Volume 10 special

Abstract In this paper, we propose a new very simple numerical method for solving liquid-gas compressible flows. Such flows are difficult to simulate because classic conservative finite volume schemes generate pressure oscillations at the liquid-gas interface. We extend to several dimensions the random choice scheme that we have constructed in [2]. The extension is performed through Strang dimensional splitting. The resulting scheme exhibits in- teresting conservation and stability properties. For achieving high perfor- mance, the scheme is tested on recent multi-core processors and GPUs, using the OpenCL environment.

Key words : OpenCL, GPU, two-fluid compressible flow, Lagrange- projection, Glimm, Strang splitting

Paper presented by JM Hérard and T. Gallouët on July, 10, 2013

Abstract Cavitation damaging is investigated by simulation of the model problem of a single gas bubble in a compressible liquid collapsing near the surface of an elastic solid wall. The three-phase system is described by the com- pressible Euler equations supplemented by the stiffened gas law for both fluids, a non-conservative evolution equation for the gas fraction charac- terizing the liquid-gas interface and the elastodynamical equations for a linear-elastic solid. The fluid model and the solid model are coupled by transition conditions at the fluid-structure interface. The fluid equations are discretized according to Saurel and Abgrall and for the elastodynamical equations a finite volume discretization is applied. These numerical methods are coupled by a weak coupling strategy. The numerical results exhibit von Schmidt waves in the structure as well as the fluid. As possible explanation for cavitation damaging the von Mises yield criterion is evaluated.

Key words : compressible two-phase flow, linear-elastic solid, weak coupling.

Paper presented by JM Hérard and T. Gallouët on June, 26, 2013

Abstract This article analyses the convergence of the Vertex Approximate Gradi- ent (VAG) scheme recently introduced in Eymard et al. 2012 for the discretization of multiphase Darcy flows on general polyhedral meshes. The convergence of the scheme to a weak solution is shown in the par- ticular case of an incompressible immiscible two-phase Darcy flow model with capillary diffusion using a global pressure formulation. A remarkable property in practice is that the convergence is proven whatever the dis- tribution of the volumes at the cell centers and at the vertices used in the control volume discretization of the saturation equation. The numerical experiments carried out for various families of 2D and 3D meshes confirm this result on a one-dimensional Buckley Leverett solution.

Key words : Finite volume, two-phase Darcy flows, diffusion fluxes, general meshes, heterogeneous anisotropic media

Paper presented by JM Hérard and T. Gallouët on June, 26, 2013

Abstract We show in this paper that the gradient schemes (which encompass a large family of discrete schemes) may be used for the approximation of the Stefan problem ∂tu ̄ − ∆ζ(u ̄) = f. The convergence of the gradient schemes to the continuous solution of the problem is proved thanks to the following steps. First, estimates show (up to a subsequence) the weak convergence to some function u of the discrete function approximating u ̄. Then Alt-Luckhaus’ method, relying on the study of the translations with respect to time of the discrete solutions, is used to prove that the discrete function approximating ζ(u ̄) is strongly convergent (up to a subsequence) to some continuous function χ. Thanks to Minty’s trick, we show that χ = ζ(u). A convergence study then shows that u is then a weak solution of the problem, and a uniqueness result, given here for fitting with the precise hypothesis on the geometric domain, enables to conclude that u = u ̄. This convergence result is illustrated by some numerical examples using the Vertex Approximate Gradient scheme.

Key words : Stefan problem, gradient schemes, uniqueness result, con- vergence study.

Paper presented by JM Hérard and T. Gallouët on June, 26, 2013

This paper is dedicated to the physical and numerical modelisation of wet steam flows with non-equilibrium condensation in power plant steam turbines. The steam, far from an ideal gas behavior is modeled thanks to real gas thermodynamic laws. The nucleation process is taken into ac- count using two models: a simple two-equation model and the Quadrature Method of Moments. The numerical scheme used to handle the complex thermodynamics is the approximate Riemann solver VFRoe for which a particular attention has been paid regarding the boundary conditions. Shock-tube verification test cases are given to check the good behavior of the scheme and validation cases in nozzles are presented to illustrate the accuracy of the physical model compared with experimental data.

Key words : condensation, nucleation, quadrature method of moments, droplets size distribution, real gas, VFRoe scheme, boundary conditions

Paper presented by JM Hérard and T. Gallouët on June, 26, 2013