A pressure‐based method for weakly compressible two‐phase flows under a Baer–Nunziato type model with generic equations of state and pressure and velocity disequilibrium

Abstract

Within the framework of diffuse interface methods, we derive a pressure-based Baer-Nunziato type model well-suited to weakly compressible multiphase flows. The model can easily deal with different equation of states and it includes relaxation terms characterized by user-defined finite parameters, which drive the pressure and velocity of each phase toward the equilibrium. There is no clear notion of speed of sound, and thus, most of the classical low Mach approximation cannot easily be cast in this context. The proposed solution strategy consists of two operators: a semi-implicit finite-volume solver for the hyperbolic part and an ODE integrator for the relaxation processes. Being the acoustic terms in the hyperbolic part integrated implicitly, the stability condition on the time step is lessened. The discretization of nonconservative terms involving the gradient of the volume fraction fulfills by construction the nondisturbance condition on pressure and velocity to avoid oscillations across the multimaterial interfaces. The developed simulation tool is validated through one-dimensional simulations of shock-tube and Riemann-problems, involving water-aluminum and water-air mixtures, vapor-liquid mixture of water and of carbon dioxide, and almost pure flows. The numerical results match analytical and reference ones, except some expected discrepancies across shocks, which however remain acceptable (errors within some percentage points). All tests were performed with acoustic CFL numbers greater than one, and no stability issues arose, even for CFL greater than 10. The effects of different values of relaxation parameters and of different amount equations of state-stiffened gas and Peng-Robinson-were investigated.

Modelling interactions between waves and diffused interfaces

“…The pressure-disequilibrium model (3) can be solved with either a finite or an infinite relaxation procedure for . At each time step we solve the non-relaxed, hyperbolic equations ( → 0) using ( 14), then we solve the system of ordinary differential equations (ODE) = ( ) , (32) that relaxes the disequilibrium pressures for a given or → +∞. When multi-stage time integration is used, these procedures are performed at each stage.…”

“…We also note that finite relaxation rates have already been considered in previous work. In particular, Re and Abgrall 32 proposed a pressure‐based method for weakly compressible two‐phase flows under a Baer–Nunziato‐type model, 33 which also considers velocity disequilibrium. And Chiocchetti and Müller 34 proposed an integration technique for the finite relaxation rates for the Baer–Nunziato model and the pressure‐disequilibrium model based on total energies, 18 but they only considered mixture test cases (no pure phases).…”