On the behavior of upwind schemes in the low Mach number limit. IV : P0 Approximation on triangular and tetrahedral cells

Finite Volume upwind schemes for the Euler equations in the low Mach number regime face a problem of lack of convergence toward the solutions of the incompressible system. However, if applied to cell centered triangular grid, this problem disappears and convergence toward the incompressible solution is recovered. We give here a general proof of this fact for arbitrary unstructured meshes. In addition, we also show that this result is equally valid for unstructured three dimensional tetrahedral meshes

On the Behavior of Upwind Schemes in the Low Mach Number Limit: II. Godunov Type Schemes

This paper presents an analysis of Godunov scheme in the low Mach number regime. We study the Riemann problem and show that the interface pressure contains acoustic waves of order Mach even if the initial data are well prepared and contain only pressure fluctuations of order Mach squared. We then propose to modify the fluxes computed by Godunov type schemes by solving a preconditioned Riemann problem instead of the original one. This strategy is applied to VFRoe solvers where we show that it allows to recover a correct scaling of the pressure fluctuations. Numerical experimen- ts confirm these theoretical results

An automatic mesh coarsening technique for three dimensional anisotropic meshes

This work is devoted to the design of a mesh generation technique able to produce a sequence of 3-D coarsened unstructured meshes from an initial anisotropic one. The coarsening algorithm uses an initial mesh and a metric field obtained from an analysis of the natural metric of this initial mesh. First, an initial natural metric (i.e a metric into which each simplicial element of the mesh is equilateral) is produced from the initial anisotropic mesh. Then the eigenvalues of this metric are modified and used together with the initial mesh to produce a coarsened mesh. In this way, the directions of anisotropies of the initial mesh are respected while the mesh spacing can be increased. This procedure can be repeated in order to produce a sequence of semi-coarsened meshes suitable for multigrid acceleration. The efficiency of this procedure is shown on examples of anisotropic meshes involving element aspect ratio as high as $10^{5}$

Shock Structure in a Two-phase Isothermal Euler Model

We have performed a traveling wave analysis of a two phase isothermal Euler model to exhibit the inner structure of shock waves in two-phase flows. In the model studied in this work, the dissipative regularizing term is not of viscous type but instead comes from relaxation phenomena toward equilibrium between the phases. This gives an unusual structure to the diffusion tensor where dissipative terms appear only in the mass conservation equations. We show that this implies that the mass fractions are not constant inside the shock although the Rankine-Hugoniot relations give a zero jump of the mass fraction through the discontinuities. We also show that there exists a critical speed for the traveling waves above which no C 1 solutions exist. Nevertheless for this case, it is possible to construct traveling solutions involving single phase shocks

Behavior of upwind scheme in the low Mach number limit : III. Preconditioned dissipation for a five equation two phase model

For single phase fluid models, like the Euler equations of compressible gas dynamics, upwind finite volume schemes suffer from a loss of accuracy when computing flows in the near incompressible regime. Preconditioning of the numerical dissipation is necessary to recover results consistent with the asymptotic behaviour of the continuous model. In this paper, we examine this situation for a two phase flow model. We show that as in the single phase case, the numerical approximation has to be done carefully in the near incompressible regime. We propose to adapt the preconditioning strategy used for single phase problems and present numerical results that show the efficiency of this approac

An automatic mesh coarsening technique for three dimensional anisotropic meshes

This work is devoted to the design of a mesh generation technique able to produce a sequence of 3-D coarsened unstructured meshes from an initial anisotropic one. The coarsening algorithm uses an initial mesh and a metric field obtained from an analysis of the natural metric of this initial mesh. First, an initial natural metric (i.e a metric into which each simplicial element of the mesh is equilateral) is produced from the initial anisotropic mesh. Then the eigenvalues of this metric are modified and used together with the initial mesh to produce a coarsened mesh. In this way, the directions of anisotropies of the initial mesh are respected while the mesh spacing can be increased. This procedure can be repeated in order to produce a sequence of semi-coarsened meshes suitable for multigrid acceleration. The efficiency of this procedure is shown on examples of anisotropic meshes involving element aspect ratio as high as $10^{5}$

A five equation reduced Model for compressible two phase flow problems

This paper presents an Euleriean model for the simulation of compressible two-phase flow problems. The starting point of the study is a seven equation, two pressure, two velocity model. This model contains relaxation terms that drive the systems toward pressure and velocity equilibrium. We perform an asymptotic analysis of this system in the limit of zero relaxation time and derive a five equation hyperbolic reduced system. We study the mathematical properties of the system, the structure of the waves and the expression of the Riemann's invariants. We then describe two different numerical approximation schemes for this system. The first one relies on a linearized Riemann solver while the second uses more heavily the mathematical structure of the system and relies on a linearization of the characteristic relations. Finally, we present some numerical experiments and comparison with the results obtained by the two pressure, two velocity model as well as some test cases in interface computations

Behavior of upwind scheme in the low Mach number limit : III. Preconditioned dissipation for a five equation two phase model

For single phase fluid models, like the Euler equations of compressible gas dynamics, upwind finite volume schemes suffer from a loss of accuracy when computing flows in the near incompressible regime. Preconditioning of the numerical dissipation is necessary to recover results consistent with the asymptotic behaviour of the continuous model. In this paper, we examine this situation for a two phase flow model. We show that as in the single phase case, the numerical approximation has to be done carefully in the near incompressible regime. We propose to adapt the preconditioning strategy used for single phase problems and present numerical results that show the efficiency of this approac