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A simple two-phase method for the simulation of complex free surface flows. (English) Zbl 1225.76210
Summary: We propose a simple and efficient diffuse interface (interface-capturing) two-phase algorithm for the simulation of complex non-hydrostatic free surface flows in general geometries. The physical model is given by a special case of the more general Baer – Nunziato model for compressible multi-phase flows. In the applications considered here, the relative pressure of the gas phase with respect to atmospheric reference conditions is assumed to be zero everywhere and the momentum of the gas phase can be neglected compared to the one of the liquid. The reduced system is closed by the Tait equation of state, which is a widespread model for water. The resulting PDE system is solved by a high order path-conservative WENO finite volume scheme on unstructured triangular meshes, to be applicable also in complex geometries. To assure low numerical dissipation at the free surface, which actually is crucial for the applications under consideration here, we use the new generalized Osher-type scheme of the author and E. F. Toro [Sci. Comput. 48, No. 1-3, 70–88 (2011; Zbl 1220.65110)], which resolves steady shear and contact waves exactly, in contrast to the simpler centered path-conservative FORCE schemes presented in [M. Dumbser et al., Comput. Methods Appl. Mech. Eng. 199, 625–647 (2010)].
The model derives directly from first principles, namely the conservation of mass and momentum, hence it does not make any of the classical simplifications inherent in the commonly used shallow water models, which are based on depth-averaging, neglecting accelerations in gravity directions and on the resulting hydrostatic pressure distribution. We validate the new two-phase model against available analytical, numerical and experimental reference solutions and we also show some comparisons with the classical shallow water model for typical dambreak-type problems.

76M12 Finite volume methods applied to problems in fluid mechanics
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
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