Abstract
A two-dimensional non-hydrostatic shallow-water model for weakly dispersive waves is developed using the least-squares finite-element method. The model is based on the depth-averaged, nonlinear and non-hydrostatic shallow-water equations. The non-hydrostatic shallow-water equations are solved with the semi-implicit (predictor-corrector) method and least-squares finite-element method. In the predictor step, hydrostatic pressure at the previous step is used as an initial guess and an intermediate velocity field is calculated. In the corrector step, a Poisson equation for the non-hydrostatic pressure is solved and the final velocity and free-surface elevation is corrected for the new time step. The non-hydrostatic shallow-water model is verified and applied to both wave and flow driven fluid flows, including solitary wave propagation in a channel, progressive sinusoidal waves propagation over a submerged bar, von Karmann vortex street, and ocean circulations of Dongsha Atolls. It is found hydrostatic shallow-water model is efficient and accurate for shallow water flows. Non-hydrostatic shallow-water model requires 1.5 to 3.0 more cpu time than hydrostatic shallow-water model for the same simulation. Model simulations reveal that non-hydrostatic pressure gradients could affect the velocity field and free-surface significantly in case where nonlinearity and dispersion are important during the course of wave propagation.
Highlights
An accurate prediction of multi-scale ocean waves and flows is important in physical oceanography, marine hydrodynamics, as well as ocean and coastal engineering
One is the Boussinesq-type model attributed to the pioneer work of [9,10], and another approach is proposed by using the Reynolds-Averaged Navier–Stokes (RANS) equations with the non-hydrostatic
Hydrostatic shallow-water models (SWMs) is efficient for modeling shallow water flows where variation of vertical scale is much smaller than the variation of horizontal scale
Summary
An accurate prediction of multi-scale ocean waves and flows is important in physical oceanography, marine hydrodynamics, as well as ocean and coastal engineering. The hydrostatic shallow-water models (SWMs) have been widely used in studying flows in rivers, lakes, estuaries, and oceans [1,2,3]. The hydrostatic assumption is valid in many cases and hydrostatic SWMs have been successfully used in numerous engineering applications. Due to the hydrostatic and non-dispersive assumption of the hydrostatic SWMs, their application to short period waves, abrupt changes in bed topographies, and stratification due to strong density gradients become limited and result in inaccurate predictions [4,5,6,7,8]. Two groups of numerical methods for relaxing the limitations of the hydrostatic and non-dispersive assumption and simulating deep water waves have been developed. One is the Boussinesq-type model attributed to the pioneer work of [9,10], and another approach is proposed by using the Reynolds-Averaged Navier–Stokes (RANS) equations with the non-hydrostatic
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