Efficient solution of mechanical and thermo-chemical finite-rate relaxation for compressible liquid-vapor flows under arbitrary equations of state

We consider the full disequilibrium Baer-Nunziato (BN) model with finite relaxation parameters for pressure, velocity, Gibbs free energy, and temperature. This formulation enables accurate modeling of complex two-phase flows, but requires a considerable computational cost. The BN model is typically solved by operator splitting, where the relaxation step involves integrating a stiff, highly nonlinear system of ordinary differential equations (ODEs), for which standard, general-purpose solvers are inadequate. We develop a time-accurate and efficient solver for simultaneous mechanical and thermo-chemical relaxation under a generic equation of state, based on an analytical solution of a linearized version of the original ODE system. In verification test cases involving condensation, evaporation, and mixed conditions, the presented relaxation solver achieves errors comparable to a state-of-the-art ODE integrator while requiring only 3%-10% of its time steps. The BN solver equipped with arbitrary-rate relaxation terms reproduces experimental data and results from alternative models in the literature, provided that the relaxation parameters are properly defined. Nonlinear closures of these terms were also used in a two-dimensional two-phase nozzle flow to model the particle dynamics as a function of droplet diameter. The proposed relaxation operator extends the applicability of the BN model in two ways: by reducing the computational cost without compromising accuracy, and by enabling application-specific closures of the relaxation parameters for complex two-phase flow problems.