The level set method is one of the most successful methods for the simulation of multi-phase flows. To keep the level set function close the signed distance function, the level set function is constantly reinitialized by solving a Hamilton-Jacobi type of equation during the simulation. When the fluid interface intersects with a solid wall, a moving contact line forms and the reinitialization of the level set function requires a boundary condition in certain regions on the wall. In this work, we propose to use the dynamic contact angle, which is extended from the contact line, as the boundary condition for the reinitialization of the level set function. The reinitialization equation and the equation for the normal extension of the dynamic contact angle form a coupled system and are solved simultaneously. The extension equation is solved on the wall and it provides the boundary condition for the reinitialization equation; the level set function provides the directions along which the contact angle is extended from the contact line. The coupled system is solved using the 3rd order TVD Runge-Kutta method and the Godunov scheme. The Godunov scheme automatically identifies the regions where the angle condition needs to be imposed. The numerical method is illustrated by examples in three dimensions.

This paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model. The HLL Riemann solver is applied to solve the Riemann problem for the model equations. This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture. Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results. To demonstrate the robustness, effectiveness and capability of these methods, the model results are compared with reference solutions. In addition to that, these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature. The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.

We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phaseflow model, that is able to cope with arbitrarily small values of the statistical phasefractions. The solver relies on a relaxation approximation of the model for which theRiemann problem is exactly solved for subsonic relative speeds. In an original manner, theRiemann solutions to the linearly degenerate relaxation system are allowed to dissipatethe total energy in the vanishing phase regimes, thereby enforcing the robustness andstability of the method in the limits of small phase fractions. The scheme is proved tosatisfy a discrete entropy inequality and to preserve positive values of the statisticalfractions and densities. The numerical simulations show a much higher precision and a morereduced computational cost (for comparable accuracy) than standard numerical schemes usedin the nuclear industry. Finally, two test-cases assess the good behavior of the schemewhen approximating vanishing phase solutions.

In the present work we investigate the numerical simulation of liquid-vapor phase changein compressible flows. Each phase is modeled as a compressible fluid equipped with its ownequation of state (EOS). We suppose that inter-phase equilibrium processes in the mediumoperate at a short time-scale compared to the other physical phenomena such as convectionor thermal diffusion. This assumption provides an implicit definition of an equilibriumEOS for the two-phase medium. Within this framework, mass transfer is the result of localand instantaneous equilibria between both phases. The overall model is strictlyhyperbolic. We examine properties of the equilibrium EOS and we propose a discretizationstrategy based on a finite-volume relaxation method. This method allows to cope with theimplicit definition of the equilibrium EOS, even when the model involves complex EOS’s forthe pure phases. We present two-dimensional numerical simulations that shows that themodel is able to reproduce mechanism such as phase disappearance and nucleation.

We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media. Some of the methods developed using the framework are already known [20]; others are new. New insight is gained for the known methods and extra flexibility is provided by the new methods. We give as an example a mixed MsFV on uniform mesh in 2-D. This method uses novel multiscale velocity basis functions that are suited for using global information, which is often needed to improve the accuracy of the multiscale simulations in the case of continuum scales with strong non-local features. The method efficiently captures the small effects on a coarse grid. We analyze the new mixed MsFV and apply it to solve two-phase flow equations in heterogeneous porous media. Numerical examples demonstrate the accuracy and efficiency of the proposed method for modeling the flows in porous media with non-separable and separable scales.

This paper is concerned with the numerical approximation of the solutions of a two-fluid two-pressure model used in the modelling of two-phase flows.We present a relaxation strategy for easily dealing with both the nonlinearities associated with the pressure laws and the nonconservative termsthat are inherently present in the set of convective equations and that couple the two phases.In particular, the proposed approximate Riemann solver is given by explicit formulas, preservesthe natural phase space, and exactly captures the coupling waves between the two phases.Numerical evidences are given to corroborate the validity of our approach.

We study a depth-averaged model of gravity-driven flows made ofsolid grains and fluid, moving over variable basal surface. In particular, we are interested in applicationsto geophysical flows such as avalanches and debris flows,which typically contain both solid material and interstitial fluid.The model system consists of mass and momentum balance equations for thesolid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns,and by interphase drag source terms. The system is hyperbolic at leastwhen the difference between solid and fluid velocities is sufficiently small.We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Well-balancing oftopography source terms is obtained via a technique that includesthese contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problemsof perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.

This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.