Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures. (English) Zbl 1409.76105
Summary: Numerical approximation of the five-equation two-phase flow of A. K. Kapila et al. [Phys. Fluids 13, No. 10, Paper No. 3002, 23 p. (2001; Zbl 1184.76268)] is examined. This model has shown excellent capabilities for the numerical resolution of interfaces separating compressible fluids as well as wave propagation in compressible mixtures [A. Murrone and H. Guillard, J. Comput. Phys. 202, No. 2, 664–698 (2005; Zbl 1061.76083); R. Abgrall and V. Perrier, Multiscale Model. Simul. 5, No. 1, 84–115 (2006; Zbl 1236.76053); F. Petitpas et al., J. Comput. Phys. 225, No. 2, 2214–2248 (2007; Zbl 1183.76831)].
However, its numerical approximation poses some serious difficulties. Among them, the non-monotonic behavior of the sound speed causes inaccuracies in wave’s transmission across interfaces. Moreover, volume fraction variation across acoustic waves results in difficulties for the Riemann problem resolution, and in particular for the derivation of approximate solvers. Volume fraction positivity in the presence of shocks or strong expansion waves is another issue resulting in lack of robustness. To circumvent these difficulties, the pressure equilibrium assumption is relaxed and a pressure non-equilibrium model is developed. It results in a single velocity, non-conservative hyperbolic model with two energy equations involving relaxation terms. It fulfills the equation of state and energy conservation on both sides of interfaces and guarantees correct transmission of shocks across them. This formulation considerably simplifies numerical resolution.
Following a strategy developed previously for another flow model [R. Saurel and R. Abgrall, J. Comput. Phys. 150, No. 2, 425–467(1999; Zbl 0937.76053)]], the hyperbolic part is first solved without relaxation terms with a simple, fast and robust algorithm, valid for unstructured meshes. Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness. The algorithm and model are compared to exact solutions of the Euler equations as well as solutions of the five-equation model under extreme flow conditions, for interface computation and cavitating flows involving dynamics appearance of interfaces. In order to deal with correct dynamic of shock waves propagating through multiphase mixtures, the artificial heat exchange method of Petitpas et al. [loc. cit.] is adapted to the present formulation.
However, its numerical approximation poses some serious difficulties. Among them, the non-monotonic behavior of the sound speed causes inaccuracies in wave’s transmission across interfaces. Moreover, volume fraction variation across acoustic waves results in difficulties for the Riemann problem resolution, and in particular for the derivation of approximate solvers. Volume fraction positivity in the presence of shocks or strong expansion waves is another issue resulting in lack of robustness. To circumvent these difficulties, the pressure equilibrium assumption is relaxed and a pressure non-equilibrium model is developed. It results in a single velocity, non-conservative hyperbolic model with two energy equations involving relaxation terms. It fulfills the equation of state and energy conservation on both sides of interfaces and guarantees correct transmission of shocks across them. This formulation considerably simplifies numerical resolution.
Following a strategy developed previously for another flow model [R. Saurel and R. Abgrall, J. Comput. Phys. 150, No. 2, 425–467(1999; Zbl 0937.76053)]], the hyperbolic part is first solved without relaxation terms with a simple, fast and robust algorithm, valid for unstructured meshes. Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness. The algorithm and model are compared to exact solutions of the Euler equations as well as solutions of the five-equation model under extreme flow conditions, for interface computation and cavitating flows involving dynamics appearance of interfaces. In order to deal with correct dynamic of shock waves propagating through multiphase mixtures, the artificial heat exchange method of Petitpas et al. [loc. cit.] is adapted to the present formulation.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76B10 | Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing |
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
Software:
HE-E1GODFReferences:
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