为提高Boussinesq水波方程中的速度精度,以最高空间导数为2的双层Boussinesq方程为研究对象,提出增加带有常系数的三阶项以修正速度公式.适用水深在0<kh<8(k为波数,h为静水深)范围内,以方程的水平速度和垂向速度与Stokes线性波速度解析解的积分误差最小为目标,优化系数取值.在1%误差内,改进公式水平速度和垂向速度的适用水深kh分别为7.34和7.83,均比原计算公式适用范围大.利用数值模型对稳态波和聚焦波演化进行计算,将最大波峰下的水平速度分别与流函数解析解和试验结果进行对比,发现改进后的吻合程度更高,验证了改进公式的有效性.研究表明,改进公式的速度精度有较大幅度提高,该方法可为其他Boussinesq模型的速度场改进提供重要参考.

In order to improve the accuracy of velocity formulation in a Boussinesq-type wave model, with a two-layer Boussinesq-type model with the highest spatial derivative of 2 being chosen as the research object, a third-order term with constant coefficient is proposed to modify the velocity formulation. The coefficient is optimized by minimizing the error between the summation of the integration of horizontal and vertical velocities of the equation and that of the analytical linear Stokes wave velocity components in the range of 0<kh< 8 (where k is wave number, h is still water depth). At a 1% tolerance error, the applicable water depths of the modified formulations for horizontal and vertical velocities are up to kh=7.34 and kh=7.83, respectively, which are larger than those of the original formulations. The evolution of the steady-state wave and the focused wave is numerically simulated by using the numerical model. The horizontal velocity under the maximum surface elevation crest is in good agreements with the analytical solution of stream function and published experimental data, which verifies the effectiveness of the modified formulations. The studies show that the velocity accuracy of the improved equation is greatly improved. This method provides an important reference for the improvement of velocity field of other Boussinesq-type models.