## 1 Introduction  
With the expanding scale of urbanization-related construction in China, the construction of groups of bridges across rivers is increasing, and the number of bridges within urban river channels is increasing, resulting in serious pressure on each river's flooding cross-section, resulting in greater backwater and uneven flow velocity distribution, affecting the two banks of the embankment, the upstream water resource infrastructures, and the surrounding city's flood control and drainage safety [1]. In these scenarios, multiple bridges are densely arranged in the river. The pier types, sizes, quantities, and layouts are complex and changeable; this especially affects the parallel construction of bridges, as it is prone to produce the superposition effect of waterback. This phenomenon causes higher channel water levels, weakening the river's flood discharge capacity, and then has an additional impact on the riverbed [2; 3; 4]. As early as the 19 th and 20 th centuries, scholars from various regions around the world conducted analyses and research on bridge pier backwaters. In 1922, the German scientist [PERSON] and, in 1934, the American scientist [PERSON] carried out extensive physical model experiments on bridge backwater, proposing empirical formulas for bridge pier backwater. At present, river simulation research mainly involves river model tests, numerical simulations, and empirical calculations, as well as other methods [5]; the physical model test is one of the important means used to predict the movements of water, sand, and sediment, and the evolution of the riverbed in natural conditions and after building hydraulic structures. Especially in some three-dimensional problems, theoretical calculations can be very difficult, and observations made in the method of a model test can be more effective [6]. Numerous domestic and international studies have been conducted by scholars and engineers on the simulation of river flow characteristics during concurrent bridge construction. These investigations primarily focus on the effects of merging bridges on river flow dynamics, the methodologies and techniques of simulations, and the ecological preservation of river channels, encompassing a wide array of disciplines, including hydrology and hydraulic engineering. In recent years, an increasing number of scholars have utilized numerical simulation methods in order to study river hydrodynamics [7]. The MIKE ZERO 2014 software was employed to simulate flow conditions before and after engineering projects. [PERSON] conducted numerical simulations on the flow field and sediment erosion-deposition characteristics of the bridge cluster section in Yixian County along the Daling River before and after bridge construction, with the research results providing references for flood control in the river section [8]. [PERSON] et al. constructed a two-dimensional comprehensive mathematical model of the water-related effects of a bridge group in the Qinhuai New River and evaluated the superimposed obstruction effect of the bridge group under different hydrological conditions on the flood discharge capacity of the river [9]. [PERSON] et al. used a two-dimensional hydrodynamic model to investigate the transformation of hydrodynamic characteristics such as waterback height and flow velocity during the construction of bridge clusters in a plain river channel and clarified the relationships between these changes and the number and spacing of the bridges [10]. [PERSON] et al. took the special bridge of the Lunan High-Speed Railway that crossed the Yi River as a case study and explored the potential influence of the layout of large-span bridge piers on the flow patterns of the river [11]. In addition to studying the impacts of cross-river bridge projects with relatively small spans, numerical modeling has also been applied to the Yangtze River, Asia's longest river and the world's third-largest river by water discharge. [PERSON] et al.