A homogeneous model is proposed in order to deal with the simulation of fast transient three-phase ows involving heat and mass transfer. The model accounts for the full thermodynamical disequilibrium between the three phases in terms of pressure, temperature and Gibbs enthalpy. The heat and mass transfer between the phases is modeled in agreement with the second law of thermodynamics, which ensures a stable return to the thermodynamical equilibrium. The set of partial differential equations associated with this model is based on the Euler set of equations supplemented by a complex pressure law, and by six scalar-equations that allow to account for the thermodynamical disequilibrium. It therefore inherits a simple wave structure and possesses important mathematical properties such as: hyperbolicity, unique shock definition through Rankine-Hugoniot relations, positivity of the mixture fractions. Hence the computation of approximated solutions is possible using classical algorithms, which is illustrated by an example of simulation of a steam-explosion.
Some accidental scenarii studied in the framework of the nuclear safety analysis involve liquids undergoing strong pressure drops at high temperature. In order to perform realistic simulations of such situations, a code based on a model that can handle both the thermodynamical disequilibrium between liquid and vapor and complex equations of state is required. We propose herein to test a homogeneous model built on the basis of the Euler system of equations and complemented by a mixture pressure law. The latter is defined in accordance with the Gibbs relation on the basis of the phasic pressures which are defined through a look-up table based on the IAPWS-97 formulation. A wide range of verification problems (Riemann problems) is then studied to assess the behavior of the numerical schemes for this complex equation of state. The tested relaxation scheme is the best compromise between accuracy and stability. At last, a simple test case of vaporization near a wall is investigated in order to test some return to thermodynamical-equilibrium timescale based on the nucleation theory.
A three-phase flow model with hybrid miscibility constraints is proposed: three immiscible phases are considered (liquid, solid and gas) but the gaseous phase is composed with two miscible components (steam water and non-condensable gas). The modelling approach is based on the building of an entropy inequality for the system of partial dfferential equations: once an interfacial velocity is given by the user, the model is uniquely defined, up to some relaxation time scales, and source terms complying with the second principle of thermodynamics can then be provided. The convective part of the system is hyperbolic when fulfilling a non-resonance condition and classical properties are studied (Riemann invariants, symmetrization). A key property is that the system possesses uniquely defined jump conditions. Last, preservation of thermodynamically admissible states and pressure relaxation are investigated.
Two phase flows including liquid water, vapor and non-condensable gases are often encountered in industrial applications. In this paper, an extension of the classical Baer-Nunziato two-fluid model is presented in order to account for the non-condensable gases. In this model, the vapor and non-condensable gases form a miscible mixture of gases at temperature equilibrium and kinematic equilibrium. The complete model then allows to account for the full kinematic and thermodynamic disequilibrium between the liquid phase and the gas phase. Moreover, a new form for the masstransfer source-term has been retained. It leads to a more straightforward numerical integration. When focusing on the numerical scheme, a new algorithm is proposed here for the pressure relaxation effect. At last, some numerical experiments are performed in order to assess the behavior of the model with respect to the relaxation time-scales.
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