Hunter Rouse was the father of the Fluid Mechanics of Open Channel Flow. In this review article the fundamental advances on the non-hydrostatic modeling of shallow open channel flow over topography, since the publication of his 1938 book, are discussed. Flows over uneven topography, like at open channel controls, occurs in a short length, thus rendering ideal fluid flow theory an adequate mathematical tool. This paper emphasis is on depth-averaged modeling of these flows using Boussinesq equations, given the significant advances basically since the 80’s. The one-dimensional steady flow model from Picard iteration is reviewed in detail, including energy and momentum formulations, flow at a weir crest, and on a slope. A new numerical scheme is developed for the solution of transcritical steady weir flow, showing excellent match with experiments. A new derivation for the two-dimensional depth-averaged unsteady ideal fluid model is presented based on a continuum mechanics formulation. The result is the Serre–Green–Naghdi equations (SGNE). It is demonstrated that the extended Bernoulli equation derived from Picard iteration is an integral form of the SGNE for 1D steady flow. A robust MUSCL-Hancock finite volume model is developed to solve the unsteady 1D SGNE, showing excellent agreement with the steady transcritical flow solutions previously constructed. The accuracy of the MUSCL-Hancock solver is critically assessed using a higher-order solver for the SGNE.

中文翻译：

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基于深度平均势方程的地表非线性浅水流模拟

亨特·劳斯（Hunter Rouse）是明渠流动的流体力学之父。在这篇综述文章中，讨论了自从他的1938年著作出版以来，浅水通道在地形上的非静力学建模的基本进展。像在明渠控制处一样，在不平坦地形上的流动以较短的长度发生，因此使理想的流体流动理论成为适当的数学工具。本文的重点是使用Boussinesq方程对这些流进行深度平均建模，因为自80年代以来已取得了重大进展。详细回顾了Picard迭代的一维稳态流模型，包括能量和动量公式，堰顶处和斜坡上的流。为解决跨临界稳定堰流问题，提出了一种新的数值方案，与实验结果非常吻合。基于连续力学公式，提出了二维深度平均非稳态理想流体模型的新推导。结果是Serre-Green-Naghdi方程（SGNE）。证明了从Picard迭代得到的扩展的Bernoulli方程是SGNE对于一维稳定流的积分形式。开发了鲁棒的MUSCL-Hancock有限体积模型来解决非稳态一维SGNE问题，与先前构造的稳定跨临界流解决方案具有极好的一致性。MUSCL-Hancock求解器的精度使用SGNE的高阶求解器进行了严格评估。证明了从Picard迭代得到的扩展的Bernoulli方程是SGNE的一维稳态流的积分形式。开发了鲁棒的MUSCL-Hancock有限体积模型来解决非稳态一维SGNE问题，与先前构造的稳定跨临界流解决方案具有极好的一致性。MUSCL-Hancock求解器的精度使用SGNE的高阶求解器进行了严格评估。证明了从Picard迭代得到的扩展的Bernoulli方程是SGNE的一维稳态流的积分形式。开发了鲁棒的MUSCL-Hancock有限体积模型来解决非稳态一维SGNE问题，与先前构造的稳定跨临界流解决方案具有极好的一致性。MUSCL-Hancock求解器的精度使用SGNE的高阶求解器进行了严格评估。