject. In his influential monograph from 2006, Prof. Fred Culick writes about the role of CFD in thermoacoustic modeling:
"The main reason that CFD has otherwise been relatively helpless in this subject is that problems of combustion instabilities involve physical and chemical matters that are still not well understood. Moreover, they exist in practical circumstances which are not readily approximated by models suitable to formulation within CFD. Hence, the methods discussed and developed in this book will likely be useful for a long time to come, in both research and practice. [...] It seems to me that eventually the most effective ways of formulating predictions and theoretical interpretations of combustion instabilities in practice will rest on combining methods of the sort discussed in this book with computational fluid dynamics, the whole confirmed by experimental results." (Culick, Fred: Unsteady Motions in Combustion Chambers for Propulsion Systems. NATO Research and Technology Organisation, 2006)
Despite advances in CFD and large-eddy simulation (LES) in particular, unsteady simulations for more than a few selected operating points are computationally infeasible. The 'methods discussed in this book' refer to reduced-order models of thermoacoustic oscillations. Whether intentional or not, the last sentence anticipates the advent of data-driven methods, and encapsulates the philosophy behind this work.
This work brings together two workhorses of the design process: physics-informed reduced-order models and data from higher-fidelity sources such as simulations and experiments. The three building blocks to all our statistical inference frameworks are: (i) a hierarchical view of reduced-order models consisting of states, parameters and governing equations; (ii) probabilistic formulations with random variables and stochastic processes; and (iii) efficient algorithms from statistical learning theory and machine learning. While leveraging advances in statistical and machine learning, we demonstrate the feasibility of Bayes? rule as a first principle in physics-informed statistical inference. In particular, we discuss two types of inverse problems in thermoacoustics: (i) implicit reduced-order models representative of nonlinear eigenproblems from linear stability analysis; and (ii) time-dependent reduced-order models used to investigate nonlinear dynamics. The outcomes of statistical inference are improved predictions of the state, estimates of the parameters with uncertainty quantification and an assessment of the reduced-order model itself.
This work highlights the role that data can play in the future of combustion modeling for thermoacoustics. It is increasingly impractical to store data, particularly as experiments become automated and numerical simulations become more detailed. Rather than store the data itself, the techniques in this work optimally assimilate the data into the parameters of a physics-informed reduced-order model. With data-driven reduced-order models, rapid prototyping of combustion systems can feed into rapid calibration of their reduced-order models and then into gradient-based design optimization. While it has been shown, e.g. in the context of ignition and extinction, that large-eddy simulations become quantitatively predictive when augmented with data, the reduced-order modeling of flame dynamics in turbulent flows remains challenging. For these challenging situations, this work opens up new possibilities for the development of reduced-order models that adaptively change any time that data from experiments or simulations becomes available.
The reader in mind is a scientist or engineer with an interest in data-driven methods. For readers mostly interested in the results, we provide references to our ideally more self-contained publications where available. For the more methodological chapters, we provide JUPYTER notebooks so that inclined readers are able to familiarize themselves with the statistical and numerical concepts of this work. They are either available on GITLAB2 for download or as a BINDER3 executed within the browser. More information on JUPYTER notebooks are found online.
Sometimes we must accept that we do not recognise or cannot model the influential physical mechanisms in a system we are observing. In these circumstances, physics-agnostic neural networks are an ideal tool because they can learn to recognise features that humans will miss.
We pulsed a turbulent combustor and measured the decay rate of thermoacoustic oscillations within it. This decay rate approaches zero as the combustor approaches the edge of its stable operating window. We wanted to identify the edge of the stable operating window without pulsing the combustor so we trained a Bayesian ensemble of Neural Networks (BayNNE) to learn the decay rate from the sound of the combustor before the pulse was applied.
This works, and the use of a Bayesian ensemble of Neural Networks (rather than a single NN) means that it also outputs the uncertainty in its prediction.
Perhaps the most striking finding was that the BayNNE recognised not only the decay rate, but also the operating point (the mass flowrates into the combustor). This showed that every operating point had a different sound and that a Neural Network could recognise the operating point just from that sound. A human may suspect this but would be unable to remember them all.
This is an interesting study for aircraft engines because fleets contains thousands of nominally-identical but slightly different engines. The signs of impending thermoacoustic instability could therefore be learned from the sound on a handful of engines and applied confidently to the others. This gives a way to avoid thermoacoustic instability, even if it has been impossible to design it out.
In the above experiments, we find that this method gives around 0.5 seconds of warning of impending thermoacoustic instability. We obtained a similar result in a rocket engine testbed:
