Introduction

At present, there are more than 500,000 bridges of various types, and the number continues to increase at a rate of 10,000 bridges per year. The scale of bridge engineering projects is also increasing, with an investment usually ranging from tens of millions to tens of billions Chinese Yuan. However, bridges involve various risks during construction and operation (Khandel and Soliman 2019; Tubaldi et al. 2022). In particular, frequent flood disasters usually lead to periodic bridge damage and traffic interruptions (Loli et al. 2022a). Subsequently, the flood causes serious loss of human life, property and environmental pollution (Kim et al. 2017; Wang et al. 2019; Loli et al. 2022b). In bridge engineering, the multi-bridge system with short distance can more likely cause engineering accidents and serious accidents consequences caused by the more complex flow field and the structure stress. Therefore, it is particularly important to study the impact of short-distance multi-bridge system on river flood risk to ensure the safety of bridges and eliminate the threat of flood disasters.

Studies on the assessment and analysis of flood risk control have been conducted for many years, but most of these studies have been based on theoretical analysis methods. Chen et al. (2020); Wang et al. (2021); Apel et al. (2004) established a stochastic flood risk assessment model based on the stochastic theories, stochastic differential equations and stochastic simulation methods. The assessment model was applied to analyze the random flood process and flood risk control of reservoir dams, reservoir flood evolution process and various uncertainties in the flood control system of the Xiluodu–Xiangjiaba–Three Gorges Dam cascade reservoir, as well as assess the risks of the reservoir overtopping, flood overrun and flood diversion. The uncertainty of flood risk was also analyzed using this model on the Rhine river downstream of Cologne considering intuitive and cognitive uncertainties. Li et al. (2013); Chen et al. (2011) established a flood disaster risk assessment model using an analytic hierarchy process (AHP) to determine the index weight based on the formation mechanism of flood disasters combined with geographic information system (GIS). The model was applied to comprehensively assess the flood risks, exposure and vulnerability in the Yangtze River Delta, Fushin Township, and the floodplain of the Fazih River in Taiwan. Kubal et al. (2009) established a multi-criteria flood risk assessment method by using Flood Calc and GIS software considering the economic, social and ecological flood risk criteria, and analyzed the flood risk of Leipzig, Germany. Wu et al. (2011) proposed a risk analysis model that considered hydrological, hydraulic and geomorphic uncertainties to assess the flood control structure risk of the Keelung River. Skakun et al. (2014) conducted a flood risk analysis on the Katima Mulilo region and determined the cities and villages with the highest risk using the time series of satellite images from 1989 to 2012 (to draw flood disaster map). Dou et al. (2018) proposed a multivariate flood hazard index model by considering rainfall intensity, flow accumulation, distance from river network, elevation, land use, surface slopes and geology. The model was applied to assess potential flood risks in the urban area of Guanzhong, China. Lyu et al. (2019) established a flood risk control assessment method for the subway systems based on the regional flood risk control assessment method and perspective. They also conducted a qualitative analysis to study the risk of flood inundation of the subway system in Guangzhou. Waghwala and Agnihotri (2019) used GIS to draw flood management maps based on satellite images to analyze the flood risk in Surat, India. They also found that the change from low urbanization to high urbanization pattern was the main driving force for the increase of flood risk in the study area.

With the rapid development of the computer, more and more researchers have applied numerical simulation method to study engineering problems (Gao et al. 2020, 2021; Lin et al. 2023; Ding et al. 2022; Mao et al. 2021). The application of numerical simulation methods based on hydrodynamic equations in flood control risk assessment was also gradually increasing. Pelletier et al. (2005) established a comprehensive assessment model for alluvial fan flood disasters by solving the continuity and Manning equations (based on an implicit numerical method). The two extreme flooding events in Tortolita and Haquahalla Piedmont in southern Arizona were simulated and the results were compared with field and satellite flood maps. Qi and Altinakar (2011) developed an integrated flood risk management system based on a two-dimensional (2D) numerical model, GIS, remote sensing technology, Monte Carlo method and an event tree analysis to study the flood risk management strategies of the Sinclair Dam failure in Georgia, USA. Tsakiris (2014) established a 2D hydrodynamic flood risk analysis model based on FLOW-R2D software to analyze the non-stationarity of flood events and the uncertainty of flood damage calculations in flood prone built up areas in mild terrain. Gain et al. (2015) established a comprehensive flood risk assessment model based on REC-RAS considering physical dimensions in hazard, exposure and social dimensions in vulnerability to analyze the flood risk of Dhaka, Bangladesh. Vojtek and Vojteková (2016) analyzed the flood risk on a local spatial scale by using HEC-RAS, GIS and remote sensing. Li et al. (2016) established a comprehensive analysis model of urban flood risk based on the urban flood simulation model (UFSM) and the urban flood damage assessment model (UFDAM) based on the Triangle Theory of Disaster Risk to numerically simulate the flood risks in the flood control area in Pudong, Shanghai. Lyu et al. (2019) proposed a flood risk management system using SWMM, GIS, Global Positioning System (GPS), BIM and an analytic hierarchy process to evaluate and manage inundation risk for metro systems. Tang et al. (2020) simulated the hydrodynamic process of the Huaihe–Hongze Lake system using a FVCOM software package based on the finite volume method and unstructured grids. They also analyzed the influence of lakes, connecting rivers and sharp bends in lakes on river flooding. Shrestha and Kawasaki (2020) used a HEC-RAS software package to simulate the flood risks and the influence of dam structures on flood control in the Bago River Basin, Myanmar. Mei et al. (2020) used a TELEMAC–MASCARET software package to establish an analysis model for the Xiamen Island based on the finite element method. The design flood risks of Xiamen Island with 12 different rainfall patterns and durations in different periods were simulated and the influencing factors of flood risk were analyzed. However, the performance of most flood forecasting models were suboptimal and often below expectation (Arduino et al. 2005).

Although flood risk analysis had been investigated, mostly studies aimed at flood risk of a single bridge in large-scale areas such as rivers, reservoirs or urban areas. The research on multi-bridge systems mainly focused on navigational capacity and the structural vulnerability exposed to hurricanes (Sang et al. 2017; Balomenos et al. 2020; Shang et al. 2022) while, to the author's knowledge, there has not been analysis of the flood risks of short-distance multi-bridge system due to the relatively rare construction of the multi-bridge system with short distance. Another reason for the lack of relevant research is that the time consumption of numerical simulation would intolerant long result from the small mesh size and large mesh number that was induced by the large number and small size of piers. Therefore, it was difficult to evaluate the flood risk of multi-bridge system with short distance based on numerical model. However, the flood risk analysis of the multi-bridge system with short distance is very necessary due to the complex and dangerous flow field inducing by multi-bridge.

In this study, an efficient flood risk analysis model of multi-bridge system with short distance was established based on the finite volume method combined with the Delaunay mesh adaptive refinement scheme using the HydroInfo software (Yu et al. 2022) which is developed independently by the Dalian University of Technology. The software can be used to simulate 1D and 2D water flow in rivers, estuaries and oceans on the computer or network server (Zhao et al. 2019). The accuracy and efficiency of the model was verified by comparing the calculated water level with the experimental data of partial failure dam break test case and Shukry open channel bend flow experiment. Then, the numerical model was used to calculate the river flood flow field of a short-distance multi-bridge system under various conditions. The impact of bridge on river flood flow field under different bridge combinations was analyzed. Meanwhile, the flood risk of multi-bridge system with short distance was evaluated to provide theoretical and technical reference for similar short-distance multi-bridge system flood risk assessment.

Numerical method

Governing equations

The governing equation can be simplified as the two-dimensional shallow water equations due to that the plane scale of a river is much larger than the vertical scale (Namin et al. 2004; Cea et al. 2006). The basic conservation equations are simplified using the shallow water assumption. Simultaneously, the distribution of hydrostatic pressure is used to represent the variation of pressure in the vertical direction (Namin et al. 2004). By averaging the basic momentum and mass conservation equations (integrating along the water depth direction), the governing equations of the 2D model for shallow water can be obtained (Li and Fan 2017; García‑Navarro et al. 2019; Lin et al. 2020):

$$\frac{\partial h}{{\partial t}} + \frac{\partial hu}{{\partial x}} + \frac{\partial hv}{{\partial y}} = q$$
(1)
$$\frac{\partial hu}{{\partial t}} + \frac{\partial }{\partial x}\left( {hu^{2} + \frac{1}{2}gh^{2} } \right) + \frac{\partial huv}{{\partial y}} = s_{x}$$
(2)
$$\frac{\partial hv}{{\partial t}} + \frac{\partial }{\partial y}\left( {hv^{2} + \frac{1}{2}gh^{2} } \right) + \frac{\partial hvu}{{\partial x}} = s_{y}$$
(3)

where u and v (m/s) represent the velocity in x and y direction, respectively; g (m/s2) represents the gravitational acceleration; h (m) represents the water depth; q (m3/s) represents the flow rate; n is the roughness coefficient; \(s_{x} = - gh\frac{{\partial z_{b} }}{\partial x} - \frac{{\tau_{bx} }}{h}\) and \(s_{y} = - gh\frac{{\partial z_{b} }}{\partial y} - \frac{{\tau_{by} }}{h}\) represent the source items in x and y directions, respectively; and \(\tau_{bx} = \frac{{n^{2} u\sqrt {u^{2} + v2} }}{{h^{1/3} }}\) and \(\tau_{by} = \frac{{n^{2} v\sqrt {u^{2} + v2} }}{{h^{1/3} }}\).

Equation discretization

The continuity equation is discretized by a finite volume integral numerical method to ensure the conservation of the scheme. The 2D shallow water equations have characteristics of rotation invariance. Therefore, the momentum equations are discretized in local coordinate systems composed of element normal directions and edges. The details of the equation discretization are described in work by Wang et al. (2005).

Equations (1)–(3) can be written as follows (Zhao et al. 2019; Yu et al. 2022):

$$U_{t} + E{(}U{)}_{x} + H{(}U{)}_{y} = S{(}U{)}$$
(4)

where the subscripts t, x and y represent the first derivatives.

$$U{ = }\left[ {\begin{array}{*{20}c} h \\ {hu} \\ {hv} \\ \end{array} } \right], \, E = \left[ {\begin{array}{*{20}c} {hu} \\ {hu^{{2}} + \frac{{1}}{{2}}gh^{2} } \\ {huv} \\ \end{array} } \right], \, H = \left[ {\begin{array}{*{20}c} {hv} \\ {huv} \\ {hv^{{2}} + \frac{{1}}{{2}}gh^{2} } \\ \end{array} } \right], \, S = \left[ {\begin{array}{*{20}c} {0} \\ {s_{x} } \\ {s_{y} } \\ \end{array} } \right]$$
(5)

The average value of the conserved variable is stored in the center of the element. Each element edge is used as the surface of the control body. Equation (4) can be rewritten as:

$$\frac{\partial U}{{\partial t}} + \nabla F = S{(}U{)}$$
(6)

where F = (E, H). Integrating on any unit Vi we obtain the following:

$$\int\limits_{{V_{i} }} {\left( {\frac{\partial U}{{\partial t}} + \nabla F} \right)dV} = \int\limits_{{V_{i} }} {S{(}U{)}dV}$$
(7)

Let Ui be the unit average value stored in the unit center, Eq. (7) can then be written as:

$$\frac{{\partial U_{i} }}{\partial t}\Delta V_{i} + \int\limits_{{\partial V_{i} }} {F \cdot {\mathbf{n}}ds} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{S}$$
(8)

where \(\Delta V_{i} , \, \partial V_{i}\) are the area and boundary of element Vi and n is the unit normal vector on the boundary. Equation (8) is discretized by approximating the surface integral with the flux vector on the edge of the element using the following:

$$\int\limits_{\partial V} {F \cdot {\mathbf{n}}ds} = \sum\limits_{j = 1}^{{N_{e} }} {F_{ij} \Delta l_{ij} }$$
(9)

where \(\Delta l_{ij}\) represents the length of the jth side of the element Vi; Fij is the numerical flux through edge Lij; and Ne represents the side number of cell Vi.

The flux Fij is calculated by the ROE approximate solution using the following (Wang et al. 2005):

$$F_{ij} = F\left[ {{(}U_{L} {)}_{ij} ,{ (}U_{R} {)}_{ij} } \right] = \frac{{1}}{{2}}\left\{ {\left[ {F{(}U_{R} {)}_{ij} + F{(}U_{L} {)}_{ij} } \right] \cdot {\mathbf{n}} - \left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{A} } \right|\left[ {{(}U_{R} {)}_{ij} - {(}U_{L} {)}_{ij} } \right]} \right\}$$
(10)

where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{A}\) represents the flux Jacobian matrix obtained by the ROE average.

$$A{ = }\frac{{\partial {(}F \cdot {\mathbf{n}}{)}}}{\partial U} = \left[ {\begin{array}{*{20}c} {0} & {n_{x} } & {n_{y} } \\ {{(}c^{2} - u^{2} {)}n_{x} - uvn_{y} } & {2un_{x} + vn_{y} } & {un_{y} } \\ { - uvn_{x} + {(}c^{2} - u^{2} {)}n_{y} } & {vn_{x} } & {un_{x} + 2vn_{y} } \\ \end{array} } \right]$$
(11)

The value of the ROE average can be given by the following equations:

Eigenvalue of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{A}\):

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a}_{1} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u} n_{x} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{v} n_{y} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a}_{2} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u} n_{x} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{v} n_{y} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a}_{3} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u} n_{x} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{v} n_{y} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c}$$
(12)

Right eigenvector of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{A}\):

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{e}_{{1}} = \left[ {\begin{array}{*{20}c} {1} \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} n_{x} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{v} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} n_{y} } \\ \end{array} } \right]{ , }\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{e}_{{2}} = \left[ {\begin{array}{*{20}c} {{0} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} n_{y} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} n_{x} } \\ \end{array} } \right]{ , }\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{e}_{{3}} = \left[ {\begin{array}{*{20}c} {1} \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} n_{x} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{v} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} n_{y} } \\ \end{array} } \right].$$
(13)

Decomposing the UR-UL to the right eigenvector of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{A}\):

$$U_{R} - U_{L} = \sum\limits_{{k = {1}}}^{3} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{k} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{e}_{k} }$$
(14)

where

$$\begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{1}} { = }\frac{\Delta h}{2} + \frac{{1}}{{{2}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} }}\left[ {\Delta {(}hu{)}n_{x} + \Delta {(}hv{)}n_{y} - {(}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u} n_{x} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{v} n_{y} {)}\Delta h} \right],} \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{2}} { = }\frac{{1}}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} }}\left\{ {\left[ {\Delta {(}hv{)} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{v} \Delta h} \right]n_{x} - \left[ {\Delta {(}hu{)} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u} \Delta h} \right]n_{y} } \right\},} \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{3}} { = }\frac{\Delta h}{2} - \frac{{1}}{{{2}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} }}\left[ {\Delta {(}hu{)}n_{x} + \Delta {(}hv{)}n_{y} - {(}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u} n_{x} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{v} n_{y} {)}\Delta h} \right].} \\ \end{array}$$
(15)

The numerical flux of Eq. (10) can be expressed as:

$$F\left[ {{(}U_{L} {)}_{ij} ,{ (}U_{R} {)}_{ij} } \right] = \frac{{1}}{{2}}\left\{ {\left[ {F{(}U_{R} {)}_{ij} + F{(}U_{L} {)}_{ij} } \right] \cdot {\mathbf{n}} - \left( {\sum\limits_{{k = {1}}}^{{3}} {\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{k} } \right|\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{k} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{e}_{k} } } \right)_{ij} } \right\}$$
(16)

Grid division

Given that the piers of bridges were quite small as compared to the size of real river channels, a uniform mesh could lead to an extremely large number of meshes in the computational domain causing long calculation times. This could make the flood risk analysis impractical for engineering applications. Therefore, an adaptive grid refinement method (Watson 1981) was used to refine the meshes near the piers while a larger mesh size was adopted for the large-scale computational domain of the river channels (to substantially reduce the number of meshes in the computational domain and improving the computational efficiency). The adaptive mesh refinement method automatically performs mesh refinement based on the size of small-scale structures, ensuring a smooth transition from small to large mesh size while maintaining mesh quality. At the beginning of mesh dividing, the model will automatically detect the length of each edge of the encrypted boundary (small-scale structures) and determine the minimum side length for each structure. The mesh size around the structure will be set to the minimum side length value of each structure. The mesh size gradually increases as it moves away from the structures, until it reaches to a larger mesh size which is set during initialization. The mesh size of the large-scale computational domain far away from the structure will be directly set to the initial given value. The model adopted the Delaunay mesh generation method, which triangulated the convex hull of the scatter dataset on the plane according to relevant criteria. The meshes generated by the Delaunay method were distributed flexibly. Therefore, the adaptive grid refinement method adapted to complex boundary conditions and allowed arbitrary local mesh refinement. The first step of mesh generation was to construct the initial mesh in which a super-triangle was constructed that contained all the boundary points. The boundary meshes were refined according to the set scale; afterward, the initial boundary scatter points were obtained. Each boundary scatter point was sequentially inserted into the super-triangle. Subsequently, the boundaries were restored. The triangle outside the computational domain was deleted to obtain the correct initial triangulation. The second step was the refinement of the initial meshes. The Bowyer–Watson method (Bowyer 1981; Watson 1981) was adopted to generate internal points until the meshes met certain division requirements. Finally, smoothing and topology optimization were conducted to refine the generated meshes (Shewchuk 2002). The detailed mesh generation method is as follows:

  1. (1)

    Construct the initial triangular mesh. The initial grid needs to meet the following requirements: a. Cover the entire computing domain; b. meet the Delaunay criterion (there are no other nodes included except for the three nodes contained on the circumscribed circle in the circumscribed circle of a triangle); and c. describe the surface features as much as possible.

  2. (2)

    A new mesh node (the circumscribed circle center of the existing mesh) is introduced in an area where the mesh size does not meet the requirements.

  3. (3)

    If the circumscribed circle of a triangle in the initial mesh contains new points, the triangle is marked as the one to be deleted (Fig. 1a).

  4. (4)

    A Delaunay void is formed by all the triangles to be deleted (Fig. 1b).

  5. (5)

    A new mesh system is formed by connecting the new points with the boundary vertices of the Delaunay cavity (Fig. 1c).

  6. (6)

    Repeat steps (2)–(5) until all triangle mesh sizes reach the desired size.

Fig. 1
figure 1

New node insertion process of the Delaunay triangulation

Full size image

Model validation

Partial failure dam break

To validate the model and the efficiency of the mesh generation scheme, the calculation results of the refined and unrefined mesh models with the experimental data were compared using a partial failure dam break test case (Chang et al. 2011). The horizontal flume was 3 m long, 2 m wide and 0.7 m high, as shown in Fig. 2. There was a vertical dam 1 m away from the upstream wall. The width of the dam was set as 0.01 m. A gate with 0.4 m wide was located at center of the dam. The gate was removed instantaneously at the t = 0 s. The initial water depth was 0.6 m in the upstream reservoir. The downstream channel was dry with open side and outlet boundaries. The history of water level was measured at four gages (Table 1). G8A was placed downstream channel of the dam. G0 was placed at the dam site. Gages G-5A and GC were placed inside the upstream reservoir.

Fig. 2
figure 2

Layout of dam break flow with partial failure

Full size image
Table 1 Gages position
Full size table

To verify the accuracy of the model and analyze the sensitivity of the mesh, the uniform mesh model without mesh refinement was established with the mesh size 0.04, 0.02 and 0.01. Figure 3a and c shows the model mesh with mesh size of 0.02 m. The time step of the model was 0.001 s. The number of calculation steps was 10,000. Figure 4 depicts the comparison between the calculated water level with different mesh scales and the experimental data. In Fig. 4, the error of calculation results was large with the mesh size 0.04. The calculated results of the mesh size 0.02 and 0.01 models were basically consistent with the experimental data. A terrain interpolation discontinuous (Fig. 5) of the dam was induced by the large mesh size of 0.04 m. Therefore, the water level at each measuring point dropped rapidly due to the water in the reservoir drain from the dam gap. On the contrary, the calculation results with mesh size 0.02 m had a good agreement with the experimental data due to the terrain interpolation of the dam was continuous (Fig. 5) that was basically conforms to the experiment.

Fig. 3
figure 3

Comparison of water level between calculated results and experimental data

Full size image
Fig. 4
figure 4

3D view of model with mesh size of 0.04 and 0.02

Full size image
Fig. 5
figure 5

Mesh of Model 1 and Model 2

Full size image

To further evaluate the computational errors of numerical model with different mesh size, the Grid Convergence Index (GCI) (Roache 1997) was used to evaluate the numerical errors. The GCI method requires that the monotonic convergence condition (0 < (f3−f2) / (f2−f1) < 1, where f is the numerical solution, and subscripts 1, 2 and 3 represent three sets of meshes from sparse to dense) of the calculation results be satisfied. The details of GCI are as follows.

For k (k = 1, 2, 3) meshes, the calculation results satisfy:

$$f_{1} = f_{{{\text{exact}}}} + g_{p} h_{1}^{p} + O\left( {h_{1}^{p + 1} } \right)$$
(17)
$$f_{2} = f_{{{\text{exact}}}} + g_{p} h_{2}^{p} + O\left( {h_{2}^{p + 1} } \right)$$
(18)
$$f_{3} = f_{{{\text{exact}}}} + g_{p} h_{3}^{p} + O\left( {h_{3}^{p + 1} } \right)$$
(19)

where fexact is the accurate solutions of numerical model. h represents mesh size. p represents the convergence accuracy. gp is the coefficient of the p-order error term that does not change with the mesh. The mesh refinement ratio is as \(r_{k,k + 1} = h_{k} /h_{k + 1}\). The total volume of the three model meshes is unchanged, so \(r_{k,k + 1} = h_{k} /h_{k + 1} = (N_{k + 1} /N_{k} )^{1/3}\), where N is the numbers of mesh units. The relative error between the two meshes is defined as:

$$\delta_{r(k,k + 1)} = \left| {\frac{{f_{k} - f_{k + 1} }}{{f_{k + 1} }}} \right| = \left| {\frac{{\delta_{k,k + 1} }}{{f_{k + 1} }}} \right|$$
(20)

where \(\delta_{k,k + 1} = f_{k} - f_{k + 1}\).

The convergence rate p can be calculated using the following equation by ignoring the higher-order term in Eqs. (17)–(19).

$$\frac{{\delta_{12} }}{{r_{12}^{p} - 1}} = \frac{{\delta_{23} r_{23}^{p} }}{{r_{23}^{p} - 1}}$$
(21)

The GCI can be defined as:

$$GCI_{k + 1} = F_{s} \frac{{\delta_{r(k,k + 1)} }}{{r_{k,k + 1}^{p} - 1}}, \, GCI_{k} = F_{s} \frac{{\delta_{r(k,k + 1)} r_{k,k + 1}^{p} }}{{r_{k,k + 1}^{p} - 1}}$$
(22)

where Fs is the safety factor ranging from 1.25 to 3.0. In this paper, it is taken as 3.0.

Table 2 shows the GCI of the numerical model with three mesh sizes. The GCI of all three meshes were less than 3%. Meanwhile, the GCI decreased with the increase in the number of meshes and tended to 0.

Table 2 GCI of the numerical model with three mesh sizes
Full size table

To analyze the accuracy and efficiency of the local refined mesh model, the mesh size was set to 0.2 m with a local refined mesh size 0.01 m (Model 2). The calculation results were compared with the calculation results of the uniform mesh model with mesh size 0.02 m (Model 1) and experimental data. Figure 3a and b shows the mesh of Model 1 and Model 2, respectively. Figure 3c and d shows the local refined mesh near the dam breach of Model 1 and Model 2. A mesh transition from 0.01 m near the dam to 0.2 m near the boundary is clearly shown in Fig. 3b and d. The mesh number of Model 1 and Model 2 was 34,860 and 6236, respectively. The time step was set to 0.001 s. The number of calculation steps was 10,000.

Figure 6 depicts the comparison between the calculated results of Model 1, Model 2 and the experimental data. The calculated results of Model 1 and Model 2 were basically consistent. To quantitatively analyze the accuracy of Model 2, Table 3 shows the L2 error [34] of the calculated water level of Model 1 and Model 2.

$$L_{2} = \sqrt {\frac{1}{N}\sum\limits_{t = 0}^{N} {\left( {\frac{{h_{t}^{n} - h_{t}^{e} }}{{h_{t}^{e} }}} \right)^{2} } }$$
(23)

where N is the simple numbers; and \(h_{t}^{n}\) and \(h_{t}^{e}\) are the calculated result and the experimental data at time t, respectively.

Fig. 6
figure 6

Comparison of calculated water level between Model 1 and Model 2

Full size image
Table 3 L2 errors between the numerical and experimental water level
Full size table

The L2 errors of Model 2 and Model 1 were largest at G8A. The L2 error of Model 2 was 0.03 less than Model 1 at this point. At GC, the L2 errors of Model 2 was 0.01 greater than Model 1. The L2 errors of Model 2 and Model 1 were consistent at G-5A and G0. Accordingly, both the calculated results of Model 2 and Model 1 were in good agreement with the experimental data. Table 2 also shows the time consumption of Model 1 and Model 2. The value was 1116 s and 306 s, respectively. The efficiency of Model 2 was 73% faster than Model 1. The local refined mesh model reduced the number of meshes and save the calculation time while ensuring the calculation accuracy. With the increase of the project scale and the extension of the simulation time, the mesh refinement model would save more time and cost. In a word, the Delaunay mesh adaptive refinement model was more suitable for practical engineering applications.

Open channel flow

To validate the accuracy of open channel flow, the Shukry open channel bend flow experiment (Shukry 1950) was simulated to compare the numerical results with the experimental data. The layout of the experimental channel is shown in Fig. 7. The length of the straight section at the inlet and outlet of the rectangular channel is 1.07 m. The width of the channel is 0.3 m. The height is 0.5 m. The inner diameter of the channel is 0.15 m while the outer diameter is 0.45 m. The slope of the bottom is 0. The inlet flow rate is 0.072. The corresponding outlet water depth is 0.28 m. The model mesh size is 0.01 m. The time step is 0.01 s.

Fig. 7
figure 7

Layout of Shukry open channel bend experiment

Full size image

Figure 8 compares the calculated water surface at Sects. 0°, 45°, 90° and 180° with experimental data. h represents the water level. B represents the width of the section from the concave bank to the convex bank. The calculated water surface is in good agreement with the experimental data. The L2 error of all sections is relatively small with a maximum L2 error value of 0.03. In summary, the model can accurately simulate the open channel flow (Table 4).

Fig. 8
figure 8

Comparison of free surface between experimental and calculated value of Shukry experiment

Full size image
Table 4 Free surface \(L_{{2}}\) errors of Shukry experiment
Full size table

Flood risk analysis of a real multi-bridge system with short distance

Basic data

There were three bridges built on the Laoha River, including the Kaji Highway Bridge (Bridge 1), the Yechi Railway Bridge (Bridge 3—Existing Laoha River Bridge) and the Tianping Line Expressway Bridge (Bridge 4—Erlong Bridge), as shown in Fig. 9. Bridge 3 was built nearly 90 years since 1935 and had been repaired for many times. However, the steel beam still had serious problems such as cracks, rivet falling off and bearing damage. Therefore, it was necessary to reconstruct the Bridge 3. The position of the reconstructed bridge was Bridge 2 in Fig. 9. The bridge parameters are shown in Table 5. Piers of Bridge 1 and 2 were round-ended shape which was simplified to rectangular section for convenience of calculation. The piers of Bridge 3 and 4 were circular shape with 3 m and 2 m, diameter, respectively, that were simplified to 4 × 2 m and 2 × 2 m rectangular section.

Fig. 9
figure 9

Layout of the bridges

Full size image
Table 5 Detail of bridges
Full size table

Model layout

The numerical model of multi-bridge system with short distance was established for different bridge combinations (Table 6) considering the bridges condition of present, reconstruction and demolition. Model 1 was a no bridge model of natural river. Model 2 was the present bridge condition model built for Bridges 1, 3 and 4. Model 3 was the construction model of the proposed Bridge 2. Model 4 was the Bridge 3 removed model after reconstruction.

Table 6 Bridge combination
Full size table

The size of the grids near the bridge piers is consistent with the short side length of rectangular bridge pier. The length of each bridge pier side will be automatically detected to determine the short side length of the bridge pier when the calculation area is divided. Then, the mesh size around the bridge pier is set to the short side length of the pier. The mesh size away from the pier is set to the given value. The mesh size gradually increases away from the pier. Mesh sizes of Models 1–4 were 10 m for the large-scale river channel while it was refined around the piers according to the size of the pier. The mesh gradually increased along the direction away from the pier and transited to 10 m mesh size (Fig. 10). The mesh numbers in the river channels of Models 1–4 were 31,600, 59,372, 61,162 and 57,324, respectively. The upstream boundary condition was the design flood (100-year flood) of the proposed bridge. The flow rate was 4410 m3/s. The downstream boundary condition was the water level boundary with a value of 538.4 m. The time step was 0.36 s. The initial water level was 539.78 m. The roughness was 0.025. The models are shown in Fig. 11.

Fig. 10
figure 10

Model mesh

Full size image
Fig. 11
figure 11

Time histories of jump toe

Full size image

Results analysis

Figure 12 shows the calculated flow field of Models 1–4. The color represented the water level, and the arrow represented the velocity vector. In Fig. 12b–d, the arrows were relatively dense near the bridge result from the small size and large numbers of mesh near the pier. The water level of Model 2 was significantly higher than Model 1 while the change of velocity was not obvious. Compared with Model 2, the water level of Model 3 rose slightly at the upstream of Bridge 4 and decreased slightly at the downstream of Bridge 4 with little change in velocity. On the contrary, the water level upstream Bridge 4 in Model 4 decreased compared with Model 3 while the water level downstream of the Bridge 4 rose with little changes in velocity too.

Fig. 12
figure 12

Comparison of calculated flow field for Models 1–4

Full size image

To quantitatively analyze the flood risk of the multi-bridge system with short distance, Fig. 13 compared the average water level and velocity at Sect. 1–10 (Fig. 14) of Model 1–4. In Fig. 14, S1–S10 represented Sect. 1–10. L was the Mileage of the section distance from the inlet boundary of the model. H was the water level. U was the average velocity of the section. Compared to Model 1, the water levels at Sects. 9 and 10 of Models 2—4 were lower with a maximum difference value 0.122 m at Sect. 9 while the water levels of Sects. 1 to 8 were higher with a maximum difference value 0.839 m at Sect. 2. The velocities from Sects. 1–10 of Models 2—4 were all lower than Model 1 with maximum 1.349 m/s at Sect. 8 of Model 3.

Fig. 13
figure 13

Comparison of calculated water level and velocity for Sects. 1–10

Full size image
Fig. 14
figure 14

Position of the sections

Full size image

The water levels of Sects. 9–10 of Model 2 were lower than Model 1 with a maximum difference value 0.122 m (Sect. 9) while the water levels of Sects. 1–8 were higher than Model 1 with a maximum difference value 0.534 m (Sect. 2). The water levels of Sects. 8 and 10 of Model 3 were lower than Model 2 with maximum 0.05 m (Sect. 8) while the water levels of Sects. 1–7 and 9 were higher than Model 2 with maximum 0.326 m (Sect. 3). The water levels of Sects. 7–10 of Model 4 were higher than Model 3 with a maximum difference value 0.034 m (Sect. 7) while the water levels of Sects. 1–6 were lower than Model 3 with a maximum difference value 0.295 m (Sect. 6).

The velocities at Sects. 1 to 10 of Model 2 were all lower than Model 1 with maximum 1.189 m/s (Sect. 8). The velocities at Sects. 1–5 and 8–10 of Model 3 were lower than Model 2 with maximum 0.167 m/s (Sect. 9) while the velocities at Sects. 6—7 were greater than Model 2 with maximum 0.013 m/s (Sect. 6). The velocities at Sects. 9–10 of Model 4 were lower than that Model 3 with a maximum difference value 0.028 m/s (Sect. 10) while the velocities of Sects. 1–8 were greater than Model 3 with a maximum difference value 0.227 m/s (Sect. 6).

In the multi-bridge system with short distance, the water levels at the upstream sections of the most downstream Bridge 4 were raised, which increased the flood risk of river channel and was detrimental to river flood control, resulting from the construction of bridges. The maximum value was 0.839 m. The water level downstream of Bridge 4 was decreased which was beneficial to the river flood risk control. The maximum value was 0.122 m. The bridges reduced the velocity in the river with a maximum reduction of 1.349 m/s. The water level downstream of the Bridge 4 dropped while the water level upstream of the Bridge 4 upraised caused by the construction of Bridges 1, 3 and 4. The maximum drop and upraise value were 0.122 m and 0.534 m, respectively. The notable rising in water level increases the flood risk. The velocity dropped with a maximum difference value 1.189 m/s. The water level around and upstream of Bridge 3 rose again while the water level around and downstream of Bridge 4 basically declined due to the construction of Bridge 2 on the basis of existing Bridges 1, 3 and 4. The maximum rise and decline value were 0.326 m and 0.05 m. Compared to the three Bridges 1, 3 and 4, the construction of the single Bridge 2 results in a little rising of the water level, which has a lower adverse impact on the flood risk compared to the three bridges. The velocity was basically reduced with maximum 0.167 m/s while the velocity around Bridge 3 was increased with maximum 0.013 m/s. The demolition of Bridge 3 caused the water level downstream of Bridge 3 to ascend and the water level in upstream river to descend. The maximum ascend value was 0.034 m, and the maximum descend value was 0.295 m. The decrease in upstream water level is much greater than the increase in downstream water level, which is favorable for the control of the river flood risk. The velocity decreased downstream of Bridge 4 with maximum 0.028 m/s and increased upstream of Bridge 4 with maximum 0.227 m/s.

Conclusion

An efficient flood risk assessment model of short-distance multi-bridge system was established based on finite volume method combined Delaunay mesh generation scheme achieve adaptive grid refinement. The model substantially reduced the mesh numbers and time consumption to overcome the problems of significantly large mesh numbers and calculation time induced by small pier sizes and large pier numbers. To validate the accuracy and efficiency of the model, the calculated results with refined and non-refined mesh were compared with the experimental data by using a partial failure dam break test case. The results had shown that the local adaptive mesh refinement could reduce the mesh numbers and save the calculation time while ensuring the calculation accuracy. The efficiency of the model was improved by more than 73%. With the increase of the project scale and the extension of the simulation time, the model efficiency advantage would be more obvious. Therefore, the model is more suitable for practical engineering applications. In addition, to validate the model for open channel flow, the Shukry experiment was adopted to compare the calculated free surface with the experimental data. The calculated water surface is in good agreement with the experimental data with a maximum L2 error value of 0.03.

The model was used to analyze the flood risk of a real multi-bridge system with short distance. The results had shown that, in general, the water level upstream of the bridge was upraised while the water level downstream of the bridge was dropped, and the velocity in the river channel was decreased result from the construction of the bridge. Therefore, the construction of the bridges is unfavorable for the control of the river flood risk. However, in the short-distance multi-bridge system, the flow field was more complex caused by the multiple bridges. The water level upstream of Bridge 2 and downstream at a certain distance (around Bridge 3) of Bridge 2 raised with maximum 0.326 m while the water level far downstream (downstream of Bridge 4) of Bridge 2 dropped due to the construction of Bridge 2 on the basis of Bridges 1, 3 and 4. Different from the general rule, the construction of Bridge 2 also increased the velocity around Bridge 3 with maximum increase value 0.013 m/s. The increment of velocity had adverse effects on the safety of bridges, which should be concerned. This study provided a new tool and technical reference for flood risk analysis for similar multi-bridge system with short distance.