The complicated turbulence structures in wall-bounded flows require accurate subgrid scale, SGS, modelling and realistic inlet boundary conditions for Large Eddy Simulation, LES. The present study focused on the investigation and development of transport equation SGS models and on the development of inlet conditions generation algorithms specialised for LES of wall-bounded flows. The investigation of SGS models has been carried out in two stages. In the first stage, models based on resolved scales and models based on subgrid scales were tested on a series of channel flow cases. Among the second group of models, there was a new SGS model whose development was based on the concept of dissipation calculated from the energy spectrum. The results indicated the superiority of the models based on subgrid scales, with the new model providing the most accurate flow field in general. (Continues...).

Theoretical models for the electrokinetics of weakly permeable porous colloidal particles are absent from the literature. The understanding of this topic will be advanced through a systematic analysis of the standard electrokinetic equations, resulting in a theory for the electrophoretic mobility of weakly permeable porous colloidal particles. / The standard electrokinetic equations are employed to model the flux of solvent and ions outside the porous particle. To be consistent with this approach, the flux of solvent and ions in the pores must also be governed by the standard electrokinetic equations. However, in practice, only transport phenomena on the particle scale are observed and it is sufficient for information regarding pore-scale behaviour to be retained purely in the form of averaged quantities. To complete the theoretical description, the standard electrokinetic equations outside the particle must be coupled to particle-scale transport equations inside the particle via boundary conditions at the porous/free-fluid interface. / It has been shown experimentally and theoretically for coupled Stokes and Darcy flows, that the correct interfacial boundary condition for the tangential external flow is given by the Beavers-Joseph-Saffman (BJS) condition. The effect of the BJS boundary condition on the hydrodynamic drag on an oscillating porous particle is investigated. It is found that the particle may be regarded as impermeable with a slip length independent of frequency, and the resulting drag is significantly reduced in comparison with an equivalent impermeable particle that does not exhibit a slip length. / The transport of a general electrolyte solution through a rigid porous body subjected to a static (d.c.) electric field is studied. The pore-scale description is given by the standard electrokinetic equations, including the effects of ion diffusion, electromigration and convection. Homogenization theory is used to derive transport equations that capture the particle-scale behaviour. It is proven that the transport coefficient tensors obey Onsager’s reciprocal relations and the diagonal coefficient tensors are positive definite. / New interfacial boundary conditions are derived using conservation arguments supplemented by Stern-layer theory. When combined with the particle-scale transport equations, these boundary conditions incorporate four principal effects into the standard electrokinetic model: solvent slip and Stern-layer ionic conduction at the interface, and macroscopic ionic conduction together with the electroosmotic flow of solvent through the particle. / The method of matched asymptotic expansions is then used to construct an approximate solution to the aforementioned system, in the thin double-layer limit. An expression for the electrophoretic mobility of a weakly permeable colloidal sphere is produced that consists of a generalization of Smoluchowski’s formula to encompass porous particles, and a next order correction. For the first time, the effects of solvent slip and Stern-layer ionic conduction within the porous/free-fluid interface, in conjunction with macroscopic ionic conduction and electroosmosis through the particle, are exhibited. It is shown that solvent slip at the porous interface is overwhelmingly the dominant effect on the mobility of weakly permeable porous colloidal particles.

Steel is a material commonly used in various constructions such as high-rise buildings, sport arenas, ships etc. Steel is a versatile building material due to its isotropic characteristics, e.g. both high tensile- and compressive strength. This allows steel to be formed into open section profiles which reduces material usage but simultaneously allows the tensile- and compressive stress resistance to be high in directions were loads are applied. Although steel has a high stress resistance its sensitivity to fire is larger than other building materials due to its high thermal conductivity. The strength of the material is reduced at higher temperatures and thereby makes the dimensioning of beams in fire cases vital in fire safety design of structural elements. An aspect to consider when dimensioning open section building elements in steel is the shadow effect. The shadow effect is the result of the open cross-section geometrical shape of beams and columns, e.g. H-profiles. The interior of the profile is screened from thermal radiation caused by fire which makes the characteristics of the thermal exposure different from closed cross-section profiles. A common way to estimate the temperatures of steel after a certain time of fire exposure is to use numerical calculations described in Eurocode. In these calculations the shadow effect is applied as a reduction of the total heat exchange, i.e. both convection and thermal radiation, from the fire exposure. A more realistic approach is to separate these boundary conditions and treat them as independent quantities. Wickström (2001) argues that a void is created within the flanges and that reduction factor thereby only should be applied to the radiative part of the total heat exchange, acting as a reduction of surface emissivity within the profile. This, since the convection is not affected by the shadow effect. Wickströms (2001) suggestion of application has been investigated in this thesis and has showed a better correlation than the approach suggested in Eurocode when compared to experimental tests. Shadow effects calculated on the premises of separated boundary conditions for the total heat exchange has of yet only been investigated in detail with TASEF+-simulations, but these simulations predicts steel temperatures with satisfactory results. It is possible to reproduce a similar setup in the program COMSOL Multiphysics in two-dimensional simulations, and further three-dimensional simulations. This possibility has been investigated in this thesis. COMSOL Multiphysics has proven to be an adequate tool when it comes to simulate fire exposure on slender steel beam with shadow effects considered. Both three- and two-dimensional models produced simulation results correlating well to simulations conducted in TASEF. Additionally, adequate correlations with experimental tests were obtained for COMSOL Multiphysics as well. Further work regarding fire simulations with the utilisation of COMSOL Multiphysics is thereby suggested.

Ce travail porte sur l’étude du transfert de chaleur entre un milieu poreux et un milieu libre en utilisant une approche multi-échelle. Dans un premier temps, nous dérivons les conditions limites à imposer à une interface libre-poreux dans le cas des transferts de chaleur à l’équilibre thermique local puis dans le cas du déséquilibre thermique local. Pour les transferts de chaleur turbulent, une simulation numérique directe est réalisée afin d’apporter une meilleur compréhension de la physique existant à l’interface libre poreux. Puis, nous déterminons un modèle turbulent avec des conditions de saut. Ces études répondent à des questions fondamentales liées à la compréhension physique des conditions de saut, des valeurs des paramètres des sauts et de la position de l’interface dans le cadre des transferts de chaleur. / This work deals with the study of heat transfer between a porous medium and a free medium, using multi scale approaches. First, we derive the boundary conditions that must be applied at a free-porous interface for laminar heat transfer at local thermal equilibrium and, then, at local thermal non-equilibrium. For turbulent heat transfer, a direct numerical simulation is performed supplying a better understanding of the physic at the free-porous interface. Then, we determine a turbulent model with associated jump conditions. These studies answer fundamental questions regarding the physical meaning of the jump conditions, the values of the jump parameters and the location of the interface for heat transfer.

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.

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.

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.

This paper examines different combinations of close-field and far-field boundary conditions for solving the 2D Black-Scholes model using finite difference methods in space. The combinations were also tested for different parameter settings. The research showed that in the area close to the strike price, the error was not particularly affected by the boundary conditions but rather by the characteristics of the problem itself. The main differences in error for the combinations of conditions are located close to the boundaries. However, if the computational domain for some reason has to be reduced, e.g. to save computational time, the boundary conditions will play an important role on the error in the area close to the strike price. Based on the findings presented in this report, Dirichlet boundary condition on the far- field boundary together with no boundary condition on the close-field is the best combination. If any of those are not applicable, the linearity condition should be used on that boundary instead.

Stochastic differential equations are becoming a popular tool for modeling the transport and acceleration of cosmic rays in the heliosphere. In diffusive shock acceleration, cosmic rays diffuse across a region of discontinuity where the up- stream diffusion coefficient abruptly changes to the downstream value. Because the method of stochastic integration has not yet been developed to handle these types of discontinuities, I utilize methods and ideas from probability theory to develop a conceptual framework for the treatment of such discontinuities. Using this framework, I then produce some simple numerical algorithms that allow one to incorporate and simulate a variety of discontinuities (or boundary conditions) using stochastic integration. These algorithms were then modified to create a new algorithm which incorporates the discontinuous change in diffusion coefficient found in shock acceleration (known as Skew Brownian Motion). The originality of this algorithm lies in the fact that it is the first of its kind to be statistically exact, so that one obtains accuracy without the use of approximations (other than the machine precision error). I then apply this algorithm to model the problem of diffusive shock acceleration, modifying it to incorporate the additional effect of the discontinuous flow speed profile found at the shock. A steady-state solution is obtained that accurately simulates this phenomenon. This result represents a significant improvement over previous approximation algorithms, and will be useful for the simulation of discontinuous diffusion processes in other fields, such as biology and finance.

This thesis investigates turbulent wall-bounded flows using the Smoothed Particle Hydrodynamics (SPH) method. The first part focuses on the SPH method itself in the context of the Navier-Stokes equations with a special emphasis on wall boundary conditions. After discussing classical wall boundary conditions a detailed introduction to unified semi-analytical wall boundary conditions is given where the key parameter is a renormalization factor that accounts for the truncated kernel support in wall-bounded flows. In the following chapter it is shown that these boundary conditions fulfill energy conservation only approximately. This leads to numerical noise which, interpreted as form of Brownian motion, is treated using an additional volume diffusion term in the continuity equation where it is shown to be equivalent to an approximate Riemann solver. Two extensions to the boundary conditions are presented dealing with variable driving forces and a generalization to Robin type and arbitrary-order interpolation. Two modifications for freesurface flows are then presented, one for the volume diffusion term and the other for the algorithm that imposes Robin boundary conditions. The variable driving force is validated using a Poiseuille flow and the results indicate an error which is five orders of magnitude smaller than with the previous formulation. Discretising the wave equation with Robin boundary conditions proves that these are correctly imposed and that increasing the order of the interpolation decreases the error. The two modifications for flows under the influence of external forces significantly reduce the error at the free-surface. Finally, a dam break over a wedge demonstrates the capabilities of all the proposed modifications. With the aim of simulating turbulent flows in channels, the thesis moves on to extending the unified semi-analytical wall-boundary conditions to three dimensions. The thesis first presents the consistent computation of the vertex particle mass. Then, the computation of the kernel renormalization factor is considered, which in 3-D consists of solving an integral over a two dimensional manifold where the smoothing kernel intersects the boundary. Using a domain decomposition algorithm special integration areas are obtained for which this integral can be solved for the 5 th -order Wendland kernel. This algorithm is successfully applied to several validation cases including a dam break with an obstacle which show a significant improvement compared to other approximative methods and boundary conditions. The second part of this thesis investigates turbulent flows, in particular turbulent channel flow. This test case is introduced in detail showing both the physical properties as well as established numerical methods such as direct numerical simulation (DNS) and large eddy simulation (LES). In the penultimate chapter several SPH simulations of the turbulent channel flow are shown. The first section deals with a quasi DNS of the minimal-flow unit, a channel flow with a minimal domain size to sustain turbulent flow structures. The Eulerian statistics are compared to literature and show good agreement except for some wall-normal quantities. Furthermore, preliminary Lagrangian statistics are shown and compared to results obtained from a mesh-based DNS. The final simulation shows a LES of a full-sized channel at Reynolds number Re τ = 1000. The Eulerian statistics are compared to literature and the discrepancies found are explained using simulations of the Taylor-Green vortex, indicating that the momentum is not transferred appropriately due to an unresolved velocity-pressure-gradient tensor.