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.

Les courants turbulents de bord ouest sont l'un des phénomènes les plus dominants des océans, il en existe aux faibles latitudes aussi. Ils sont caractérisés par une dynamique très turbulente avec une forte production d'énergie cinétique, et une forte variabilité interne. Plusieurs régions existent où les courants de bord ouest se rétrofléchissent (décollage de la côte) pour former des structures cohérentes: des anticyclones, des bursts (arrachements) et des dipoles. Circulant le long de la côte, les courants de bord ouest interagissent très fortement avec le bord ouest et la bathymétrie et sont donc un problème de couche limite. Cependant aucune étude du point de vue de la théorie de couche limite n'a été jamais été faite. Cette thèse aborde le problème d'un point de vue de couche limite par l'utilisation d'un modèle idéalisé "shallow water" à très haute résolution (2.5km) afin d'isoler et de comprendre les processus. Les résultats sont ensuite appliqués à des sorties de modèle réaliste Drakkar (~10km) basé sur le code NEMO. Le courant de Somali est ensuite pris pour cette application. / Strong western boundary currents are one of dominant features of the world oceans, also at low latitudes. They exhibit a turbulent dynamics and their region is a source of strong kinetic energy production and internal variability of the worlds oceans. Several places exists where the western boundary currents retrofect (i.e separation from the coast) and generate coherent structures as anticyclonic eddies, bursts and dipoles. The dynamics of turbulent western boundary currents has so far not been extensively studied in the viewpoint of turbulent boundary-layer theory. The approach followed in this thesis is to use a fine resolution (2.5km) reduced-gravity shallow water model to understand the turbulent boundary-layer processes and then apply these findings to the Ocean General Circulation Model NEMO in the Drakkar configuration (~10km). The case of the Somali Currentis considered for this application.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Uma simulação numérica que leva em conta os efeitos de estratificação e mistura escalar (como a temperatura, salinidade ou substância solúvel em água) é necessária para estudar e prever os impactos ambientais que um reservatório de usina hidrelétrica pode produzir. Este trabalho sugere uma metodologia para o estudo de escoamentos ambientais, principalmente aqueles em que o conhecimento da interação entre a estratificação e mistura pode dar noções importantes dos fenômenos que ocorrem. Por esta razão, ferramentas de simulação numérica 3D de escoamento ambiental são desenvolvidas. Um gerador de malha de tetraedros do reservatório e o modelo de turbulência algébrico baseado no número de Richardson são as principais ferramentas desenvolvidas. A principal dificuldade na geração de uma malha de tetraedros de um reservatório é a distribuição não uniforme dos pontos relacionada com a relação desproporcional entre as escalas horizontais e verticais do reservatório. Neste tipo de distribuição de pontos, o algoritmo convencional de geração de malha de tetraedros pode tornar-se instável. Por esta razão, um gerador de malha não estruturada de tetraedros é desenvolvido e a metodologia utilizada para obter elementos conformes é descrita. A geração de malha superficial de triângulos utilizando a triangulação Delaunay e a construção do tetraedros a partir da malha triangular são os principais passos para o gerador de malha. A simulação hidrodinâmica com o modelo de turbulência fornece uma ferramenta útil e computacionalmente viável para fins de engenharia. Além disso, o modelo de turbulência baseado no número de Richardson leva em conta os efeitos da interação entre turbulência e estratificação. O modelo algébrico é o mais simples entre os diversos modelos de turbulência. Mas, fornece resultados realistas com o ajuste de uma pequena quantidade de parâmetros. São incorporados os modelos de viscosidade/difusividade turbulenta para escoamento estratificado. Na aproximação das equações médias de Reynolds e transporte de escalar é utilizando o Método dos Elementos Finitos. Os termos convectivos são aproximados utilizando o método semi-Lagrangeano, e a aproximação espacial é baseada no método de Galerkin. Os resultados computacionais são comparados com os resultados disponíveis na literatura. E, finalmente, a simulação de escoamento em um braço de reservatório é apresentada. / To study and forecast the environmental impacts that a hydroelectric power plant reservoir may produce, a numerical simulation that takes into account the effects of stratification and scalar mixing (such as temperature, salinity or water-soluble substance) is required. This work proposes a methodology for the study of the environmental fluid flow phenomena, mainly for flows in which the knowledge of the interaction between stratification and mixing can give important notions of the phenomena that occur. For this, a numerical simulation tool for 3D environmental flow is developed. A tetrahedral mesh generator of the reservoir based on the terrain topology and an algebraic turbulence model based on the Richardson number are the main tools developed. The main difficulty in tetrahedral mesh generation of a reservoir is nonuniform distribution of the points related to the huge ratio between the horizontal and vertical scales of the reservoir. In this type of point distributions, conventional tetrahedron mesh generation algorithm may become unstable. For this reason, a unstructured tetrahedral mesh generator is developed and the methodology used to obtain conforming elements is described. Triangular surface mesh generation using the Delaunay triangulation and the construction of the tetrahedra from the triangular surface mesh are the main steps to the mesh generator. The hydrodynamic simulation of reservoirs with a turbulence model provides a useful tool that is computationally viable for engineering purposes. Furthermore, the turbulence model based on the Richardson number takes into account the effects of interaction between turbulence and stratification. The algebraic model is the simplest among the various models of turbulence, but provides realistic results with the fitting of a small amount of parameters. Eddy-Viscosity/Diffusivity models for stratified turbulent flows models are incorporated. Using the Finite Element Method (FEM) approximation the Reynolds-averaged Navier-Stokes (RANS) and mean scalar transport equations are approximated. The convective terms are discretized employing the Semi-Lagrangian method, and the spatial discretization is based on the Galerkin method. The computational results are compared with the results available in the literature. Finally, the simulation of the flow in a branch of a reservoir is presented.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Uma simulação numérica que leva em conta os efeitos de estratificação e mistura escalar (como a temperatura, salinidade ou substância solúvel em água) é necessária para estudar e prever os impactos ambientais que um reservatório de usina hidrelétrica pode produzir. Este trabalho sugere uma metodologia para o estudo de escoamentos ambientais, principalmente aqueles em que o conhecimento da interação entre a estratificação e mistura pode dar noções importantes dos fenômenos que ocorrem. Por esta razão, ferramentas de simulação numérica 3D de escoamento ambiental são desenvolvidas. Um gerador de malha de tetraedros do reservatório e o modelo de turbulência algébrico baseado no número de Richardson são as principais ferramentas desenvolvidas. A principal dificuldade na geração de uma malha de tetraedros de um reservatório é a distribuição não uniforme dos pontos relacionada com a relação desproporcional entre as escalas horizontais e verticais do reservatório. Neste tipo de distribuição de pontos, o algoritmo convencional de geração de malha de tetraedros pode tornar-se instável. Por esta razão, um gerador de malha não estruturada de tetraedros é desenvolvido e a metodologia utilizada para obter elementos conformes é descrita. A geração de malha superficial de triângulos utilizando a triangulação Delaunay e a construção do tetraedros a partir da malha triangular são os principais passos para o gerador de malha. A simulação hidrodinâmica com o modelo de turbulência fornece uma ferramenta útil e computacionalmente viável para fins de engenharia. Além disso, o modelo de turbulência baseado no número de Richardson leva em conta os efeitos da interação entre turbulência e estratificação. O modelo algébrico é o mais simples entre os diversos modelos de turbulência. Mas, fornece resultados realistas com o ajuste de uma pequena quantidade de parâmetros. São incorporados os modelos de viscosidade/difusividade turbulenta para escoamento estratificado. Na aproximação das equações médias de Reynolds e transporte de escalar é utilizando o Método dos Elementos Finitos. Os termos convectivos são aproximados utilizando o método semi-Lagrangeano, e a aproximação espacial é baseada no método de Galerkin. Os resultados computacionais são comparados com os resultados disponíveis na literatura. E, finalmente, a simulação de escoamento em um braço de reservatório é apresentada. / To study and forecast the environmental impacts that a hydroelectric power plant reservoir may produce, a numerical simulation that takes into account the effects of stratification and scalar mixing (such as temperature, salinity or water-soluble substance) is required. This work proposes a methodology for the study of the environmental fluid flow phenomena, mainly for flows in which the knowledge of the interaction between stratification and mixing can give important notions of the phenomena that occur. For this, a numerical simulation tool for 3D environmental flow is developed. A tetrahedral mesh generator of the reservoir based on the terrain topology and an algebraic turbulence model based on the Richardson number are the main tools developed. The main difficulty in tetrahedral mesh generation of a reservoir is nonuniform distribution of the points related to the huge ratio between the horizontal and vertical scales of the reservoir. In this type of point distributions, conventional tetrahedron mesh generation algorithm may become unstable. For this reason, a unstructured tetrahedral mesh generator is developed and the methodology used to obtain conforming elements is described. Triangular surface mesh generation using the Delaunay triangulation and the construction of the tetrahedra from the triangular surface mesh are the main steps to the mesh generator. The hydrodynamic simulation of reservoirs with a turbulence model provides a useful tool that is computationally viable for engineering purposes. Furthermore, the turbulence model based on the Richardson number takes into account the effects of interaction between turbulence and stratification. The algebraic model is the simplest among the various models of turbulence, but provides realistic results with the fitting of a small amount of parameters. Eddy-Viscosity/Diffusivity models for stratified turbulent flows models are incorporated. Using the Finite Element Method (FEM) approximation the Reynolds-averaged Navier-Stokes (RANS) and mean scalar transport equations are approximated. The convective terms are discretized employing the Semi-Lagrangian method, and the spatial discretization is based on the Galerkin method. The computational results are compared with the results available in the literature. Finally, the simulation of the flow in a branch of a reservoir is presented.