Flerfasestrømning er av betydning i en lang rekke anvendelser, blant annet i olje- og gassindustrien, i den kjemiske og i prosessindustrien, inkludert i varmepumpende systemer, samt i sikkerhetsanalyse av kjernekraftverk. Denne avhandlingen analyserer modeller for tofasestrømning, og metoder for numerisk løsning av disse modellene. Den er derfor ett bidrag til å utvikle pålitelige ingeniørverktøy for flerfase-anvendelser. Slike verktøy trengs og forventes av ingeniører i industrien. Den tilnærmede Riemann-løseren framsatt av Roe har blitt studert. Roe-skjema for tre ulike modeller for tofasestrømning har blitt implementert i rammen av en standard numerisk algoritme for løsning av hyperbolske bevaringslover. Disse skjemaene har blitt analysert ved hjelp av referanse-regnestykker fra litteraturen, og ved sammenlikning med hverandre. Et Roe-skjema for den fire-liknings ettrykks tofluid-modellen har blitt implementert, og det har blitt vist at en andreordens utvidelse basert på bølge-dekomponering og fluksdifferanse-splitting virker godt, og gir forbedrede resultater sammenliknet med det førsteordens skjemaet. Det har blitt foreslått et Roe-skjema for en fem-liknings totrykks tofluid-modell med trykkrelaksering. Bruken av analoge numeriske metoder for fire-liknings- og fem-liknings-modellene gjorde det mulig med en direkte sammenlikning av en metode med og uten trykkrelaksering. Numeriske eksperiment demonstrerte at de to framgangsmåtene konvergerte til samme resultat, men at den fem-liknings trykkrelakserings-metoden var betydelig mer dissipativ, særlig for kontakt-diskontinuiteter. Videre viste beregninger at selv om fem-liknings-modellen har reelle egenverdier, så produserte den oscillasjoner for tilfeller der fire-liknings-modellen hadde komplekse egenverdier. Et Roe-skjema har blitt konstruert for driftfluks-modellen med generelle lukningslover. Roe-matrisen er helt analytisk for det tilfellet at man kan anvende Zuber-Findlay-slipp-loven som beskriver boblestrømning. Dermed er dette Roe-skjemaet mer effektivt enn tidligere fullt numeriske Roe-skjema for driftfluks-modellen. En isentropisk diskret-nivå-flerfasemodell har blitt presentert. En diskusjon av hvordan man kan ta hensyn til ulike interfase-trykkmodeller har blitt gitt. Med de passende modellene for interfase-trykk og -fart, var samsvaret svært godt mellom diskret-nivå-modellen og det fem-liknings Roe-skjemaet. Multi-steg- (MUSTA) metoden har som siktemål å komme nær oppstrøms-metodene i nøyaktighet, samtidig som den bevarer enkelheten til sentrerte skjema. Her har metoden blitt brukt på driftfluks-modellen. Når antallet steg økes, nærmer resultatene fra MUSTA-metoden seg det man får med Roe-metoden. De gode resultatene til MUSTA-metoden er avhengige av at man bruker et stort nok lokalt grid. Derfor er hovedfordelen med MUSTA-metoden at den er enkel, snarere enn at man sparer regnetid. En karakteristikk-basert metode for å spesifisere grensebetinglser for flerfase-modeller har blitt testet, og funnet å virke godt for transiente problem. / This thesis analyses models for two-phase flows and methods for the numerical resolution of these models. It is therefore one contribution to the development of reliable design tools for multiphase applications. Such tools are needed and expected by engineers in a range of fields, including in the oil and gas industry. The approximate Riemann solver of Roe has been studied. Roe schemes for three different two-phase flow models have been implemented in the framework of a standard numerical algorithm for the solution of hyperbolic conservation laws. The schemes have been analysed by calculation of benchmark tests from the literature, and by comparison with each other. A Roe scheme for the four-equation one-pressure two-fluid model has been implemented, and a second-order extension based on wave decomposition and flux-difference splitting was shown to work well and to give improved results compared to the first-order scheme. The convergence properties of the scheme were tested on smooth and discontinuous solutions. A Roe scheme has been proposed for a five-equation two-pressure two-fluid model with pressure relaxation. The use of analogous numerical methods for the five-equation and four-equation models allowed for a direct comparison of a method with and without pressure relaxation. Numerical experiments demonstrated that the two approaches converged to the same results, but that the five-equation pressure-relaxation method was significantly more dissipative, particularly for contact discontinuities. Furthermore, even though the five-equation model with instantaneous pressure relaxation has real eigenvalues, the calculations showed that it produced oscillations for cases where the four-equation model had complex eigenvalues. A Roe scheme has been constructed for the drift-flux model with general closure laws. For the case of the Zuber-Findlay slip law describing bubbly flows, the Roe matrix is completely analytical. Hence the present Roe scheme is more efficient than previous fully numerical Roe schemes for the drift-flux model. An isentropic discrete-equation multiphase model has been presented. The incorporation of different interfacial-pressure models was discussed, and examples were given. With the adequate models for the interfacial pressure and velocity, the agreement was very good between the discrete-equation model and the five-equation Roe scheme. The flux-limiter centred (FLIC) scheme was tested for the four-equation two-fluid model. Only the first-order version (FORCE) of the scheme was found to work well, but it was rather diffusive. The purpose of the multi-stage (MUSTA) method is to come close to the accuracy of upwind schemes while retaining the simplicity of centred schemes. Here it has been applied to the drift-flux model. As the number of stages was increased, the results of the MUSTA scheme approached those of the Roe method. The good results of the MUSTA scheme were dependent on the use of a large-enough local grid. Hence, the main advantage of the MUSTA scheme is its simplicity. A multiphase characteristic-based boundary-condition method has been tested, and it was shown to be workable for transient problems.

Es wurde ein neues Zweifluidmodell entwickelt, um Nichtgleichgewichtseffekte in kritischen und nahkritischen Ein-komponenten-Zweiphasenströmungen von anfänglich unter-kühlten oder siedenden Fluiden durch Rohre und Düsen zu untersuchen. Das vorgeschlagene Sechs-Gleichungsmodell besteht aus den phasenbezogenen Erhaltungsgleichungen für Masse und Impuls, der Bilanzgleichung für die thermische Energie der flüssigen Phase sowie einer zusätzlichen Transport-gleichung für die volumetrische Blasenanzahl. Zur Lösung des Systems aus partiellen Differentialgleichungen wird ein semi-implizites Finite-Differenzen-Zeitschrittverfahren angewendet. Die Schließung des Gleichungssystems wird durch Einbindung thermodynamischer Beziehungen und konstitutiver Gleichungen, die den strömungsformabhängigen Impuls-, Wärme- und Stofftransport beschreiben, erreicht. Für Strömungssysteme mit spontaner Entspannungsverdampfung aus dem rein flüssigen Zustand (Flashing) werden verschiedene Keimbildungsmodelle eingesetzt, die den Anfangszustand der verzögerten Dampfbildung beschreiben. Auf diese Weise werden thermodynamische Nichtgleichgewichtszustände als Folge von Zuständen mit für die Aktivierung von Keimstellen benötigtem Energieüberschuß, eingeschränkt vorhandener Phasengrenzfläche sowie begrenzter Wärmeübertragung zwischen den Phasen betrachtet. Abweichungen vom fluid-dynamischen Gleichgewicht (Phasenschlupf) ergeben sich aufgrund unterschiedlicher Trägheitseigenschaften und verschieden stark ausgeprägter mechanischer Kopplung zwischen den Phasen. Die mit diesem Modell erhaltenen numerischen Ergebnisse stimmen gut mit experimentellen Werten für Zweiphasen-strömungen mit unterschiedlichen Eintrittsbedingungen und Kanalgeometrien überein. / A new two-fluid flow model has been developed in order to examine non-equilibrium effects in critical and near-critical one-component two-phase flows of initially subcooled or saturated fluids through pipes and nozzles. The six-equation model proposed consists of the phasic conservation equations of mass and momentum, the liquid thermal energy, and of an additional transport equation for the bubble number density. To solve for the unknowns of the system of partial differential equations, a semi-implicit finite difference time-marching method is utilized. The closure of the set of equations is accomplished by thermodynamic relationships and additional constitutive equations describing momentum transport, interphase heat, and mass transfer which account for different flow regimes. For fluid flow systems undergoing a sudden change of phase from the pure liquid state (flashing), distinct nucleation models are included to describe the initial state of delayed vapor generation. In this way thermal non-equilibrium states are considered to be the consequence of excessive energy states required to activate nucleation sites, of restricted interfacial area and limited heat transfer between the phases. Deviation from fluid-dynamic equilibrium (phasic slip) results from different inertial properties and from distinct strength of mechanical coupling between the phases. The numerical results obtained with this model agree quite well with experimental data for two-phase flows with various inlet conditions and channel geometries.

No description available.

[pt] O atual esgotamento de campos de petróleo tradicionais tem aumentado a demanda pela produção e transporte óleos não convencionais, que podem possuir uma alta viscosidade dinâmica. Neste contexto, o estudo do escoamento simultâneo de gás e óleos viscosos em tubulações é de grande importância para a indústria de Óleo e Gás. Simulações numéricas uni-dimensionais desempenham um papel essencial nestes estudos, especialmente aquelas baseadas no Modelos de Dois-Fluidos 1D, cuja solução numérica em malhas refinadas consiste na Metodologia de Captura de Regimes. O propósito deste trabalho é utilizar esta abordagem para reproduzir dados experimentais de escoamentos óleo viscoso-gás em golfadas e estratificado ondulado em um duto horizontal em escala laboratorial. Para aprimorar os resultados desta metodologia, dados experimentais foram usados conjuntamente com um procedimento de otimização e uma versão simplificada do Modelos de Dois-Fluidos 1D para criar duas novas expressões para o fator de atrito interfacial, as quais mostraram maior eficiência que correlações padrão da literatura. O efeito da introdução da pressão dinâmica, difusão axial de quantidade de movimento e tensão interfacial dinâmica no Modelo de Dois-Fluidos 1D foi analisado. Resultados de gradiente de pressão e de fração volumétrica de líquido (histogramas, valores médios e perfis transientes) foram comparados com dados experimentais. Observou-se, com o auxílio de análises de boa-colocação, que a pressão dinâmica e as novas expressões para o fator de atrito interfacial fornecem resultados satisfatórios. / [en] The current depletion of traditional oil fields is increasing the demand for the production and transport of unconventional oils, which might possess a high dynamic viscosity. In this context, the study of the simultaneous flow of gas and viscous oils in pipelines is of paramount importance for the Oil and Gas industry. One-dimensional numerical simulations play a key role in such studies, especially the ones based on the 1D Two-Fluid Model, whose numerical solution in fine meshes consists in the Regime Capturing Methodology. The purpose of this work is to use this approach for reproducing the experimental data of isothermal slug and stratified wavy viscous oil-gas flows in a horizontal laboratory-scale pipe. For improving the results of the methodology, experimental data were used together with an optimization procedure and a simplified version of the 1D Two-Fluid Model for successfully creating two new expressions for the interfacial friction factor, which showed better efficiency than standard literature correlations. The effect of introducing a dynamic pressure, axial momentum diffusion and dynamic interfacial shear in the 1D Two-Fluid Model was examined. Results of pressure gradient and liquid holdup (histograms, mean values and transient profiles) were compared against experimental data. It was seen, with the aid of well-posedness analyses, that the dynamic pressure and the new expressions for the interfacial shear stress provided satisfactory results.

Gas-solid fluidized bed reactors are widely applied in chemical and energy industries, and their design and scale-up are virtually empirical, extremely expensive and time consuming. This scenario has motivated the development of alternative theoretical tools, and two-fluid modeling, where gas and particulate are both treated as interpenetrating continuum phases, has appeared as a most promising approach. Owing to the large domains to be resolved in real-scale fluidized bed reactors, only filtered modeling approaches are feasible, and closure models become necessary to recover sub-grid effects that are filtered by the very coarse numerical grids that are imposed owing to computational limitations. Those closure models, which in hydrodynamic formulations account for filtered interphase momentum exchanges and filtered and residual stresses in the phases, can be derived from results of highly resolved simulations (HRS) performed over small size domains under refined numerical grids. One widely practiced approach consists of applying two-fluid modeling under micro-scale defined closures, generally known as microscopic two-fluid modeling. This approach includes microscopic closures for solid phase stresses derived from the kinetic theory of granular flows (KTGF), which accounts for kinetic-collisional effects only, and is adequate to dilute flows. Otherwise, the conventional KTGF does not account for interparticle friction effects, and its application to dense flow conditions is quite questionable. In this work a literature available modified version of KTGF is applied which also accounts for interparticle friction, and highly resolved simulations are performed for dense flow conditions in order to evaluate the effects of friction over relevant filtered parameters (namely effective drag coefficient, filtered and residual stresses). Ranges of domain average solid volume fractions and gas Reynolds numbers are considered (macro-scale conditions) embracing dense gas-solid fluidized flows from suspensions up to pneumatic transport. The MFIX open source code is used in all the simulations, which are performed over 2D periodical domains for a unique monodisperse particulate. The HRS results (i.e. meso-scale flow fields) are filtered over regions compatible with grid sizes in large scale simulations, and the relevant filtered parameters of concern are derived and classified by ranges of other filtered parameters taken as independent variables (filtered solid volume fraction, filtered slip velocity, and filtered kinetic energy of solid velocity fluctuations, which are referred to as markers). Results show that the relevant filtered parameters of concern are well correlated to all of those filtered markers, and also to all of the imposed macro-scale conditions. Otherwise, interparticle friction showed no significant effects over any filtered parameter. It is recognized that this issue clearly requires further investigation notably regarding the suitability of the markers that were assumed for classifying the filtered results. The current work is intended as a contribution for future developments of more accurate closure models for large scale simulations of gas-solid fluidized flows. / Reatores de leito fluidizado de escoamento gás-sólido são largamente utilizados nas indústrias química e de energia, e o seu projeto e escalonamento são virtualmente empíricos, extremamente caros e demorados. Este cenário tem motivado o desenvolvimento de ferramentas teóricas alternativas, e a modelagem de dois fluidos, onde gás e particulado são ambos tratados com fases contínuas interpenetrantes, tem surgido como uma aproximação muito promissora. Devido aos grandes domínios a serem resolvidos em reatores de leito fluidizado de escala real, apenas aproximações de modelagem filtradas são viáveis, e modelos de fechamento tornam-se necessários para recuperar efeitos sub-malha que são filtrados pelas malhas numéricas grosseiras que são impostas devido as limitações computacionais. Estes modelos de fechamento, que em formulações hidrodinâmicas respondem principalmente por trocas de momentum filtradas entre fases e tensões filtradas e residuais nas fases, podem ser obtidos de resultados de simulações altamente resolvidas (SAR) realizadas em domínios de dimensões reduzidas sob malhas numéricas refinadas. Uma aproximação largamente praticada consiste na aplicação de modelagem de dois fluidos sob fechamentos definidos na micro-escala, genericamente conhecida como modelagem microscópica de dois fluidos. Esta aproximação inclui fechamentos microscópicos para tensões da fase sólida obtidos da teoria cinética dos escoamentos granulares (TCEG), que considera apenas efeitos cinéticos-colisionais, e é adequada para escoamentos diluídos. Por outro lado, a TCEG convencional não leva em conta efeitos de fricção interpartículas, e sua aplicação para condições densas de escoamento é bastante questionável. Neste trabalho aplica-se uma versão modificada da TCEG disponível na literatura que também leva em conta fricção interpartículas, e simulações altamente resolvidas são realizadas para condições de escoamentos densos visando avaliar os efeitos da fricção sobre os parâmetros filtrados relevantes (coeficiente de arrasto efetivo, tensões filtradas e residuais). Considera-se faixas de frações volumétricas de sólido e números de Reynolds do gás médios no domínio (condições de macro-escala) abrangendo escoamentos gás-sólido fluidizados densos desde suspensões até transporte pneumático. O código aberto MFIX é utilizado em todas as simulações, que foram executadas sobre domínios periódicos 2D para um único particulado monodisperso. Os resultados das SAR (i.e., campos de escoamento de meso-escala) foram filtrados sobre regiões compatíveis com tamanhos de malha praticados em simulações de grandes escalas, e os parâmetros filtrados relevantes de interesse são calculados e classificados por faixas de outros parâmetros filtrados tomados como variáveis independentes (fração volumétrica de sólido filtrada, velocidade de deslizamento filtrada, e energia cinética das flutuações de velocidade da fase sólida filtrada, que são referidos como marcadores). Os resultados mostram que os parâmetros filtrados relevantes de interesse são bem correlacionados com todos os marcadores, e também com todas as condições de macro-escala impostas. Por outro lado, a fricção interpartículas não mostrou efeitos significativos sobre qualquer parâmetro filtrado. Reconhece-se que este aspecto claramente requer investigações adicionais, notadamente com respeito à adequação dos marcadores que foram considerados para classificação dos resultados filtrados. O trabalho corrente é posto como uma contribuição para o desenvolvimento futuro de modelos de fechamento mais acurados para simulações de grandes escalas de escoamentos gás-sólido fluidizados.

A presente pesquisa tem como objetivo realizar um estudo teórico e desenvolver simulações computacionais envolvendo a dinâmica de sistemas gás-sólido. O foco principal do trabalho é a modelagem do particulado através da análise das forças de contato entre partículas de materiais granulares utilizando modelos contínuos baseados na mecânica dos solos e na teoria cinética dos escoamentos granulares (sistemas grandes com muitas partículas, formulação Euleriana - Volumes Finitos) e modelos discretos baseados nas características físicas dos materiais (sistemas intermediários e número limitado de partículas, formulação Lagrangeana - Método dos Elementos Discretos). Investigam-se os modelos existentes na literatura com intuito de melhorar os modelos contínuos e discretos baseados na interação entre as partículas que caracterizam a dinâmica do particulado em sistemas gás-sólido. Propõe-se uma nova abordagem para a determinação do coeficiente de rigidez da mola baseada em uma equivalência entre os modelos lineares e não-lineares. Utiliza-se o código fonte MFIX para realizar simulações computacionais da dinâmica de sistemas gás-sólido, analisando o processo de fluidização, mistura e segregação de partículas, influência das correlações de arrasto, e análise das forças de contato entre as partículas através do novo método para a determinação do coeficiente de rigidez da mola . Os resultados obtidos são comparados com dados numéricos e experimentais da literatura. / The purpose of the present study is to perform a theoretical study and develop numerical simulations involving dynamic in gas-solid systems. The focus of the work is the modeling of particulate matter using continuous models based on soil mechanics and the kinetic theory of granular flows (large systems with many particles, Eulerian formulation - Finite Volume) and discrete models based on physical characteristics of the particles (intermediate systems and limited number of particles, Lagrangian formulation - Discrete Element Method). It is proposed a new approach to determine the normal spring stiffness coefficient of the linear model through the numerical solution for the overlap between particles in non-linear models. The linear spring stiffness is determined using an equivalence between the linear and the non-linear models. It is used the MFIX computational code to perform numerical simulations of the dynamics of gas-solid systems. It is analyzed the processes of fluidization, mixing and particle segregation and the influence of drag correlations. The proposed approach for normal spring stiffness coefficient is applied in the numerical simulations of two problems: single freely falling particle and bubbling fluidized bed. The results were compared with numerical and experimental data from literature.

Foram desenvolvidos estudos de modelagem e simulação numérica de escoamentos bifásicos gás-sólido na coluna ascendente de leitos fluidizados circulantes utilizando um modelo Euleriano de duas fases separadas. O sistema de equações diferenciais parciais conservativas governantes foi obtido através de um procedimento tradicional. Ambas as fases foram assumidas como meio contínuo. Aplicou-se o procedimento de médias estatísticas de Euler, enfatizando a obtenção dos modelos hidrodinâmicos A e B desenvolvidos no IIT/ANL. Realizou-se análise comparativa de correlações para transferência de quantidade de movimento na interface. Discutiu-se a formulação de condições de contorno apropriadas. As equações diferenciais parciais médias foram discretizadas em volumes de controle Eulerianos. As equações de continuidade foram resolvidas implicitamente. As equações de quantidade de movimento foram resolvidas através de um procedimento explícito-implícito. Foram desenvolvidas simulações numéricas para uma coluna ascendente típica de leitos fluidizados circulantes. Desenvolveu-se análise paramétrica da influência de vários aspectos físicos e matemáticos sobre o escoamento. Avaliou-se resultados de simulação através de metodologia de identificação e caracterização de estruturas coerentes. Estudou-se o efeito da função de arrasto na interface sobre os processos dinâmicos que caracterizam estas estruturas coerentes. Foram realizados estudos numéricos de turbulência a partir de resultados de simulação direta. Várias conclusões e recomendações para futuros trabalhos foram propostas com base nas análises realizadas. Foram apresentadas algumas considerações gerais relativas a aspectos críticos na modelagem e simulação com modelo das duas fases separadas. / Studies were carried out on modeling and numerical simulation of gas-solid two-phase flows in the riser of circulating fluidized beds using an Eulerian two-fluids model. The system of conservative partial differential governing equations was derived through a traditional procedure. Both phases were assumed as a continuum. The Euler averaging procedure was applied emphasizing the derivation of the so called hydrodynamic models A and B developed at IIT/ANL. A comparative analysis was performed among correlations for momentum transfer at the interface. The formulation of suitable boundary conditions was discussed. The average partial differential conservative equations were discretized on Eulerian control volumes. The continuity equations were solved implicitly. The momentum equations were solved through an explicit-implicit procedure. Numerical simulation was performed for a typical circulating fluidized bed riser. A parametric analysis was carried out regarding the influence on the flow of various physical and mathematical aspects. Results of simulation were evaluated through a methodology of identification and characterization of coherent structures. The effect of the interface drag function on dynamic features of those coherent structures was addressed. Numerical studies on turbulence were performed from results of direct simulation. Several conclusions and recommendations for future work were put forward on the basis of the performed analyses. Some general considerations were presented regarding critical features of modeling and simulation through Eulerian two-fluids models.

Cette thèse traite de la modélisation et de l'approximation numérique des écoulements liquide-gaz compressibles. La difficulté centrale est la modélisation et l'approximation de l'interface liquide-gaz. Le modèle bifluide est constitué d'un système de lois de conservation fermé par une loi d'état du mélange. La loi d'état conditionne les bonnes propriétés (hyperbolicité, existence d'une entropie de Lax) du système. Les schémas classiques de type Godunov conduisent à des imprécisions les rendant inutilisables en pratique. L'existence de solutions discontinues rend difficile la construction de schémas d'ordre élevé et nécessite des maillages très fins pour une précision acceptable. Il est indispensable de proposer des algorithmes performants pour les calculateurs parallèles les plus récents. Nous aborderons chacune de ces problématiques: construction d'une "bonne" loi de pression, construction de schémas numériques adaptés, programmation sur calculateur massivement multicoeur. / This thesis deals with the modeling and numerical approximation of compressible gas-liquid flows. The main difficulty lies in modeling and approximation of the liquid-gas interface. The two-fluid model is a system of conservation laws closed with a mixture pressure law. The law has to be chosen carefully, it conditions good properties of the system as hyperbolicity or existence of a Lax entropy. Classic conservative Godunov-type schemes lead to inaccuracies that make them unusable inpractice. The existence of discontinuous solutions makes it difficult to build high order schemes and requires very fine meshes to an acceptable accuracy. It is therefore essential to provide efficient algorithms for the High Performance Computing. In this thesis, we will partially treat each of these issues : construction of a "good" pressure law, building adapted numerical schemes, programming on GPU or GPU cluster.

<div>The current study is aimed at developing a well-posed and objective, i.e., frame invariant, Eulerian one-dimensional (1D) Two-Fluid Model (TFM) to predict flow regime transition from dispersed to clustered bubbly and slug flow for vertical adiabatic two-phase flows. Two-phase flows in general are characterized by local material wave or void fraction wave instabilities and flow regime transitions are one of the important consequences of these instabilities. The physical mechanism of wake entrainment for clustering of dispersed bubbles is proposed, leading to formation of bubble clusters and Taylor bubbles. The focus of the work is on simulation of the local interfacial structures for bubble clusters and Taylor bubbles, using a well-posed, unstable and non-linearly bounded 1D Shallow Water TFM.</div><div><br></div><div>The first part of the current study investigates the dynamic behavior of the well posed 1D mechanistic TFM obtained from the averaging approach of Ishii [1], due to wake entrainment instability. For this, a 1D Shallow Water TFM derived from the 1D mechanistic TFM is used, which retains the same dynamic behavior as that of the latter at short wavelengths and the required wake entrainment force is derived mechanistically. Three stability approaches are followed to study the dynamic behavior of the 1D Shallow Water TFM: characteristics, dispersion analysis, and nonlinear numerical simulations. An in-house code is used for the 1D numerical simulations of the growth of void fraction waves due to wake entrainment. The simulation results are validated with the experimental data of Cheng and Azzopardi [2] and Song et al. [3] To conclude the first part, the 1D results of the two-equation Shallow Water TFM are carried over to the complete four-equation TFM for quasi 1D simulations using the commercial CFD code of ANSYS Fluent.</div><div><br></div>As an alternative to the mechanistic approach, which is based on Newtonian mathematics, a variational approach based on Lagrangian and Hamiltonian mathematics is used in the second part of the thesis. While the mechanistic approach operates in terms of forces acting on the two-phase mixture, the variational approach operates in terms of energies of the two-phase system. To derive the equations of motion using the variational approach, the extended Hamilton principle of least action is applied to the Lagrangian density of the two-phase mixture. One of the appealing features<br>17<br>of this procedure is that the derived equations of motion are objective (Geurst [4]), in particular the added mass terms.<br>Thus, the second part of the current study focuses on deriving an objective, well-posed and unstable 1D TFM as well as developing a constitutive model for the wake entrainment effect using the variational method. Additional momentum transfer terms present in both the liquid phase and gas phase momentum equations, which render the variational TFM objective, are discussed. The variational method is then used to derive the 1D Shallow Water TFM using the fixed flux assumption. The conservative interfacial momentum transfer terms require formulation of the inertial coupling between the phases. Potential flow theory is first used to derive the inertial coupling coefficient for a single bubble and then for a pair of bubbles to consider interaction between the two bubbles. Then, a lumped parameter model is used to derive the inertial coupling coefficient for the wake entrainment effect. A local drag coefficient is obtained for the non-conservative interfacial drag force from the experimental data using kinematic approximation, i.e., force balance between drag and gravity. The linear and non-linear stability analyses are used to address the stability of the 1D variational Shallow Water TFM. The presence of appropriate short-wave physics makes the 1D Shallow Water TFM hyperbolic well-posed and kinematically unstable. Finally, numerical simulations are performed to demonstrate the development of void fraction waves due wake entrainment. The growth of void fraction waves is non-linearly bounded, i.e., Lyapunov stable. The simulation results are compared with the experimental data to validate the propagation properties of void fraction waves for bubble clusters and Taylor bubbles. This work illustrates the short-wave two-phase flow simulation capability of the TFM for the bubbly to slug flow regime transition.

Les travaux de thèse sont axés sur les méthodes numériques pour les écoulements diphasiques, compressibles, à faible vitesse, avec apparition soudaine de forts gradients de pression. La vitesse matérielle de chacune des phases étant très petite devant la célérité des ondes acoustiques, le régime d'écoulement est dit à faible nombre de Mach. Dans ce travail, la loi d'état de la phase considérée contient toujours une information mesurant sa plus ou moins grande compressibilité. Ainsi, la faible compressibilité de l'eau peut produire un régime d'écoulement où des sauts de pression importants apparaissent même si le nombre de Mach est très faible. La première partie de la thèse s'est focalisée sur un modèle diphasique dit homogène-équilibré. Les deux phases de l'écoulement ont alors la même vitesse, pression, température et même potentiel chimique. Un premier travail a été la construction de solveurs de Riemann approchés dits tout-nombre-de-Mach. En l'absence de transitoire rapide, ces solveurs basent leur contrainte de pas de temps sur la vitesse des ondes matérielles lentes et sont donc précis pour suivre ces dernières. En revanche, lorsqu'une onde de choc rapide traverse l'écoulement, ces solveurs s'adaptent automatiquement afin de la capturer. La seconde partie de la thèse s'est focalisée sur la prise en compte du couplage convection-source dans le cadre des modèles en approche bi-fluides avec effets de relaxation pression-vitesse. Dans ces modèles, les deux phases de l'écoulement possèdent leur propre jeu de variables. Dans ce travail, un schéma implicite à mailles décalées, basé sur l'influence des termes sources dans des problèmes de Riemann linéaires, a été proposé / The present work focuses on numerical methods for low-material velocity compressible two-phase flows with high pressure jumps. In this context, the material velocity of both phases is small compared with the celerity of the acoustic waves. The flow is said to be a low-Mach number flow. In this work, the equation of state of the considered phase always contains information relative to its compressibility. For example, the low-compressibility of liquid water may lead to fast transients in which high pressure jumps are produced even if the flow Mach number is low. The first part of this work has leaned on two-phase homogeneous-equilibrium models. Thus, both phases have the same velocity, pressure, temperature and the same chemical potential. The construction of what is called an all-Mach-number approximate Riemann solver has been conducted. When no fast transients come through the flow, the above solvers enable computations with CFL conditions based on low-material velocities. As a result, they remain accurate to follow slow material interfaces, or subsonic contact discontinuities. However, when fast shock waves propagate, these solvers automatically adapt in order to capture them. The second part of the thesis has been dedicated to the design of numerical methods enhancing the coupling between convection and relaxation for two-fluid models containing pressure-velocity relaxation effects. In such models, both phases have their own set of variables. A time-implicit staggered scheme, based on the influence of relaxation source terms on linear Riemann problems has been proposed.