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J. Glimm, D. Saltz, and D. H. Sharp, Two phase flow modelling of a fluid mixing layer, Journal of Fluid Mechanics, vol.378, pp.119-143, 1999.

E. Godlewski, M. Parisot, J. Saint-marie, and F. Wahl, Congested shallow water type model: roof modelling in free surface flow, 2017.

J. Haack, S. Jin, and J. G. Liu, An all-speed asymptotic preserving method for the isentropic Euler and navierstokes equations, Communications in Computational Physics, vol.12, pp.955-980, 2012.

J. Hérard and O. Hurisse, A fractional step method to compute a class of compressible flows with microinertia, Computers & Fluids, vol.55, pp.57-69, 2012.

D. Iampietro, F. Daude, P. Galon, and J. Hérard, A Mach-sensitive implicit-explicit scheme adapted to compressible multi-scale flows, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01531306

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F. Kerger, Modelling transient air-water flows in civil and environmental engineering, 2010.

F. Kerger, P. Archambeau, E. S. , B. J. Dewals, and M. Pirotton, An exact Riemann solver and a Godunov scheme for simulating highly transient mixed flows, Journal of Computational and Applied Mathematics, vol.235, issue.8, pp.2030-2040, 2011.

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L. Ramezani, B. Karney, and A. Malekpour, Encouraging effective air management in water pipelines: A critical review, Journal of Water Resources Planning and Management, vol.142, issue.12, 2016.

V. H. Ransom and D. L. Hicks, Hyperbolic two-pressure models for two-phase flow, Journal of Computational Physics, vol.53, pp.124-151, 1984.

V. V. Rusanov, Calculation of interaction of non-steady shock waves with obstacles, Zh. Vychisl. Mat. Mat. Fiz, vol.1, issue.2, pp.267-279, 1961.

Y. Taitel and A. E. Dukler, A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow, AIChE J, vol.22, pp.47-55, 1976.

S. Tokareva and E. Toro, HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow, Journal of Computational Physics, vol.229, issue.10, pp.3573-3604, 2010.

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B. C. Trindade, Air pocket modeling in water mains with an air valve, 2012.

B. C. Trindade and J. G. Vasconcelos, Modeling of water pipeline filling events accounting for air phase interactions, Journal of Hydraulic Engineering, vol.139, issue.9, pp.921-934, 2013.

J. G. Vasconcelos, S. J. Wright, and P. L. Roe, Improved simulation of flow regime transition in sewers: Twocomponent pressure approach, Journal of Hydraulic Engineering, vol.132, issue.6, pp.553-562, 2006.

J. G. Vasconcelos, S. J. Wright, and P. L. Roe, Numerical oscillations in pipe-filling bore predictions by shockcapturing models, Journal of Hydraulic Engineering, vol.135, issue.4, pp.296-305, 2009.

F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, 2004.

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G. Dimarco, R. Loubère, and M. Vignal, Study of a new asymptotic preserving scheme for the Euler system in the low Mach number limit, 2016.
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J. Haack, S. Jin, and J. G. Liu, An all-speed asymptotic preserving method for the isentropic Euler and navierstokes equations, Communications in Computational Physics, vol.12, pp.955-980, 2012.

D. Iampietro, F. Daude, P. Galon, and J. Hérard, A weighted splitting approach adapted to low Mach number flows, Springer Proceedings in Mathematics and Statistics, vol.200, pp.3-11, 2017.

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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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, 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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