Three‐dimensional numerical modelling of free surface flows with non‐hydrostatic pressure

Abstract

A three-dimensional numerical model is developed for incompressible free surface ows. The model is based on the unsteady Reynolds-averaged Navier-Stokes equations with a non-hydrostatic pressure distribution being incorporated in the model. The governing equations are solved in the conventional sigma co-ordinate system, with a semi-implicit time discretization. A fractional step method is used to enable the pressure to be decomposed into its hydrostatic and hydrodynamic components. At every time step one ÿve-diagonal system of equations is solved to compute the water elevations and then the hydrodynamic pressure is determined from a pressure Poisson equation. The model is applied to three examples to simulate unsteady free surface ows where non-hydrostatic pressures have a considerable e ect on the velocity ÿeld. Emphasis is focused on applying the model to wave problems. Two of the examples are about modelling small amplitude waves where the hydrostatic approximation and long wave theory are not valid. The other example is the wind-induced circulation in a closed basin. The numerical solutions are compared with the available analytical solutions for small amplitude wave theory and very good agreement is obtained.

A nonhydrostatic version of FVCOM: 1. Validation experiments

“…For example, the nonhydrostatic pressure equation in UNTRIM [Casulli, 1999;Casulli and Zanolli, 2002] is solved with an assumption that q = 0 in the computational cell nearest the free surface. This simplification is also used in many other nonhydrostatic models [e.g., Zhou and Stansby, 1999;Stansby and Zhou, 1998;Koçyigit et al, 2002;Deponti et al, 2006]. This approach, as pointed out by Yuan and Wu [2004] and also confirmed in our experiment, can lead to a phase error in the free surface elevation.…”

“…The surface standing wave problem has been a benchmark test case for many other nonhydrostatic models [Casulli and Stelling, 1998;Koçyigit et al, 2002;Zijlema and Stelling, 2005;Kanarska et al, 2007]. Casulli [1999] found that the projection method used in his semi-implicit nonhydrostatic model can cause severe free surface damping.…”

Application of detached-eddy simulation to free surface flow over dunes

“…It is well known that the hydrostatic assumption is invalid when the vertical acceleration induced by flow over a sharp bed is no longer much less than the gravity; instead, the model must be extended to a fully hydrodynamic mode. To improve the predictions of shallow flows, fully hydrodynamic models are often implemented by decomposing the pressure into hydrostatic and non-hydrostatic components via means of a predictor-corrector method (Casulli, 1999;Casulli & Zanolli, 2002;Chen, 2003;Fringer, Gerritsen, & Street, 2006;Jankowski, 1999;Kocyigit, Falconer, & Lin, 2002;Engineering Applications of Computational Fluid Mechanics 557 Zhang, Liu, & Xue, 2006). In the present work, the fully hydrodynamic model with free surface has been switched from RANS to DES.…”

Development of a 3D non-hydrostatic pressure model for free surface flows

“…However, in some flows in which the ratio of the wave length to the depth is small, this approximation is inaccurate. More recently, as computer power has increased dramatically, a few numerical models have been developed that determine the non-hydrostatic pressure by solving a pressure-Poisson equation [2,6,10]. The numerical techniques for the pressure-Poisson equation are usually either the semi-implicit method for the pressure-linked equation (simple)-family methods [8] or fractional time step methods [6].…”

“…In other non-hydrostatic models [2,6], only parts of the equations are treated implicitly and then the resulting matrix inverted inexpensively. For example, the water surface elevation and the vertical diffusion terms in the momentum equations are discretised implicitly in Casulli [2].…”