曲线坐标系下波浪传播的数学模型及其比较与验证

引入自适应网格技术，建立了曲线坐标系下考虑能耗影响的缓变水深水域波浪传播的数值模拟模型，通过在模型中引入能量耗散项以更好地处理全透射边界。基于原型缓坡方程，完善了等水深的轴对称双导堤水域内波浪传播的精确解表达式及其计算结果；提供了等水深的环形河道水域内较为详细的精确解。在极坐标系下就等水深的轴对称双导堤水域内波浪的传播，推导了缓坡方程的小角度抛物近似模型的解析解表达式。就上述两水域内波浪传播的数值模拟模型的数值解、抛物近似模型的解、精确解进行了详细比较，说明数值解与精确解相吻合、数值模拟模型比小角度和大角度抛物近似模型具有更高的精度。数值模拟模型能较好地模拟水深复杂变化的地形上波浪的传播，能有效反映波浪传播过程中的多种物理现象。

Mathematical models for wave propagation in curvilinear coordinates and comparisons among different models

In curvilinear coordinates, a numerical simulation model for wave propagation including energy dissipation over a slowly varying topography is presented employing self-adaptive grids. The energy dissipation term is used to treat the outgoing boundary in the numerical model. Based on the original mild-slope equation, the expression and the calculated results of the exact solution for waves between symmetric diverging breakwaters in uniform water depth are improved; and the detailed exact solutions for waves in a circular channel are also derived. An analytical solution is also deduced from the small-angle parabolic approximation of the mild-slope equation in polar coordinate system. For the above two cases the results from numerical simulation, exact solution and parabolic approximation and compared. Systematical tests show that the results of the numerical simulation are in good agreement with those of the original mild-slope equation, and that the numerical simulation gives better results than the small-angle parabolic approximation. The numerical simulation model is able to give good results of simulating the wave propagation in waters of complicated topography and effectively predict essential processes of wave propagation.