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, Elementary mixed flow: a pipe filling

. Aureli, 143 6.2.1 Global setting and objectives, 2015.

.. .. Case, Mixed flow with air pocket entrapment: a U-Tube test

P. .. Conclusion,

, A Sloping pipes and wall friction

, B Estimation of the pressure jump for the pipe filling test case

, C Period of pressure waves oscillations for the pipe filling test case

. .. References,

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, En utilisant une approche 1D, de nombreux modèles ont été proposés dans la littérature. Ils sont essentiellement monophasiques avec pour objectif de décrire les régimes à surface libre, en charge et les transitions associées. En pratique, les interactions eau-air peuvent modifier significativement la dynamique de l'écoulement, particulièrement en présence de poches d'air piégées. Certaines approches monophasiques ont alors été étendues à la prise en compte de ces poches sur des configurations simplifiées mais la communauté scientifique souligne la nécessité de développer un modèle capable de rendre compte des interactions eau-air dans tous les régimes. Les contributions de cette thèse s, Les travaux menés au cours de cette thèse portent sur la modélisation des écoulements mixtes eau-air en conduite. Ces derniers présentent un enjeu industriel important puisqu'ils sont au coeur de nombreuses installations telles que les centrales de production d'énergie ou les réseaux urbains d'assainissement

P. Le-modèle, Il est composé de cinq équa-tions dont les cinq inconnues principales correspondent à la hauteur d'une des phases en plus de la vitesse et de la pression de chaque phase. La configuration bicouche est pertinente pour les écoulements mixtes en conduite puisqu'elle permet de décrire naturellement le régime stratifié, le régime en charge (ou sec) et les poches d'air piégées. On parle alors d'un modèle bicouche compressible qui appartient par nature à la classe des modèles bifluide bipression ini, dénommé CTL pour Compressible Two-Layer, est un modèle bifluide résultant de l'intégration des équations d'Euler isentropiques sur chaque phase dans une configuration bicouche

, Ce type de modèle nécessite des lois de fermeture pour les variables d'interface (vitesse, pression) et les termes sources. Dans notre contexte d'écoulements mixtes eau-air, des lois de fermetures originales sont proposées, notamment pour la pression d'interface. Elles s'appuient sur la contrainte hydrostatique imposée sur la phase eau et sur une caractérisation entropique du modèle comme initialement suggéré par Coquel

, En effet, il dégénère par construction vers une description monophasique compressible adaptée aux régimes en charge et sec. De plus, des propriétés mathématiques notables sont obtenues telles que l'hyperbolicité, l'unicité des relations de saut, la positivité des hauteurs et des densités. La formulation du modèle est également étendue pour traiter des conduites circulaires à section variable. La proposition résultante est en rupture par rapport aux modèles 1D d'écoulements mixtes disponibles dans la littéra-ture. Au-delà de l'approche diphasique qui permet de rendre compte des interactions entre les phases, l'originalité tient dans la modélisation compressible de la phase eau à surface libre, Le modèle ainsi fermé répond aux spécificités d'un écoulement mixte, à savoir l'importance des effets gravitaires à surface libre et l'importance des effets acoustiques en charge

, La simulation de ce modèle bicouche compressible soulève de nombreux challenges pour les applications visées

, Les difficultés additionnelles relatives aux écoulements mixtes correspondent à la coexistence d'une dynamique lente à surface libre avec une dynamique rapide en charge, et à la gestion de phases évanescentes inhérentes aux régimes en charge et sec. Afin de répondre à ces problématiques, on s'est intéressé au développement d'un schéma implicite-explicite (IMEX) avec splitting d'opérateur. Une méthode performante a ainsi été proposée. Elle repose sur une gestion explicite de la dynamique gravitaire et une gestion implicite de la dynamique acoustique via un splitting approprié. Une méthode originale de relaxation a notamment été développée pour la partie implicite en proposant une stabilisation adaptée au régime d'écoulement. Enfin, Le développement de méthodes numériques pour les modèles bifluide est intrinsèquement source de difficultés en raison de la structure complexe du système d'ondes sous-jacent et des termes sources de relaxation en interaction forte avec la partie convective

, Une configuration canonique de remplissage de conduite est d'abord étudiée. Elle permet de démontrer la capacité du modèle et du schéma associé à gérer une configuration impliquant les spécificités monophasiques d'un écoulement mixte. De plus, une comparaison avec un modèle monophasique référence de la littérature, i.e. le modèle PFS [3], est proposée. Un comportement plus robuste dans les transitions entre régimes est alors obtenu tout en garantissant un temps de calcul raisonnable malgré la complexité accrue du système à résoudre, Une démarche de vérification et de validation de l'approche générale a été élaborée. Le schéma proposé est stable et converge vers les bonnes solutions de choc

, La qualité des résultats obtenus sur ces deux cas tests permet de placer le modèle CTL à l'état de l'art dans la modélisation 1D des écoulements mixtes monophasiques. Enfin, une configuration répondant à l'influence de poches d'air piégées est considérée. En particulier, une solution de référence est construite analytiquement. Le très bon accord avec les résultats numériques obtenus en utilisant le modèle CTL illustre sa capacité à rendre compte des interactions eau-air pouvant fortement

, en l'occurrence le développement d'un modèle, sa discrétisation et sa validation, constituent une contribution originale pour la modélisation des écoulements mixtes eau-air en conduite. Des perspectives sont envisageables sur ces trois aspects, Les trois volets abordés dans cette thèse

, Cette étape de validation supplémentaire doit être menée. De plus, l'utilisation du modèle dans un contexte industriel nécessite le développement de conditions limites adaptées. Pour les écoulements en conduite, il semble en effet essentiel de pouvoir imposer un débit ou une pression en entrée/sortie. Dans le cadre des modèles bifluide bipression, le développement de ces conditions limites de type Dirichlet est très peu abordé dans la littérature en raison de la complexité du système d'ondes sousjacent. Une stratégie envisageable est de résoudre un demi-problème de Riemann aux cellules frontières en appliquant une série d'hypothèses pour faciliter sa résolution. En particulier, l'hypothèse ? x h k = 0 appliquée localement à la frontière permet de découpler les deux phases en se ramenant à deux systèmes d'Euler isentropique. Des méthodes classiques pourraient alors être utilisées. Cette stratégie suppose toutefois une méthode numérique explicite appliquée au système non découpé. Son extension au schéma SPR développé dans cette thèse, La formulation du modèle proposé n'impose pas de restrictions quant à la gestion de configurations diphasiques complexes telles que le transport forcé ou la coalescence de poches d'air

, Cette perspective s'appuie sur la structure commune des modèles bifluide bipression indépendamment du type de moyenne utilisée (spatiale, temporelle, statistique). Ainsi, une interprétation appropriée du taux de présence ? k = h k H ouvre la porte vers un modèle 1D multi-régime traduisant effectivement les effets gravitaires à surface libre. Dans un registre similaire, la présence de contraintes géométriques dans les installations industrielles, telles que des coudes, peut engendrer des dynamiques intrinsèquement 2D alors que le niveau de description du modèle CTL est par construction 1D. En s'appuyant également sur la structure commune des modèles bifluide bipression, Le modèle CTL a été établi à partir d'une description bicouche des écoulements eau-air en conduite

, Une autre extension naturelle en terme de représentativité physique consiste à intégrer des équations de conservation d'énergie au modèle. Comme pour les équations de conservation de masse et de quantité de mouvement, elles résultent de l'intégration des équations de conservation d'énergie locales écrites pour chaque phase. Elles sont notamment présentes dans le modèle introduit par Ransom

, Il s'agira alors d'établir des lois de fermetures en accord avec la contrainte hydrostatique et d'étudier l'adaptabilité du schéma SPR. Ainsi, des dynamiques complexes eau-vapeur impliquant des transferts de masse et

. D'un-point-de-vue-analytique, En effet, il s'agit de passer d'une description diphasique compressible à une description monophasique incompressible. Cela impose d'établir un adimensionnement bien choisi afin de prendre en compte d'une part une asymptotique de relaxation entre les phases et d'autre part une asymptotique bas Mach. En particulier, se pose la question de la compatibilité de l'équation de transport sur la hauteur d'eau. Enfin, un travail de modélisation doit être mené afin d'établir une expression analytique du temps de relaxation associé à la relaxation en pression, il serait intéressant d'étudier le lien entre les équations de la phase eau dans le modèle CTL et les équations de Saint-Venant

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