One-dimensional and two-dimensional Green-Naghdi equation solvers for shallow flow over uniform and non-uniform beds
Jalali, Mohammad Reza
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Numerical simulation of wave behaviour in shallow and deep water is often a key aspect of ocean, coastal, and river hydrodynamic studies. This thesis derives nonlinear one- and two-dimensional level I Green-Naghdi (GN) equations that model the motions of free surface waves in shallow water over non-uniform bed topography. By assuming fitted velocity profiles through the depth, GN equations are simpler than Boussinesq equations, while retaining the wave dispersion property. Implicit matrix solvers are used to solve the spatially discretised 1D and 2D GN equations, with a 4th order Runge Kutta scheme used for time integration. To verify the developed numerical solvers of 1D GN equations, a series of simulations are undertaken for standard benchmark tests including sloshing in a tank and solitary wave propagation over a flat bed. In all cases, grid convergence tests were conducted. In the sloshing test, both numerical schemes and the analytical solution were in complete agreement for small-amplitude free surface motions. At larger values of initial sloshing amplitude, the nonlinear effects caused the free surface waves to steepen, and eventually the numerical simulations became unstable. This could be resolved in future using a shock-capturing scheme. Excellent agreement was achieved between the numerical predictions and analytical solution for solitary waves propagating. The 2D GN equation solver was then verified for the benchmark tests of Gaussian hump sloshing and solitary wave propagation in closed basin. The predicted free surface motions for Gaussian hump sloshing were in good agreement with linear Fourier analytical solutions for a certain initial period, after which nonlinear effects started to dominate the numerical solution. A reversibility check was undertaken. Nonlinear effects were investigated by increasing the amplitude of the hump, and applying harmonic separation (by comparison against slosh predictions for a corresponding Gaussian trough). It was found that the even harmonic components provided a useful indication of the nonlinear behaviour of the 2D GN equations. 2D GN simulations of a 0.6 m amplitude solitary wave propagation in 1 m deep water over a flat, horizontal bed confirmed that nonlinear interaction was correctly modelled, when the solitary wave hit a solid wall and its runup reached 2.36 m which was 0.36m more than the linear analytical solution and almost identical to a second order solution.