Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space
Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space
复制标题
明渠中经过扰动的稳定二维自由表面流：Korteweg-De Vries 方程的解和弱非线性相空间的分析
作者：
B. Binder
二维自由表面流在明渠中通过扰动是流体力学中的经典问题--在过去的两个世纪中受到了相当大的关注（例如，参见Lamb's Approximation，1879）。追溯到罗素在1834年对平移巨浪的实验观测，Korteweg和de弗里斯（1895年）等人推导出一个非受迫方程来描述模拟孤立波所需的非线性和色散之间的平衡。最近，Akylas（1984）推导出了一个强迫KdV方程来模拟自由表面上的压力分布（例如，船）。从那时起，强迫KdV方程已被证明是一个有用的模型近似的二维流动过去的扰动在一个开放的渠道。在本文中，我们审查的定常解的强制KdV方程的四种类型的本地化干扰：（i）平板分离两个自由表面;（ii）紧凑的隆起，或在通道底部地形下降;（iii）紧凑的压力分布在自由表面上和（iv）一个阶梯式的变化，否则恒定的水平的通道底部地形。此外，Dias和Vanden-Broeck（2002）开发了一种相平面方法来分析凸块上的流动，并且这种一般方法也被应用于其他三种类型的强迫（参见Binder等人，2005年至2015年）。在本研究中，我们使用相平面法，以十一种基本的流动型态来分类强迫KdV方程的定常解。此外，考虑到解决方案是无波的远上游和远下游，我们比较KdV模型近似的均匀流条件下的远场与完整的问题的精确解。特别是，我们推导出一个新的KdV模型近似的上游无量纲流量，这是方便地给出已知的下游无量纲流量。
Two-dimensional free-surface flow past disturbances in an open channel is a classical problem in hydrodynamics—a problem that has received considerable attention over the last two centuries (e.g., see Lamb’s Treatise, 1879). With traces back to Russell’s experimental observations of the Great Wave of Translation in 1834, Korteweg and de Vries (1895), and others, derived an unforced equation to describe the balance between nonlinearity and dispersion required to model the solitary wave. More recently, Akylas (1984) derived a forced KdV equation to model a pressure distribution on the free-surface (e.g., a ship). Since then, the forced KdV equation has been shown to be a useful model approximation for two-dimensional flow past disturbances in an open channel. In this paper, we review the stationary solutions of the forced KdV equation for four types of localised disturbances: (i) a flat plate separating two free surfaces; (ii) a compact bump, or dip in the channel bottom topography; (iii) a compact distribution of pressure on the free surface and (iv) a step-wise change in the otherwise constant horizontal level of the channel bottom topography. Moreover, Dias and Vanden-Broeck (2002) developed a phase plane method to analyse flow over a bump, and this general approach has also been applied to the three other types of forcing (see Binder et al., 2005–2015, and others). In this study, we use eleven basic flow types to classify the steady solutions of the forced KdV equation using the phase plane method. Additionally, considering solutions that are wave-free both far upstream and far downstream, we compare KdV model approximations of the uniform flow conditions in the far-field with exact solutions of the full problem. In particular, we derive a new KdV model approximation for the upstream dimensionless flow-rate which is conveniently given in terms of the known downstream dimensionless flow-rate.