Application of higher-level GN theory to some wave transformation problems

Recently derived (Webster et al., 2011), simplified higher-level Green-Naghdi equations (GN-3, GN-5 and GN-7) are used in this work to simulate the transformation of two-dimensional, shallow-water wave problems. The spatial derivatives are discretized through a five-point difference scheme. A new algorithm is developed to solve the resulting block-pentadiagonal matrix. These high-level GN equations are then utilized to develop a numerical wave tank. A wave-maker is placed at the forcing boundary of the tank that uses the stream-function theory to generate nonlinear incident waves. The numerical wave tank is used to analyze the effects of large-amplitude waves passing over a submerged bar. A damping zone is placed near the wave-maker (up-wave side) to absorb the reflected waves from the front side of the submerged bar. Another damping zone is placed at the down-wave side of the computational domain to absorb the radiated waves. In the first test case, the front and back slopes of the bar are both mild (Luth et al., 1994). The waves that evolved over the bar are simulated by using the GN-3, GN-5 and GN-7 equations. The GN-3 equations provide time histories that compare well with the experimental data at different wave gauges, except at the ones behind the bar. The results of the GN-5 and GN-7 equations compare very well with all the experimental data considered here. In the second test case, the front and back slopes of the bar are both steep (Ohyama et al., 1995). The GN-5 equations predict the wave elevation well. In the third test case, the front and back slopes of the bar alternate, one of them being mild and the other one being steep (Zou et al., 2010). Again, the predictions of the GN-5 equations agree with the experimental data well. In all the test cases considered in this work, there are some differences between the GN-3 and GN-5 results after the crest of the bar. Numerical results obtained by the GN-5 and GN-7 equations are almost the same along the wave flume, but the GN-7 equations require more computational time. Therefore, the GN-5 results are accepted here as the converged GN theory results. The numerical validations show that the GN-5 equations can simulate the strongly nonlinear and dispersive waves observed behind the submerged bar crest satisfactorily.