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Reduced-order models are frequently used in the simulation of complex flows to overcome the high computational cost of direct numerical simulations, especially for three-dimensional nonlinear problems. Proper orthogonal decomposition, as one of the most commonly used tools to generate reduced-order models, has been utilized in many engineering and scientific applications. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. In this dissertation, we put forth two new closure models for the proper orthogonal decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, are carefully derived and numerically investigated. Since modern closure models for turbulent flows generally have non-polynomial nonlinearities, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This dissertation proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear proper orthogonal decomposition closure models. This method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter, a two-dimensional flow past a cylinder at Reynolds number Re = 200, and a three-dimensional flow past a cylinder at Reynolds number Re = 1000. With the help of the two-level algorithm, the new nonlinear proper orthogonal decomposition closure models (i.e., the dynamic subgrid-scale model and the variational multiscale model), together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of a three-dimensional turbulent flow past a cylinder at Re = 1000. Five criteria are used to judge the performance of the proper orthogonal decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the proper orthogonal decomposition basis coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate. We present a rigorous numerical analysis for the discretization of the new models. As a first step, we derive an error estimate for the time discretization of the Smagorinsky proper orthogonal decomposition reduced-order model for the Burgers equation with a small diffusion parameter. The theoretical analysis is numerically verified by two tests on problems displaying shock-like phenomena. We then present a thorough numerical analysis for the finite element discretization of the variational multiscale proper orthogonal decomposition reduced-order model for convection-dominated convection-diffusion-reaction equations. Numerical tests show the increased numerical accuracy over the standard reduced-order model and illustrate the theoretical convergence rates. We also discuss the use of the new reduced-order models in realistic applications such as airflow simulation in energy efficient building design and control problems as well as numerical simulation of large-scale ocean motions in climate modeling. Several research directions that we plan to pursue in the future are outlined. / Ph. D.

In this dissertation, we consider two different strategies for improving the projection-based reduced order model (ROM) accuracy: (I) adding closure terms to the standard ROM; (II) using Lagrangian data to improve the ROM basis. Following strategy (I), we propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $nu = 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs. Furthermore, we propose a new hybrid ROM framework for the numerical simulation of fluid flows. This hybrid framework incorporates two closure modeling strategies: (i) A structural closure modeling component that involves the recently proposed data-driven variational multiscale ROM approach, and (ii) A functional closure modeling component that introduces an artificial viscosity term. We also utilize physical constraints for the structural ROM operators in order to add robustness to the hybrid ROM. We perform a numerical investigation of the hybrid ROM for the three-dimensional turbulent channel flow at a Reynolds number $Re = 13,750$. In addition, we focus on the mathematical foundations of ROM closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. For strategy (II), we propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis. / Doctor of Philosophy / Reduced order models (ROMs) are popular in physical and engineering applications: for example, ROMs are widely used in aircraft designing as it can greatly reduce computational cost for the aircraft's aeroelastic predictions while retaining good accuracy. However, for high Reynolds number turbulent flows, such as blood flows in arteries, oil transport in pipelines, and ocean currents, the standard ROMs may yield inaccurate results. In this dissertation, to improve ROM's accuracy for turbulent flows, we investigate three different types of ROMs. In this dissertation, both numerical and theoretical results show that the proposed new ROMs yield more accurate results than the standard ROM and thus can be more useful.

Over the past century, great strides have been made in the advancement of mine fire knowledge since the 1909 Cherry Mine Fire Disaster, one of the worst in U.S. history. However, fire hazards remain omnipresent in underground coal mines in the U.S. and around the world. A precise fire numerical analysis (simulation) before any fire events can give a broad view of the emergency scenarios, leading to improved emergency response, and better health and safety outcomes. However, the simulation cost of precise large complex dynamical systems such as fire in underground mines makes practical and even theoretical application challenging. This work details a novel methodology to reduce fire and airflow simulation costs in order to make simulation of complex systems around fire and mine ventilation systems viable. This study will examine the development of a Reduced Order Model (ROM) to predict the flow field of an underground mine geometry using proper orthogonal decomposition (POD) to reduce the airflow simulation cost in a nonlinear model. ROM proves to be an effective tool for approximating several possible solutions near a known solution, resulting in significant time savings over calculating full solutions and suitable for ensemble calculations. In addition, a novel iterative methodology was developed based on the physics of the fluid structure, turbulent kinetic energy (TKE) of the dynamical system, and the vortex dynamics to determine the interface boundary in multiscale (3D-1D) fire simulations of underground space environments. The proposed methodology was demonstrated to be a useful technique for the determination of near and far fire fields, and could be applied across a broad range of flow simulations and mine geometries. Moreover, this research develops a methodology to analyze the tenable limits in a methane fire event in an underground coal mine for bare-faced miners, mine rescue teams, and fire brigade teams in order to improve safety and training of personnel trained to fight fires. The outcomes of this research are specific to mining although the methods outlined might have broader impacts on the other fields such as tunneling and underground spaces technology, HVAC, and fire protection engineering industries. / Ph. D.

Cette thèse concerne le développement, l'extension et l'application d'une formulation stochastique des équations de la mécanique des fluides introduite par Mémin (2014). La vitesse petite échelle, non-résolue, est modélisée au moyen d'un champ aléatoire décorrélé en temps. Cela modifie l'expression de la dérivée particulaire et donc les équations de la mécanique des fluides. Les modèles qui en découlent sont dénommés modèles sous incertitude de position. La thèse s'articulent autour de l'étude successive de modèles réduits, de versions stochastiques du transport et de l'advection à temps long d'un champ de traceur par une vitesse mal résolue. La POD est une méthode de réduction de dimension, pour EDP, rendue possible par l'utilisation d'observations. L'EDP régissant l'évolution de la vitesse du fluide est remplacée par un nombre fini d'EDOs couplées. Grâce à la modélisation sous incertitude de position et à de nouveaux estimateurs statistiques, nous avons dérivé et simulé des versions réduites, déterministe et aléatoire, de l'équation de Navier-Stokes. Après avoir obtenu des versions aléatoires de plusieurs modèles océaniques, nous avons montré numériquement que ces modèles permettaient de mieux prendre en compte les petites échelles des écoulements, tout en donnant accès à des estimés de bonne qualité des erreurs du modèle. Ils permettent par ailleurs de mieux rendre compte des évènements extrêmes, des bifurcations ainsi que des phénomènes physiques réalistes absents de certains modèles déterministes équivalents. Nous avons expliqué, démontré et quantifié mathématiquement l'apparition de petites échelles de traceur, lors de l'advection par une vitesse mal résolu. Cette quantification permet de fixer proprement des paramètres de la méthode d'advection Lagrangienne, de mieux le comprendre le phénomène de mélange et d'aider au paramétrage des simulations grande échelle en mécanique des fluides. / This thesis develops, analyzes and demonstrates several valuable applications of randomized fluid dynamics models referred to as under location uncertainty. The velocity is decomposed between large-scale components and random time-uncorrelated small-scale components. This assumption leads to a modification of the material derivative and hence of every fluid dynamics models. Through the thesis, the mixing induced by deterministic low-resolution flows is also investigated. We first applied that decomposition to reduced order models (ROM). The fluid velocity is expressed on a finite-dimensional basis and its evolution law is projected onto each of these modes. We derive two types of ROMs of Navier-Stokes equations. A deterministic LES-like model is able to stabilize ROMs and to better analyze the influence of the residual velocity on the resolved component. The random one additionally maintains the variability of stable modes and quantifies the model errors. We derive random versions of several geophysical models. We numerically study the transport under location uncertainty through a simplified one. A single realization of our model better retrieves the small-scale tracer structures than a deterministic simulation. Furthermore, a small ensemble of simulations accurately predicts and describes the extreme events, the bifurcations as well as the amplitude and the position of the ensemble errors. Another of our derived simplified model quantifies the frontolysis and the frontogenesis in the upper ocean. This thesis also studied the mixing of tracers generated by smooth fluid flows, after a finite time. We propose a simple model to describe the stretching as well as the spatial and spectral structures of advected tracers. With a toy flow but also with satellite images, we apply our model to locally and globally describe the mixing, specify the advection time and the filter width of the Lagrangian advection method, as well as the turbulent diffusivity in numerical simulations.

O objetivo principal desta tese consiste no desenvolvimento de ferramentas matemáticas e computacionais dedicadas a um sistema de diagnóstico de barras quebradas no rotor de Motores de Indução Trifásicos. O sistema proposto é baseado em um método matemático de decomposição de sinais elétricos, denominado de Decomposição em Componentes Ortogonais, e ferramentas de aprendizagem de máquinas. Como uma das principais contribuições desta pesquisa, realizou-se um aprofundamento do entendimento da técnica de Decomposição em Componentes Ortogonais e de sua aplicabilidade como ferramenta de processamento de sinais para sistemas elétricos e eletromecânicos. Redes Neurais Artificiais e Support Vector Machines, tanto para classificação multi-classes quanto para detecção de novidades, foram configurados para receber índices advindos do processamento de sinais elétricos de motores, e a partir deles, identificar os padrões normais e os padrões com falhas. Além disso, a severidade da falha também é diagnosticada, a qual é representada pelo número de barras quebradas no rotor. Para a avaliação da metodologia, considerou-se o acionamento de motores de indução pela tensão de alimentação da rede e por inversores de frequência, operando sob diversas condições de torque de carga. Os resultados alcançados demonstram a eficácia das ferramentas matemáticas e computacionais desenvolvidas para o sistema de diagnóstico, sendo que os índices criados se mostraram altamente correlacionados com o fenômeno da falha. Mais especificamente, foi possível criar índices monotônicos com a severidade da falha e com baixa variabilidade, demonstrando-se que as ferramentas são eficientes extratores de características. / This doctoral thesis consists of the development of mathematical and computational tools dedicated to a diagnostic system for broken rotor bars in Three Phase Induction Motors. The proposed system is based on a mathematical method for decomposing electrical signals, named the Orthogonal Components Decomposition, and machine learning tools. As one of the main contributions of this research, an in-depth investigation of the decomposition technique and its applicability as a signal processing tool for electrical and electromechanical systems was carried-out. Artificial Neural Networks and Support Vector Machines for multi-class classification and novelty detection were configured to receive indices derived from the processing of electrical signals and then identify normal motors and faulty motors. In addition, the fault severity is also diagnosed, which is represented by the number of broken rotor bars. Experimental data was tested in order to evaluate the proposed method. Signals were obtained from induction motors operating with different torque levels and driven either directly by the grid or by frequency inverters. The results demonstrate the effectiveness of the mathematical and computational tools developed for the diagnostic system since the indices created are highly correlated with the fault phenomenon. More specifically, it was possible to create monotonic indices with the fault severity and with low variability, what supports that the solution is an efficient fault-specific feature extractor.

O objetivo principal desta tese consiste no desenvolvimento de ferramentas matemáticas e computacionais dedicadas a um sistema de diagnóstico de barras quebradas no rotor de Motores de Indução Trifásicos. O sistema proposto é baseado em um método matemático de decomposição de sinais elétricos, denominado de Decomposição em Componentes Ortogonais, e ferramentas de aprendizagem de máquinas. Como uma das principais contribuições desta pesquisa, realizou-se um aprofundamento do entendimento da técnica de Decomposição em Componentes Ortogonais e de sua aplicabilidade como ferramenta de processamento de sinais para sistemas elétricos e eletromecânicos. Redes Neurais Artificiais e Support Vector Machines, tanto para classificação multi-classes quanto para detecção de novidades, foram configurados para receber índices advindos do processamento de sinais elétricos de motores, e a partir deles, identificar os padrões normais e os padrões com falhas. Além disso, a severidade da falha também é diagnosticada, a qual é representada pelo número de barras quebradas no rotor. Para a avaliação da metodologia, considerou-se o acionamento de motores de indução pela tensão de alimentação da rede e por inversores de frequência, operando sob diversas condições de torque de carga. Os resultados alcançados demonstram a eficácia das ferramentas matemáticas e computacionais desenvolvidas para o sistema de diagnóstico, sendo que os índices criados se mostraram altamente correlacionados com o fenômeno da falha. Mais especificamente, foi possível criar índices monotônicos com a severidade da falha e com baixa variabilidade, demonstrando-se que as ferramentas são eficientes extratores de características. / This doctoral thesis consists of the development of mathematical and computational tools dedicated to a diagnostic system for broken rotor bars in Three Phase Induction Motors. The proposed system is based on a mathematical method for decomposing electrical signals, named the Orthogonal Components Decomposition, and machine learning tools. As one of the main contributions of this research, an in-depth investigation of the decomposition technique and its applicability as a signal processing tool for electrical and electromechanical systems was carried-out. Artificial Neural Networks and Support Vector Machines for multi-class classification and novelty detection were configured to receive indices derived from the processing of electrical signals and then identify normal motors and faulty motors. In addition, the fault severity is also diagnosed, which is represented by the number of broken rotor bars. Experimental data was tested in order to evaluate the proposed method. Signals were obtained from induction motors operating with different torque levels and driven either directly by the grid or by frequency inverters. The results demonstrate the effectiveness of the mathematical and computational tools developed for the diagnostic system since the indices created are highly correlated with the fault phenomenon. More specifically, it was possible to create monotonic indices with the fault severity and with low variability, what supports that the solution is an efficient fault-specific feature extractor.