The results presented in this dissertation concern two different types of tomographic imaging. The first part of the dissertation is devoted to the time reversal method for approximate reconstruction of images in thermoacoustic tomography. A thorough numerical study of the method is presented. Error estimates of the time reversal approximation are provided. In the second part of the dissertation a type of emission tomography, called Compton camera imaging is considered. The mathematical problem arising in Compton camera imaging is the inversion of the cone transform. We present three methods for inversion of this transform in two dimensions. Numerical examples of reconstructions by these methods are also provided. Lastly, we turn to a problem of significance in homeland security, namely the detection of geometrically small, low emission sources in the presence of a large background radiation. We consider the use of Compton type detectors for this purpose and describe an efficient method for detection of such sources. Numerical examples demonstrating this method are also provided.

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This dissertation presents a formulation of the Continuous Sensitivity Equation Method (CSEM) applied to the Computational Fluid Dynamics (CFD) simulation of thermoacoustic instability problems. The proposed sensitivity analysis approach only requires a single run of the CFD simulation. Moreover, the sensitivities of field variables, pressure, velocity and temperature to boundary-condition parameters are directly obtained from the solution to sensitivity equations. Thermoacoustic instability is predicted by the Rayleigh criterion. The sensitivity of the Rayleigh index is computed utilizing the sensitivities of field variables. The application of the CSEM to thermoacoustic instability problems is demonstrated by two classic examples. The first example explores the effects of the heated wall temperature on the one-dimensional thermoacoustic convection. The sensitivity of the Rayleigh index, which is the indicator of thermoacoustic instabilities, is computed by the sensitivity of field variables. As the heat wall temperature increases, the sensitivity of the Rayleigh index decreases. The evolution from positive to negative sensitivity values suggests the transition from a destabilizing trend to stabilizing trend of the thermoacoustic system. Thermoacoustic instabilities in a self-excited Rijke tube are investigated following the relatively simple thermoacoustic convection problem. The complexity of simulating the Rijke tube increases in both dimensions and mechanisms which incorporate the species transport process and chemical reactions. As a representative model of the large lean premixed combustor, Rijke tube has been extensively studied. Quantitative sensitivity analysis sets the present work apart from previous research on the prediction and control of thermoacoustic instabilities. The effects of two boundary-condition parameters, i.e. the inlet mass flow rate and the equivalence ratio, are tested respectively. Small variations in both parameters predict a rapid change in sensitivities of field variables in the early stage of the total time length of 1.2s. The sensitivity of the Rayleigh index "blows up" at a specific time point of the early stage. In addition, variations in the inlet mass flow rate and the equivalence ratio lead to opposite effects on the sensitivity of the Rayleigh index. There exist some common findings on the application of the CSEM. For both thermoacoustic problems, the sensitivities of field variables and the Rayleigh index exhibit oscillatory nature, confirming that thermoacoustic instability is an overall effect of the coupling process between fluctuations of pressure and heat release rate. All the sensitivities of the Rayleigh index show rapid changes and "blow up" in the early stage. Although the numerical errors could influence the fidelity of computational results, it is believed that the rapid changes reflect the susceptibility to thermoacoustic instabilities in the studied systems. It should also be noted that the sensitivities are obtained for small variations in influential parameters. Therefore, the resulting sensitivities do not predict the occurrence of thermoacoustic instabilities under a condition that is far from the reference state determined by either CFD simulation results (employed in this dissertation) or experimental data. The sensitivity solver developed for the present research has the feature of flexibility. Additional mechanisms and more complicated instability criteria could be easily incorporated into the solver. Moreover, the sensitivity equations formulated in this dissertation are derived from the full set of nonlinear governing equations. Therefore, it is possible to extend the use of the sensitivity solver to other CFD problems. The developed sensitivity solver needs to be optimized to gain better performance, which is considered to be the primary future work of this research. / Ph. D.

Thermoacoustic instabilities in gas turbine operation arise due to unsteady fluctuations in heat release coupled with acoustic oscillations, often caused by varying equivalence ratio perturbations within the flame field. These instabilities can cause irreparable damage to critical turbine components, requiring an understanding of the spatial/temporal variations in equivalence ratio values to predict flame response. The technique of computed tomography for flame chemiluminescence emissions allows for 3D spatially resolved flame measurements to be acquired using a series of integral projections (camera images). High resolution tomography reconstructions require a selection of projection angles around the flame, while captured chemiluminescence of radical species intensity fields can be used to determine local fuel-air ratios. In this work, a tomographic reconstruction algorithm program was developed and utilized to reconstruct the intensity fields of CH* and OH*, and these reconstructions were used to quantify local equivalence ratios in an acoustically forced flame. A known phantom function was used to verify and validate the tomography algorithm, while convergence was determined by subsequent monitoring of selected iterative criteria. A documented method of camera calibration was also reproduced and presented here, with suggestions provided for future calibration improvement. Results are shown to highlight fluctuating equivalence ratio trends while illustrating the effectiveness of the developed tomography technique, providing a firm foundation for future study regarding heat release phenomena. / Master of Science / Acoustic sound amplification occurs in the combustion chamber of a gas turbine due to the machine ramping up in operation. These loud sound oscillations continue to grow larger and can damage the turbine machinery and even threaten the safety of the operator. Because of this, many researchers have attempted to understand and predict this behavior in hopes of ending them altogether. One method of studying these sound amplifications is looking at behaviors in the turbine combustion flame so as to potentially shed light on how these large disturbances form and accumulate. Both heat release rate (the steady release of energy in the form of heat from a combustion flame) and equivalence ratio (the mass ratio of fuel to air burned in a combustion process) have proven viable in illustrating oscillatory flame behavior, and can be visualized using chemiluminescence imaging paired with computed tomography. Chemiluminescence imaging is used to obtain intensity fields of species from high resolution camera imaging, while computed tomography techniques are capable of reconstructing these images into a three-dimensional volume to represent and visualize the combustion flame. These techniques have been shown to function effectively in previous literature and were further implemented in this work. A known calibration technique from previous work was carried out along with reconstructing a defined phantom function to show the functionality of the developed tomography algorithm. Results illustrate the effectiveness of the tomographic reconstruction technique and highlight the amplified acoustic behavior of a combustion flame in a high noise environment.

Thermoacoustic engines convert heat into acoustic pressure waves with no moving parts; this inherently results in high reliability, low maintenance and low manufacturing costs. Significant increases in the performance of these devices have enabled rivalry with more mature energy conversion methods in both efficiency and power output. This optimal production of acoustic power can be ultimately used to achieve cryogenic temperatures in thermoacoustic refrigerators, or can be interfaced with reciprocating electro-acoustic power transducers to generate electricity. This thesis describes the design, fabrication and testing of a Thermoacoustic Power Converter. The system interfaces a thermoacoustic-Stirling heat engine with a pair of linear alternators to produce 100 watts of electricity from a heat input. It operates with helium at 450 psig internal pressure and a hot side temperature of 1200F. Through thermoacoustic phenomena, these conditions sustain a powerful pressure wave at a system specific 100 Hz. This pressure wave is used to drive the two opposed linear alternators in equal and opposite directions to produce a single phase AC electrical output at that same system frequency. The opposing motion of the two alternators enables a vibration-balanced system. The engine has created 110 watts of acoustic power and the complete Thermoacoustic Power Converter system has produced 70 watts of AC electricity. Compensating for some heat leaks, the converter reaches 26.3% heat to acoustic power efficiency and 16.8% heat to electric efficiency when those maximum values are achieved. This conversion of heat to acoustic power is 40% of the Carnot thermodynamic efficiency limit.

In this dissertation, we first derive exact reconstuction algorithms for thermoacoustic tomography (TAT) and broadband diffraction tomography (a linearized inverse scattering problem) using derived time-reversal formulas. Then we focus on some important practical problems of TAT including the limited-view problem, the effects of acoustic heterogeneity, and fast reconstruction algorithms. In Chapter II, we propose time-reversal methods and apply them to tomography with diffrating sources. We first develop time-domain methods to time-reverse a transient scalar wave using only the field measured on an arbitrary closed surface enclosing the initial sources. Under certain conditions, a time-reversed field can be obtained with the delay-and-sum algorithm (backprojection to spheres) used in synthetic aperture imaging.Consequently, the physicalmeaningandthe validconditions of this widely used algorithm are revealed quantitatively for the first time from basic physics. Then exact reconstruction for TAT and broadband diffraction tomography is proposed by time-reversing the measured field back to the time when each source or secondary source is excited. The theoretical conclusions are supported by a numerical simulation ofthree-dimensional diffraction tomography.The extension ofour time-reversal methods to the case using Green function in a heterogeneous medium is also discussed. In Chapter III, the limited-view problem is studied for TAT. We define a "detection region," within which all points have sufficient detection views. It is explained analytically and shown numerically that the boundaries of any object inside this region can be recovered stably.Otherwise some sharp details become blurred.One can identify in advance the parts of the boundaries that will be affected if the detection view is insufficient. Computations are conducted for both numerically simulated and experimental data. The reconstructions confirm our theoretical predictions. In Chapter IV, the effects of wavefront distortions induced by acoustic heterogeneities in breast TAT are studied. Amplitude distortions are shown to be insignificant for different scales of acoustic heterogeneities. After that we consider the effects of phase distortions (errors in time-of-flight) in our numerical studies. The numerical results on the spreads of point sources and boundaries caused by the phase distortions are in good agreement with the proposed formula. We also demonstrate that the blurring of images can be compensated for by using the distribution of acoustic velocityin the tissues in the reconstructions. In Chapter V, we discuss exact and fast Fourier-domain reconstruction algorithms for TAT in planar and circular configurations.

An exogenous contrast agent has been developed to enhance the contrast achievable in Thermoacoustic Tomography (TAT). TAT utilizes the penetration depth of microwave energy while producing high resolution images through acoustic waves. A sample irradiated by a microwave source expands due to thermoelastic expansion. The acoustic wave created by this expansion is recorded by an ultrasonic transducer. The water content in biological samples poses an obstacle, as it is the primary absorber of microwave radiation. The addition of an exogenous contrast agent improves image quality by more effectively converting microwave energy to heat. The use of iron oxide nanoparticles in MRI applications has been explored but super paramagnetic iron oxide nanoparticles (SPION) have benefits in microwave applications, as well. Through ferromagnetic resonance, SPION samples more effectively convert microwave energy into heat. This transduction to heat creates significantly larger thermoacoustic waves than water, alone. Characterization of the SPION samples is executed through TAT, TEM, XPS, EDS, and a vector network analyzer with a dielectric probe kit. Onedimensional and phantom model imaging with an iron oxide nanoparticle contrast agent provide a two-fold improvement in contrast at current system configurations. Further enhancement is possible through adjustments to the nanoparticles and TAT system.

The spherical Radon transform (SRT) integrates a function over the set of all spheres with a given set of centers. Such transforms play an important role in some newly developing types of tomography as well as in several areas of mathematics including approximation theory, integral geometry, inverse problems for PDEs, etc. In Chapter I we give a brief description of thermoacoustic tomography (TAT or TCT) and introduce the SRT. In Chapter II we consider the injectivity problem for SRT. A major breakthrough in the 2D case was made several years ago by M. Agranovsky and E. T. Quinto. Their techniques involved microlocal analysis and known geometric properties of zeros of harmonic polynomials in the plane. Since then there has been an active search for alternative methods, which would be less restrictive in more general situations. We provide some new results obtained by PDE techniques that essentially involve only the finite speed of propagation and domain dependence for the wave equation. In Chapter III we consider the transform that integrates a function supported in the unit disk on the plane over circles centered at the boundary of this disk. As is common for transforms of the Radon type, its range has an in finite co-dimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography, for instance when dealing with incomplete data, error correction, and other issues. A complete range description for the circular Radon transform is obtained. In Chapter IV we investigate implementation of the recently discovered exact backprojection type inversion formulas for the case of spherical acquisition in 3D and approximate inversion formulas in 2D. A numerical simulation of the data acquisition with subsequent reconstructions is made for the Defrise phantom as well as for some other phantoms. Both full and partial scan situations are considered.

In this dissertation, we first derive exact reconstuction algorithms for thermoacoustic tomography (TAT) and broadband diffraction tomography (a linearized inverse scattering problem) using derived time-reversal formulas. Then we focus on some important practical problems of TAT including the limited-view problem, the effects of acoustic heterogeneity, and fast reconstruction algorithms. In Chapter II, we propose time-reversal methods and apply them to tomography with diffrating sources. We first develop time-domain methods to time-reverse a transient scalar wave using only the field measured on an arbitrary closed surface enclosing the initial sources. Under certain conditions, a time-reversed field can be obtained with the delay-and-sum algorithm (backprojection to spheres) used in synthetic aperture imaging.Consequently, the physicalmeaningandthe validconditions of this widely used algorithm are revealed quantitatively for the first time from basic physics. Then exact reconstruction for TAT and broadband diffraction tomography is proposed by time-reversing the measured field back to the time when each source or secondary source is excited. The theoretical conclusions are supported by a numerical simulation ofthree-dimensional diffraction tomography.The extension ofour time-reversal methods to the case using Green function in a heterogeneous medium is also discussed. In Chapter III, the limited-view problem is studied for TAT. We define a "detection region," within which all points have sufficient detection views. It is explained analytically and shown numerically that the boundaries of any object inside this region can be recovered stably.Otherwise some sharp details become blurred.One can identify in advance the parts of the boundaries that will be affected if the detection view is insufficient. Computations are conducted for both numerically simulated and experimental data. The reconstructions confirm our theoretical predictions. In Chapter IV, the effects of wavefront distortions induced by acoustic heterogeneities in breast TAT are studied. Amplitude distortions are shown to be insignificant for different scales of acoustic heterogeneities. After that we consider the effects of phase distortions (errors in time-of-flight) in our numerical studies. The numerical results on the spreads of point sources and boundaries caused by the phase distortions are in good agreement with the proposed formula. We also demonstrate that the blurring of images can be compensated for by using the distribution of acoustic velocityin the tissues in the reconstructions. In Chapter V, we discuss exact and fast Fourier-domain reconstruction algorithms for TAT in planar and circular configurations.

Les instabilités thermo-acoustiques résultent de l'interaction entre les oscillations de pression acoustique et les fluctuations du taux de dégagement de chaleur de la flamme. Ces instabilités de combustion sont particulièrement préoccupantes en raison de leur fréquence dans les turbines à gaz modernes et à faible émission. Leurs principaux effets indésirables sont une réduction du temps de fonctionnement du moteur en raison des oscillations de grandes amplitudes ainsi que de fortes vibrations à l'intérieur de la chambre de combustion. La simulation numérique est maintenant devenue une approche clé pour comprendre et prédire ces instabilités dans la phase de conception industrielle. Cependant, la prédiction de ce phénomène reste difficile en raison de sa complexité; cela se confirme lorsque les paramètres physiques du processus de modélisation sont incertains, ce qui est pratiquement toujours le cas pour des systèmes réels.Introduire la quantification des incertitudes pour la thermo-acoustique est le seul moyen d'étudier et de contrôler la stabilité des chambres de combustion qui fonctionnent dans des conditions réalistes; c'est l'objectif de cette thèse.Dans un premier temps, une chambre de combustion académique (avec un seul injecteur et une seule flamme) ainsi que deux chambres de moteurs d'hélicoptère (avec N injecteurs et des flammes) sont étudiés. Les calculs basés sur un solveur de Helmholtz et un outil quasi-analytique de bas ordre fournissent des estimations appropriées de la fréquence et des structures modales pour chaque géométrie. L'analyse suggère que la réponse de la flamme aux perturbations acoustiques joue un rôle prédominant dans la dynamique de la chambre de combustion. Ainsi, la prise en compte des incertitudes liées à la représentation de la flamme apparaît comme une étape nécessaire vers une analyse robuste de la stabilité du système.Dans un second temps, la notion de facteur de risque, c'est-à-dire la probabilité pour un mode thermo-acoustique d'être instable, est introduite afin de fournir une description plus générale du système que la classification classique et binaire (stable / instable). Les approches de modélisation de Monte Carlo et de modèle de substitution sont associées pour effectuer une analyse de quantification d'incertitudes de la chambre de combustion académique avec deux paramètres incertains (amplitude et temps de réponse de la flamme). On montre que l'utilisation de modèles de substitution algébriques réduit drastiquement le nombre de calculs initiales, donc la charge de calcul, tout en fournissant des estimations précises du facteur de risque modal. Pour traiter les problèmes multidimensionnel tels que les deux moteurs d'hélicoptère, une stratégie visant à réduire le nombre de paramètres incertains est introduite. La méthode <<Active Subspace>> combinée à une approche de changement de variables a permis d'identifier trois directions dominantes (au lieu des N paramètres incertains initiaux) qui suffisent à décrire la dynamique des deux systèmes industriels. Dès lors que ces paramètres dominants sont associés à des modèles de substitution appropriés, cela permet de réaliser efficacement une analyse de quantification des incertitudes de systèmes thermo-acoustiques complexes.Finalement, on examine la perspective d'utiliser la méthode adjointe pour analyser la sensibilité des systèmes thermo-acoustiques représentés par des solveurs 3D de Helmholtz. Les résultats obtenus sur des cas tests 2D et 3D sont prometteurs et suggèrent d'explorer davantage le potentiel de cette méthode dans le cas de problèmes thermo-acoustiques encore plus complexes. / Thermoacoustic instabilities result from the interaction between acoustic pressure oscillations and flame heat release rate fluctuations. These combustion instabilities are of particular concern due to their frequent occurrence in modern, low emission gas turbine engines. Their major undesirable consequence is a reduced time of operation due to large amplitude oscillations of the flame position and structural vibrations within the combustor. Computational Fluid Dynamics (CFD) has now become one a key approach to understand and predict these instabilities at industrial readiness level. Still, predicting this phenomenon remains difficult due to modelling and computational challenges; this is even more true when physical parameters of the modelling process are uncertain, which is always the case in practical situations. Introducing Uncertainty Quantification for thermoacoustics is the only way to study and control the stability of gas turbine combustors operated under realistic conditions; this is the objective of this work.First, a laboratory-scale combustor (with only one injector and flame) as well as two industrial helicopter engines (with N injectors and flames) are investigated. Calculations based on a Helmholtz solver and quasi analytical low order tool provide suitable estimates of the frequency and modal structures for each geometry. The analysis suggests that the flame response to acoustic perturbations plays the predominant role in the dynamics of the combustor. Accounting for the uncertainties of the flame representation is thus identified as a key step towards a robust stability analysis.Second, the notion of Risk Factor, that is to say the probability for a particular thermoacoustic mode to be unstable, is introduced in order to provide a more general description of the system than the classical binary (stable/unstable) classification. Monte Carlo and surrogate modelling approaches are then combined to perform an uncertainty quantification analysis of the laboratory-scale combustor with two uncertain parameters (amplitude and time delay of the flame response). It is shown that the use of algebraic surrogate models reduces drastically the number of state computations, thus the computational load, while providing accurate estimates of the modal risk factor. To deal with the curse of dimensionality, a strategy to reduce the number of uncertain parameters is further introduced in order to properly handle the two industrial helicopter engines. The active subspace algorithm used together with a change of variables allows identifying three dominant directions (instead of N initial uncertain parameters) which are sufficient to describe the dynamics of the industrial systems. Combined with appropriate surrogate models construction, this allows to conduct computationally efficient uncertainty quantification analysis of complex thermoacoustic systems.Third, the perspective of using adjoint method for the sensitivity analysis of thermoacoustic systems represented by 3D Helmholtz solvers is examined. The results obtained for 2D and 3D test cases are promising and suggest to further explore the potential of this method on even more complex thermoacoustic problems.