A maioria dos líquidos encontrados na natureza são não-Newtonianos e o estudo do seu comportamento tem uma importância significante em diferentes áreas da engenharia. Entre eles, uma larga classe de materiais que exibem pequena ou nenhuma deformação quando sujeitos a um nível de tensões inferiores a uma tensão limite de escoamento – chamado de comportamento viscoplástico. A presente Dissertação tem como objetivo o estudo numérico de escoamentos bidimensionais em regime permanente de fluidos viscoplásticos não-lineares em uma cavidade forçada. O modelo mecânico é definido pelas equações de conservação de massa e de balanço de momentum acopladas ao modelo viscoplástico recentemente introduzido por Souza Mendes e Dutra – SMD – e é aproximado por um método de elementos finitos multi-campos estabilizado baseado na metodologia de Galerkin mínimos-quadrados que possui como variáveis primais os campos de tensão-extra, velocidade e pressão. As condições de compatibilidade entre os subespaços de elementos finitos para tensão-extra-velocidade e velocidade-pressão são violadas, permitindo assim a utilização de interpolações de igual ordem. O método estabilizado foi implementado no código de elementos finitos para fluidos não-Newtonianos em desenvolvimento no Laboratório de Mecânica dos Fluidos Aplicada e Computacional (LAMAC) da UFRGS. Em diversos trabalhos encontrados na literatura, a superfície de escoamento do material é definida como a região onde o módulo da tensão-extra é igual à tensão limite de escoamento. É mostrado nesta Dissertação que esta metodologia pode conduzir à alguns erros, dado ao grande aumento experimentado pela taxa de cisalhamento em uma pequena faixa de tensões próximas à tensão limite de escoamento. Assim, foi adotada outra metodologia, definindo a superfície de escoamento como a linha onde a taxa de cisalhamento é igual a um valor dado pela relação de parâmetros reológicos do fluido, especificamente a tensão limite de escoamento e a viscosidade Newtoniana para baixas taxas de cisalhamento. Nas simulações numéricas realizadas, o número de salto, J, o coeficiente de power-law, n, e a vazão adimensional, U*, são variados de forma a avaliar de que modo influenciam na dinâmica de escoamentos viscoplásticos. Os resultados obtidos estão de acordo com a literatura e atestam a estabilidade da formulação empregada. / Non-Newtonian fluids are the majority of liquids found on the nature and the study of their behavior has a significant importance on different areas of engineering. Among them, there is a wide class of materials that exhibits little or no deformation when subjected to a stress level behind an apparent yield stress – called the viscoplastic behavior. The present thesis aimed to a numerical study of two dimensional steady state laminar flows of non-linear viscoplastic fluids in a lid-driven cavity. The mechanical model was defined by the mass conservation and momentum balance equations coupled to the recently introduced Souza Mendes and Dutra – SMD – viscoplastic model and has been approximated by a stabilized multi-field finite element method based on the Galerkin least-squares methodology, having as primal variables the extra-stress, velocity and pressure fields. In this way, the compatibility conditions between the extra-stressvelocity and pressure-velocity (Babuška-Brezzi condition) finite element subspaces are violated, allowing to use equal-order finite element interpolations. The stabilized method has been implemented in the finite element code for non-Newtonian fluids under development at the Laboratory of Applied and Computational Fluid Mechanics (LAMAC) of UFRGS. In several works found on the literature, the yield surface of the material is defined as the region where the stress modulus is equal to the yield stress. Is shown in this work that this methodology can lead to some errors, due to the large strain rate increasing in a small range of values of stress on the vicinity of the yield stress. Therefore, it was adopted another approach, defining the yield surface as the line where the strain rate is equal to a value given by the relation of the rheological parameters of the fluid, namely the yield stress and the viscosity at low shear rates. In the performed numerical simulations, the jump number, J, the the power-law coefficient, n,and the non-dimensional flow rate, U*, are ranged in order to evaluate how they the influence on the viscoplastic fluid dynamics have been investigated. All results found were in accordance with the affine literature and attests the good stability features of the formulation.

Comprendre, prédire et finalement retarder la transition vers la turbulence dans les écoulements sont d'importants problèmes posés aux scientifiques depuis les travaux pionniers d'Osborne Reynolds en 1883. Ces questions ont été principalement adressées à l'aide de la théorie des instabilités hydrodynamiques. A cause des ressources informatiques limitées, les analyses de stabilité linéaire reposent essentiellement sur d'importantes hypothèses simplificatrices telles que celle d'un écoulement parallèle. Dans ce cadre, connu sous le nom de stabilité locale, seule la stabilité d'écoulement ayant un fort intérêt académique mais relativement peu d'applications pratiques a pu être étudiée. Néanmoins, au cours de la décennie passée, l'hypothèse d'écoulement parallèle a été relaxée au profit de celle d'un écoulement bidimensionnel conduisant alors à ce que l'on appelle la stabilité globale. Ce nouveau cadre permet alors d'étudier les mécanismes d'instabilité et de transition ayant lieu au sein d'écoulements plus réalistes. Plus particulièrement, la stabilité d'écoulements fortement non-parallèles pouvant présenter des décollements massifs, une caractéristique fréquente dans les écoulements d'intérêt industriel, peut maintenant être étudiée. De plus, avec l'accroissement constant des moyens de calcul et le développement de nouveaux algorithmes de recherche de valeurs propres itératifs, il est aujourd'hui possible d'étudier la stabilité d'écoulements pleinement tridimensionnels pour lesquels aucune hypothèse simplificatrice n'est alors nécessaire. Dans la continuité des travaux présentés par Bagheri et al. en 2008, le but de la présente thèse est de développer les outils nécessaires à l'analyse de la stabilité d'écoulements 3D. Trois écoulements ont été choisis afin d'illustrer les nouvelles capacités de compréhension apportées par l'analyse de la stabilité globale appliquée à des écoulements tridimensionnels réels : i) l'écoulement au sein d'une cavité entraînée 3D, ii) l'écoulement se développant dans un tuyau sténosé, et enfin iii) l'écoulement de couche limite se développant au passage d'une rugosité cylindrique montée sur une plaque plane. Chacun de ces écoulements a différentes applications pratiques allant d'un intérêt purement académique à une application biomédicale et aérodynamique. Ce choix d'écoulements nous permet également d'illustrer les différents aspects des outils développés au cours de cette thèse ainsi que les limitations qui leur sont inhérentes. / Understanding, predicting and eventually delaying transition to turbulence in fluid flows have been challenging issues for scientists ever since the pioneering work of Osborne Reynolds in 1883. These problems have mostly been addressed using the hydrodynamic linear stability theory. Yet, due to limited computational resources, linear stability analyses have essentially relied until recently on strong simplification hypotheses such as the “parallel flow” assumption. In this framework, known as “local stability theory”, only the stability of flows with strong academic interest but limited practical applications can be investigated. However, over the course of the past decade, simplification hypotheses have been relaxed from the “parallel flow” assumption to a two-dimensionality assumption of the flow resulting in what is now known as the “global stability theory”. This new framework allows one to investigate the instability and transition mechanisms taking place in more realistic flows. More particularly, the stability of strongly non-parallel flows exhibiting separation, a common feature of numerous flows of practical interest, can now be studied. Moreover, with the continuous increase of computational power available and the development of new iterative eigenvalue algorithms, investigating the global stability of fully three-dimensional flows, for which no simplification hypothesis is necessary, is now feasible. Following the work presented in 2008 by Bagheri et al., the aim of the present thesis is thus to develop the tools mandatory to investigate the stability of 3D flows. Three flow configurations have been chosen to illustrate the new investigation capabilities brought by global stability theory when it is applied to realistic three-dimensional flows: i) the flow within a cuboid lid-driven cavity, ii) the flow within an asymmetric stenotic pipe and iii) the boundary layer flow developing over a cylindrical roughness element mounted on a flat plate. Each of these flows have different practical applications ranging from purely academic interests to biomedical and aerodynamical applications. They also allow us to put in the limelight different aspects and possible limitations of the various tools developed during this PhD thesis.

In this report the regular finite differences method (FDM) and a least-squares radial basis function-generated finite differences method (RBF-FD-LS) is used to solve the two-dimensional incompressible Navier-Stokes equations for the lid driven cavity problem. The Navier-Stokes equations is solved using stream function-vorticity formulation. The purpose of the report is to compare FDM and RBF-FD-LS with respect to accuracy and computational cost. Both methods were implemented in MATLAB and the problem was solved for Reynolds numbers equal to 100, 400 and 1000. In the report we present the solutions obtained as well as the results from the comparison. The results are discussed and conclusions are drawn. We came to the conclusion that RBF-FD-LS is more accurate when the stepsize of the grids used is held constant, while RBF-FD-LS costs more than FDM for similar accuracy.

Ce travail de thèse présente l'implémentation de la simulation d'écoulements diphasiques dans des conditions de réacteurs nucléaires à caloporteur eau, à l'échelle de bulles individuelles. Pour ce faire, nous étudions plusieurs modèles d'écoulements thermohydrauliques et nous focalisons sur une technique de capture d'interface mince entre phases liquide et vapeur. Nous passons ainsi en revue quelques techniques possibles de maillage adaptatif (AMR) et nous fournissons des outils algorithmiques et informatiques adaptés à l'AMR par patchs dont l'objectif localement la précision dans des régions d'intérêt. Plus précisément, nous introduisons un algorithme de génération de patchs conçu dans l'optique du calcul parallèle équilibré. Cette approche nous permet de capturer finement des changements situés à l'interface, comme nous le montrons pour des cas tests d'advection ainsi que pour des modèles avec couplage hyperbolique-elliptique. Les calculs que nous présentons incluent également la simulation du système de Navier-Stokes incompressible qui modélise la déformation de l'interface entre deux fluides non-miscibles. / This PhD work presents the implementation of the simulation of two-phase flows in conditions of water-cooled nuclear reactors, at the scale of individual bubbles. To achieve that, we study several models for Thermal-Hydraulic flows and we focus on a technique for the capture of the thin interface between liquid and vapour phases. We thus review some possible techniques for Adaptive Mesh Refinement (AMR) and provide algorithmic and computational tools adapted to patch-based AMR, which aim is to locally improve the precision in regions of interest. More precisely, we introduce a patch-covering algorithm designed with balanced parallel computing in mind. This approach lets us finely capture changes located at the interface, as we show for advection test cases as well as for models with hyperbolic-elliptic coupling. The computations we present also include the simulation of the incompressible Navier-Stokes system, which models the shape changes of the interface between two non-miscible fluids.

Este trabalho tem como principal objetivo o desenvolvimento de um método numérico para simulação das grandes escalas de escoamentos turbulentos tridimensionais utilizando uma modelagem de turbulência baseada em filtragem temporal (denominada TLES - Temporal Large Eddy Simulation). O método desenvolvido combina discretizações temporais com ordem de mínima precisão 2 (Adams-Bashforth, QUICK, Runge-Kutta), um método de projeção de ordem 2, com discretizações espaciais também de ordem 2 obtidas pelo método de volumes finitos. Esta metodologia foi empregada na simulação de problemas teste turbulentos como o canal e a cavidade impulsionada, sendo este último resultado simulado pela primeira vez com modelagem TLES. Os resultados mostram uma excelente concordância quando comparado com resultados de simulações diretas (DNS) e dados experimentais, superando resultados clássicos obtidos com formulação LES com filtragem espacial. / The main objective of this work is to develop a numerical method for large eddy simulation of tridimensional turbulent flows using a model based on temporal filtering (TLES - Temporal Large Eddy Simulation). The developed method combines at least 2nd order temporal discretizations (Adams-Bashforth, QUICK, Runge-Kutta), a 2nd order projection method, and 2nd order spatial discretizations obtained by the finite volume method. This methodology was employed to the simulation of turbulent benchmark problems such as channel and lid-driven cavity flows. The latter is simulated for the first time using a TLES turbulence modelling. Results show excellent agreement when compared to Direct Numerical Simulations (DNS) and experimental data, with better results than classical results produced by standard LES formulation with spatial filtering.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Modern engineering increasingly requires the comprehension of phenomena related to combustion, aeroacustics, turbulence transition, among others. For these purposes the Computational Fluids Dynamics (CFD) requires the used high order methods. One of these methods is the Fourier pseudo-spectral method, that provides an excellent numerical accuracy, and with the use of the Fast Fourier Transform algorithm (FFT), it presents a low computational cost in comparison to anothers high-order methods. Another important issue is the projection method of the pression term, which does not require the pressure computation from the Navier-Stokes equations. The procedure to calculate the pression field is usually the most onerous in classical methodologies. Nevertheless, the pseudo-spectral method can be only applied to periodic boundary flows, thus limiting its use. Aiming to solve this restriction, a new methodology is proposed at the present work, which has the objective of simulating nonperiodic flows using the Fourier pseudo-spectral method. For this purpose the immersed boundary method, that represents the boundary conditions through a force field imposed at Navier-Stokes equations is used. As a test to this new methodology, a classic problem of Computational Fluid Dynamics, The Lid Driven Cavity was simulated. The obtained results are promising and demonstrate the possibility to simulating non-periodic flows making use of the Fourier pseudo-spectral method. / Para compreender fenômenos relacionados à combustão, aeroacústica, transição a turbulência entre outros, a Dinâmica de Fluídos Computacional (CFD) utiliza os métodos de alta ordem. Um dos mais conhecidos é o método pseudo-espectral de Fourier, o qual alia: alta ordem de precisão na resolução das equações, com um baixo custo computacional. Este está ligado à utilização da FFT e do método da projeção do termo da pressão, o qual desvincula os cálculos da pressão da resolução das equações de Navier-Stokes. O procedimento de calcular o campo de pressão, normalmente é o mais oneroso nas metodologias convencionais. Apesar destas vantagens, o método pseudo-espectral de Fourier só pode ser utilizado para resolver problemas com condições de contorno periódicas, limitando o seu uso no campo da dinâmica de fluídos. Visando resolver essa restrição uma nova metodologia é proposta no presente trabalho, que tem como objetivo simular escoamentos não-periódicos utilizando o método pseudo-espectral de Fourier. Para isso, é utilizada a metodologia da Fronteira Imersa, a qual representa as condições de contorno de um escoamento através de um campo de força imposto nas equações de Navier-Stokes. Como teste, para essa nova metodologia, foi simulada uma cavidade com tampa deslizante (Lid Driven Cavity), problema clássico da mecânica de fluídos, que objetiva validar novas metodologias e códigos computacionais. Os resultados obtidos são promissores e demostram que é possível simular um escoamento não-periódico com o método pseudo-espectral de Fourier. / Mestre em Engenharia Mecânica

The development of numerical methods for the incompressible Navier-Stokes equations received much attention in the past 50 years. Finite element methods emerged given their robustness and reliability. In our work, we choose the P2-P1 finite element for space approximation which gives 2nd-order accuracy for velocity and 1st-order accuracy for pressure. Our research focuses on the development of several high-order semi-implicit time-stepping methods to compute unsteady flows. The methods investigated include backward difference formulae (SBDF) and defect correction strategy (DC). Using the defect correction strategy, we investigate two variants, the first one being based on high-order artificial compressibility and bootstrapping strategy proposed by Guermond and Minev (GM) and the other being a combination of GM methods with sequential regularization method (GM-SRM). Both GM and GM-SRM methods avoid solving saddle point problems as for SBDF and DC methods. This approach reduces the complexity of the linear systems at the expense that many smaller linear systems need to be solved. Next, we proposed several numerical improvements in terms of better approximations of the nonlinear advection term and high-order initialization for all methods. To further minimize the complexity of the resulting linear systems, we developed several new variants of grad-div splitting algorithms besides the one studied by Guermond and Minev. Splitting algorithm allows us to handle larger flow problems. We showed that our new methods are capable of reproducing flow characteristics (e.g., lift and drag parameters and Strouhal numbers) published in the literature for 2D lid-driven cavity and 2D flow around the cylinder. SBDF methods with grad-div stabilization terms are found to be very stable, accurate and efficient when computing flows with high Reynolds numbers. Lastly, we showcased the robustness of our methods to carry 3D computations.

Este trabalho tem como principal objetivo o desenvolvimento de um método numérico para simulação das grandes escalas de escoamentos turbulentos tridimensionais utilizando uma modelagem de turbulência baseada em filtragem temporal (denominada TLES - Temporal Large Eddy Simulation). O método desenvolvido combina discretizações temporais com ordem de mínima precisão 2 (Adams-Bashforth, QUICK, Runge-Kutta), um método de projeção de ordem 2, com discretizações espaciais também de ordem 2 obtidas pelo método de volumes finitos. Esta metodologia foi empregada na simulação de problemas teste turbulentos como o canal e a cavidade impulsionada, sendo este último resultado simulado pela primeira vez com modelagem TLES. Os resultados mostram uma excelente concordância quando comparado com resultados de simulações diretas (DNS) e dados experimentais, superando resultados clássicos obtidos com formulação LES com filtragem espacial. / The main objective of this work is to develop a numerical method for large eddy simulation of tridimensional turbulent flows using a model based on temporal filtering (TLES - Temporal Large Eddy Simulation). The developed method combines at least 2nd order temporal discretizations (Adams-Bashforth, QUICK, Runge-Kutta), a 2nd order projection method, and 2nd order spatial discretizations obtained by the finite volume method. This methodology was employed to the simulation of turbulent benchmark problems such as channel and lid-driven cavity flows. The latter is simulated for the first time using a TLES turbulence modelling. Results show excellent agreement when compared to Direct Numerical Simulations (DNS) and experimental data, with better results than classical results produced by standard LES formulation with spatial filtering.

Tato diplomová práce prezentuje metodu spektrálních prvků. Tato metoda je použita k řešení stacionárního 2-D laminárního proudění Newtonovské nestlačitelné tekutiny. Proudění je popsáno stacionarní Navier-Stokesovou rovnicí. Dohromady s okrajovou pod- mínkou tvoří Navier-Stokesův problém. Na slabou formulaci této úlohy je aplikována metoda spektrálních prvků. Touto discretizací se získá soustava nelineárních rovnic. K obrdžení lineární soustavy je použita Newtonova iterační metoda. Podorobný algorit- mus tvoří jádro Navier-Stokeseva solveru, který je naprogramován v Matlabu. Na závěr jsou pomocí tohoto solveru řešeny dva příklady: proudění v kavitě a obtékání válce. Přík- lady jsou řešeny pro různé Reynoldsovy čísla. První od 1 do 1000 a druhý od 1 do 100.

This thesis considers the stability and transition of incompressible boundary layers. In particular, the Falkner–Skan–Cooke boundary layer subject to a cylindrical surface roughness, and the Blasius boundary layer with applied localized suction are investigated. These flows are of great importance within the aviation industry, feature complex transition scenarios, and are strongly three-dimensional in nature. Consequently, no assumptions regarding homogeneity in any of the spatial directions are possible, and the stability of the flow is governed by an extensive three-dimensional eigenvalue problem. The stability of these flows is addressed by high-order direct numerical simulations using the spectral element method, in combination with a Krylov subspace projection method. Such techniques target the long-term behavior of the flow and can provide lower limits beyond which transition is unavoidable. The origin of the instabilities, as well as the mechanisms leading to transition in the aforementioned cases are studied and the findings are reported. Additionally, a novel method for computing the optimal forcing of a dynamical system is developed. This type of analysis provides valuable information about the frequencies and structures that cause the largest energy amplification in the system. The method is based on the inverse power method, and is discussed in the context of the one-dimensional Ginzburg–Landau equation and a two-dimensional flow case governed by the Navier–Stokes equations. / <p>QC 20151015</p>