Transcritical shallow-water flow past topography: finite-amplitude theory
Transcritical shallow-water flow past topography: finite-amplitude theory
复制标题
经过地形的跨临界浅水流：有限振幅理论
DOI：
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发表时间：
2009
影响因子：
3.7
通讯作者：
N. F. Smyth
中科院分区：
文献类型：
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作者：
G. El;R. Grimshaw;N. F. Smyth
我们使用强迫Su-Gardner （SG）方程组的框架考虑了在流动方向上局部化的经过宽底脊的浅水流动，主要关注了当迎面流的弗劳德数接近于1时的跨临界状态。这些方程是全欧拉系统的渐近长波近似，在波浪幅度没有同时展开的情况下获得，因此在再现完全非线性浅水流动的定量特征方面，有望优于通常的弱非线性boussinesq型模型。局部地形上的局部跨临界水力解（产生上游和下游的水力跳变）和描述这些水力跳变分辨率的非定常波纹孔解的组合，用于描述取决于地形高度和弗劳德数组合的各种流动状态。利用近年来发展起来的SG波孔调制理论，推导了跨临界全非线性浅水流的主要参数，如波孔上游和下游的先导孤立波幅值、波孔边缘的速度和阻力。我们的研究结果证实，以前在单向强迫Korteweg-de Vries （KdV）模型框架下发展的描述的大多数特征定性地适用于有限振幅波，而在双向强迫SG系统框架下可以获得定量描述。我们的解析解与强制SG方程的数值模拟结果在这些方程的适用范围内是一致的。
We consider shallow-water flow past a broad bottom ridge, localized in the flow direction, using the framework of the forced Su–Gardner (SG) system of equations, with a primary focus on the transcritical regime when the Froude number of the oncoming flow is close to unity. These equations are an asymptotic long-wave approximation of the full Euler system, obtained without a simultaneous expansion in the wave amplitude, and hence are expected to be superior to the usual weakly nonlinear Boussinesq-type models in reproducing the quantitative features of fully nonlinear shallow-water flows. A combination of the local transcritical hydraulic solution over the localized topography, which produces upstream and downstream hydraulic jumps, and unsteady undular bore solutions describing the resolution of these hydraulic jumps, is used to describe various flow regimes depending on the combination of the topography height and the Froude number. We take advantage of the recently developed modulation theory of SG undular bores to derive the main parameters of transcritical fully nonlinear shallow-water flow, such as the leading solitary wave amplitudes for the upstream and downstream undular bores, the speeds of the undular bores edges and the drag force. Our results confirm that most of the features of the previously developed description in the framework of the unidirectional forced Korteweg–de Vries (KdV) model hold up qualitatively for finite amplitude waves, while the quantitative description can be obtained in the framework of the bidirectional forced SG system. Our analytic solutions agree with numerical simulations of the forced SG equations within the range of applicability of these equations.