Depth-integrated free-surface flow with non-hydrostatic formulation

Field: Earth and Planetary Sciences (Physical Sciences)

This dissertation presents the formulation of depth-integrated wave propagation and runup models from a system of governing equations for two-layer non-hydrostatic flows. The conventional two-layer non-hydrostatic formulation is re-derived from the continu-ity and Euler equations in non-dimensional form to quantify contributions from nonlin-earity and dispersion and transformed into an equivalent integrated system, which sepa-rately describes the flux and dispersion-dominated processes. The formulation includes interfacial advection to facilitate mass and momentum exchange over the water col-umn. This equation structure allows direct implementation of a momentum conserving scheme and a moving waterline technique to model wave breaking and runup without in-terference from the dispersion processes. The non-hydrostatic pressure, however, must be solved at the layer interface and the bottom simultaneously from the pressure Poisson equation, which involves a non-symmetric 9-band sparse matrix for a two-dimensional vertical plane problem. A parameterized non-hydrostatic pressure distribution is intro-duced to reduce the computational costs and maintain essential dispersion properties for modeling of coastal processes. The non-hydrostatic pressure at mid flow depth is expressed in terms of the bottom pressure with a free parameter, which is optimized to match the exact linear dispersion relation for the water depth parameter up to kd = 3. This reduces the integrated two-layer formulation to a hybrid system with unknown non-hydrostatic pressure at the bottom only and a tridiagonal matrix in the pressure Poisson equation. The hybrid system reduces to a one-layer model for a linear distribution of the non-hydrostatic pressure. Fourier analysis of the governing equations of the two-layer, hybrid, and one-layer systems yield analytical expressions of the linear dispersion and shoaling gradient as well as the super and sub-harmonics transfer functions. The two-layer system reproduces the linear dispersion relation within a 5% error for water depth parameter up to kd = 11. The hybrid system with an optimized free parameter yields the same dispersion relation as the extended Boussinesq equations. The one-layer system shows a major improvement of the dispersion properties in comparison to the classical Boussinesq equations, but is not sufficient to model coastal wave transformation. The linear shoaling gradient serves as analytical tool to measure wave transformation over a plane slope although it is secondary compared to the linear dispersion relation. In comparison to second-order wave theory, the two-layer system shows overall underestimation of the nonlinearity, while the hybrid system reasonably describes the super and sub-harmonics for kd ranging from 0 to 3. The two-layer, hybrid, and one-layer systems share common numerical procedures. A staggered finite difference scheme discretizes the governing equations in the horizontal dimension and the Keller box scheme reconstructs the non-hydrostatic terms in the vertical direction. A semi-implicit scheme integrates the depth-integrated flow in time with the non-hydrostatic pressure determined from a Poisson-type equation. Numerical results are verified and validated through a series of numerical and laboratory experiments selected to measure model capabilities in wave dispersion, shoaling, breaking, runup, drawdown, and overtopping. The two-layer model shows good performance in handling these processes through its integrated structure, but slightly underestimates the wave height in shoaling. The hybrid model provides comparable results with the twolayer system in general and slightly improved performance in shoaling calculations due to better approximation of nonlinearity. The one-layer model exhibits stable and robust performance even when the wave characteristics are beyond its applicable range.