Graph-Theoretical Formulation of Gradually Varied Flow in Open Channel Networks - MATLAB Code

Gradually varied flow frequently occurs in various open channel networks, including irrigation systems, canal networks, river systems, stormwater networks, and sewer systems. This paper presents a novel method to formulate this problem using graph theory. The characteristics and topology of an arbitrary channel network can first be defined using matrix and vector formats. The governing equations for all nodes and channel sections are then elegantly integrated using these matrices and vectors. A compact formulation comprising just two matrix equations is obtained to represent the gradually varied flow in an open channel network. Compared with traditional methods, this new approach eliminates the need to list separate equations for different types of nodes and to sequentially formulate equations for each channel. As a result, it provides a generalized process for modelling any channel network. The method also treats internal nodes as inflow/outflow boundaries, enabling more flexible applications—such as modelling complex drainage systems with manholes receiving runoff. Validation of the proposed method is demonstrated by comparing results from various channel networks with those obtained using traditional approaches.

明渠渐变流（Gradually Varied Flow）广泛存在于各类明渠管网中，涵盖灌溉系统、渠网、河网、雨水管网与污水管网。本文提出了一种基于图论（Graph Theory）的明渠渐变流问题建模新方法。首先可通过矩阵（matrix）与向量（vector）形式定义任意渠网的拓扑结构与水力特性；随后借助上述矩阵与向量，可将所有节点与渠段的控制方程进行统一整合，最终得到仅包含两个矩阵方程的紧凑表述形式，用以表征明渠管网内的明渠渐变流。相较于传统方法，该新方法无需针对不同类型节点单独列写控制方程，也无需逐一对各渠段进行建模推导，因此可形成一套通用的渠网建模流程。此外，该方法可将内部节点视为入流/出流边界，从而支持更灵活的应用场景，例如对带有汇流检查井的复杂排水系统进行建模。最后通过将该方法在各类渠网中的计算结果与传统方法的求解结果进行对比，验证了所提方法的有效性。