The handling of gradient calculations on non-orthogonal meshes in the Finite Volume Method (FVM) is important in the modelling of complex geometries, since different implementation methods have an influence on the accuracy and the stability of the solution. The application in the current study is the numerical solution of heat conduction in a complex geometry. It finds relevance in many engineering applications such as the Micro-Channel Heat Exchanger (MCHE) that acts as a recuperator in a High Temperature Reactor (HTR) power generation cycle. A program based on the FVM was developed in Excel for the solution of the diffusion equation on a non-orthogonal mesh. A test case of heat conduction in a rectangular block, meshed with a tetrahedral mesh, was solved with the Excel code. The same test case was solved with OpenFOAM. The results of the two codes were compared. Small differences were found and their origins were traced to slightly different implementation methods. Knowledge of the differences in implementation between the two codes resulted in a better understanding of the aspects that influenced accuracy and stability. Computations on meshes with the presence of mesh skewness and non-orthogonal mesh lines at boundaries were performed and an accompanying decrease in accuracy was observed. The results showed that the standard FVM as implemented in the Excel code and in OpenFOAM will need advanced methods to compensate for mesh skewness and non-orthogonality found at boundaries. During the study, a deeper knowledge and understanding was gained of the challenge of obtaining accurate solutions of heat conduction on non-orthogonal meshes. This knowledge may lead to the overall improvement of the simulation of heat transfer models in general and for the MCHE specifically. / Thesis (MIng (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2013.

The handling of gradient calculations on non-orthogonal meshes in the Finite Volume Method (FVM) is important in the modelling of complex geometries, since different implementation methods have an influence on the accuracy and the stability of the solution. The application in the current study is the numerical solution of heat conduction in a complex geometry. It finds relevance in many engineering applications such as the Micro-Channel Heat Exchanger (MCHE) that acts as a recuperator in a High Temperature Reactor (HTR) power generation cycle. A program based on the FVM was developed in Excel for the solution of the diffusion equation on a non-orthogonal mesh. A test case of heat conduction in a rectangular block, meshed with a tetrahedral mesh, was solved with the Excel code. The same test case was solved with OpenFOAM. The results of the two codes were compared. Small differences were found and their origins were traced to slightly different implementation methods. Knowledge of the differences in implementation between the two codes resulted in a better understanding of the aspects that influenced accuracy and stability. Computations on meshes with the presence of mesh skewness and non-orthogonal mesh lines at boundaries were performed and an accompanying decrease in accuracy was observed. The results showed that the standard FVM as implemented in the Excel code and in OpenFOAM will need advanced methods to compensate for mesh skewness and non-orthogonality found at boundaries. During the study, a deeper knowledge and understanding was gained of the challenge of obtaining accurate solutions of heat conduction on non-orthogonal meshes. This knowledge may lead to the overall improvement of the simulation of heat transfer models in general and for the MCHE specifically. / Thesis (MIng (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2013.

This thesis studies advection modeling for heat transfer and fluid flow problems using a new Non--Inverted Skew Upwind Scheme (called NISUS). Variants of the new scheme are formulated and developed with 8-noded hexahedral elements using the Finite Element Method (FEM)and rectangular elements based on a Finite Volume Method (FVM). A new method of mass weighting to predict convective fluxes of each scalar from the nodal point values is developed. Due to an explicit representation in terms of nodal variables, local inversion of the upwind coefficient matrix is not needed. Also, this thesis evaluates two variants of the new scheme (i.e., 3-node / 3-point and 4-node / 8-point formulations) within a 3--D FEM and a third variant within a 2--D FVM. The 3--D FEM variants are applied to a variety of test problems involving the transport of a scalar variable, while the 2--D FVM variant is applied to fluid flow problems including natural convection in an enclosure and micro--channel flow simulations. The promising performance of NISUS, as compared with exact and previous solutions, is demonstrated both in terms of accuracy and stability. Furthermore, a new data storage format called Compressed Banded Data (CBD) is developed for sparse banded matrices generated by the control volume finite element method (CVFEM). The platform of the new CBD structure permits dynamic switching between various solvers, without any procedural change in the implementation of existing simulation software. The performance of different Krylov techniques with an ILU(0) preconditioner is observed and compared in three test problems with a direct solver.

This thesis studies advection modeling for heat transfer and fluid flow problems using a new Non--Inverted Skew Upwind Scheme (called NISUS). Variants of the new scheme are formulated and developed with 8-noded hexahedral elements using the Finite Element Method (FEM)and rectangular elements based on a Finite Volume Method (FVM). A new method of mass weighting to predict convective fluxes of each scalar from the nodal point values is developed. Due to an explicit representation in terms of nodal variables, local inversion of the upwind coefficient matrix is not needed. Also, this thesis evaluates two variants of the new scheme (i.e., 3-node / 3-point and 4-node / 8-point formulations) within a 3--D FEM and a third variant within a 2--D FVM. The 3--D FEM variants are applied to a variety of test problems involving the transport of a scalar variable, while the 2--D FVM variant is applied to fluid flow problems including natural convection in an enclosure and micro--channel flow simulations. The promising performance of NISUS, as compared with exact and previous solutions, is demonstrated both in terms of accuracy and stability. Furthermore, a new data storage format called Compressed Banded Data (CBD) is developed for sparse banded matrices generated by the control volume finite element method (CVFEM). The platform of the new CBD structure permits dynamic switching between various solvers, without any procedural change in the implementation of existing simulation software. The performance of different Krylov techniques with an ILU(0) preconditioner is observed and compared in three test problems with a direct solver.

In order to handle turbulent flow problems, one equation turbulence models are implemented in to a previously developed explicit, Reynolds averaged Navier-Stokes solver. Discretization of Navier-Stokes solver is based on cell-vertex finite volume formulation combined with single step Lax-Wendroff numerical method which is second order accurate in space. Turbulent viscosity is calculated by using one equation Spalart-Allmaras and Baldwin-Barth turbulence transport equations. For the discretization of Spalart-Allmaras and Baldwin-Barth equations, both finite volume scheme which is used for Navier-Stokes equation in this work and explicit finite difference discretization method are used. In order to increase the convergence rate of the solver, local time stepping technique is applied. Stabilization of non-physical oscillations resulting from the numerical scheme is maintained by adding second and fourth order artificial smoothing terms. Three test cases are considered. In order to validate the accuracy of the Navier-Stokes solver, solver is tested over a laminar flat plate. The results are compared with analytical solutions. Later, in order to check the performance of the turbulence models, turbulent flow over flat plate and turbulent transonic flow over NACA-0012 airfoil are handled. For turbulent flow over flat plate obtained results are compared with analytical and empirical solutions, whereas for transonic turbulent flow obtained results are compared with numerical and experimental solutions.

A method to solve the Navier-Stokes equations for incompressible viscous flows is proposed. This method is SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm to iteratively solve the two-dimensional laminar steady momentum equations and based upon finite volume method on staggered grids. Numerical tests are performed on several cases of the flow in the lid-driven cavity, as well as of the flow after a backward-facing step with SIMPLE and SIMPLER (SIMPLE Revised) methods. Finally, results are compared qualitatively and quantitatively with numerical and experimental results available in the literature for different Reynolds numbers to validate the methods.

The objective of this PhD research programme is to investigate the effectiveness of a finite volume method incorporating radial basis functions for simulating nonlinear transport processes. The finite volume method is the favoured numerical technique for solving the advection-diffusion equations that arise in transport simulation. The method transforms the original problem into a system of nonlinear, algebraic equations through the process of discretisation. The accuracy of this discretisation determines to a large extent the accuracy of the final solution. A new method of discretisation is presented that employs radial basis functions (rbfs) as a means of local interpolation. When combined with Gaussian quadrature integration methods, the resulting finite volume discretisation leads to accurate numerical solutions without the need for very fine meshes, and the additional overheads they entail. The resulting nonlinear, algebraic system is solved efficiently using a Jacobian-free Newton-Krylov method. By employing the new method as an extension of existing shape function-based approaches, the number of nonlinear iterations required to obtain convergence can be reduced. Furthermore, information obtained from these iterations can be used to increase the efficiency of subsequent rbf-based iterations, as well as to construct an effective parallel reconditioner to further reduce the number of nonlinear iterations required. Results are presented that demonstrate the improved accuracy offered by the new method when applied to several test problems. By successively refining the meshes, it is also possible to demonstrate the increased order of the new method, when compared to a traditional shape function basedmethod. Comparing the resources required for both methods reveals that the new approach can be many times more efficient at producing a solution of a given accuracy.

Dans le cadre de la conception des réacteurs, les schémas de calculs utilisant des codes de cal- culs neutroniques déterministes sont validés par rapport à un calcul stochastique de référence. Les biais résiduels sont dus aux approximations et modélisations (modèle d'autoprotection, développement en polynômes de Legendre des lois de choc) qui sont mises en oeuvre pour représenter les phénomènes physiques (absorption résonnante, anisotropie de diffusion respec- tivement). Ce document se penche sur la question de la pertinence de la modélisation de la loi de choc sur une base polynômiale tronquée. Les polynômes de Legendre sont utilisés pour représenter la section de transfert multigroupe dans les codes déterministes or ces polynômes modélisent mal la forme très piquée de ces sections, surtout dans le cadre des maillages énergétiques fins et pour les noyaux légers. Par ailleurs, cette représentation introduit aussi des valeurs négatives qui n'ont pas de sens physique. Dans ce travail, après une brève description des lois de chocs, les limites des méthodes actuelles sont démontrées. Une modélisation de la loi de choc par une fonction constante par morceaux qui pallie à ces insuffisances, a été retenue. Cette dernière nécessite une autre mod- élisation de la source de transfert, donc une modification de la méthode actuelle des ordonnées discrètes pour résoudre l'équation du transport. La méthode de volumes finis en angle a donc été développée et implantée dans l'environ- nement du solveur Sn Snatch, la plateforme Paris. Il a été vérifié que ses performances étaient similaires à la méthode collocative habituelle pour des sections représentées par des polynômes de Legendre. Par rapport à cette dernière, elle offre l'avantage de traiter les deux représenta- tions des sections de transferts multigroupes : polynômes de Legendre et fonctions constantes par morceaux. Dans le cadre des calculs des réacteurs, cette méthode mixte a été validée sur différents motifs : des cellules en réseau infini, des motifs hétérogènes et un calcul de réflecteur. Les principaux résultats sont : - un développement polynômial à l'ordre P 3 est suffisant par rapport aux biais résiduels dus aux autres modélisations (autoprotection, méthode de résolution spatiale). Cette modéli- sation est convergée au sens de l'anisotropie du choc sur les cas représentatifs des réacteurs à eau légère. - la correction de transport P 0c n'est pas adaptée, notamment sur les calculs d'absorbant B4 C. / In reactor physics, calculation schemes with deterministic codes are validated with respect to a reference Monte Carlo code. The remaining biases are attributed to the approximations and models induced by the multigroup theory (self-shielding models and expansion of the scattering law using Legendre polynomials) to represent physical phenomena (resonant absorption and scattering anisotropy respectively). This work focuses on the relevance of a polynomial expansion to model the scattering law. Since the outset of reactor physics, the latter has been expanded on a truncated Legendre polynomial basis. However, the transfer cross sections are highly anisotropic, with non-zero values for a very small range of the cosine of the scattering angle. Besides, the finer the energy mesh and the lighter the scattering nucleus, the more exacerbated is the peaked shape of this cross section. As such, the Legendre expansion is less suited to represent the scattering law. Furthermore, this model induces negative values which are non-physical. In this work, various scattering laws are briefly described and the limitations of the existing model are pointed out. Hence, piecewise-constant functions have been used to represent the multigroup scattering cross section. This representation requires a different model for the dif- fusion source. The discrete ordinates method which is widely employed to solve the transport equation has been adapted. Thus, the finite volume method for angular discretisation has been developed and imple- mented in Paris environment which hosts the Sn solver, Snatch. The angular finite volume method has been compared to the collocation method with Legendre moments to ensure its proper performance. Moreover, unlike the latter, this method is adapted for both the Legendre moments and the piecewise-constant functions representations of the scattering cross section. This hybrid-source method has been validated for different cases: fuel cell in infinite lattice, heterogeneous clusters and 1D core-reflector calculations. The main results are given below : - a P 3 expansion is sufficient to model the scattering law with respect to the biases due to the other approximations used for calculations (self-shielding, spatial resolution method). This order of expansion is converged for anisotropy representation in the modelling of light water reactors. - the transport correction, P 0c is not suited for calculations, especially for B4 C absorbant.

L’objectif de cette thèse est de répondre au besoin d’accélérer la restitution des résultats de calcul d’un code CFD pour la simulation d’écoulements hydrodynamiques quasi-incompressibles. Ce code présente l’originalité de résoudre explicitement les équations de Navier-Stokes sous l’hypothèse de faible compressibilité avec des schémas numériques d’ordre élevé. Les développements effectués visent à réduire les temps de calcul à précision équivalente.Une première partie est consacrée à l’implémentation d’une formulation purement incompressible avec une résolution implicite de la pression par un schéma de projection. La formulation incompressible autorise des pas de temps plus grand en s’affranchissant de la vitesse du son, mais au prix d’une algorithmique plus complexe et de la nécessité de résoudre un système linéaire. La comparaison des deux formulations,faiblement-compressible et incompressible, tend à montrer la pertinence du schéma de projection pour les écoulements laminaires instationnaires.Un deuxième axe de développement a consisté en la proposition d’une amélioration de la méthode de frontière immergée initialement présente dans le code.Si les résultats obtenus ne sont pas encore pleinement satisfaisants, ils montrent que la montée en ordre d’une méthode de frontière immergée peut être moins contraignante en formulation incompressible.Enfin la dernière partie traite de l’immersion rapide et robuste de géométries complexes telles qu’elles peuvent être rencontrées dans l’industrie. La localisation géométrique par arbre octal permet d’évaluer rapidement une fonction de distance signée indispensable pour la méthode de frontière immergée. / An original strategy to address hydrodynamic flow was recently proposed through a high-order weakly-compressible Cartesian grid approach. The method is based on a fully-explicit temporal scheme for solving the Navier-Stokes equations. The present thesis aims to reduce the computational time required to obtain the results without deteriorating the accuracy.A first part is dedicated to the implementation of a truly incompressible formulation with an implicit solution for the pressure field through a projection scheme. The incompressible solver allows larger time step size for time integration since the speed of sound tends to infinity. In return the algorithms are no longer straight forward and a linear system has to be solved through the Pressure Poisson Equation. Comparisons carried out between both formulations show that the projection scheme can be better adapted to efficiently simulate unsteady viscous flows. Then an improvement of the immersed boundary method has been proposed. Results are not fully satisfactory for now. However, it seems easier to develop a numerical scheme for the incompressible approach rather than the weakly-compressible one.Finally, the last part addresses the setup up of complex triangulations in immersed boundary simulations. A fast and robust procedure is developed for distance computation with an octree data structure.

O princípio operacional do dispositivo de galgamento consiste de uma estrutura que utiliza uma rampa para direcionar as ondas incidentes para o reservatório. A água armazenada retorna para o oceano após a passagem por uma turbina que está acoplada a um gerador de energia elétrica. O presente trabalho propõe dois estudos numéricos a respeito de um conversor de energia das ondas do mar do tipo galgamento. Para ambos os casos, o objetivo do estudo é a aplicação do método Design Construtal na definição da melhor forma para a rampa de modo a maximizar a massa de água que entra no reservatório, conduzindo a uma maior geração de energia elétrica. O grau de liberdade b/B, isto é, a razão entre a base superior e a base inferior da rampa trapezoidal, foi otimizado, mantendo-se fixos a área total do tanque de ondas, a área da rampa e as características da onda. Para a análise numérica do princípio de funcionamento deste dispositivo foi empregado um domínio computacional tridimensional (3D), gerado através do software GAMBIT, onde o conversor é acoplado a um tanque de ondas regulares. A solução das equações de conservação e a equação do transporte da fração volumétrica foi realizada com o código comercial de Dinâmica dos Fluidos Computacional FLUENT, que é baseado no Método de Volumes Finitos (MVF). Aplica-se o modelo multifásico Volume of Fluid (VOF) no tratamento da interação água-ar. Para o primeiro estudo, as características da onda regular empregada estavam em escala de laboratório. Os resultados mostraram que houve uma razão ótima (b/B)o = 0.43, que maximiza a quantidade de água que entra no reservatório para o caso estudado. Para ambos os casos, a razão ótima foi encontrada para o extremo inferior do grau de liberdade, além dos resultados apontarem um aumento significativo na massa admitida no reservatório e, por consequente, um maior aproveitamento das ondas incidentes. / The operational principle of an overtopping device consists of a structure which utilizes a ramp to direct incident waves to the reservoir. The stored water returns to the ocean after passing through a turbine that is coupled to an electric generator. This work proposes two numerical studies of a WEC of sea waves of the type overtopping. In both cases, the objective of the study is the application of Constructal Design method to define the best geometry of the ramp which maximizes the mass of water entering the reservoir, leading to increase the generation of electricity. The degree of freedom b/B, that is, the ratio between the upper base and the lower base of the trapezoidal ramp, has been optimized, keeping fixed the total area of the wave tank, the area of the ramp and the wave characteristics. For the numerical analysis of the working principle of this device it was used a three-dimensional computational domain (3D) generated by GAMBIT software where the device is inserted to a tank of regular waves. The solution of conservation equations and equation of transport of the volumetric fraction was carried out with the Commercial Code of Computational Fluid Dynamics FLUENT, which is based on Finite Volume Method (FVM). It was applied the multiphase model Volume of Fluid (VOF) in the treatment of the interaction water-air. For the first study, the characteristics of the employed regular wave were on a laboratory scale. The results showed that there were an optimal ratio (b/B)o = 0.43, which maximizes the amount of water entering the reservoir for the case study. For the second study, the characteristics of the regular wave were employed at actual scale and the results showed that there was an optimum ratio (b/B)o = 0.38, which maximizes the amount of water entering the reservoir for the case study. In both cases, the optimum ratio is found for the extreme lower of freedom of degree and the results showed a significant increase in the mass allowed in the reservoir and, consequently, larger use of the incident waves.