A Fully Non-hydrostatic Three-Dimensional Model with an Implicit Algorithm for Free-Surface Flows

A three-dimensional (3D) implicit numerical model based upon the Navier-Stokes equations (NSE) for free-surface flows is developed. To avoid solving the fully 3D Pressure Poisson Equation (PPE), an algorithm is developed by decoupling the original matrix system into a series of sub-matrix systems corresponding to two-dimensional (2D) vertical planes. The features of the model include (1) a novel treatment for the non-hydrostatic pressure at the top-layer cell in a staggered grid, consistent with the overall non-hydrostatic pressure discretization; (2) the option of partial-cell treatment on bottom topographies, enabling the model to simulate free-surface flows interacting with uneven bottoms with a reasonable vertical resolution; (3) simultaneously solving all flow-field components (e.g., the velocities, pressure, and free-surface elevation) within each time step; and (4) capability of simulating both 3D short waves and long waves with a second-order accuracy in both time and space. The developed model is validated by two free-surface flow problems, in which the vertical accelerations are considerable and thus the pressure fields are in non-hydrostatic distributions. The numerical results are in excellent agreements with the analytical solutions and experimental data, illustrating the capability of the algorithm for simulating 3D free surface flows.