Thermo-hydro-dynamic consistency and stiffness in general compressible multiphase flows

This thesis project was carried out at the Commissariat à l'Énergie Atomique (CEA), Direction des Applications Militaire (DAM). It is a continuation of the thesis of T.Vazquez-Gonzalez(2016) and is contemporary with Paulin(2021). The aim is to address two objectives in the development of multiphase models and schemes: being thermodynamically consistent and capturing the inherent multiphase stiffness to ensure stability. In previous works (Vazquez-Gonzalezetal.,2020), a consistent numerical scheme (named GEEC) was designed using variational approaches and mimicry. As a result, a quasi-isentropic behaviour and exact conservations were obtained. However, this proximity to isentropy made the numerical scheme potentially vulnerable to numerical residuals. Moreover, as one of the objective was to capture isentropy, the internal energy discretization was developed with a mimetic approach without addressing stiffness issues. In this thesis, we propose a new discretisation of the internal energy equations, formally applicable to any other multiphase scheme, which addresses the stiffness and the thermodynamic consistency issues in the pressure coupling. The model discretized by Vazquez-Gonzalez et al.(2020), known as the 6-equation model, derives from an averaging approach stripped of all second-order correlations. Thus, all dissipation and higher order potentials hidden in the fluctuations are eliminated. Consequently, the model represents only ideal situations and is therefore seldom applicable to real multi-phase flows without being complemented by fluctuation terms. Now,as it is the basis for all other models, it was crucial to ensure its correct discretization from the start. As follow up to Vazquez-Gonzalez et al.(2020), we now propose a method to introduce higher-order isentropic effects (surface tension, turbulence, etc.) while maintaining a model formalization that captures stiffness and thermodynamic consistency. To show the interest of the method, four way couplings are considered in dispersed particle-laden flows. Collisions are introduced through variational approaches and the ensuing equations are discretized by mimicking the GEEC scheme. Dissipative couplings with the carrier phase are embodied in drag forces. Numerical simulations of crossing jets validate the approach. The last part of the thesis is an exploratory work that focuses on thermodynamic consistency issues in Lagrange-Euler (LE) modeling of dispersed particles-laden flows. A new description of particles is coupled to the least action principle, leading to the coupled dynamics of dispersed and carrier phases. Discrete dynamics derived from the same procedure. This approach is extended to compressible particles. Numerical simulations lead to contrasted results, showing the possibilities of the method but also the need of many improvements.Predicting the behaviour of multi- phase flows where many contrasted phases coexist is a challenge that has mobilized numericists and physicists since the development of nuclear safety codes. The challenge is especially tough when flows feature strong shocks, phase changes and transport over long distances. Simulation must then incorporate the compressibility of all phases, their different dynamics and the strong and various couplings occurring between them. Because of this complexphysics, the mathematical structure of the models often departs from the Euler classical hyperbolic equations. New numerical methods must be then designed in order to solve these models with finite computational resources and strong robustness which constrained numerical schemes in terms of stability and thermodynamic consistency.