An innovative phase transition modeling for reproducing cavitation through a five-equation model and theoretical generalization to six and seven-equation models

This work is devoted to model the phase transition for two-phase flows with a mechanical equilibrium model. First, a five-equation model is obtained by means of an asymptotic development starting from a non-equilibrium model (seven-equation model), by assuming a single-velocity and a single pressure between the two phases, and by using the Discrete Equation Method (DEM) for the model discretization. Then, a splitting method is applied for solving the complete system with heat and mass transfer, \textit{i.e.}, the solution of the model without heat and mass transfer terms is computed and, then, updated by supposing a heat and mass exchange between the two phases. Heat and mass transfer is modeled by applying a thermo-chemical relaxation procedure allowing to deal with metastable states. The interest of the proposed approach is to preserve the positivity of the solution, and to reduce at the same time the computational cost. Moreover, it is very flexible since, as it is shown in this paper, it can be extended easily to six (single velocity) and seven-equation models (non-equilibrium model).Several numerical test-cases are presented, \textit{i.e.} a shock-tube and an expansion tube problems, by using the five-equation model coupled with the cavitation model. This enables us to demonstrate, using the standard cases for assessing algorithms for phase transition, that our method is robust, efficient and accurate, and provides results at a lower CPU cost than existing methods. The influence of heat and mass transfer is assessed and we validate the results by comparison with experimental data and to the existing state-of-art methods for cavitation simulations