Nessa tese, mostramos que a Hamiltoniana H e o operador helicidade de uma partícula de Dirac que se movimenta em duas dimensões na presença de um tubo de fluxo magnético infinitamente fino na origem admitem, cada um, uma família de quatro parâmetros de extensões auto-adjuntas. Para cada extensão correspondem condições de contorno a serem satisfeitas pelas auto-fuções na origem. Apesar dos operadores H e formalmente comutarem antes da especificação das condições de contorno, para garantirmos a conservação da helicidade, não é suficiente obtermos as mesmas condições de contorno para ambos os operadores, ou seja, não é suficiente a determinação de um domínio comum a ambos. Mostramos que, para certas relações entre os parâmetros das extensões satisfeitas, é possível a determinação dos domínios mais gerais onde ambos os operadores H e são auto-adjuntos e onde a helicidade é conservada, simultaneamente com a preservação da simetria de Aharonov-Bohm ( + 1), onde é o fluxo magnético em unidades naturais. Nossos resultados implicam que, nem a conservação da helicidade nem a simetria de Aharonov-Bohn, resolvem o problema da escolha da condição de contorno fisicamente correta. / We show that both the Hamiltonian H and the helicity operator of a Dirac particle moving in two dimension in the presence of an infinitely thin magnetic flux tube admit each a four- parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BC\'s) to be satisfied by the eigenfunctions at the origin. Althou- gh the actions af these two operators commute before specification of boundary conditions, to ensure helicity conservation it is not sufficient to take the same BC\'s for both operators. We show that, given certain relations between the parameters of the extensions it is possible to write down the most general domain where both operators H and are self-adjoint with heli- city conservation and also Aharonov-Bohm symmetry ( + 1) preserved, where is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results im- ply that neither helicity conservation nor Aharonov-Bohm symmetry by themselves solves the problem of choosing the \"physical \"boundary conditions for this system.

Em mecânica dos fluidos, especificamente em processos turbulentos, o problema de fechamento representa um dos maiores desafios para qualquer pessoa interessada nesta área. Durante décadas, cientistas vêm usando abordagens estatísticas com o objetivo de \"fechar\" o problema ou, pelo menos, diminuir as dificuldades inerentes. Assim, o presente trabalho apresenta uma criteriosa análise com base em ferramentas estatísticas em que ondas quadradas aleatórias, aliadas a um número fixo de parâmetros, foram utilizadas para criar equações paramétricas para representar um fluxo turbulento unidimensional com uma abordagem a priori, diferenciando de outras abordagens aplicadas amplamente na área, que utilizam uma abordagem a posteriori. Em seguida, simulações foram realizadas, a fim de avaliar o comportamento do modelo. Nas simulações pôde-se reproduzir o comportamento observado na literatura e estipular a abrangência do método. Além disso, uma importante discussão acerca das condições de contorno foi desenvolvida. / In fluid mechanics, specifically in turbulent processes, the closure problem represents one of the biggest challenges for anyone interested in this area. For decades, scientists have been using statistical approaches aiming to close the problem or, at least, decrease the inherent difficulties. So, the present project presents a judicious analyze based on statistical tools in which random square waves, allied with a fixed numbers of parameters, were used to create parametric equations to represent a turbulent flow with an a priori approach, differentiating from other approaches broadly applied in the area, which use an a posteriori approach. Then simulations were done, in order to evaluate the behavior of the model. In the simulations, the behavior of some data from the literature could be followed and the scope of the method was stipulated. Besides this, an important discussion about boundary conditions was developed.

The purpose of this work is to study the effects of boundaries on granular flows down vibrating inclines, on segregation in granular mixtures induced by boundary vibrations, and on flows of granular mixtures through vibrating sieves. In each case, we employ techniques borrowed from the kinetic theory to derive an appropriate set of boundary conditions, and combine them with existing flow theories to calculate the profiles of solid volume fraction, mean velocity, and granular temperature throughout the flows. The boundaries vibrate with full three-dimensional anisotropy in a manner that can be related to their amplitudes, frequencies, and phase angles in three independent directions. At impenetrable surfaces (such as those on the inclines), the conditions derived ensure that momentum and energy are each balanced at the boundary. At penetrable surfaces (such as sieves), the conditions also ensure that mass is balanced at the boundary. In these cases, the momentum and energy balances also are modified to account for particle transport through the boundary. Particular interest in all the applications considered here is in how the details of the boundary geometry and the nature of its vibratory motion affect the resulting flows. In one case, we derive conditions that apply to a monosized granular material that interacts with a bumpy, vibrating, impenetrable boundary, and predict how such boundaries affect steady, fully developed unconfined inclined flows. Results indicate that the flows can be significantly enhanced by increasing the total energy of vibration and are more effectively enhanced by normal vibration than by tangential vibration. Regardless of the direction of vibration, the bumpiness of the boundary has a profound effect on the flows. In a second case, we derive conditions that apply to a binary granular mixture that interacts with a flat, vibrating, penetrable sieve-like boundary, and predict how such boundaries affect the process in which the particles pass through the sieve. In the special case in which the particles are all the same size, the results make clear that energy is more effectively transmitted to the assemblies when either the total vibrational energy or the normal component of the vibrational energy is increased, but that an increase in the energy transferred to the material can sometimes actually decrease the flow rates through the sieve. Consequently, at any instant of time in the sieving process, there is an optimum level of vibrational energy that will maximize the flow rate. For the sieving of binary granular assemblies, the physics associated with the effects of energy transfer on the flow rates still applies. However, in these cases, the flows through the sieve are also profoundly affected by segregation that occurs while the particles reside on sieve before the pass through. For this reason, we also isolate the segregation process from the sieving process by considering the special case in which the holes in the vibrating sieve are too small to allow any particles to pass through. In this case, the results show that under most circumstances the region immediately adjacent to the vibrating surface will be populated almost entirely by the smaller particles or by the more dissipative particles if there is no size disparity, and that the reverse is true in a second region above the first.

<div>Incompressible single-phase and two-phase flows are widely encountered in and underlie many engineering applications. In this thesis, we aim to develop efficient methods and algorithms for numerical simulations of these classes of problems. Specically, we present two schemes: (1) a modied consistent splitting scheme for incompressible single-phase flows with open/out flow boundaries; (2) a three-dimensional hybrid spectral element-Fourier spectral method for wall-bounded two-phase flows.</div><div><br></div><div><div>In the first part of this thesis, we present a modied consistent splitting type scheme together with a family of energy stable outflow boundary conditions for incompressible single-phase outflow simulations. The key distinction of this scheme lies</div><div>in the algorithmic reformulation of the viscous term, which enables the simulation of outflow problems on severely-truncated domains at moderate to high Reynolds numbers. In contrast, the standard consistent splitting scheme is observed to exhibit a numerical instability even at relatively low Reynolds numbers, and this numerical instability is in addition to the backflow instability commonly known to be associated with strong vortices or backflows at the outflow boundary. Extensive numerical experiments are presented for a range of Reynolds numbers to demonstrate the effectiveness and accuracy of the proposed algorithm for this class of flows.</div></div><div><br></div><div><div>In the second part of this thesis, we present a numerical algorithm within the phase-field framework for simulating three-dimensional (3D) incompressible two-phase flows in flow domains with one homogeneous direction. In this numerical method, we represent the flow variables using Fourier spectral expansions along the homogeneous direction and C0 spectral element expansions in the other directions. This is followed by using fast Fourier transforms so that the solution to the 3D problem is obtained by solving a set of decoupled equations about the Fourier modes for each flow variable. The computations for solving these decoupled equations are performed in parallel to effciently simulate the 3D two-phase</div><div>ows. Extensive numerical experiments are presented to demonstrate the performance and the capabilities of the scheme in simulating this class of flows.</div></div>

To increase our understanding of how ice sheets and glaciers interact with the climate system, numerical models have become an indispensable tool. However, the complexity of these systems and the natural limitation in computational power is reflected in the simplifications of the represented processes and the spatial and temporal resolution of the models. Whether the effect of these limitations is acceptable or not, can be assessed by theoretical considerations and by validating the output of the models against real world data. Equally important is to verify if the numerical implementation and computational method accurately represent the mathematical description of the processes intended to be simulated. This thesis concerns a set of numerical models used in the field of glaciology, how these are applied and how they relate to other study areas in the same field. The dynamical flow of glaciers, which can be described by a set of non-linear partial differential equations called the Full Stokes equations, is simulated using the finite element method. To reduce the computational cost of the method significantly, it is common to lower the order of the used elements. This results in a loss of stability of the method, but can be remedied by the use of stabilization methods. By numerically studying different stabilization methods and evaluating their suitability, this work contributes to constraining the values of stabilization parameters to be used in ice sheet simulations. Erroneous choices of parameters can lead to oscillations of surface velocities, which affects the long term behavior of the free-surface ice and as a result can have a negative impact on the accuracy of the simulated mass balance of ice sheets. The amount of basal sliding is an important component that affects the overall dynamics of the ice. A part of this thesis considers different implementations of the basal impenetrability condition that accompanies basal sliding, and shows that methods used in literature can lead to a difference in velocity of 1% to 5% between the considered methods. The subglacial hydrological system directly influences the glacier's ability to slide and therefore affects the velocity distribution of the ice. The topology and dominant mode of the hydrological system on the ice sheet scale is, however, ill constrained. A third contribution of this thesis is, using the theory of R-channels to implement a simple numerical model of subglacial water flow, to show the sensitivity of subglacial channels to transient processes and that this limits their possible extent. This insight adds to a cross-disciplinary discussion between the different sub-fields of theoretical, field and paleo-glaciology regarding the characteristics of ice sheet subglacial hydrological systems. In the study, we conclude by emphasizing areas of importance where the sub-fields have yet to unify: the spatial extent of channelized subglacial drainage, to what degree specific processes are connected to geomorphic activity and the differences in spatial and temporal scales. As a whole, the thesis emphasizes the importance of verification of numerical models but also acknowledges the natural limitations of these to represent complex systems. Focusing on keeping numerical ice sheet and glacier models as transparent as possible will benefit end users and facilitate accurate interpretations of the numerical output so it confidently can be used for scientific purposes. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.</p> / Greenland Analogue Project

O método adjunto tem sido extensivamente utilizado como ferramenta de síntese no projeto de aeronaves por permitir que se obtenham sensibilidades de distintas medidas de mérito com relação a parâmetros que controlam a geometria de superfícies aerodinâmicas. O presente trabalho visa uma ampliação das aplicações da formulação contínua do método, ao utilizar propriedades físicas do escoamento nas fronteiras permeáveis do domínio computacional como parâmetros de controle de uma particular medida de mérito. Desse modo é possível, entre muitas possibilidades, determinar a sensibilidade de integrais como sustentação ou arrasto de uma aeronave com relação às condições de cruzeiro, por exemplo. Mais do que isso, essa informação pode ser obtida com a mesma solução adjunta computada para realizar otimização de forma. Vale destacar, ainda, que para que se consiga obter essa informação a partir das equações adjuntas, é necessário que se implemente condições de contorno baseadas em equações diferenciais características, resolvendo o problema de Riemann completo nas fronteiras do domínio. A implementação das usuais condições de contorno homogêneas, vastamente difundidas na literatura, resultaria em gradientes nulos. Esta nova abordagem do método é então aplicada a escoamentos modelados pelas equações de Euler 2-D compressíveis em estado estacionário. Ambos os problemas, físico e adjunto, são resolvidos numericamente com um código computacional que utiliza o método dos volumes finitos com segunda ordem de precisão no espaço e discretização centrada com dissipação artificial. As soluções estacionárias são obtidas ao se postular um termo tempo-dependente e integra-lo com um esquema Runge-Kutta de 5 passos e 2a ordem de precisão. As simulações são realizadas em malhas não-estruturadas formadas por elementos triangulares em 4 geometrias distintas: um bocal divergente, um perfil diamante, um aerofólio simétrico (NACA 0012) e o outro assimétrico (RAE 2822). Os gradientes adjuntos são então validados por meio da comparação com os obtidos pelo método de diferenças finitas nos regimes de escoamento subsônico, supersônico e transônico. / The adjoint method has been extensively used as an aircraft design tool, since it enables one to obtain sensitivities of many different mesures of merit with respect to parameters that control the aerodynamic surface geometry. This works aims to open up the possibilities of the method\'s applications by using flow physical properties at the permeable boundaries of the computational domain as control parameters of a particular measure of merit. This way it is possible, among many possibilities, to compute lift or drag sensitivities of an aircraft with respect to cruise conditions, for instance. Moreover, this information can be obtained with the same adjoint solution used to perform shape optimization. It is also worth noting that in order to obtain this information from the adjoint equations it is necessary to implement characteristics-based boundary conditions, resolving the complete Riemann problem at the boundaries of the computational domain. The use of the traditional homogeneous boundary conditions, widely spread in the literature, would lead the gradient to vanish. This new approach of the method is, then, applied to flows modeled by the 2-D steady state compressible Euler equations. Both, physical and adjoint problems are numerically solved with a computational code that makes use of a 2nd order finite volume method and central differences with artifficial dissipation. The steady solutions are obtained by postulating a time-dependent term and integrating it with a 5-stage 2nd order Runge-Kutta scheme. The simulations are performed on unstructured triangular meshes to 4 different geometries: a divergent nozzle, a diamond profile, a symmetric airfoil (NACA 0012) and a assymmetric airfoil (RAE 2822). The adjoint gradients are then validated by comparison with those obtained by finite differences method in subsonic, supersonic and transonic flow regimes.

acase@tulane.edu

<p>In this thesis a set of boundary conditions for multiphase flow is suggested.</p><p>Characteristic-based boundary conditions are reviewed for single-phase flow. The problem of open-boundary conditions is investigated, and to avoid drifting values, the use of control functions is proposed.</p><p>The use of control functions is also verified with a new test which assesses the quality of the boundary conditions. Particularly, P- and PI-control functions are examined. PI-controllers have the ability to specify a given variable exactly at the outlet as well as at the inlet, without causing spurious reflections which are amplified.</p><p>Averaged multiphase flow equations are reviewed, and a simplified model is established. This model is used for the boundary analysis and the computations. Due to the averaging procedure, signal speeds are reduced to the order of the flow speed. This leads to numerical challenges. For a horizontal channel flow, a splitting of the interface pressure model is suggested. This bypasses the numerical problems associated with separation by gravity, and a physical realistic model is used. In this case, the inviscid model is shown to possess complex eigenvalues, and still the characteristic boundary conditions give reasonable results.</p><p>The governing equations are solved with a Runge-Kutta scheme for the time integration. For the spatial discretisation, a finite-volume and a finite-difference method are used. Both implementations give equivalent results. In single-phase flow, the results improve significantly when a numerical filter is applied. For two-dimensional two-phase flow, the computations are unstable without a numerical filter.</p>

<p>Conformal field theories (CFT) constitute an interesting class of twodimensionalquantum field theories, with applications in string theoryas well as condensed matter physics. The symmetries of a CFT can beencoded in the mathematical structure of a conformal vertex algebra.The rational CFT’s are distinguished by the property that the categoryof representations of the vertex algebra is a modular tensor category.The solution of a rational CFT can be split off into two separate tasks, apurely complex analytic and a purely algebraic part.</p><p>The TFT-construction gives a solution to the second part of the problem.This construction gets its name from one of the crucial ingredients,a three-dimensional topological field theory (TFT). The correlators obtainedby the TFT-construction satisfy all consistency conditions of thetheory. Among them are the factorization constraints, whose implicationsfor boundary conditions are the main topic of this thesis.</p><p>The main result reviewed in this thesis is that the factorization constraintsgive rise to a semisimple commutative associative complex algebrawhose irreducible representations are the so-called reflection coefficients.The reflection coefficients capture essential information aboutboundary conditions, such as ground-state degeneracies and Ramond-Ramond charges of string compactifications. We also show that the annuluspartition function can be derived fromthis classifying algebra andits representation theory.</p>

River dynamic behaviour is affected by variations both in the water phase and in the transported sediment phase. A change in the water regime may lead to significant morphological changes in the bed profile, which in turn may strongly influence the flow conditions. Transcritical flows over mobile beds are particularly challenging to model due to the rapid variation in space and time of the solid transport, and to the specific treatment required for boundary conditions. The one-dimensional numerical model presented in this dissertation divides the flow in two fully coupled layers: a water layer and a water-sediment transport layer. This model was initially designed to depict dam-break flows, which does not require a specific treatment of boundary conditions. An extension of the two-layer approach is proposed in order to properly take into account boundary conditions. The treatment of boundary conditions commonly relies on characteristics. Within a two-layer model, which embodies five governing equations, an appropriate eigenstructure analysis is developed based on numerical estimations. This novel approach results in a new characterization of the critical stage by defining a specific two-layer Froude number. The model is compared to the classical Saint-Venant – Exner approach and favourably applied to several typical situations: uniform flow, which allows a straightforward calibration of the model parameters; regressive erosion around a mild-to-steep slope transition; evolution of a mobile bed under a hydraulic jump; and scour hole formation downstream of a fixed bed.