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Contribution to the simulation of low-velocity compressible two-phase flows with high pressure jumps using homogeneous and two-fluid approaches

Abstract : The present work focuses on the development of 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. By making the Mach number tend towards zero, many authors in the literature perform an asymptotic analysis of the conservation laws at stake. Incompressible models are then derived. In this work, compressibility of both phases is systematically considered. Nevertheless, either analytical or tabulated, the equation of state always contains information related to the thermodynamical stiffness of the fluid. Such a stiffness can be seen as a measure of the compressibility of the considered phase. Thus, for two-phase flows involving sub-cooled liquid water and over-heated vapor, the liquid phase is known to be slightly compressible. When sudden events occur such as valve closures, 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. This is often observed experimentally through the study of water hammer events. The first part of this work has leaned on two-phase homogeneous-equilibrium models. Thus, both phases have the same velocity, pressure, temperature as well as the same chemical potential. The evolution of the flow is observed only through mixture variables. The construction and the evaluation 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 associated with high pressure jumps propagate, these solvers automatically adapt in order to capture the position of the shock front as well as the right pressure levels. The second part of the thesis has been dedicated to the design of numerical methods enhancing the coupling between convection and relaxation in two-fluid models. In such models, both phases have their own set of variables: velocity, pressure, temperature and void fraction. Contrary to homogeneous models, the mechanical and thermodynamical equilibriums are not imposed, and the value of velocity, pressure, temperature and chemical potential between both phases may differ. Convergence towards equilibrium is ensured by adding relaxation source terms in these models. For the sake of simplicity, the framework has been restricted to the isentropic Baer-Nunziato model with velocity-pressure relaxations. A time-implicit staggered scheme, based on the influence of relaxation source terms on linear Riemann problems has been proposed. What is more, an asymptotic analysis dealing with the integration of relaxation source terms into the convective part of the two-phase flow model has been carried out.
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David Iampietro. Contribution to the simulation of low-velocity compressible two-phase flows with high pressure jumps using homogeneous and two-fluid approaches. Mathematical Physics [math-ph]. Aix-Marseille Université, 2018. English. ⟨tel-01919156⟩

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