Journal of Computational Physics 155, 439–455 (1999) Article ID jcph.1999.6346, available online at http://www.idealibrary.com on Numerical Simulation of Free Surface Flows V. Maronnier, M. Picasso, and J. Rappaz D´ epartement de Math´ ematiques, Ecole Polytechnique F´ ed´ erale de Lausanne, 1015 Lausanne, Switzerland Received October 26,1998; revised April 9, 1999 A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the math- ematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, un- structured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain. c  1999 Academic Press 1. INTRODUCTION Numerical simulation of free surface flows is of great interest for industrial applications such as casting, welding, injection, or extrusion processes. In many interesting situations, the motion of the free surface is complex, making front-tracking methods [15] or Lagrangian techniques [10, 11, 22] difficult to handle. Indeed, in the frame of Lagrangian methods, the nodes of the mesh are moved according to the fluid velocity, the mesh is severely distorted, and remeshing becomes unavoidable. Arbitrary Lagrangian–Eulerian methods [13, 16, 20, 27, 32] remedy this situation by allowing the internal nodes to move independently from the fluid velocity. However, the selection of the mesh velocity is nontrivial for complex flows. An alternative is to consider the Eulerian approach, which consists in using a fixed mesh but adding an unknown function ϕ whose values characterize the volume fraction of liquid and which satisfies an advection equation ∂ϕ ∂ t + v ·∇ϕ = 0, 439 0021-9991/99 $30.00 Copyright c  1999 by Academic Press All rights of reproduction in any form reserved.