The solution to partial differential equations generally requires approximations that result in numerical error in the final solution. Of the different types of numerical error in a solution, discretization error is the largest and most difficult error to estimate. In addition, the accuracy of the discretization error estimates relies on the solution (or multiple solutions used in the estimate) being in the asymptotic range. The asymptotic range is used to describe the convergence of a solution, where an asymptotic solution approaches the exact solution at a rate proportional to the change in mesh spacing to an exponent equal to the formal order of accuracy. A non-asymptotic solution can result in unpredictable convergence rates introducing uncertainty in discretization error estimates. To account for the additional uncertainty, various discretization uncertainty estimators have been developed. The goal of this work is to evaluation discretization error and discretization uncertainty estimators based on Richardson extrapolation for computational fluid dynamics problems. In order to evaluate the estimators, the exact solution should be known. A select set of solutions to the 2D Euler equations with known exact solutions are used to evaluate the estimators. Since exact solutions are only available for trivial cases, two applications are also used to evaluate the estimators which are solutions to the Navier-Stokes equations: a laminar flat plate and a turbulent flat plate using the k-Ï SST turbulence model. Since the exact solutions to the Navier-Stokes equations for these cases are unknown, numerical benchmarks are created which are solutions on significantly finer meshes than the solutions used to estimate the discretization error and uncertainty. Metrics are developed to evaluate the accuracy of the error and uncertainty estimates and to study the behavior of each estimator when the solutions are in, near, and far from the asymptotic range. Based on the results, general recommendations are made for the implementation of the error and uncertainty estimators. In addition, a new uncertainty estimator is proposed with the goal of combining the favorable attributes of the discretization error and uncertainty estimators evaluated. The new estimator is evaluated using numerical solutions which were not used for development and shows improved accuracy over the evaluated estimators. / Master of Science

L’objectif de cette étude est de simuler l’érosion d’un sol cohésif sous l’effet d’un écoulement incompressible. Le modèle élaboré décrit une vitesse d’érosion interfaciale qui dépend de la contrainte de cisaillement de l’écoulement. La modélisation numérique proposée est une approche eulérienne, où une méthode de pénalisation de domaines est utilisée pour résoudre les équations de Navier-Stokes autour d’un obstacle. L’interface eau/sol est décrite par une fonction Level Set couplée à une loi d’érosion à seuil.L’approximation numérique est basée sur un schéma DDFV (Discrete Duality Finite Volume) autorisant des raffinements locaux sur maillages non-conformes et non-structurés. L’approche par pénalisation a mis en évidence une couche limite d'inconsistance à l'interface fluide/solide lors du calcul de la contrainte de cisaillement. Deux approches sont proposées pour estimer précisément la contrainte de ce problème à frontière libre. La pertinence du modèle à prédire l’érosion interfaciale du sol est confirmée par la présentation de plusieurs résultats de simulation, qui offrent une meilleure évaluation et compréhension des phénomènes d'érosion / This study focuses on the numerical modelling of the interfacial erosion occurring at a cohesive soil undergoing an incompressible flow process. The model assumes that the erosion velocity is driven by a fluid shear stress at the water/soil interface. The numerical modelling is based on the eulerian approach: a penalization procedure is used to compute Navier-Stokes equations around soil obstacle, with a fictitious domain method, in order to avoid body- fitted unstructured meshes. The water/soil interface’s evolution is described by a Level Set function coupled to a threshold erosion law.Because we use adaptive mesh refinement, we develop a Discrete Duality Finite Volume scheme (DDFV), which allows non-conforming and non-structured meshes. The penalization method, used to take into account a free velocity in the soil with non-body-fitted mesh, introduces an inaccurate shear stress at the interface. We propose two approaches to compute accurately the erosion velocity of this free boundary problem. The ability of the model to predict the interfacial erosion of soils is confirmed by presenting several simulations that provide better evaluation and comprehension of erosion phenomena.

A numerical code for solving the time-dependent incompressible 3D Navier-Stokes equations with finite volumes on overlapping staggered grids in cylindrical and rectangular geometry is developed. In the code, written in FORTRAN, the momentum equation for the velocity is solved by projection method and Poisson equation for the pressure is solved by ADI implicit method in two directions combined with discrete fast Fourier transform in the third direction. A special technique for overcoming the singularity on the cylinder's axis is developed. This code, taking into account dependence upon temperature of the viscosity, density and surface tension of the liquid, is used to study the fluid motion in a cylinder with free cylindrical surface (under normal and zero-gravity conditions); and in a rectangular closed cell with a source of thermocapillary convection (bubble inside attached to one of the cell's faces). They are significant problems in crystal growth and in general experiments in fluid dynamics respectively. Nevertheless, the main study is dedicated to the liquid bridge problem.<p><p>The development of thermocapillary convection inside a cylindrical liquid bridge is investigated by using a direct numerical simulation of the 3D, time-dependent problem for a wide range of Prandtl numbers, Pr = 0.01 - 108. For Pr > 0.08 (e.g. silicon oils), above the critical value of temperature difference between the supporting disks, two counter propagating hydrothermal waves bifurcate from the 2D steady state. The existence of standing and traveling waves is discussed. The dependence of viscosity upon temperature is taken into account. For Pr = 4, 0-g conditions, and for Pr = 18.8, 1-g case with unit aspect ratio an investigation of the onset of chaos was numerically carried out. <p><p>For a Pr = 108 liquid bridge under terrestrial conditions ,the appearance and the development of thermoconvective oscillatory flows were investigated for different ambient conditions around the free surface.<p><p>Transition from 2D thermoconvective steady flow to a 3D flow is considered for low-Prandtl fluids (Pr = 0.01) in a liquid bridge with a non-cylindrical free surface. For Pr < 0.08 (e.g. liquid metals), in supercritical region of parameters 3D but non-oscillatory convective flow is observed. The computer program developed for this simulation transforms the original non-rectangular physical domain into a rectangular computational domain.<p><p>A study of how presence of a bubble in experimental rectangular cell influences the convective flow when carrying out microgravity experiments. As a model, a real experiment called TRAMP is numerically simulated. The obtained results were very different from what was expected. First, because of residual gravity taking place on board any spacecraft; second, due to presence of a bubble having appeared on the experimental cell's wall. Real data obtained from experimental observations were taken for the calculations.<p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished

The advance in computational fluid dynamics in recent years has provided the opportunity for many fluid dynamic problems to be analyzed numerically. One such problem concerns the modeling of plunging water into a still water body, often encountered in pump stations. Air bubbles introduced into the system by the plunging jet can be a significant problem, especially when consumed into operating pumps. The classical approach to investigate the hydrodynamics of plunging jet in pump stations is by physical model studies. This approach is time consuming, tedious and costly. The availability of computational power today, along with appropriate numerical techniques, allows such phenomenon to be studied in a greater level of detail and more cost efficient. Despite the advantages of numerical studies, little attention has been devoted to solve the plunging jet and air transport problem numerically. In this current work, a 3-dimensional finite volume, Large Eddy Simulation (LES) code is developed to simulate these flow conditions. For turbulent flow, the large scale quantities were numerically resolved while the dynamic sub-grid scale model is used to model the small scale energy dissipations. The code also has the capability to handle free surface deformation, an important aspect in simulating the impact section of an impinging jet. Modeling of the air entrainment is performed numerically utilizing the information obtained from the hydrodynamics. Migration of air bubbles is modeled using the scalar transport equation, modified to account for the buoyancy of the bubbles. Instead of the typical Lagrangian schemes, which track individual air bubbles, air bubble dynamics are modeled in the form of concentrations. Modeling air bubbles in this manner is computational efficient and simpler to implement. For the air entrainment simulations, standard numerical boundaries conditions and empirical entrainment equations are used to provide the necessary boundary conditions. The developed model is compared with the literature, producing satisfactory results, suggesting that the code has an excellent potential of extending its application to practical industry practices.

Despite the superior properties of Fibre Reinforced Polymer (FRP) materials, the use of FRPs in buildings is limited. A key cause of concern for their use in buildings arises from their poor performance in fire occurrences. This thesis presents the results of fire performance of Reinforced Concrete (RC) beams and slabs strengthened with externally bonded FRP sheets. The performance and effectiveness of insulation materials and techniques are also investigated in this thesis. Two full-scale reinforced concrete T-beams and two intermediate-scale slabs were strengthened in flexure with carbon and glass fibre reinforced polymer sheets and insulated with a layer of spray-on material. The T-beams and slabs were then exposed to a standard fire. Fire test results show that fire endurances of more than 4 h can be achieved using an appropriate insulation system. Tests were performed in order to understand the behaviour of FRP concrete bond at high temperatures. An empirical model was then formulated to describe the bond strength deterioration due to temperature rise. Innovative measurement techniques were employed throughout the experiments to measure important observables like strain and temperature. Meanwhile, the effectiveness and practicality of techniques such as Fibre Optic Sensing (FOS) and Particle Image Velocimetry (PIV) for high temperature applications were investigated. A numerical finite-volume heat transfer model was developed to simulate the heat transfer phenomenon. The validity of the numerical model was verified by comparing the results with the results from the fire tests. By using this model, parametric analyses were performed to investigate the effect of different fire scenarios on the performance of the insulated beams. To simulate the structural performance of the T-beams a numerical model which was capable of predicting stresses and strains and deflections of a heated beam was developed. The model is capable of incorporating the effects of axial forces in the response of a restrained beam. This model was verified and used in combination with the thermal model to simulate the deflections of T-beams in fire. / Thesis (Ph.D, Civil Engineering) -- Queen's University, 2013-09-17 15:11:16.185

The successful application of Computational Aeroacoustics (CAA) requires high accuracy numerical schemes with good dissipation and dispersion characteristics. Unstructured meshes have a greater geometrical flexibility than existing high order structured mesh methods. This work investigates the suitability of unstructured mesh techniques by computing a two-dimensionallinearised Euler problem with various discretisation schemes and different mesh types. The goal of the present work is the development of an unstructured numerical method with the high accuracy, low dissipation and low dispersion required to be an effective tool in the study of aeroacoustics. The suitability of the unstructured method is investigated using aeroacoustic test cases taken from CAA Benchmark Workshop proceedings. Comparisons are made with exact solutions and a high order structured method. The higher order structured method was based upon a standard central differencing spatial discretisation. For the unstructured method a vertex-based data structure is employed. A median-dual control volume is used for the finite volume approximation with the option of using a Green-Gauss gradient approximation technique or a Least Squares approximation. The temporal discretisation used for both the structured and unstructured numerical methods is an explicit Runge-Kutta method with local timestepping. For the unstructured method, the gradient approximation technique is used to compute gradients at each vertex, these are then used to reconstruct the fluxes at the control volume faces. The unstructured mesh types used to evaluate the numerical method include semi-structured and purely unstructured triangular meshes. The semi-structured meshes were created directly from the associated structured mesh. The purely unstructured meshes were created using a commercial paving algorithm. The Least Squares method has the potential to allow high order reconstruction. Results show that a Weighted Least gradient approximation gives better solutions than unweighted and Green-Gauss gradient computation. The solutions are of acceptable accuracy on these problems with the absolute error of the unstructured method approaching that of a high order structured solution on an equivalent mesh for specific aeroacoustic scenarios.

Les écoulements diphasiques en milieu poreux sont des phénomènes complexes et qui concernent de nombreux problèmes industriels. EDF travaille sur la faisabilité et la sécurité d'un stockage en couche géologique profonde de déchets nucléaires. Dans ce domaine la simulation des écoulements diphasiques en milieu poreux est particulièrement importante dans au moins trois domaines : tout d'abord lors de la phase de ventilation des galeries du stockage qui pourrait désaturer la roche présente et ainsi en modifier ses propriétés de rétention, mais également lors de la phase de resaturation des matériaux et enfin lors de l'arrivée de l'eau sur les parties métalliques contenues dans le stockage qui entraînera alors des phénomènes de corrosion et un dégagement d'hydrogène. Dans ce contexte, EDF souhaite se doter de méthodes numériques performantes et robustes ne nécessitant pas de conditions restrictives sur la forme des mailles. Ce travail s'inscrivant dans cette problématique, est consacré dans un premier temps au développement du schéma volumes finis SUSHI (Scheme Using Stabilization and Hybrid Interfaces) dans le code de mécanique d'EDF, Code_Aster afin de modéliser les écoulements diphasique en milieu poreux. Ce schéma a été développé en 2D et en 3D. Parallèlement une nouvelle formulation qui permet de traiter de manière uniforme les écoulements en milieu saturé et insaturé pour des problèmes miscibles et immiscibles est proposée. Différentes études modélisant des difficultés liées aux problématiques du stockage de déchets radioactifs en couches géologiques profondes ont été traitées. On peut citer l'étude d'un bi-matériau qui met en avant le ré-équilibrage capillaire d'un matériau par un autre possédant des propriétés et des conditions initiales en saturation très hétérogènes. On citera également l'étude de l'injection d'hydrogène dans un milieu initialement saturé en eau pure qui est tirée du benchmark « Ecoulement diphasique » proposé par le GNR MOMAS. Cette étude avait pour objectif de mettre en évidence le bon traitement de l'apparition d'une phase dans un milieu saturé et donc la pertinence de notre nouvelle formulation à traiter d'une manière unifié un problème d'écoulement saturé et un problème d'écoulement insaturé / The two-phase flow in porous media is a complex phenomenon and which relate to many industrial problems. EDF works on the feasibility and the safety of a storage in deep geologic layer of nuclear waste. In this domain the simulation of the two-phase flow in porous media is particularly important in at least three domains : first of all during the phase of ventilation of the galleries of the storage which could desaturate the rock and so modify its properties, but also during the phase of resaturation of the materials and finally during the arrival of the water on the metal parts contained in the storage which will then involve phenomena of corrosion and a hydrogen release. In this context, EDF wishes to obtain robust numerical methods without restrictive condition on the mesh. This work is dedicated at first to the development of the finite volume scheme SUSHI (Scheme Using Stabilization and Hybrid Interfaces) in the code of mechanics of EDF, Code_Aster in order to simulate the two-phase flow in porous media. This scheme was developed in 2D and in 3D. At the same time a new formulation which allows to simulate in a uniform way the flows in saturated and unsaturated porous media for miscible and immiscible problems is proposed. Various studies simulating difficulties related to the problems of the storage of nuclear waste in deep geological layers were study. We can quote the study of a bi-material which advances the capillary rebalancing of a material by an other one possessing properties and initial very heterogeneous conditions in saturation. We will also quote the study of the injection of hydrogen in an porous media initially saturated in pure water which is proposed by the benchmark "two-phase Flow " proposed by the GNR MOMAS. This study had for objective to bring to light the good treatment of the appearance of a phase in a saturated porous media and thus the relevance of our new formulation to study with a way unified a problem of saturated flow and a problem of unsaturated flow

No description available.

The world population is expected to increase to approximately 11 billion by 2100. The ageing population (aged 60 and over) is projected to exceed the number of children in 2047. This will be a situation without precedent. The number of citizens with disorders of old age like Dementia will rise to 115 million worldwide by 2050. The estimated cost of Dementia will also increase, from $604 billion in 2010, to $1,117 billion by 2030. At the same time, medical expertise, evidence-driven policymaking and commissioning of services are increasingly evolving the definitive architecture of comprehensive long-term care to account for these changes. Technological advances, such as those provided by computational science and biomedical engineering, will allow for an expansion in our ability to model and simulate an almost limitless variety of complex problems that have long defied traditional methods of medical practice. Numerical methods and simulation offer the prospect of improved clinically relevant predictive information, and of course optimisation, enabling more efficient use of resources for designing treatment protocols, risk assessment and urgently needed management of a long term care system for a wide spectrum of brain disorders. Within this paradigm, the importance of the relationship of senescence of cerebrospinal fluid transport to dementia in the elderly make the cerebral environment notably worthy of investigation through numerical and computational modelling. Hydrocephalus can be succinctly described as the abnormal accumulation (imbalance between production and circulation) of cerebrospinal fluid (CSF) within the brain. Using hydrocephalus as a test bed, one is able to account for the necessary mechanisms involved in the interaction between cerebral fluid production, transport and drainage. The current state of knowledge about hydrocephalus, and more broadly integrative cerebral dynamics and its associated constitutive requirements, advocates that poroelastic theory provides a suitable framework to better understand the disease. In this work, Multiple-network poroelastic Theory (MPET) is used to develop a novel spatio-temporal model of fluid regulation and tissue displacement in various scales within the cerebral environment. The model is discretised in a variety of formats, through the established finite difference method, finite difference – finite volume coupling and also the finite element method. Both chronic and acute hydrocephalus was investigated in a variety of settings, and accompanied by emerging surgical techniques where appropriate. In the coupled finite difference – finite volume model, a key novelty was the amalgamation of anatomically accurate choroid plexuses with their feeding arteries and a simple relationship relaxing the constraint of a unique permeability for the CSF compartment. This was done in order to account for Aquaporin-4 sensitisation. This model is used to demonstrate the impact of aqueductal stenosis and fourth ventricle outlet obstruction. The implications of treating such a clinical condition with the aid of endoscopic third (ETV) and endoscopic fourth ventriculostomy (EFV) are considered. It was observed that CSF velocity in the aqueduct, along with ventricular displacement, CSF pressure, wall shear stress and pressure difference between lateral and fourth ventricles increased with applied stenosis. The application of ETV reduced the aqueductal velocity, ventricular displacement, CSF pressure, wall shear stress and pressure difference within nominal levels. The greatest reversal of the effects of atresia come by opting for ETV rather than the more complicated procedure of EFV. For the finite difference model incorporating nonlinear permeability, qualitatively similar results were obtained in comparison to the pertinent literature, however, there was an overall amplification of ventriculomegaly and transparenchymal pressure difference using this model. A quantitative and qualitative assessment is made of hydrocephalus cases involving aqueductal stenosis, along with the effects to CSF reabsorption in the parenchyma and subarachnoid space. The finite element discretisation template produced for the n^th- dimensional transient MPET system allowed for novel insight into hydrocephalus. In the 1D formulation, imposing the breakdown of the blood-CSF barrier responsible for clearance resulted in an increase in ventricular displacement, transparenchymal venous pressure gradient and transparenchymal CSF pressure gradient, whilst altering the compliance proved to markedly alter the rate of change of displacement and CSF pressure gradient. The influence of Poisson's ratio was investigated through the use of the dual-grid solver in order to distinguish between possible over or under prediction of the ventricular displacement. In the 2D model based on linear triangles, the importance of the MPET boundary conditions is acknowledged, along with the quality of the underlying mesh. Interesting results include that the fluid content is highest in the periventricular region and the skull, whilst after longer time scales, the peak CSF content becomes limited to the periventricular region. Venous fluid content is heavily influenced by the Biot-Willis constant, whilst both the venous and CSF/ISF compartments show to be strongly influenced by breakdown in the blood-CSF barrier. Increasing the venous compliance effects the arterial, capillary and venous compartments. Decreasing the venous compliance shows an accumulation of fluid, possibly helping to explain why the ventricles can be induced to compress rather than expand under decreased compliance. Finally, a successful application of the 3D-MPET template is shown for simple geometries. It is envisaged that future observations into the biology of cerebral fluid flow (such as perivascular CSF-ISF fluid exchange) and its interaction with the surrounding parenchyma, will demand the evolution of the MPET model to reach a level of complexity that could allow for an experimentally guided exploration of areas that would otherwise prove too intricate and intertwined under conventional settings.

Dans cette thèse, nous proposons un modèle pour le calcul des écoulements le long des failles. Ce modèle, baptisé modèle double interface permet de traiter deux difficultés majeures rencontrées lors de la modélisation des failles. Tout d'abord, l'utilisation d'un modèle interface, dans lequel les failles sont représentées par des éléments de dimension inférieure permet de s'affranchir du problème d'échelle spatiale. Ensuite, l'utilisation de deux interfaces pour représenter chaque faille permet de traiter naturellement les maillages non-conformes apparaissant dans ce type de problème. Les questions de failles non-planes et de réseaux de failles sont aussi abordées. Ce modèle est validé numériquement sur différents cas tests académiques et un cas synthétique inspiré du stockage du CO2 a aussi été réalisé. Finalement, une étude théorique a été menée afin de confirmer mathématiquement l'approche retenue. / In this thesis, we are interested in the modelisation of fluid flow along conductive faults. This model, so-called double interface model tackles two majors difficulties encountered when modelising faults. First of all, the use of an interface model, in which the faults are represented by lower dimension elements allows to treat the problem of space scale. Then, the use of two interfaces to modelise each fault allows to handle quite naturally the non-matching grid problem arising from this kind of problem. The question of non-planar fault and fault networks is also addressed. This model is then validated on several academic test cases and a synthetic case inspire by CO2 storage is also performed. Finally, a theoric study is also conducted in order to validate our approach.