近似到O（μ~2）阶完全非线性的Boussinesq水波方程

Boussinesq wave equations with fully nonlinear characteristics at order O（μ~2）

为了获得具有更优性能的波浪传播数学模型,对一组近似到O(μ2)阶完全非线性的Boussinesq方程进行了改进,加强过程保留高阶非线性项,改进后的方程色散性能和非线性性能均有提高.该方程可以简化为多个以水深平均速度表达的二阶Boussinesq类水波方程.理论分析了加强对方程二阶非线性和三阶非线性的影响,并将其同传统加强方式进行了比较,结果表明,含高阶非线性项的加强方式得到的方程性能更好.基于该方程,在非交错网格下建立了基于有限差分方法的高精度数值模型.利用数值模型模拟波浪在潜堤上的传播变形,探讨了2种不同加强方式以及非线性对数值结果的影响,结果表明,加强过程保留高阶非线性的方程模拟结果更佳.

To obtain the mathematical model of wave propagation in a better performance,a new set of Boussinesq equations with full nonlinearity approximating to the order O（μ2） was developed by enhancing an existing set of Boussinesq equations.Different from the traditional enhancement of Boussinesq-type equations,all the nonlinear terms were retained,improving both the dispersion performance and nonlinear performance of the improved equation.This equation could be simplified to a set of second-order Boussinesq-type wave equations expressed by depth-averaged velocity.The effect of the enhancement on the second-order nonlinearity and third-order nonlinearity of the equation was analyzed theoretically and compared with that of the traditional enhancement method.Theoretical results reveal that the Boussinesq model enhanced with higher order nonlinear terms has better nonlinear properties.Next,a high-precision numerical equation model was built based on a non-staggered grid system using the finite difference method.The numerical simulation was conducted for regular wave propagation over a submerged breakwater,and the numerical results were compared with the experimental data to demonstrate the influences of the nonlinearity and two types of enhancements on the numerical result.Results show that the Boussinesq model with higher order nonlinear terms can more accurately simulate the nonlinear evolution of water waves,and the demonstrated enhancement can be used as a reference for improving the nonlinear properties of similar Boussinesq-type models.