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. (2nd May 2022)

• Record Type: Journal Article
• Authors: Re, Barbara; Abgrall, Rémi
• Is Part Of: International journal for numerical methods in fluids. Volume 94:Number 8(2022)
• Page Start: 1183
• Page End: 1232
• Publication Date: 2022-05-02
• Subjects: Baer–Nunziato type model -- compressible two‐phase flows -- Peng–Robinson equation of state -- pressure and velocity relaxation with finite parameters -- pressure formulation -- semi‐implicit finite‐volume scheme; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.5087
• Languages: English
• ISSNs: 0271-2091

Abstract

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. Abstract : A simulation method for weakly compressible two‐phase flows is derived starting from the Baer–Nunziato model. Generic equation of states and finite pressure and velocity relaxation parameters provide a wide applicability. To deal with low Mach numbers, a special pressure scaling is used and the acoustic terms are integrated implicitly. A proper discretization of the non‐conservative terms prevents oscillations across multimaterial interfaces. The method is verified through several two phase flows, for example, a shock‐tube involving carbon dioxide at saturation conditions.