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An implicit finite volume model in sigma coordinate system is developed to simulate two-dimensional (2D) vertical free surface flows, deploying a non-hydrostatic pressure distribution. The algorithm is based on a projection method which solves the complete 2D Navier-Stokes equations in two steps. First the pressure term in the momentum equations is excluded and the resultant advection-diffusion equations are solved. In the second step the continuity and the momentum equation with only the pressure terms are solved to give a block tri-diagonal system of equation with pressure as the unknown. This system can be solved by a direct matrix solver without iteration. A new implicit treatment of non-hydrostatic pressure, similar to the lower layers is applied to the top layer which makes the model free of any hydrostatic pressure assumption all through the water column. This treatment enables the model to evaluate both free surface elevation and wave celerity more accurately. A series of numerical tests including free-surface flows with significant vertical accelerations and nonlinear behaviour in shoaling zone are performed. Comparison between numerical results, analytical solutions and experimental data demonstrates a satisfactory performance.A. AHMADI, P. BADIEI AND M. M. NAMIN computer power in recent years, three-dimensional (3D) models are extensively developed and applied for such problems. One of the main difficulties of such models is the proper handling of free surface moving boundary, especially for the cases such as short period waves. The moving free surface forms the upper boundary of the computational domain; however, its position also constitutes a part of the solution yet to be defined. In many 3D free-surface models, it is assumed that the vertical acceleration is small so that the idea of hydrostatic pressure can be applied. However, this statement is only applicable for simulating flows where the horizontal scale of motion is much larger than its vertical scale. For cases such as short period waves, rapidly changing bed topographies, and stratification due to strong density gradients, the hydrostatic pressure distribution is no longer valid and employing a dynamic pressure distribution, based on the complete form of NSE, is indispensable.In recent years, the development of non-hydrostatic models has been the topic of many research activities. Chorin [1] developed an explicit projection method that obtains the projected intermediate velocities after solving advection and diffusion terms explicitly at each time step, and then corrects the projected velocities by solving the pressure Poisson equation (PPE). Although the projection method may cause splitting errors in some applications [2], it has the advantage of using different schemes for different stages. Mahadevan et al. [3] proposed a non-hydrostatic ocean model using a semi-implicit control volume method. Stansby and Zhou [4] developed a two-dimensional (2D) vertical flow model for simulating the non-hydrostatic problem, using a semi-implicit time ...

SUMMARYA non-hydrostatic finite volume model is presented to simulate three-dimensional (3D) free-surface flows on a vertical boundary fitted grid system. The algorithm, which is an extension to the previous two dimensional vertical (2DV) model proposed by Ahmadi et al. (Int. J. Numer. Meth. Fluids 2007; 54(9):1055-1074), solves the complete 3D Navier-Stokes equations in two major steps based on projection method. First, by excluding the pressure terms in momentum equations, a set of advection-diffusion equations are obtained. In the second step, the continuity and the momentum equations with the remaining pressure terms are solved which yields a block tri-diagonal system of equations with pressure as the unknown. In this step, the 3D system is decomposed into a series of 2DV plane sub-systems which are solved individually by a direct matrix solver. Iteration is required to ensure convergence of global 3D system. To minimize the number of vertical layers and subsequently the computational cost, a new top-layer pressure treatment is proposed which enables the model to simulate a range of surface waves using only 2-5 vertical layers.