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Lifting surfaces of unmanned aerial vehicles (UAV) are often operated in low Reynolds number (Re) ranges, wherein the transition of boundary layer from laminar-to-turbulent plays a more significant role than in high-Re aerodynamics applications. This poses a challenge for traditional computational fluid dynamics (CFD) simulations, since typical modeling approaches assume either fully laminar or fully turbulent flow. In particular, the boundary layer state must be accurately predicted to successfully determine the separation behavior which significantly influences the aerodynamic characteristics of the airfoil. Reynolds-averaged Navier-Stokes (RANS) based CFD simulations of an elliptic airfoil are performed for time-varying angles of attack, and results are used to elucidate relevant flow physics and aerodynamic data for an elliptic airfoil under realistic operating conditions. Results are also used to evaluate the performance of several different RANS-based turbulence modeling approaches for this class of flowfield.

This dissertation presents results from a thorough study of the tip-gap turbulent flow structure in a low-speed linear compressor cascade wind tunnel at Virginia Tech that includes a moving belt system to simulate the relative motion between the tip and the casing. The endwall pressure measurements and the surface oil flow visualizations were made on a stationary endwall to obtain the flow features and to determine the measurement profiles of interest. A custom-made miniature 3-orthogonal-velocity-component fiber-optic laser-Doppler velocimetry (LDV) system was used to measure all three components of velocity within a 50 mm spherical measurement volume within the gap between the endwall and the blade tip, mainly for the stationary wall with 1.65% and 3.30% tip gaps as well as some initial experiments for the moving wall. Since all of the vorticity in a flow originates from the surfaces under the action of strong pressure gradient, it was very important to measure the nearest-wall flow on the endwall and around the blade tip. The surface skin friction velocity was measured by using viscous sublayer velocity profiles, which verified the presence of an intense lateral shear layer that was observed from surface oil flow visualizations. All second- and third-order turbulence quantities were measured to provide detailed data for any parallel CFD efforts. The most complete data sets were acquired for 1.65% and 3.30% tip gap/chord ratios in a low-speed linear compressor cascade. This study found that tip gap flows are complex pressure-driven, unsteady three-dimensional turbulent flows. The crossflow velocity normal to the blade chord is nearly uniform in the mid tip-gap and changes substantially from the pressure to suction side. The crossflow velocity relies on the local tip pressure loading that is different from the mid-span pressure loading because of tip leakage vortex influence. The tip gap flow is highly skewed three-dimensional flow throughout the full gap. Normalized circulation within the tip gap is independent of the gap size. The tip gap flow interacts with the primary flow, separates from the endwall, and rolls up on the suction side to form the tip leakage vortex. The tip leakage vortex is unsteady from the observation of the TKE transport vector and oil flow visualizations. The reattachment of tip separation vortex on the pressure side strongly depends on the blade thickness-to-gap height ratio after the origin of tip leakage vortex but is weakly related to it before the origin of tip leakage vortex for a moderate tip gap. Other than the nearest endwall and blade tip regions, the TKE does not vary much in tip gap. The tip leakage vortex produces high turbulence intensities. The tip gap flow correlations of streamwise and wall normal velocity fluctuations decrease significantly from the leading edge to the trailing edge of the blade due to flow skewing. The tip gap flow is a strongly anisotropic turbulent flow. Rapid distortion ideas can not apply to it. A turbulence model based on stress transport equations and experimental data is necessary to reflect the tip gap flow physics. For the moving endwall, relative motion skews the inner region flow and is decorrelated with the outer layer flow. Hence, the TKE and correlations of streamwise and wall normal velocity fluctuations decrease. / Ph. D.

[pt] Doenças cardiovasculares são responsáveis por um elevado número de óbitos em seres humanos. Muitas dessas patologias são dependentes do ciclo cardíaco e estão localizadas na aorta, a maior e principal artéria do nosso corpo. O conhecimento dos padrões de escoamento e distribuições de tensões nas paredes da aorta podem auxiliar no diagnóstico e prevenção de algumas dessas doenças. Dessa forma, estudou-se numericamente o escoamento do sangue, durante o ciclo cardíaco, em um modelo 3D da aorta de um paciente específico, após a implantação de TAVI (Transcatheter Aortic Valve Implantation). O ciclo cardíaco é formado por dois períodos chamados de sístole e diástole. Durante a sístole, sangue é bombeado do coração para a aorta, apresentado altos valores de vazão, resultando em escoamento turbulento. Por outro lado, na diástole, com o fechamento da válvula aórtica, o sangue escoa com baixas velocidades em regime laminar. Até hoje, cientistas enfrentam um desafio na modelagem da turbulência, pois não existe uma única modelagem que forneça previsibilidade para todas as situações envolvendo o regime turbulento, com esforço computacional razoável. Para seleção do modelo de turbulência mais adequado para análise do escoamento no interior da aorta, na presença da transição de regimes de escoamento durante o ciclo cardíaco, com um custo razoável, selecionou-se a metodologia baseada na Média de Reynolds. Diferentes modelos foram comparados com dados experimentais extraídos do mesmo modelo aórtico em escala real, porém em regime permanente, com vazão correspondente ao pico da sístole. Por fim, avaliou-se o impacto das condições de contorno e dos modelos de turbulência durante o ciclo cardíaco na distribuição e valores de tensões e grandezas turbulentas no endotélio vascular. Mostrou-se que a distribuição espacial das médias temporais de tensão foram qualitativamente e quantitativamente similares, para os dois ciclos cardíacos representativos de diferentes pacientes, porém com pequenas mudanças locais para cada caso. Em termos dos modelos de turbulência, observou-se que o modelo SAS (Scale Adaptive Simulation) foi capaz de representar a relaminarização do escoamento sanguíneo no período diastólico. / [en] Cardiovascular diseases are responsible for a high number of deaths in humans. Many of these pathologies are dependent on the cardiac cycle and are located in the aorta, the largest and main artery in our body. Knowledge of flow patterns and stress distributions in the walls of the aorta can help in the diagnosis and prevention of some of these diseases. Thus, the flow of blood during the cardiac cycle was numerically studied in a 3D model of the aorta of a specific patient, after TAVI (Transcatheter Aortic Valve Implantation) implantation. The cardiac cycle consists of two periods called systole and diastole. During the systole, blood is pumped from the heart to the aorta, presenting high flow rates, resulting in a turbulent flow. On the other hand, in diastole, with the closure of the aortic valve, the blood flows with low velocities in laminar regime. Until today, scientists face a challenge in turbulence modeling, as there is no single model that provides predictability for all situations involving the turbulent regime, with reasonable computational effort. In order to select the most suitable turbulence model for the analysis of the flow inside the aorta, in the presence of the transition of flow regimes during the cardiac cycle, with a reasonable cost, the methodology based on the Reynolds Average was selected. Different models were compared with experimental data extracted from the same real-scale aortic model, but a in steady state, with flow corresponding to the systolic peak. Finally, the impact of boundary conditions and turbulence models during the cardiac cycle on the distribution and values of stresses and turbulent quantities in the vascular endothelium were evaluated. It was shown that the spatial distribution of the temporal averages of tension was qualitatively and quantitatively similar, for the two cardiac cycles representative of different patients, but with small local changes for each case. In terms of turbulence models, it was observed that the SAS (Scale Adaptive Simulation) model was able to represent the relaminarization of blood flow in the diastolic period.

Three-dimensional (3D) turbulent boundary layers (TBLs) play a crucial role in determining the aerodynamic properties of most aero-mechanical devices. However, accurately predicting these flows remains a challenge due to the complex nonlinear and nonlocal physics involved, which makes it difficult to develop universally applicable models. This limitation is particularly significant as the industry increasingly relies on simulations to make decisions in high-consequence environments, such as the certification or aircraft, and high-fidelity simulation methods that don't rely on modeling are prohibitively expensive. To address this challenge, it is essential to gain a better understanding of the physics underlying 3D TBLs. This research aims to improve the predictive accuracy of turbulence models in 3D TBLs by examining the impact of model assumptions underpinning turbulent constitutive relations, which are fundamental building blocks of every turbulence model. Specifically, the study focuses on the relevance and necessity of nonlinear and nonlocal model assumptions for accurately predicting 3D TBLs. The study considers the attached 3D boundary layer flow over the textbf{Be}nchmark textbf{V}alidation textbf{E}xperiment for textbf{R}ANS/textbf{L}ES textbf{I}nvestiagtions (BeVERLI) Hill as a test case and corresponding particle image velocimetry data for the investigation. In a first step, the BeVERLI Hill experiment is comprehensively described, and the important characteristics of the flow over the BeVERLI Hill are elucidated, including complex symmetry breaking characteristics of this flow. Reynolds-averaged Navier-Stokes simulations of the case using standard eddy viscosity models are then presented to establish the baseline behavior of local and linear constitutive relations, i.e., the standard Boussinesq approximation. The tested eddy viscosity models fail in the highly accelerated hill top region of the BeVERLI hill and near separation. In a further step, several nonlinear and nonlocal turbulent constitutive relations, including the QCR model, the model by Gatski and Speziale, and the difference-quotient model by Egolf are used as metrics to gauge the impact of nonlinearities and nonlocalities for the modeling of 3D TBLs. It is shown that nonlinear and nonlocal approaches are essential for effective 3D TBL modeling. However, simplified reduced-order models could accurately predict 3D TBLs without high computational costs. A constitutive relation with local second-order nonlinear mean strain relations and simplified nonlocal terms may provide such a minimal model. In a final step, the structure and response of non-equilibrium turbulence to continuous straining are studied to reveal new scaling laws and structural models. / Doctor of Philosophy / Airplanes and other flying objects rely on the way air flows around them to generate lift and stay in the sky. This airflow can be very complex, especially close to the surface of the object, where it is affected by friction with the object. This friction generates a layer of air called a boundary layer, which can become turbulent and lead to complex patterns of airflow. The boundary layer is generated by the friction between the air and the surface of the object, which causes the air molecules to "stick" to the surface. This sticking creates a layer of slow-moving air that slows down the flow of air around the object. This loss of momentum creates drag, which is one of the main factors that resist the motion of objects in the air. The slowing of the air flow in the boundary layer is due to the viscosity of the air, which is a measure of how resistant the air is to deformation. The molecules in the air have a tendency to stick together, making it difficult for them to move past each other. This resistance causes the momentum of the air to be lost as it flows over the surface of the object because air molecules close to the surface "pull" on the ones farther away. Understanding how turbulent boundary layers (TBLs) work is essential to accurately predict the airflow around these objects using computer simulations. However, it's challenging because TBLs involve complex physics that are difficult to model accurately. This research focuses on a specific type of TBL called a three-dimensional (3D) TBL. This study looks at how different assumptions affect the accuracy of computer simulations that predict this type of airflow. It is found that using more complex models that take into account nonlinear and nonlocal physics can help predict 3D TBLs more accurately. However, these models are computationally expensive, and it is also found that simpler models can work well enough and are cheaper. This research further establishes important physical relations of the mechanisms pertaining 3D TBLs that could support the advancement of current models.

Este trabalho é dedicado à análise e implementação de esquemas "upwind" de alta ordem modernos e o modelo de turbulência k-epsilon padrão no Freeflow-2D; um ambiente integrado para simulação numérica em diferenças finitas de problemas de escoamentos incompressíveis com superfícies livres. O propósito do estudo é a simulação de escoamentos de fluidos newtonianos incompressíveis, bidimensionais, confinados e/ou com superfícies livres e a altos valores do número de Reynolds. O desempenho do código Freeflow-2D atual é avaliada na simulação do escoamento numa expansão brusca e de um jato livre incidindo perpendicularmente sobre uma superfície rígida impermeável. O código é então aplicado na simulação de um jato planar turbulento em uma porção de fluido com superfície livre e estacionário. Os resultados numéricos obtidos são comparados com dados experimentais, soluções analíticas e soluções numéricas de outros trabalhos. / This work is devoted to the analysis and implementation of modern high-order upwind schemes and the standard k-epsilon turbulence model into the Freeflow-2D; a finite difference integrated environment for the numerical simulation of incompressible free surface flow problems. The purpose of this study is the two-dimensional simulation of high-Reynolds incompressible newtonian confined and/or free surface flows. The performance of the current Freeflow-2D code is assessed by applying it to the simulation of flow over a backward facing step and of an impinging free jet onto an impermeable rigid surface. The code is then applied to a turbulent planar jet into a pool. The numerical results are compared with experimental data, analytical solution, and numerical simulations of other works.

The context of this thesis is the numerical simulation of turbulent flows at moderate Reynolds numbers and the improvement of the capabilities of an in-house 3D unsteady and incompressible flow solver called SFELES to simulate such flows. In addition to this abstract, this thesis includes five other chapters. The second chapter of this thesis presents the numerical methods implemented in the two CFD solvers used as part of this work, namely SFELES and PHASTA. The third chapter concentrates on the implementation of a new library called FlexMG. This library allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods, which require much less memory to solve linear systems than a direct sparse LU solver available in SFELES. Multigrid is an iterative procedure that relies on a series of increasingly coarser approximations of the original 'fine' problem. The underlying concept is the following: low wavenumber errors on fine grids become high wavenumber errors on coarser levels, which can be effectively removed by applying fixed-point methods on coarser levels. Two families of algebraic multigrid preconditioners have been implemented in FlexMG, namely smooth aggregation-type and non-nested finite element-type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non-nested finite element-type multigrid so that our approaches can be considered as hybrid. Our aggregation-type multigrid is smoothed with either a constant or a linear least square fitting function, whereas the non-nested finite element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled with a GMRES (Generalized Minimal RESidual) method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics (unsteady flow past a circular cylinder at low Reynolds number), their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in this work. The fourth chapter is devoted to the study of subgrid scale models for the large eddy simulation (LES) of turbulent flows. It is well known that turbulence features a cascade process by which kinetic energy is transferred from the large turbulent scales to the smaller ones. Below a certain size, the smallest structures are dissipated into heat because of the effect of the viscous term in the Navier-Stokes equations. In the classical formulation of LES models, all the resolved scales are used to model the contribution of the unresolved scales. However, most of the energy exchanges between scales are local, which means that the energy of the unresolved scales derives mainly from the energy of the small resolved scales. In this fourth chapter, constant-coefficient-based Smagorinsky and WALE models are considered under different formulations. This includes a classical version of both the Smagorinsky and WALE models and several scale-separation formulations, where the resolved velocity field is filtered in order to separate the small turbulent scales from the large ones. From this separation of turbulent scales, the strain rate tensor and/or the eddy viscosity of the subgrid scale model is computed from the small resolved scales only. One important advantage of these scale-separation models is that the dissipation they introduce through their subgrid scale stress tensor is better controlled compared to their classical version, where all the scales are taken into account without any filtering. More precisely, the filtering operator (based on a top hat filter in this work) allows the decomposition u' = u - ubar, where u is the resolved velocity field (large and small resolved scales), ubar is the filtered velocity field (large resolved scales) and u' is the small resolved scales field. At last, two variational multiscale (VMS) methods are also considered. The philosophy of the variational multiscale methods differs significantly from the philosophy of the scale-separation models. Concretely, the discrete Navier-Stokes equations have to be projected into two disjoint spaces so that a set of equations characterizes the evolution of the large resolved scales of the flow, whereas another set governs the small resolved scales. Once the Navier-Stokes equations have been projected into these two spaces associated with the large and small scales respectively, the variational multiscale method consists in adding an eddy viscosity model to the small scales equations only, leaving the large scales equations unchanged. This projection is obvious in the case of a full spectral discretization of the Navier-Stokes equations, where the evolution of the large and small scales is governed by the equations associated with the low and high wavenumber modes respectively. This projection is more complex to achieve in the context of a finite element discretization. For that purpose, two variational multiscale concepts are examined in this work. The first projector is based on the construction of aggregates, whereas the second projector relies on the implementation of hierarchical linear basis functions. In order to gain some experience in the field of LES modeling, some of the above-mentioned models were implemented first in another code called PHASTA and presented along with SFELES in the second chapter. Finally, the relevance of our models is assessed with the large eddy simulation of a fully developed turbulent channel flow at a low Reynolds number under statistical equilibrium. In addition to the analysis of the mean eddy viscosity computed for all our LES models, comparisons in terms of shear stress, root mean square velocity fluctuation and mean velocity are performed with a fully resolved direct numerical simulation as a reference. The fifth chapter of the thesis focuses on the numerical simulation of the 3D turbulent flow over a re-entry Apollo-type capsule at low speed with SFELES. The Reynolds number based on the heat shield is set to Re=10^4 and the angle of attack is set to 180º, that is the heat shield facing the free stream. Only the final stage of the flight is considered in this work, before the splashdown or the landing, so that the incompressibility hypothesis in SFELES is still valid. Two LES models are considered in this chapter, namely a classical and a scale-separation version of the WALE model. Although the capsule geometry is axisymmetric, the flow field in its wake is not and induces unsteady forces and moments acting on the capsule. The characterization of the phenomena occurring in the wake of the capsule and the determination of their main frequencies are essential to ensure the static and dynamic stability during the final stage of the flight. Visualizations by means of 3D isosurfaces and 2D slices of the Q-criterion and the vorticity field confirm the presence of a large meandering recirculation zone characterized by a low Strouhal number, that is St≈0.15. Due to the detachment of the flow at the shoulder of the capsule, a resulting annular shear layer appears. This shear layer is then affected by some Kelvin-Helmholtz instabilities and ends up rolling up, leading to the formation of vortex rings characterized by a high frequency. This vortex shedding depends on the Reynolds number so that a Strouhal number St≈3 is detected at Re=10^4. Finally, the analysis of the force and moment coefficients reveals the existence of a lateral force perpendicular to the streamwise direction in the case of the scale-separation WALE model, which suggests that the wake of the capsule may have some preferential orientations during the vortex shedding. In the case of the classical version of the WALE model, no lateral force has been observed so far so that the mean flow is thought to be still axisymmetric after 100 units of non-dimensional physical time. Finally, the last chapter of this work recalls the main conclusions drawn from the previous chapters.

L’interaction onde de choc/couche limite en écoulement transsonique autour de profils aérodynamiques est étudiée numériquement utilisant différentes classes de modélisation de la turbulence. Les approches utilisées sont celles de modèles URANS et de méthodes hybrides RANS-LES. L’emploi d’une correction de compressibilité pour les fermetures à une équation est aussi évalué. Premièrement, la séparation intermittente induite par le choc sur un profil supercritique en conditions d’incidence proches de l’angle critique d’apparition du tremblement est analysée. Suite à des simulations URANS, la modélisation statistique la mieux adaptée est étudiée et utilisée dans l’approche DDES (Delayed Detached-Eddy Simulation). L’étude de la topologie de l’écoulement, des pressions pariétales et champs de vitesse statistiques montrent que les principales caractéristiques de l’oscillation auto-entretenue du choc sont capturées par les simulations. De plus, la DDES prédit des fluctuations secondaires de l’écoulement qui n’apparaissent pas en URANS. L’étude de l’interface instationnaire RANS-LES montre que la DDES évite le MSD (modeled stress depletion) pour les phases de l’écoulement attaché ou séparé. Le problème de la ‘zone grise’ et de son influence sur les résultats est considéré. Les conclusions de l’étude sur le profil supercritique est ensuite appliquées à l’étude numérique d’un profil transsonique laminaire. Dans ce contexte, l’effet de la position de la transition de la couche limite sur les caractéristiques de deux régimes d’interaction choc/couche limite sélectionnés est étudié. En conditions de tremblement, les simulations montrent une forte influence du point de transition sur l’amplitude du mouvement du choc et sur l’instationnarité globale de l’écoulement. / Shock wave/boundary layer interactions arising in the transonic flow over airfoils are studied numerically using different levels of turbulence modeling. The simulations employ standard URANS models suitable for aerodynamics and hybrid RANS-LES methods. The use of a compressibility correction for one-equation closures is also considered. First, the intermittent shock-induced separation occurring over a supercritical airfoil at an angle of attack close to the buffet onset boundary is investigated. After a set of URANS computations, a scale-resolving simulation is performed using the best statistical approach in the context of a Delayed Detached-Eddy Simulation (DDES). The analysis of the flow topology and of the statistical wall-pressure distributions and velocity fields show that the main features of the self-sustained shock-wave oscillation are predicted by the simulations. The DDES also captures secondary flow fluctuations which are not predicted by URANS. An examination of the unsteady RANS-LES interface shows that the DDES successfully prevents modeled-stress depletion whether the flow is attached or separated. The gray area issue and its impact on the results are also addressed. The conclusions from the supercritical airfoil simulations are then applied to the numerical study of a laminar transonic profile. Following a preliminary characterization of the airfoil aerodynamics, the effect of the boundary layer transition location on the properties of two selected shock wave/boundary layer interaction regimes is assessed. In transonic buffet conditions, the simulations indicate a strong dependence of the shock-wave motion amplitude and of the global flow unsteadiness on the tripping location.

Turbulence is an open and challenging problem for mathematical approaches, physical modeling and numerical simulations. Numerical solutions contribute significantly to the understand of the nature and effects of turbulence. The focus of this thesis is the development of appropriate numerical methods for the computer simulation of turbulent flows. Many of the existing approaches to turbulence utilize analogies from kinetic theory. Degond & Lemou (J. Math. Fluid Mech., 4, 257-284, 2002) derived a k-✏ type turbulence model completely from kinetic theoretic framework. In the first part of this thesis, a numerical method is developed for the computer simulation based on this model. The Boltzmann equation used in the model has an isotropic, relaxation collision operator. The relaxation time in the collision operator depends on the microscopic turbulent energy, making it difficult to construct an efficient numerical scheme. In order to achieve the desired numerical efficiency, an appropriate change of frame is applied. This introduces a stiff relaxation source term in the equations and the concept of asymptotic preserving schemes is then applied to tackle the stiffness. Some simple numerical tests are introduced to validate the new scheme. In the second part of this thesis, alternative approaches are sought for more efficient numerical techniques. The Lattice Boltzmann Relaxation Scheme (LBRS) is a novel method developed recently by Rohan Deshmukh and S.V. Raghuram Rao for simulating compressible flows. Two different approaches for the construction of implicit sub grid scale -like models as Implicit Large Eddy Simulation (ILES) methods, based on LBRS, are proposed and are tested for Burgers turbulence, or Burgulence. The test cases are solved over a largely varying Reynolds number, demonstrating the efficiency of this new ILES-LBRS approach. In the third part of the thesis, as an approach towards the extension of ILES-LBRS to incompressible flows, an artificial compressibility model of LBRS is proposed. The modified framework, LBRS-ACM is then tested for standard viscous incompressible flow test cases.

Este trabalho é dedicado à análise e implementação de esquemas "upwind" de alta ordem modernos e o modelo de turbulência k-epsilon padrão no Freeflow-2D; um ambiente integrado para simulação numérica em diferenças finitas de problemas de escoamentos incompressíveis com superfícies livres. O propósito do estudo é a simulação de escoamentos de fluidos newtonianos incompressíveis, bidimensionais, confinados e/ou com superfícies livres e a altos valores do número de Reynolds. O desempenho do código Freeflow-2D atual é avaliada na simulação do escoamento numa expansão brusca e de um jato livre incidindo perpendicularmente sobre uma superfície rígida impermeável. O código é então aplicado na simulação de um jato planar turbulento em uma porção de fluido com superfície livre e estacionário. Os resultados numéricos obtidos são comparados com dados experimentais, soluções analíticas e soluções numéricas de outros trabalhos. / This work is devoted to the analysis and implementation of modern high-order upwind schemes and the standard k-epsilon turbulence model into the Freeflow-2D; a finite difference integrated environment for the numerical simulation of incompressible free surface flow problems. The purpose of this study is the two-dimensional simulation of high-Reynolds incompressible newtonian confined and/or free surface flows. The performance of the current Freeflow-2D code is assessed by applying it to the simulation of flow over a backward facing step and of an impinging free jet onto an impermeable rigid surface. The code is then applied to a turbulent planar jet into a pool. The numerical results are compared with experimental data, analytical solution, and numerical simulations of other works.