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.

The Kuroshio current-induced island wake downstream of Green Island, Taiwan, is studied using satellite imagery and a depth-averaged shallow-water model. Spatial–temporal scales, such as aspect ratio, dimensionless width, and Strouhal number, as well as propagation speed of the vortices, are quantified for Reynolds numbers 100, 200, and 500, respectively. It is found that the aspect ratio is between 1.76 and 2.72 obtained from satellite images and between 3.50 and 3.72 from numerical simulations. The dimensionless width is in the range of 1.84 to 1.92 from satellite images, and 1.29 to 1.33 from numerical modelling. The Strouhal number is between 0.17 and 0.20 from numerical simulations. Computed results are compared with theory, available satellite imagery, and in situ measurements, and show reasonable agreement. The magnitude of the propagation speed of the vortices is found to be of the same order as that of the Kuroshio, implying non-linearity as well as interactions of vortices, and the Kuroshio is strong. The Coriolis force affects the free-surface distribution, but its effect on vortices and flow field is insignificant.

Hsu, T.-W.; Doong, D.-J.; Hsieh, K.-J., and Liang, S.-J., 2015. Numerical study of monsoon effect on Green Island wake. The wind effect on Kuroshio-induced island wake downstream of Green Island, Taiwan, is studied using a depth-averaged shallow-water model. The seasonal monsoon effect on spatial–temporal scales, such as the aspect ratio, dimensionless width, and Strouhal number, as well as the relative propagation speed of Green Island vortices, is studied. It is found that a NE monsoon has a more pronounced effect on wake characteristics than a SW monsoon does in terms of net speed and vorticity, aspect ratio, and dimensionless width. Vortices are pushed back and packed close to the north of the island with a smaller area of recirculation when Kuroshio flows against the NE monsoon; vortices are pushed farther downstream with a larger area of recirculation when Kuroshio flows in favor of the SW monsoon. However, the wind effect on temporal variation of island wake seems less significant. The period of vortex shedding is between 14.2 and 15.3 hours. The corresponding Strouhal number is between 0.116 and 0.125, and the relative propagation speed of vortices is nondimensional, between 0.51 and 0.56. These values are in good agreement with the results of previous reports.

A series of experiments exploring the propagation of a solitary wave over a submerged step were performed using a flow-field visualization measurement system, an image-connection technique as well as model simulations. The experimental data were used to validate a one-layer finite-element non-hydrostatic model and a multi-layer finite-difference non-hydrostatic σ model for various submerged step configurations and wave conditions—combinations of step height ratios d/h, width ratios B/h and solitary wave height ratios H/h, where d denotes the step height, B the step width, H the solitary wave height, and h the still water depth. The main differences between the numerical results and the experimental data are highlighted. The effect of the height and width of the submerged step as well as the wave height of the solitary wave are quantified in terms of reflection (R), transmission (T), and energy dissipation (D). Through a series of numerical experiments, an optimal combination of the height ratio d/h, width ratio B/h, and solitary wave height ratio H/h for breakwater design for coastal protection is suggested.