On direct methods in water-wave theory
On direct methods in water-wave theory
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DOI：
10.1017/s0022112088003222
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发表时间：
1988-12
影响因子：
3.7
通讯作者：
J. Shields;W. Webster
中科院分区：
文献类型：
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作者：
J. Shields;W. Webster
根据Kantorovich的“直接”或变分方法，导出了任意时变材料表面之间三维无粘流的模型方程。这种方法导致了近似理论的层次结构，每个理论都具有更高的空间近似级别和复杂性。结果表明，这些方程在本质上等同于Green&Naghdi(1974,1976)的“有向流体片理论”。该理论可用于研究长波在有限水深水中的传播，因此与经典的Rayleigh-Boussinesq摄动法的理论形成了竞争。为了证明这种方法的优势，我们比较了对水平海底上稳定的二维海浪的预测。数值解表明，直接理论比摄动理论收敛得更快。此外，高阶直接理论的方程包含与极限高度的波有关的奇点，确实可以相对准确地预测这种波。最后，直接理论的适用范围要大得多：可以模拟短至水深三倍的海浪。这基本上是一种深水条件，远远超出了瑞利-布西内斯克方法的收敛范围。
Model equations for three-dimensional, inviscid flow between two arbitrary, time-varying material surfaces are derived using a ‘direct’ or variational approach due to Kantorovich. This approach results in a hierarchy of approximate theories, each of a higher level of spatial approximation and complexity. It can be shown that the equations are equivalent in substance to ‘the theory of directed fluid sheets’ of Green & Naghdi (1974, 1976). The theory can be used to study the propagation of long waves in water of finite depth and, as such, competes with theories derived using the classical Rayleigh–Boussinesq perturbation methods. In order to demonstrate that there is an advantage to the present approach, we compare predictions for steady, two-dimensional waves over a horizontal bottom. Numerical solutions indicate that the direct theory converges more rapidly than the perturbation theories. Also, the equations of the higher-order direct theories contain singularities related to waves of limiting height, and indeed such waves can be predicted with relative accuracy. Finally, the range of applicability of the direct theory is far greater: waves as short as three times the water depth can be modelled. This is essentially a deep-water condition, well beyond the range of convergence of the Rayleigh–Boussinesq approach.