关于Boussinesq型水波方程理论和应用研究的综述

A review on the theory and application of Boussinesq-type equations for water waves

Boussinesq型方程是研究水波传播与演化问题的重要工具之一,本文就1967-2018年常用的Boussinesq型水波方程从理论推导和数值应用两个方面进行了回顾,以期推动该类方程在海岸(海洋)工程波浪水动力方向的深入研究和应用。此类方程推导主要从欧拉方程或Laplace方程出发。在一定的非线性和缓坡假设等条件下,国内外学者建立了多个Boussinesq型水波方程,并以Stokes波的相关理论为依据,考察了这些方程在相速度、群速度、线性变浅梯度、二阶非线性、三阶非线性、波幅离散、速度沿水深分布以及和(差)频等多方面性能的精度。将Boussinesq型水波方程分为水平二维和三维两大类,并对主要Boussinesq型水波方程的特性进行了评述。进而又对适合渗透地形和存在流体分层情况下的Boussinesq型水波方程进行了简述与评论。最后对这些方程的应用进行了总结与分析。

Boussinesq-type equation is one of the important tools for simulating the propagation and evolution of water waves. The theoretical derivation and numerical application of the Boussinesq-type water wave equation dating back to 1967 are reviewed with the hope of promoting its deep development and application in the fields of coastal and ocean engineering. From the theoretical point of view, the derivation of such equations mainly starts from Euler equations or Laplace equations. Under the conditions of certain nonlinearity and gentle slope assumptions, a variety of Boussinesq-type water wave equations have been proposed worldwide. Through the comparisons with the related theories of Stokes waves, these equations are investigated with respect to phase velocity, group velocity, linear shoaling gradient, second-order nonlinearity, third-order nonlinearity, dispersion characteristics due to amplitude dispersion, velocity distribution along the vertical column, sub-and super harmonics etc. The majority of Boussinesq-type equations in literature for waves are reviewed and grouped into two categories, namely horizontal two-dimensional type and three-dimensional type. The usage of Boussinesq-type equations involved with permeable media and the presence of fluid stratification are also briefly described and commented. Finally, the application of these equations is summarized and analyzed.