Abstract : This thesis is devoted to the modelling and numerical simulation of liquid-vapor flows. In order to describe these phase transition flows, a two-fluid two-pressure approach is considered. This description of the liquid-vapor mixing is associated to the seven-equation model introduced by Baer and Nunziato. This work investigates the properties of this model in order to simulate the phase transition flows occurring in nuclear engineering.
First, a theoretical thermodynamic framework is constructed to describe the liquid-vapor mixing. Provided with this framework, various modelling choices are suggested for the interaction terms between the phases. These closure laws comply with an entropy inequality. The mathematical properties of this model are thereafter examined. The convective part is associated to a nonconservative hyperbolic system. First, we focus on the definition of its weak solutions. Several flow regimes for the two-phase mixing derive from this analysis. Such regimes for the two-phase flows are analogous to the torrential and fluvial regimes for the shallow-water equations. Furthermore, we establish the linear and nonlinear stabilities of the liquid-vapor equilibrium. Finally, the implementation of a turbulence model and the introduction of a reconstruction process for the interfacial area are investigated in order to refine the description of the interfacial transfers.
Using a fractional step approach, a Finite Volume method is at last constructed to simulate this model. First, various nonconservative adaptations of standard Riemann solvers are developed to approach the convective part. Unlike the classic nonconservative framework, these schemes converge towards the same solution. Furthermore, a new relaxation scheme is proposed to approach the interfacial transfers. Provided with these schemes, the whole numerical method preserves the liquid-vapor equilibria. Using this numerical method, a careful comparison between the one- and two-pressure two-fluid models is presented. The numerical simulation of the strongly unbalanced liquid-vapor flows is at last applied to the safety analysis of the pressurized water nuclear reactors.