This paper deals with a numerical method based on the simulation of unsteady and laminar two-dimensional free-surface incompressible flows. Navier-Stokes equations are discretised by finite differences. The pressure is obtained by solving a Poisson equation dealing with a fictitious velocity field and solved by a finite volume method. An interface tracking method is used to evaluate the free-surface elevation. We have studied numerically the interaction between regular water waves and a submerged obstacle. Numerical damping method with stretched mesh is used for wave absorption at the outlet boundary. Mesh refinement is assured using the Adaptive Mesh Refinement method.
Numerical method used to solve this kind of problem can be divided into three families: Eulerian methods, Lagrangian methods and mixed Eulerian/Lagrangian methods in order to take advantages of each method and to avoid their disadvantages (Floryan and Rasmussen, 1989).
The Eulerian methods, the more common, can be divided into four types: with a fixed grid (Hyman, 1984; Harlow and Welch, 1965; Welch & al., 1965; Noh and Woodward, 1976 or Hirt and Nichols, 1981); the adaptative methods (Thompson & al., 1985); the third group is concerned by the mapping methods (Loh and Rasmussen, 1987); finally, the last group are applied to particular flows where simplified Navier-Stokes equations are used (Youngreen and Acrivos, 1976; Geller & al., 1986).
Concerning the Lagrangian methods, the frame of reference follows the fluid motion, so the advantage is that the interface motion is easy to describe and that the boundary conditions on this interface are easy to implement. The disadvantage is that in some case the cells are greatly distorted.
Some methods present a mixing of the two previous methods
In our study, a fixed classical Eulerian method with a staggered variable system is used, the fluid moving through the grid.