Esta tese apresenta um modelo matemático para simulação numérica do escoamento com turbilhonamento (swirl) em tochas de plasma térmico de arco não transferido que operam em corrente contínua, assim como os resultados obtidos com as simulações para estudo de casos. O modelo magneto-hidrodinâmico (modelo MHD) bidimensional permitiu simular a interação entre o escoamento e o arco elétrico usando uma configuração axissimétrica, que abrange as seguintes regiões: entrada do gás; interior da tocha; jato de plasma livre no ambiente. O modelo foi implementado num código numérico baseado no Método dos Volumes Finitos para a solução numérica das equações governantes. Para os estudos foram simulados casos com diferentes condições operacionais (vazão de gás; intensidade de corrente elétrica; gases plasmogênicos: ar e argônio; intensidade de turbilhonamento). A fim de verificar a qualidade do modelo, alguns resultados foram comparados com a literatura e apresentaram boa concordância: a maior diferença obtida entre valores de temperatura experimentais e valores calculados foi -10%, e a média das diferenças obtidas nas comparações foi de aproximadamente ±3,2%. Os perfis de temperatura e de velocidade obtidos para a região do arco e para o jato de plasma resultante permitiram o estudo do comportamento do escoamento na tocha de plasma em diferentes condições. Conclui-se que o modelo desenvolvido é apto à realização de investigações numéricas do escoamento em tochas de plasma e dos efeitos do turbilhonamento na interação arco/escoamento. / This thesis presents a mathematical model for numerical simulation of swirling flow in DC non-transferred arc thermal plasma torches, as well as the results obtained from simulations to case studies. The two-dimensional magnetohydrodynamic model (MHD model) allowed simulate the interaction between the flow and the electric arc using an axisymmetric configuration, covering the following areas: gas inlet; inside the torch; free jet of plasma in the environment. The model was implemented in a computer code based on the Finite Volume Method (FVM) to enable the numerical solution of the governing equations. For the study, cases were simulated with different operating conditions (gas flow rate; electric current intensity; plasmogenic gases: air and argon; swirl intensity). In order to verify the quality of the model, some results were compared with the literature and showed good agreement: the biggest difference between experimental temperature values and calculated values was 10%, and the average of the differences obtained in the comparisons was approximately ±3.2%. The resulting profiles of temperature and velocity obtained for the region of the arc and the plasma jet allowed the study of the flow behavior in the plasma torch in different conditions. It is concluded that the model developed is able to carry out numerical investigations of the flow in plasma torches and the effects of swirl in the interaction arc/flow.

Ce travail s'inscrit dans l'étude mathématique d'écoulements de fluides environnementaux. Nous en abordons deux aspects, à travers deux contextes distincts d'application.En lien avec la simulation des écoulements en milieux poreux, on s'intéresse dans une première partie à la discrétisation d'opérateurs de diffusion anisotropes hétérogènes par des méthodes de volumes finis sur des maillages généraux. Dans le but d'obtenir des solutions approchées qui respectent les bornes physiques des modèles, notre attention se porte sur la conservation du principe du maximum pour les opérateurs elliptiques. Nous présentons des mécanismes généraux permettant de corriger tout schéma volumes finis afin de garantir un principe du maximum discret tout en préservant certaines de ses propriétés principales. On étudie en particulier les propriétés de coercivité et de convergence des schémas corrigés.La deuxième partie est consacrée à la construction de modèles approchés pour la propagation des vagues en eaux peu profondes et sur des topographies irrégulières. A cet effet, nous proposons tout d'abord une adaptation de la démarche d'étude classique à des écoulements bidimensionnels sur des topographies polygonales. Dans un cadre plus général, nous développons ensuite une démarche formelle qui débouche sur des alternatives non locales à quelques modèles classiques (équations de Saint-Venant, équations de Serre, système de Boussinesq). Ces nouveaux modèles contiennent des termes régularisants pour les contributions du fond. / This work investigates two research questions associated with environmental flows and their mathematical modeling.The first part is devoted to the development of finite volume methods for anisotropic and heterogeneous diffusion operators arising in models of porous media flows. To ensure that the approximate solutions lie within physical bounds, we aim at maintaining a discrete analogous of the maximum principle for elliptic operators. Starting from any given cell-centered finite volume scheme, we present a general approach to devise non-linear corrections providing a discrete maximum principle while retaining some main properties of the scheme. In particular, we study the coercivity and convergence properties of the modified schemes.The second part of this work focuses on the derivation of approximate models for shallow water wave propagation over rough topographies. In the particular case of one-dimensional polygonal bottom profiles, we first propose an adaptation of the usual derivation method using complex analysis tools. We then develop a formal approach to account for more general topographies. We propose nonlocal alternatives to some classical models (namely Saint-Venant equations, Serre equations and Boussinesq system). All these alternative models only involve smoothing contributions of the bottom.

A medição de vazão de escoamentos multifásicos é uma necessidade constante em diversas atividades industriais como exploração de óleo e gás, controle de linhas de transporte de vapor e monitoramento de sistemas de resfriamento de usinas nucleares. Dentre os meios disponíveis para a realização da medição de vazão mássica, os dispositivos de medição de pressão diferencial constituem um dos métodos mais simples, sendo sua construção, aplicação e operação em escoamentos monofásicos bem conhecidas e definidas por normas técnicas. No entanto, sua aplicação tem sido estendida a escoamentos multifásicos, geralmente estando aliada a uma técnica adicional de medição de fração de vazio ou fração volumétrica das fases. Este trabalho descreve o estudo numérico de escoamentos multifásicos através de medidores de vazão baseados em pressão diferencial como placas de orifício e bocais de vazão de raio longo. Para tal, primeiramente foram conduzidas simulações de escoamentos monofásicos através de placas de orifício e bocais de vazão de raio longo na faixa de número de Reynolds 15.000 500.000. Os resultados de coeficiente de descarga obtidos foram quantitativamente comparados com os valores preditos por norma ISO, apresentando desvio máximo de aproximadamente 4, 9% para as placas e de 1,0% para os bocais. Em uma segunda etapa, escoamentos do tipo gás úmido (wet gas) através de placas de orifício foram simulados através de três abordagens diferentes. Os resultados de vazão mássica total obtidos foram comparados com dados experimentais fornecidos pela PETROBRAS. As abordagens que consideram o escorregamento entre as fases apresentaram previsões mais próximas dos experimentos, com desvio relativo médio de 3,9%, enquanto a modelagem homogênea apresentou um desvio médio de 6, 6%. Nestes estudos, foram também avaliadas as estruturas desenvolvidas no escoamento através de visualizações da distribuição de fases. São também apresentadas duas sugestões para complementação da caracterização de um escoamento multifásico: (1) a introdução da informação de fração de vazio na formulação apresentada por Paz (2011) e (2) a análise estatística do sinal de pressão diferencial em placas de orifício. Com relação ao primeiro item, comparações quantitativas com dados experimentais sugeriram que a alternativa apresentada é viável para operações de monitoramento da produção. Já o último estudo mostrou qualitativamente a influência da quantidade de líquido na flutuação da pressão diferencial / The flowrate measurement of multiphase flows is a constant need at many industrial activities such as oil and gas exploration, steam transport lines control and monitoring of nuclear plants cooling systems. Within the available means for performing flowrate measurement, the differential pressure devices constitute one of the simplest methods, with their construction, application and operation in single phase flows being well known and defined by technical standards. However, their application has been extended to multiphase flows, usually being allied to a void fraction or phase volume fraction measurement technique. This work describes a numerical study of multiphase flows through differential pressure-based flowrate meters such as orifice plates and long radius nozzles. Firstly simulations of single phase flows through orifice plates and long radius nozzles were conducted in the Reynolds number range 15.000500.000. The obtained results of discharge coefficients were quantitatively compared to ISO Standard predicted values, showing a maximum deviation of approximately 4,9% for the orifice plates and of 1,0% for the nozzles. In a second stage, wet gas flows through orifice plates were simulated by means of three approaches. The calculated results of total mass flowrate were compared to experimental data provided by PETROBRAS. The approaches that considered the slip between phases provided the closest results to the experiments, with a mean relative error of 3, 9%, while the homogeneous modeling presented an error of 6, 6%. In these studies, the structures developed within the domain were also evaluated through the visualization of the phases distribution. Two suggestions for complementing the characterization of a multiphase flow are presented: (1) the introduction of void fraction information into the formulation presented by Paz (2011) and (2) the statistical analysis of the orifice plate pressure drop signal. Regarding the first item, quantitative comparison with experimental data suggested that the presented alternative is viable for production monitoring operations. The last study qualitatively revealed the influence of the liquid loading in the pressure drop fluctuation.

Lors de la rentrée atmosphérique nous sommes amenés à modéliser trois phénomènes physiques différents. Tout d'abord, l'écoulement autour du véhicule entrant dans l'atmosphère est hypersonique,il est caractérisé par la présence d'un choc fort et provoque un fort échauffement du véhicule. Nous modélisons l'écoulement par les équations de Navier-Stokes compressibles et l'échauffement du véhicule au moyen de la thermique anisotrope. De plus le véhicule est protégé par un bouclier thermique siège de réactions chimiques que l'on nomme communément ablation.Dans le premier chapitre de cette thèse nous présentons le schéma numérique de diffusion CCLAD (Cell-Centered LAgrangian Diffusion) que nous utilisons pour résoudre la thermique anisotrope. Nous présentons l'extension en trois dimensions de ce schéma ainsi que sa parallélisation.Nous continuons le manuscrit en abordant l'extension de ce schéma à une équation de diffusion tensorielle. Cette équation est obtenue en supprimant les termes convectifs de l'équation de quantité de mouvement des équations de Navier-Stokes. Nous verrons qu'une pénalisation doit être introduite afin de pouvoir inverser la loi constitutive et ainsi appliquer la méthodologie CCLAD. Nous présentons les propriétés numériques du schéma ainsi obtenu et effectuons des validations numériques.Dans le dernier chapitre, nous présentons un schéma numérique de type Volumes Finis permettant de résoudre les équations de Navier-Stokes sur des maillages non-structurés obtenu en réutilisant les deux schémas de diffusion présentés précédemment. / When studying the problem of atmospheric reentry we need to model three different physical phenomenons. First, the ow around the atmospheric reentry vehicle is hypersonic, it is characterized by the presence of a strong shock which leads to a rapid heating of the vehicle. We model the ow using the compressible Navier-Stokes equations and the heating of the vehicle is modeled with the anisotropic heat transfer equation. Furthermore the vehicle is protected by an heat shield, where thermochemical reactions, commonly named ablation, occurs.In the first chapter of this thesis we introduce the numerical diffusion scheme CCLAD (Cell-Centered LAgrangian Diffusion) that we use to solve the anisotropic heat diffusion. We develop its non trivial extension to three-dimensional geometries and present its parallelization. We continue this thesis by the presentation of the extension of this scheme to tensorial diffusion. This equation is obtained by suppressing the convective terms of the momentum equation of the Navier-Stokes equations. We show that we need to introduce a penalization term in order to be able to invert the constitutive law. The invertibility of the constitutive law allows us to apply the CCLAD methodology to this equation straightforwardly. We present the numerical properties of this scheme and show numerical validations.In the last chapter, we present a Finite Volume scheme on unstructured grids that solves the compressible Navier-Stokes equations. This numerical scheme is mainly obtained by gathering the contributions of the two diffusion schemes we developed in the previous chapters.

Dans ce travail, on s'intéresse au développement d'une méthode hybride qui couple spatialement les schémas d'intégration temporelle explicite et implicite. Afin de répondre aux contraintes du solveur industriel FLUSEPA, les schémas explicite Heun et implicite Crank-Nicolson ont été hybridés via un paramètre de transition : l'approche mise en place est appelée schéma AION. Cette dernière est étudiée en détails avec une attention particulière sur son comportement spectral et sa capacité à maintenir l'ordre de précision. On montre que le traitement hybride a d'intéressants comportements dissipatif et dispersif tout en empêchant la réflexion d'ondes parasites et en maintenant la précision attendue. De plus, l'approche hybride est validée sur plusieurs cas académiques à la fois pour les flux convectifs et pour les flux diffusifs. Et comme espéré, la méthode est plus intéressante en terme de temps de calcul que les méthodes standards d'intégration temporelle. Pour l'extension de cette approche à la méthode temporelle adaptative présente dans FLUSEPA, il a été nécessaire d'améliorer le traitement qui permet à la méthode d’être conservative tout en obtenant des propriétés spectrales acceptables. Finalement l'approche hybride a été aussi étendue pour la modélisation RANS/LES de la turbulence avec des temps de calcul intéressants tout en capturant la physique de l'écoulement / This work deals with the design of a hybrid time integrator that couples spatially explicit and implicit time integrators. In order to cope with the industrial solver of Ariane Group called FLUSEPA, the explicit scheme of Heun and the implicit scheme of Crank-Nicolson are hybridized using the transition parameter : the whole technique is called AION time integration. The latter is studied into details with special focus on spectral behaviour and on its ability to keep the accuracy. It is shown that the hybrid technique has interesting dissipation and dispersion properties while maintaining precision and avoiding spurious waves. Moreover, this hybrid approach is validated on several academic test cases for both convective and diffusive fluxes. And as expected the method is more interesting in term of computational time than standard time integrators. For the extension of this hybrid approach to the temporal adaptive method implemented in FLUSEPA, it was necessary to improve some treatments in order to maintain conservation and acceptable spectral properties. Finally the hybrid time integration was also applied to a RANS/LES turbulent test case with interesting computational time while capturing the flow physics.

La thèse constitue une juxtaposition de plusieurs contributions originales à lélaboration et lanalyse de modèles numériques capables de décrire une vaste gamme découlements à surface libre ainsi que les phénomènes de transport associés. Deux axes principaux sous-tendent les recherches entreprises. Il sagit, dune part, dune contribution à létude de modèles visant à reproduire adéquatement les interactions du fluide avec divers constituants transportés, tels que de lair entraîné, un polluant ou des sédiments. Cette phase du travail inclut également la caractérisation et la prédiction du comportement de lécoulement en présence dune topographie mobile ou érodable, y compris dans le cas dun barrage en remblai subissant une surverse. Dautre part, partant du constat quune modélisation fidèle des processus de transport, notamment hydrosédimentaires, passe inévitablement par un raffinement du calcul hydrodynamique proprement dit, une partie des travaux effectués dans le cadre de cette thèse est spécifiquement orientée vers un enrichissement de la connaissance des champs hydrodynamiques au sein du modèle.

Ideal magnetohydrodynamics (MHD) simulations are known to have problems in satisfying the solenoidal constraint (i.e. the divergence of magnetic field should be equal to zero, $ ablacdotvec{B} = 0$). The simulations become unstable unless specific measures have been taken. In this thesis, a solenoidal constraint satisfying technique that allows discrete satisfaction of the solenoidal constraint up to the machine accuracy is presented and validated with a variety of test cases. Due to its inspiration from Chorin's artificial compressibility method developed for incompressible CFD applications, the technique was named as extit{artificial compressibility analogy (ACA)} approach. It is demonstrated that ACA is a purely hyperbolic, stable and consistent technique, which is moreover easy to implement. Unlike some other techniques, it does not pose any problems of the sort that $ ablacdotvec{B}$ errors accumulate in the vicinity of the stagnant regions of flow. With these crucial properties, ACA is thought to be a remedy to the drawbacks of the most commonly used solenoidal constraint satisfying techniques in the literature namely: Incorrect shock capturing and poor performance of the convective stabilization mechanism in regions of stagnant flow for Powell's source term method; exceedingly complex implementation for constrained transport technique due to the staggered grid representation; computationally expensive nature due to the necessity of a Poisson solver combined with hyperbolic/elliptic numerical methods for classical projection schemes. In the first chapter of the thesis, general background knowledge is given about plasmas, MHD and its history, a certain class of upwind finite volume methods, namely Riemann solvers, and their applications in MHD, the definition, constituents, formation mechanisms and effects of space weather and some of the space missions that are and will be performed in its prediction. Secondly, detailed analysis of the compressible ideal MHD equations is given in the form of the derivation of the equations, their dimensionless numbers which will be of use to specify the flows in the following chapters, and finally, the presentation of the MHD waves and discontinuities, which indicates the complexity of the system of ideal MHD equations and therefore their further numerical analysis. The next discussion is about the main subject of the thesis, namely the solenoidal constraint satisfying techniques. First of all, the definition and significance of the solenoidal constraint is given. Afterwards, the most common solenoidal constraint satisfying techniques in the literature are reviewed along with their abovementioned drawbacks. Moreover, particular emphasis is given to the Powell's source term approach which was also implemented in the upwind finite volume MHD solver developed. In addition, the hyperbolic divergence cleaning technique is presented in detail together with the resemblance and differences between it and ACA. Some other solenoidal constraint satisfying techniques are briefly mentioned at this stage. After these, ACA is presented in the following way: The point of inspiration, which is the analogy made with Chorin's artificial compressibility method developed for incompressible CFD, the introduction of the modified system of ideal MHD equations due to ACA, the derivation of the wave equation governing the propagation of $ ablacdotvec{B}$ errors and the analytical consistency proof. Having finished the core discussion of the thesis, the solver developed and its constituents are given in the fourth chapter. Furthermore, a brief overview of the platform into which this solver was implemented, namely COOLFluiD, is also given at this point. Afterwards, a thorough numerical verification of the ACA approach has been made on an increasingly complex suite of test cases. The results obtained with ACA and Powell's source term implementations are given in order to numerically analyse and verify ACA and compare the two methods and validate them with the results from literature. The sixth chapter is devoted to further validation of ACA performed with a variety of more advanced space weather-related simulations. In this chapter, also the $vec{B}_{ extrm{0}} + vec{B}_{ extrm{1}}$ splitting technique used to treat planetary magnetosphere is presented along with its application to ACA and Powell's source term approaches. This technique is utilized in obtaining the solar wind/Earth's magnetosphere interaction results and is based on suppressing the direct inclusion of the Earth's magnetic field, which is a dipole field, in the solution variables. In this way, problems are avoided with the energy equation that could arise from the drastic change of the ratio of the dipole field and the variable field computed by the solver (i.e. $frac{lvertvec{B}_{ extrm{0}}lvert}{lvertvec{B}_{ extrm{1}}lvert}$) in the computational domain. Finally, conclusions and future perspectives related to the material presented in the thesis are put forward.

Large-eddy simulation (LES) and unsteady Reynolds-averaged Navier-Stokes (URANS) calculations have been performed to investigate the effects of different mathematical models for scalar variance and its dissipation rate as applied to both a non-reacting bluff-body turbulent flow and an extension to a reacting case. In the conserved scalar formalism, the mean value of a thermo-chemical variable is obtained through the PDF-weighted integration of the local description over the conserved scalar, the mixture fraction. The scalar variance, one of the key parameters for the determination of a presumed β-function PDF, is obtained by solving its own transport equation with the unclosed scalar dissipation rate modelled using either an algebraic expression or a transport equation. The proposed approach is first applied to URANS and then extended to LES. Velocity, length and time scales associated with the URANS modelling are determined using the standard two-equation k-ε transport model. In contrast, all three scales required by the LES modelling are based on the Smagorinsky subgrid scale (SGS) algebraic model. The present study proposes a new algebraic and a new transport LES model for the scalar dissipation rate required by the transport equation for scalar variance, with a time scale consistent with the Smagorinsky SGS model.

A three dimensional, Navier-Stokes finite volume flow solver which uses Roe&rsquo / s upwind flux differencing scheme for spatial and Runge-Kutta explicit multi-stage time stepping scheme and implicit Lower-Upper Symmetric Gauss Seidel (LU-SGS) iteration scheme for temporal discretization on unstructured and hybrid meshes is developed for steady rocket internal viscous flow applications. The spatial accuracy of the solver can be selected as first or second order. Second order accuracy is achieved by piecewise linear reconstruction. Gradients of flow variables required for piecewise linear reconstruction are calculated with both Green-Gauss and Least-Squares approaches. The solver developed is first verified against the three-dimensional viscous laminar flow over flat plate. Then the implicit time stepping algorithms are compared against two rocket motor internal flow problems. Although the solver is intended for internal flows, a test case involving flow over an airfoil is also given. As the last test case, supersonic vortex flow between concentric circular arcs is selected.

This work deals with discretizing viscous fluxes in the context of unstructured data based finite volume and meshless solvers, two competing methodologies for simulating viscous flows past complex industrial geometries. The two important requirements of a viscous discretization procedure are consistency and positivity. While consistency is a fundamental requirement, positivity is linked to the robustness of the solution methodology. The following advancements are made through this work within the finite volume and meshless frameworks. Finite Volume Method: Several viscous discretization procedures available in the literature are reviewed for: 1. ability to handle general grid elements 2. efficiency, particularly for 3D computations 3. consistency 4. positivity as applied to a model equation 5. global error behavior as applied to a model equation. While some of the popular procedures result in inconsistent formulation, the consistent procedures are observed to be computationally expensive and also have problems associated with robustness. From a systematic global error study, we have observed that even a formally inconsistent scheme exhibits consistency in terms of global error i.e., the global error decreases with grid refinement. This observation is important and also encouraging from the view point of devising a suitable discretization scheme for viscous fluxes. This study suggests that, one can relax the consistency requirement in order to gain in terms of robustness and computational cost, two key ingredients for any industrial flow solver. Some of the procedures are analysed for positivity as applied to a Laplacian and it is found that the two requirements of a viscous discretization procedure, consistency(accuracy) and positivity are essentially conflicting. Based on the review, four representative schemes are selected and used in HIFUN-2D(High resolution Flow Solver on UNstructured Meshes), an unstructured data based cell center finite volume flow solver, to simulate standard laminar and turbulent flow test cases. From the analysis, we can advocate the use of Green Gauss theorem based diamond path procedure which can render high level of robustness to the flow solver for industrial computations. Meshless Method: An Upwind-Least Squares Finite Difference(LSFD-U) meshless solver is developed for simulating viscous flows. Different viscous discretization procedures are proposed and analysed for positivity and the procedure which is found to be more positive is employed. Obtaining suitable point distribution, particularly for viscous flow computations happens to be one of the important components for the success of the meshless solvers. In principle, the meshless solvers can operate on any point distribution obtained using structured, unstructured and Cartesian meshes. But, the Cartesian meshing happens to be the most natural candidate for obtaining the point distribution. Therefore, the performance of LSFD-U for simulating viscous flows using point distribution obtained from Cartesian like grids is evaluated. While we have successfully computed laminar viscous flows, there are difficulties in terms of solving turbulent flows. In this context, we have evolved a strategy to generate suitable point distribution for simulating turbulent flows using meshless solver. The strategy involves a hybrid Cartesian point distribution wherein the region of boundary layer is filled with high aspect ratio body-fitted structured mesh and the potential flow region with unit aspect ratio Cartesian mesh. The main advantage of our solver is in terms of handling the structured and Cartesian grid interface. The interface algorithm is considerably simplified compared to the hybrid Cartesian mesh based finite volume methodology by exploiting the advantage accrue out of the use of meshless solver. Cheap, simple and robust discretization procedures are evolved for both inviscid and viscous fluxes, exploiting the basic features exhibited by the hybrid point distribution. These procedures are also subjected to positivity analysis and a systematic global error study. It should be remarked that the viscous discretization procedure employed in structured grid block is positive and in fact, this feature imparts the required robustness to the solver for computing turbulent flows. We have demonstrated the capability of the meshless solver LSFDU to solve turbulent flow past complex aerodynamic configurations by solving flow past a multi element airfoil configuration. In our view, the success shown by this work in computing turbulent flows can be considered as a landmark development in the area of meshless solvers and has great potential in industrial applications.