适合渗透海床波浪运动的三维Boussinesq型方程

New 3D Boussinesq-type model for water waves on permeable seabeds

为考虑孔隙介质对波浪传播变形带来的衰减效应,在渗透介质流体的控制方程中引入线性阻力、非线性阻力和惯性力。自由表面采用精确的运动学和动力学边界条件,水底边界条件采用精确的动力学边界条件,自由水体和渗透介质水体之间满足垂向速度连续、水平速度满足动量相等条件。首先推导最高空间导数为3的以两组计算速度表达的适合单层渗透海床波浪运动的三维Boussinesq型水波方程。其次,对新给出的方程进行傅里叶分析,并将方程的相速度及衰减率与Stokes线性波的解析解进行比较。在1%误差下,相对水深h_(2)/h_(1)=0.1~10,方程的解析解在无因次水深h_(1)/L<1.0(深水波长L=gT^(2)(/2π))范围内与Stokes线性波的解析解都有较好的吻合,这超过了历史文献中任一个Boussinesq型水波方程的适用范围。进而建立立面二维水槽数值模型,并利用预报-校正-迭代的有限差分方法对数值模型进行求解,在变量对时间的导数求解中选择混合四阶AdamsBashforth-Moulton格式。最后,利用数值模拟再现波浪在渗透地形上的传播变形,并与相关试验结果进行比较,二者吻合程度较高。

In order to consider the attenuation effect of the pore medium on the wave propagation deforma⁃tion,linear resistance,nonlinear resistance and inertial force are introduced in the governing equations of the permeable medium fluid.The exact kinematic and kinetic boundary conditions are used on free surface,and the exact kinetic boundary conditions are adopted on the underwater boundary conditions,and the vertical ve⁃locity satisfies the continuity and the horizontal velocity satisfies the momentum equality condition between the free water and the water in permeable medium.Firstly,the three-dimensional Boussinesq-type waterwave equations expressed in two sets of computational velocities with the highest spatial derivative of 3 were derived to suit the wave motion of single-layer permeable seabed.Secondly,Fourier analysis was performed on the newly-presented equations,and the phase velocities and decay rates of the equations were compared with the analytical solutions of Stokes linear waves.The analytical solutions of the equation are in good agree⁃ment with the analytical solutions of Stokes linear waves in the range of a dimensionless water depth of h_(1)/L<1.0(deep water wavelength L=gT^(2)/(2π))at 1%error with a relative water depth of h_(2)/h_(1)=0.1~10=0.1-10,which exceeds the range of applicability with any Boussinesq-type model in history.Further,a numerical model of the two-dimensional flume was developed and the numerical model was solved using a prediction-correction-it⁃erative finite-difference method,and a composite fourth-order Adams-Bashforth-Moulton scheme was cho⁃sen for time iteration.Finally,the wave evolution over the permeable terrain was simulated and numerical simulations were carried out.Comparison with the relevant experimental results shows a good agreement.