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In the context of multiphase flows we are faced with vector PDE solutions combining waves whose speeds are several orders of magnitude apart. The wave of interest is the transport one, and is relatively slow. The other fast acoustic waves are not interesting but impose a very restrictive CFL condition if a fully explicit in time scheme is considered. We therefore use a time semi-implicit conservative scheme where the fast waves are handled with a linearized implicit formulation and the slow wave remains explicitly solved. The CFL condition, governed by the explicit wave speed is then optimal. We combine this method with a multiscale analysis of the vector solution which enables to use a time varying adaptive grid based on the relevant smoothness properties of the discrete solution. In this short paper we compare different strategies to evaluate the fluxes at cells interfaces on a non uniform grid.

We present an adaptive wavelet method for solving the incompressible Navier-Stokes equations in two space dimensions using the vorticity-stream function formulation. For time discretization a semi-implicit scheme of second order is used. The space discretization is based on a Petrov-Galerkin method, where orthogonal spline wavelets of 4th order are employed as trial functions and operator adapted wavelets as test functions. The no-slip boundary conditions are imposed using a volume penalisation method. As example we present adaptive simulations of vortexdipole wall interactions.

A fully adaptive multiscale finite volume scheme for solving the 2D compressible Euler equations on moving grids is presented. The scheme uses a multiscale analysis based on biorthogonal wavelets to adapt the grid in space. Refinement in time is performed using a locally varying time stepping strategy that has been recently developed. The CFL condition is satisfied locally and the number of grid adaptations is reduced. The performance of the scheme using global and local multilevel time stepping, respectively, is investigated by a flow past an oscillating boundary.

The present work is devoted to the numerical approximation of a compressible two-phase flow. The phases are non-miscible and separated by an interface where capillary effects are considered.We use diffuse interface models having a single velocity and pressure. The surface tension forces are added with the CSF method [1]. We propose a Godunov type method based on a pressure relaxation procedure for the system including surface tension terms. Two numerical illustrations are performed showing the parasitic currents reduction and a liquid break-up.

We study the coupling of two gas dynamics systems in Lagrangian coordinates at the interface = 0. The coupling condition was formalized in [9, 10] by requiring that two boundary value problems should be well-posed, and it yields as far as possible the continuity of the solution at the interface. In this work we prove that we may choose the variables we transmit and extend the theory to Lagrangian systems of different sizes. The coupling condition is expressed in terms of Riemann problems. This is well suited for the numerical methods we are implementing and adapted to Lagrangian systems since the sign of the wave speeds is known, which enables us to solve the coupled Riemann problem.

A thermodynamic model for phase transition is introduced. The equation of state (EOS) is not globally convex hence difficulties exist in solving the Riemann Problem (R.P.). This motivates another method for solving the R.P. where thermodynamically out-of-equilibrium states are taken into account in the framework of [2, 1, 9]. This method is tested numerically on several examples.

This article presents an operator splitting for solving the time-dependent incompressible Navier-Stokes equations with Finite Elements. By using a postprocessing step the splitting method shows a reduction factor higher than second order. In this algorithm a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm.

It is well know the importance of bifurcation diagrams in fluid models where any of the parameters has some kind of uncertainty (usually the Reynolds number in Navier-Stokes, or the Ekman parameter in geophysical models).

The problem considered consists of a fluid within a cylindrical annulus heated laterally. As soon as a horizontal temperature gradient is applied a convective state appears. This state becomes unstable through stationary or oscillatory bifurcations as control parameters involved in the problem reach critical values. The problem is modelled with the incompressible Boussinesq Navier-Stokes equations and appropriate boundary conditions. In particular we consider lateral conducting walls and surface tension effects. This choice presents singularities at the point where free and solid surfaces meet, which consist on discontinuities on the temperature and its derivatives. These singularities are smoothed using a polynomial filtering. The main goal of this work is the study of the effect of this filtering in the stability problem. The filter improves the convergence of the numerical method. Convergence with the filtering scale depends on the Marangoni parameter.

In this paper we address the problem of the numerical approximation of the incompressible flow around a vibrating airfoil. The robust higher order finite element method (FEM) for incompressible flow approximation is presented. The method is based on the combination of several techniques, e.g., the Arbitrary Lagrangian-Eulerian formulation of the Navier-Stokes equations, the stabilization of the finite element scheme and the linearization of the discrete nonlinear problem.