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Bayesian Methods for Gas-Phase Tomography

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Bayesian Methods for Gas-Phase Tomography by Samuel Jacobi Grauer A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Mechanical and Mechatronics Engineering Waterloo, Ontario, Canada, 2018 © Samuel Jacobi Grauer 2018 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner Ashley F. Emery Professor, University of Washington Supervisor Kyle J. Daun Associate Professor, University of Waterloo Internal Member Serhiy Yarusevych Associate Professor, University of Waterloo Internal Member Steven L. Waslander Associate Professor, University of Waterloo Internal-external Member Christopher Batty Assistant Professor, University of Waterloo ii Author’s Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract Gas-phase tomography refers to a set of techniques that determine the 2D or 3D distribution of a target species in a jet, plume, or flame using measurements of light, made around the boundary of a flow area. Reconstructed quantities may include the concentration of one or more species, temperature, pressure, and optical density, among others. Tomography is increasingly used to study fundamental aspects of turbulent combustion and monitor emissions for regulatory compliance. This thesis develops statistical methods to improve gas-phase tomography and reports two novel experimental applications. Tomography is an inverse problem, meaning that a forward model (calculating measurements of light for a known distribution of gas) is inverted to estimate the model parameters (transforming experimental data into a gas distribution). The measurement modality varies with the problem geometry and objective of the experiment. For instance, transmittance data from an array of laser beams that transect a jet may be inverted to recover 2D fields of concentration and temperature; and multiple high-resolution images of a flame, captured from different angles, are used to reconstruct wrinkling of the 3D reacting zone. Forward models for gas-phase tomography modalities share a common mathematical form, that of a Fredholm integral equation of the first- kind (IFK). The inversion of coupled IFKs is necessarily ill-posed, however, meaning that solutions are either unstable or non-unique. Measurements are thus insufficient in themselves to generate a realistic image of the gas and additional information must be incorporated into the reconstruction procedure. Statistical inversion is an approach to inverse problems in which the measurements, experimental parameters, and quantities of interest are treated as random variables, characterized by a probability distribution. These distributions reflect uncertainty about the target due to fluctuations in the flow field, noise in the data, errors in the forward model, and the ill-posed nature of reconstruction. The Bayesian framework for tomography features a likelihood probability density function (pdf), which describes the chance of observing a measurement for a given distribution of gas, and prior pdf, which assigns a relative plausibility to candidate distributions based on assumptions about the flow physics. Bayes’ equation updates information about the target iv in response to measurement data, combining the likelihood and prior functions to form a posterior pdf. The posterior is usually summarized by the maximum a posteriori (MAP) estimate, which is the most likely distribution of gas for a set of data, subject to the effects of noise, model errors, and prior information. The framework can be used to estimate credibility intervals for a reconstruction and the form of Bayes’ equation suggests procedures for improving gas tomography. The accuracy of reconstructions depends on the information content of the data, which is a function of the experimental design, as well as the specificity and validity of the prior. This thesis employs theoretical arguments and experimental measurements of scalar fluctuations to justify joint-normal likelihood and prior pdfs for gas-phase tomography. Three methods are introduced to improve each stage of the inverse problem: to develop priors, design optimal experiments, and select a discretization scheme. First, a self-similarity analysis of turbulent jets—common targets in gas tomography—is used to construct an advanced prior, informed by an estimate of the jet’s spatial covariance. Next, a Bayesian objective function is proposed to optimize beam positions in limited-data arrays, which are necessary in scenarios where optical access to the flow area is restricted. Finally, a Bayesian expression for model selection is derived from the joint-normal pdfs and employed to select a mathematical basis to reconstruct a flow. Extensive numerical evidence is presented to validate these methods. The dissertation continues with two novel experiments, conducted in a Bayesian way. Broadband absorption tomography is a new technique intended for quantitative emissions detection from spectrally-convolved absorption signals. Theoretical foundations for the diagnostic are developed and the results of a proof-of-concept emissions detection experiment are reported. Lastly, background-oriented schlieren (BOS) tomography is applied to combustion for the first time. BOS tomography employs measurements of beam steering to reconstruct a fluid’s optical density field, which can be used to infer temperature and density. The application of BOS tomography to flame imaging sets the stage for instantaneous 3D combustion thermometry. Numerical and experimental results reported in this thesis support a Bayesian approach to gas- phase tomography. Bayesian tomography makes the role of prior information explicit, which can be leveraged to optimize reconstructions and design better imaging systems in support of research on fluid flow and combustion dynamics. v Acknowledgements To begin, I would like to thank Professor Kyle James Daun, noted “really nice person,” wizard of inverse analysis, heat transfer expert, proud Manitoban, and benevolent supervisor. Professor Daun, you sold me on Waterloo during our first meeting with your warmth and enthusiasm. When I arrived in Duisburg, I would tell people “you may have met my supervisor, Kyle from Waterloo” and they would light up in recognition and relay some charming anecdote from your last visit. Ditto for the inverse conferences at Michigan State and VMI. Anyone would be glad to leave such an impression. You got me hooked on Bayesian analysis and ill-posed inverse problems in math and engineering,1 which warrants appreciation. I admire your energy and patience, your literary flair, and your deep understanding of applied research. It has been an honour and a pleasure to study under your guidance and I truly cannot thank you enough. Four other researchers played an outsized role in my trajectory. In chronological order: I thank Prof. Scott Ormiston, for his mentorship during my undergraduate studies. I thank Prof. Mary Wells, for first inviting me to Waterloo and giving me free rein to tackle a challenging problem. I thank Dr. Paul Hadwin, for entertaining my dumb questions and troubleshooting my code; for tolerating my argumentative streak; and for coaching me through my statistics coursework. And I thank Prof. Dr. Khadijeh Mohri, for bringing me to Duisburg. You believed in our project and made sure we saw it through; your determination is remarkable. I have fond memories of our daily run and hope to continue our collaboration. My labmates have been wonderful and deserve my gratitude, as well. I will name several of you without elaboration, but know that I could (and would) go on at great length, had I the time and motivation. Again chronologically: Etienne Caron, Noel Chester, Kamal Jhajj, Josh Rasera, Cangji Shi, Nigel Singh, Brad Conrad (h.c.), Roger Tsang, Natalie Field, Tim Thompson, Simon Trivett, Tim Sipkens, Sina Talebi Moghaddam, Mohit Verma, Rodrigo Miguel, Cory Yang, and Kaihsiang Lin.2 Each of you has taught me a great deal and I wish you all the best. To the 1 To the uninitiated reader, this development has been mostly positive. More hooked on phonics than hooked on drugs. 2 Past members are listed in order of their departure; current members are listed by seniority. vi Germans—Andreas3, Johannes, Torsten, Patrick, et al.—you were first-rate hosts and your refusal to jaywalk was delightful. Some elaboration is in order, of course. Roger, thanks for putting up with me, thanks for your resilience and for the good conversation; you’re a tennis trivia champ and you’ve always got a spot on my team. Natalie, much of the same—sans tennis trivia. And Josh, Alanis Morissette has nothing on you, man. Keep on rockin’ in the free world. To Tim, I owe you a swimming pool of ice cream. You have been the consummate roommate and labmate. You are a brilliant researcher and I wish you every success on your journey to Emeritus. I look forward to visiting you in Vancouver and hosting you whenever you’re able to make it out my way. To other friends from Waterloo, who know who they are, the pasta parties were a romp and your companionship will be dearly missed. And Emily, the writing cafés were a godsend. Brilliant suggestion (insert OK hand emoji). Lastly, to my parents, Hermann and Sheryl Grauer, thank you a thousand times. You taught me what is Good and showed me how to live well. You instilled in me a spirit to work hard and serve others. You fostered my curiosity and encouraged me to persist when I did not think I could. I never wanted for love nor support. In return, I love you from the bottom of my heart and it is my greatest ambition to follow in your footsteps. To my siblings, Elizabeth and Joseph (also Grauer), you guys are alright. Thanks for being there.4 3 Andreas3 refers to Andreases Unterberger, Rittler, and Kempf (Prof.
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