Research On Boussinesq Equations In Curvilinear Orthogonal Coordinate System

As surface waves propagate from deep to shallow water, the wave will take series of transformation including shoaling, refraction, diffraction, reflection, breaking and energy dissipation due to the effect of topography and various hydraulic structures. Boussinesq-type equations, which include the effect of the lowest order effects of nonlinear and frequency, has been shown to provide an accurate description of wave transformation in coastal regions. However, owing to the assumptions of weak dispersion and weak nonlinearity, the standard Boussinesq equations derived by Peregrine are restricted to shallow water areas and to small nonlinear. In this paper, based on summarizing previous numerical studies on wave transformations, several works are documented:Based on the mass conservation equation and Euler's equation, the extended form of Boussinesq equations is derived by using the velocity at an arbitrary water depth as the independent variable, and several terms are added into governing equations to model the effects of bottom friction, wave breaking and subgrid turbulent mixing. Compared with the standard Boussinesq equations (Peregrine), the new alternative form of equation significantly improves the dispersion and nonlinearity properties of equations, making them applicable to a wider range of water depths.The improved slot technique, which maintains mass conservation in the presence of artificial slots, is used to treat the problem of moving shoreline. The improved technique can simulate the wave runup more accurately. The absorbing boundary condition is tackled easily and properly by using sponge layer technique.The expression of the improved Boussinesq equations in curvilinear orthogonal coordinate system is derived. On the basis of Poission equation conversion, the methods to generate curvilinear orthogonal grids are introduced, and then the two-dimensional numerical wave model under curvilinear orthogonal coordinate system is established.A composite 4-th order Adams-Bashforth-Moulton scheme is used to solve the equations. With this higher-order scheme, the accuracy of numerical computation results is well ensured. Furthermore, a numerical filter method is applied to eliminate some undesired short waves generated as the program runs, which prevents the computed results of the numerical model from distortions.The numerical model is tested by computing wave field for several examples of laboratory experiment, and agreement between model results and availableexperimental data is found to be quite reasonable, which demonstrates the model's ability to simulate wave shoaling, refraction, diffraction and reflection.The model is also applied to study the wave set-up, wave breaking and wave-induced current, and the computed results indicate are well agree with measured data.In addition, numerical simulations of the Berkhoff classical laboratory experiment using linear mild-slop model and nonlinear mild-slop model which is developed by introducing Li's improved nonlinear dispersion relation (2003) into the mild-slop equation are undertaken, and computed results are used to compare with those of the model established in the paper. Comparison results indicate that the new model can give considerably accurate computed results. Moreover, it is confirmed that more desired results could be obtained by utilizing the Li's new dispersion relation (2003) instead of the linear form.