适合极端深水的双层高阶Boussinesq水波方程

Two-layer high-order Boussinesq model for water waves in extremely deep water

为精确描述深水强非线性波浪运动,本文推导了适用于极端水深、具有高精度色散和非线性特征的双层Boussinesq水波方程。首先把流体虚拟地划分为上下两层,对上下两层的速度势分别在静水面处和交界面处沿水深做泰勒展开,任一点速度可用此两处速度表达;其次在两层流体的中间水深位置上选择速度变量,进一步用两个计算水平速度矢量和两个垂向速度分量取代它们,依此速度表达流场内任一水深处的速度;最后结合自由表面的运动学方程和动力学方程、交界面上速度相等以及海底边界条件,推导了双层高阶Boussinesq水波方程。对该方程进行傅立叶分析,方程色散关系式为Padé(18,20),当分层位置为0.12倍静水深时,该方程具有非常优良的线性和非线性性能。在1%误差下,相速度适用水深可达kh=210,沿水深的速度剖面分布最大适用水深可达kh=114,二阶和差频最大适用水深可达kh=103。

To accurately describe strongly nonlinear wave motion in deep water, a new two-layer Boussinesq model for water waves is derived in this paper with excellent dispersive and nonlinear properties in extremely deep water. First, we separated the fluid into two parts：the upper layer and lower layer. Then, using Taylor expansion, we ex-panded the velocity potential in the vertical direction at the still water surface and interface, and the velocity at ar-bitrary water depths can be expressed by the velocities defined at z=0 and z=-h1 , respectively. Second, we re-placed these two velocities defined at z =0 and z=-h1 with two velocities defined at midwater depths within the two layers, which were further replaced by two computational velocities. Then, other velocities at arbitrary water depths could be expressed by these computational velocities. Finally, by applying this velocity information to dynamic and kinematic equations at the surface elevation to the velocity connection condition at z=-h1 and to the bottom condi-tion, we derived a two-layer Boussinesq model. A Fourier analysis is conducted to this model, and the linear dis-persion expression is Padé （18, 20）. Moreover, when the interface water depth is set to h1= 0.12 h, the model exhibits extremely dispersive and highly nonlinear properties. Within 1%error, the model can be applicable to max-imum water depths kh=210 for phase celerity, kh=114 for vertical profile of the velocities, and kh=103 for super-and sub-harmonics.