A numerical model based on the Saint-Venant equations (one-dimensional shallow water equations) is proposed to simulate shallow flows in an open channel with regular and irregular cross-section shapes. The Saint-Venant equations are solved by the finite-volume method based on Godunov-type framework with a modified Harten, Lax, and van Leer (HLL) approximate Riemann solver. Cross-sectional area is replaced by water surface level as one of primitive variables. Two numerical integral algorithms, compound trapezoidal and Gauss–Legendre integrations, are used to compute the hydrostatic pressure thrust term for natural streams with arbitrary and irregular cross-sections. The Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and second-order Runge–Kutta methods is adopted to achieve second-order accuracy in space and time, respectively. The performance of the resulting scheme is evaluated by application in rectangular channels, trapezoidal channels, and a natural mountain river. The results are compared with analytical solutions and experimental or measured data. It is demonstrated that the numerical scheme can simulate shallow flows with arbitrary cross-section shapes in practical conditions.

中文翻译：

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基于有限体积Godunov型格式的规则断面和不规则断面明渠水流数值模拟

提出了一种基于圣维南方程（一维浅水方程）的数值模型来模拟具有规则和不规则横截面形状的明渠中的浅水流。Saint-Venant 方程通过基于 Godunov 型框架的有限体积法求解，并使用改进的 Harten、Lax 和 van Leer (HLL) 近似 Riemann 求解器。横截面积被水面高度取代为原始变量之一。两种数值积分算法，复合梯形积分和高斯-勒让德积分，用于计算具有任意和不规则横截面的自然流的静水压力推力项。采用单调上游中心守恒定律（MUSCL）和二阶龙格-库塔方法分别实现空间和时间的二阶精度。通过在矩形渠道、梯形渠道和天然山区河流中的应用来评估所得方案的性能。将结果与分析解决方案和实验或测量数据进行比较。证明了该数值方案可以在实际条件下模拟任意截面形状的浅水流。