浅水非线性长波演化的高精度模拟

波-波相互作用引起不同频率波浪之间的能量交换，在浅水非线性效应更为明显。波浪演化经历非稳态与稳态过程，为了精确模拟这一过程，要求水波模型具有足够高的线性与非线性精度。基于一组具有良好色散性和非线性的双层Boussinesq水波方程，在非交错网格下建立了数值模型，模型采用混合4阶Adams-Bashforth-Moulton格式的时间步进。为了模拟造波板运动产生的波浪，数值模型中选用了适合该模型的两点造波，并在首个周期内边界两点波浪乘以从0到1缓变的正弦函数。模型计算了水槽中非线性长波的演化过程，将固定点的波面位移、不同频率的波幅和物理模型试验结果进行了比较，二者与试验结果的吻合程度均较高。最后模拟了不同波幅下的波浪演化，分析了再现长度随波幅的变化。研究表明，波浪近似线性时再现长度与理论解析基本吻合，伴随波浪非线性增强，再现长度呈现非线性减少。

Accurate numerical simulation of nonlinear long wave evolution over a shallow water depth

Wave-wave interaction induced energy transferring among different wave frequencies, and nonlinear effect is more prominent in shallow water. During the process of wave evolution, there exists unsteady and steady wave transformation. To accurately simulate the whole process, the numerical model for water waves is required to have sufficiently high accuracy in both linear and nonlinear properties. Based on a two-layer Boussinesq-type model with high accuracy in dispersion and nonlinearity, a numerical model is established on non-staggered regular grids. A composite fourth-order Adams-Bashforth-Moulton scheme is used for time integration. To simulate the wave generated by the motion of the wave-making paddle, a two-point wave-making method suitable for the numerical model is used. The waves at the two points near incident boundary are multiplied by a sinusoidal function varying from 0 to 1 in one wave period. The numerical simulations are conducted on the nonlinear wave evolution process in the flume. Surface elevations and higher-order amplitudes at fixed points are compared with the results from physical experimental data, the good agreements are found. Finally, the numerical model is applied to simulate the wave evolutions with different wave amplitudes, and the variation of the recurrence length with respect to wave amplitude is analyzed. The results show that the computed recurrence length basically coincides well with the theoretical analysis for approximate linear waves, and it decreases nonlinearly with the enhancement of wave nonlinearity.