Three-dimensional hydrodynamic model for free surface flow

Abstract

A three-dimensional mathematical model for free surface flow based on the Reynolds-averaged Navier-Stokes equations is developed in a sigma co-ordinate system. The time-splitting method is used to separate advection and diffusion terms from the pressure terms in the governing equations. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method, while the hydrostatic pressure equations are solved very efficiently by an Alternating Direction Implicit (ADI) scheme. The convection terms are discretized by the Roe's scheme of second-order accuracy. A staggered mesh is used. The model is tested against available analytical solutions and experimental data. Good agreement has been achieved. RÉSUMÉUn modèle mathématique tridimensionnel pour les écoulements a surface libre, basé sur les equations de Navier Stokes en moyenne de Reynolds, est développé dans un système de coordonnées sigma. La methode des pas fractionnaires en temps est employee pour séparer les termes de convection et de diffusion des termes de pression dans les equations. La pression est ensuite séparée en pression hydrostatique et dynamique pour éviter en grande partie les erreurs d'arrondis de Fordinateur. L'équation resultante de la pression dynamique est résolue par une methode multigrille, alors que les equations de la pression hydrostatique sont résolues tres efficacement par un schema implicite de directions alternées (ADI). Les termes de convection sont discrétisées suivant Ie schema de Roe du second ordre. Le maillage est en quinconce. Le modèle est confronté a des solutions analytiques et des données expérimentales disponibles. Un bon l'accord a été observe.

Further experiences with computing non-hydrostatic free-surface flows involving water waves

“…Because of the vertical gradient of non-hydrostatic pressure occuring in that equation, one is enable to implement condition (8). Some researchers [3, 9-11, 13-15, 24], however, assume the surface cells to be hydrostatic, i.e.…”

“…Our implementation consists of solving the momentum equation for the vertical velocity w (3) at the free surface. This will introduce virtual pressure points near the free surface that can be readily eliminated by means of a linear extrapolation using condition (8). The usual treatment, however, as presented in References [3, 9-11, 13-15, 24], is to set q = 0 inside the surface cells, and accordingly the vertical velocity w is determined by applying the continuity equation (4).…”

“…Special attention is paid to the further optimization of the preconditioning in terms of e ciency and robustness. An alternative to the solution may be a multigrid technique; this is done in References [4,7,8,20]. The papers of Stansby and Zhou [13], Zhou and Stansby [14] and Ko cyigit et al [15] do not indicate which iterative method has been employed in their codes.…”

“…Nevertheless, we argue to take into account the curvature terms in the horizontal pressure gradient due to -transformation since, experiences show that these terms appear not to be sensitive to the relative large changes in bathymetry. Other publications in this direction are [4,7,8,19].…”

“…This is the approach that we shall follow and is also presented in References [4, 12-15, 24, 25]. In Reference [8], an alternating direction implicit (ADI) scheme is employed to obtain the hydrostatic pressure. In References [5,7,[19][20][21], an explicit time discretization is used so that the time step is restricted by the deepest part of the coastal area or the smallest grid size.…”