## Abstract

The shape of bridge piers across rivers is one of the significant factors that affect the backwater in the river. The study on the influence of bridge pier shapes on the flow patterns of the river is valuable to the design of the bridge and river flooding. Based on the MIKE21 Flow Model hydrodynamic model, dynamic numerical simulations were conducted to investigate the effect of different shapes of bridge piers on water flow. The result shows that rectangular, circular and elliptical piers have a water blocking effect on the river. The flow patterns changed in local areas near the piers. Under the same flow rate, the backwater of the rectangular pier was the largest, followed by the circular piers, and the elliptical pier was the smallest. The value of backwater volume caused by the rectangular pier was 1.95 times the elliptical pier. The elliptical bridge piers basically do not change the overall flow patterns of the river, and have little effect on the river regime. This work provides a reference for the layout of river bridge piers.

## Highlights

The MIKE21 FM module is used to analyze the numerical simulation of different pier shapes under different working conditions, as well as the backwater conditions and speed changes.

The MIKE21 numerical model can simulate complex terrain well. The relative error of the model in the calibration process is less than 1%, and the Nash efficiency coefficient is 0.97.

## INTRODUCTION

Cross-river bridge piers have a disturbing effect on the water flow: the water flow is restrained, and the water depth at a certain distance upstream of the bridge gradually increases along the course, which reduces the flooding capacity of the river to a certain extent and has an impact on the bridge pier foundation and embankment flood control. Therefore, before the implementation of the project it is required to analyze the river congestion and flow pattern changes caused by the construction of the bridge project (Luo *et al.* 2015; Huang *et al.* 2017; Liu *et al.* 2020). At present, scholars at home and abroad have carried out a large number of related studies on the impact of cross-river bridges on river channels, and have achieved more fruitful results. Yu & Zhu (2020) proposed a novel inlet turbulent boundary condition to simulate the dynamic evolution of local scour around bridge piers. Milad *et al.* (2019) used computational fluid dynamics methods to simulate the flow velocity and flow field around a tandem-shaped bridge pier. Wu *et al.* (2018) investigated the effect of manifolds on changing the flow field around bridge piers and thus reducing local scour around the piers. Sulaiman *et al.* 2019 performed a three-dimensional simulation of the turbulent flow field around an inhomogeneous bridge pier and the model predicted the flow field around the bridge pier. Kocaman (2014) studied the congestion of bridge piers at different flows using FLOW-3D and obtained a significant increase in the value of bridge pier congestion with increasing flow. Using the MIKE21 hydrodynamic model, Ren *et al.* (2017) simulated the congestion of the piers of six cross-river bridges in the estuary section of the Nandu River and compared the results with the empirical formulae, suggesting the use of the MIKE21 mathematical model to calculate the congestion values. Wei *et al.* (2016) simulated the flow field before and after the implementation of the scheme in this river section by establishing a planar two-dimensional flow mathematical model to analyze the effect of tandem cylindrical piers on the flow conditions in the river section from the perspective of engineering applications. Xu *et al.* (2018) studied the structure and congestion characteristics of obliquely arranged bridge pier bypass flow using the MIKE hydrodynamic model, and analyzed and summarized the changes in the maximum congestion height and the structure of bridge pier bypass flow field with pier size, oblique intersection angle, and incoming flow velocity. Wang *et al.* (2016) analyzed the relationship between water resistance ratio and congestion characteristics of bridge piers using numerical simulation for the problem of bridge pier congestion in a plain area river. As can be seen from the above, domestic and foreign researchers have mainly focused on the impact of river bridges on river congestion and flooding by fixed-shaped piers, while the impact of different-shaped piers on river congestion by using the MIKE21 hydrodynamic model is rare. This paper constructs a hydrodynamic model of the lower Yellow River based on MIKE21 to simulate the effects of changes in pier morphology on the flow velocity, water level and flow field of the river channel. The changes of flow patterns under different flow conditions were analyzed to provide a theoretical basis for scientific evaluation of flood control effects of river-related projects.

## METHODOLOGY

MIKE21 model is a two-dimensional mathematical simulation software developed by the Danish Institute of Hydrodynamics (DHI), which is applied to numerical simulation of water flow and water environment in estuaries, bays and near-shore areas of the ocean (Wang 2019). It is one of the more advanced models internationally, whose Flow Model hydrodynamic model is based on the three-way incompressible and Reynolds-averaged Navier-Stokes equation and obeys Boussinesq assumption and hydrostatic pressure assumption, and the model takes into account the Cochrane force, eddy viscosity (Guo *et al.* 2013; Ban & Wu 2018; Luo *et al.* 2018; Wang *et al.* 2020a, 2020b; Yang *et al.* 2020).

### The governing equation

*ξ*is the free surface water level (m);

*x*and

*y*are spatial coordinates (m);

*p*and

*q*are the flux density in

*x*and

*y*direction respectively (m

^{2}/s);

*d*is the depth change with time (m);

*g*is the acceleration of gravity (m

^{2}/s), which is 9.8 m

^{2}/s;

*C*is the Chezy coefficient (m

^{1/2}/s);

*ρ i*s the water density (kg/m

^{3});

*τ*,

_{xx}*τ*and

_{xy}*τ*are the horizontal shear stress in

_{yy}*x*direction and the vertical shear stress in

*x*direction, and the vertical shear stress in

*y*direction respectively (

*P*); Ω

_{a}*q*is the Coriol coefficient;

*f*is the drag coefficient;

*V*,

*V*and

_{x}*V*are the wind velocity and the wind velocity in

_{y}*x*and

*y*direction respectively (m/s);

*P*is the atmospheric pressure (

*P*).

_{a}This study aimed at calculating the discrete solution of two-dimensional shallow water governing equations with finite volume method based on unit center.

### The simulation steps

- (1)
Import the datum points with elevation in the calculation area, generate them into an irregular triangular grid, and adapt the grid encryption method to the shape of the bridge pier near the bridge pier, control the triangular grid side length of 30 meters in the main channel part, encrypt the side length of 2 meters in the location near the bridge pier, and the number of grid is 14248 to improve the calculation accuracy (Wang & Li 2018).

- (2)
Set up each boundary code, set the upstream boundary of the river as open boundary code 1, the downstream boundary of the river as open boundary code 2, and the boundary of both banks as closed boundary code 0.

- (3)
Set each time item required for the model calculation: the simulation start time, the number of time steps experienced by the simulation, and the time step length in seconds. It should be noted that the time step set here is used to adjust the output frequency of the simulation results of each module, and is not the actual time step.

- (4)
The initial conditions of the model are set, including the water level and flow velocity conditions at the initial time of calculation. In this example, the initial conditions are as follows: initial water level 0.8 meters; the model adopts cold start, and the initial flow velocity is 0.

- (5)
The boundary conditions of the model are set, mainly including upstream and downstream open boundary and solid wall closed boundary conditions. The upstream inlet boundary sets a fixed flow rate and the downstream outlet boundary sets a fixed water level. Closed boundary according to the solid wall impenetrable requirements, takes the normal velocity to zero.

- (6)
In the bridge pier where the section set observation points, the location of the bridge pier in the third and fifth observation points at the observation point spacing of 80 meters.

- (7)
The model needs to observe the results of the export, mainly including the river surface elevation, the flow velocity, the flow velocity and flow field of the section near the bridge pier and the line data of the section where the bridge pier is located.

### The evaluation of simulation

## CASE STUDY

### Study area

A proposed bridge across the Yellow River is located in the northeast of Jinan City, between the Nanduanwang section and Guiren section of the Yellow River test section. The distance between the two banks of the river section is 2,200 meters. The bridge adopts a medium-bearing steel box bollard arch, the upper structure form bridge type is a medium-bearing steel box bollard arch, the lower structure adopts a solid pier and bearing platform group pile foundation. The total length of the bridge is 2,285 meters, the standard cross-sectional width of the bridge is 34.5 meters, and the spacing of the bridge pier is 160 meters.

According to the design scheme, the hydraulic elements of the proposed bridge section and its impact on the surrounding environment were considered comprehensively. The model calculation area was selected from 0.8 kilometers upstream to 1.8 kilometers downstream of the proposed bridge, with a total length of 4.5 kilometers and a river width of 400–1,200 meters. The location diagram is shown in Figure 1.

### The division of river reach meshes

Based on the characteristics of the simulation river reach to construct a two-dimensional model, the length of the simulation reach was 4.5 kilometers. The cross-river bridge adopted double piers symmetrically in the direction of flow, the length-to-width ratio of the rectangular pier was 1.5 to 1; the diameter of the circular pier was defined as B; the length-to-width ratio of the oval pier was 1.5 to 1. Triangular meshes were used to divide the simulated river reach. At locations where the elevation changed little and the shoreline was relatively smooth, the size of meshes could be appropriately increased to reduce model calculation time. The mesh in the river reach where the project was located, and where the elevation changed drastically or the local terrain changed suddenly, should be appropriately densified to ensure calculation accuracy. The length of the triangle side of the main channel controlled was defined as 30 meters. At the same time, considering the relatively small size of the completed bridge pier, the calculation did not consider the existence of the bridge pier on the beach. The side of meshes near the pier was reduced to 2 meters, and the number of meshes was 14,248.

### The setting of boundary conditions

The embankment boundary of the model predicts the flow pattern at the river abutment according to the MIKE21 Flow Model. The boundary conditions of free water surface were defined as pressure boundary conditions. Considering the atmospheric pressure, the pressure (P) was defined as Pa (the atmosphere pressure), and the force was zero.

The upstream entrance of the model was set as the flow boundary condition, the downstream outlet was set as the water level boundary condition, and the downstream outlet of the river channel was set as the water level boundary condition. The river can be completely inundated during large floods, and the dynamic boundary is used to simulate the inundation state of the river during the calculation, using the method of limiting the water depth to deal with the dynamic boundary problem. In other words, the meshes were divided into three types: the dry area, the wet area and the semi-dry area; when the water depth(h) was less than h_{1}, it was the dry area, and h_{1} could be 0.005 meters; when the water depth was less than h_{2} and larger than h_{1}, it was the semi-dry area, which was only flow flux but no momentum flux on the unit interface, and h_{2} was taken as 0.1 meters; when the water depth was larger than h_{1}, it was the wet area.

In order to analyze the water surface elevation changes and flow patterns of the river under different incoming water and diversion conditions, the water level flow at the lower boundary was interpolated according to the water level flow relationship at the upstream water level station with an average 1.01‰ specific drop in the river, and simulated for different piers at different flows respectively (Table 1).

Working conditions . | Upstream boundary flow/(m^{3}/s)
. | Downstream boundary water surface elevation/m . |
---|---|---|

Working condition 1 | 9,600 (the 100-year cycle) | 22.66 |

Working condition 2 | 1,390 (annual average) | 17.24 |

Working condition 3 | 295 (lowest water level, P = 95%) | 16.65 |

Working conditions . | Upstream boundary flow/(m^{3}/s)
. | Downstream boundary water surface elevation/m . |
---|---|---|

Working condition 1 | 9,600 (the 100-year cycle) | 22.66 |

Working condition 2 | 1,390 (annual average) | 17.24 |

Working condition 3 | 295 (lowest water level, P = 95%) | 16.65 |

### The calibration of variables

The water depth was the parameter related to hydrodynamics in the measured data. Therefore, the hydrodynamic model selects the river water depth for parameter rate determination and model validation. Based on the hydrodynamic model, the main parameters that had a larger impact on the simulation results were roughness, and the water depth in the dry and wet area. In the process of adjusting the parameter values, different roughnesses, dry and wet water depth, and wind resistance coefficients are selected to fit the river water level based on the monitoring data at various points in the river.

This model was calibrated to select the annual average measured flow data in 2000. Based on the water level data investigated along the river, the two-dimensional mathematical model calculation utilized the upstream flow and downstream water level as the boundary conditions to calibrate the model (Table 2).

Sections . | Observed water levels/m . | Model calculation value/m . | Nash efficiency coefficient R
. _{NS} |
---|---|---|---|

Guiren | 18.32 | 18.34 | 0.97 |

Zhangwangzhuang | 18.10 | 18.07 | |

Yangfang | 17.89 | 17.86 | |

Nanduanwang | 18.23 | 18.25 | |

Mazhazi | 17.99 | 18.00 |

Sections . | Observed water levels/m . | Model calculation value/m . | Nash efficiency coefficient R
. _{NS} |
---|---|---|---|

Guiren | 18.32 | 18.34 | 0.97 |

Zhangwangzhuang | 18.10 | 18.07 | |

Yangfang | 17.89 | 17.86 | |

Nanduanwang | 18.23 | 18.25 | |

Mazhazi | 17.99 | 18.00 |

According to the calculation results, the model calculation results are basically consistent with the river characteristics, the Nacy efficiency coefficient is 0.97 in the rate determination and validation process, and the relative errors are less than 1%, which indicates that the proposed model can better reflect the river topography and the model results have high credibility.

According to the calculation results, the model calculation results were basically consistent with the river channel characteristics. During the calibration and verification process, the Nash efficiency coefficient was 0.97, indicating that the built model could reflect the river channel topography well. Therefore, the result was reliable.

After calibration, the river bed roughness in this model calculation was 0.026, the water depth in the dry area was 0.01 meters, and the water depth in the wet area was 0.005 meters. The hydrodynamic simulation results were reliable. The simulation period was 2.5 hours, and the time interval was 30 seconds. It should be noted that this model ignored the influence of tides, precipitation, wind, evaporation, ice, seepage, and waves.

## RESULTS AND ANALYSIS

According to the calculation scheme, this study executed quantitative analysis of different bridge piers in different conditions of the river course. Seven detection points are arranged on the axis of the river bridge, of which point 3 and point 5 are the locations of the bridge piers. The distance between each point is 80 meters, as shown in Figure 2.

### The simulation of flow field

This study utilized the MIKE FM module to analyze the flow field in different working conditions. Limited by boundary conditions, the contour map of the water surface elevation and flow velocity near different bridge piers in different working conditions are shown (Figures 3–8).

The changes of the elevation contour map of the water surface near piers in different working conditions are provided in Table 3.

Shapes of bridge piers . | . | The 10-year cycle . | Annual average . | Lowest water level (P = 95%)
. |
---|---|---|---|---|

The rectangular pier | The maximum water level elevation | 27.24 | 18.71 | 16.876 |

The minimum water level elevation | 26.80 | 18.61 | 16.866 | |

The height of backwater | 0.44 | 0.10 | 0.010 | |

The round pier | The maximum water level elevation | 27.16 | 18.67 | 16.856 |

The minimum water level elevation | 26.84 | 18.60 | 16.848 | |

The height of backwater | 0.32 | 0.07 | 0.008 | |

The oval pier | The maximum water level elevation | 27.11 | 18.624 | 16.851 |

The minimum water level elevation | 26.86 | 18.576 | 16.846 | |

The height of backwater | 0.25 | 0.048 | 0.005 |

Shapes of bridge piers . | . | The 10-year cycle . | Annual average . | Lowest water level (P = 95%)
. |
---|---|---|---|---|

The rectangular pier | The maximum water level elevation | 27.24 | 18.71 | 16.876 |

The minimum water level elevation | 26.80 | 18.61 | 16.866 | |

The height of backwater | 0.44 | 0.10 | 0.010 | |

The round pier | The maximum water level elevation | 27.16 | 18.67 | 16.856 |

The minimum water level elevation | 26.84 | 18.60 | 16.848 | |

The height of backwater | 0.32 | 0.07 | 0.008 | |

The oval pier | The maximum water level elevation | 27.11 | 18.624 | 16.851 |

The minimum water level elevation | 26.86 | 18.576 | 16.846 | |

The height of backwater | 0.25 | 0.048 | 0.005 |

As can be seen from the diagram, the bridge piers in the river channel reduce the area of the water crossing section and impede the normal flow pattern of the water, resulting in congestion upstream of the piers. The elliptical piers are less obstructive to the congestion and plunge of the river, which obviously improves the congestion and plunge of the river and reduces the negative impact on the river potential.

The changes of the elevation contour map of the velocity near piers in different working conditions were provided in Table 4.

Shapes of bridge piers . | . | The 10-year cycle . | Annual average . | Lowest water level (P = 95%)
. |
---|---|---|---|---|

The rectangular pier | Maximum flow rate | 1.35 | 0.83 | 0.45 |

Minimum flow rate | 0.34 | 0.18 | 0.13 | |

Flow rate difference | 1.01 | 0.65 | 0.32 | |

The round pier | Maximum flow rate | 1.42 | 0.85 | 0.46 |

Minimum flow rate | 0.75 | 0.41 | 0.21 | |

Flow rate difference | 0.67 | 0.44 | 0.25 | |

The oval pier | Maximum flow rate | 1.47 | 0.87 | 0.48 |

Minimum flow rate | 0.82 | 0.48 | 0.27 | |

Flow rate difference | 0.65 | 0.39 | 0.21 |

Shapes of bridge piers . | . | The 10-year cycle . | Annual average . | Lowest water level (P = 95%)
. |
---|---|---|---|---|

The rectangular pier | Maximum flow rate | 1.35 | 0.83 | 0.45 |

Minimum flow rate | 0.34 | 0.18 | 0.13 | |

Flow rate difference | 1.01 | 0.65 | 0.32 | |

The round pier | Maximum flow rate | 1.42 | 0.85 | 0.46 |

Minimum flow rate | 0.75 | 0.41 | 0.21 | |

Flow rate difference | 0.67 | 0.44 | 0.25 | |

The oval pier | Maximum flow rate | 1.47 | 0.87 | 0.48 |

Minimum flow rate | 0.82 | 0.48 | 0.27 | |

Flow rate difference | 0.65 | 0.39 | 0.21 |

From the chart, it shows that there is an angle between the axis of the bridge pier and the streamline of the water flow in the simulated river. Due to the blocking action of the bridge pier, the flow direction through the bridge pier changes significantly, the vortex and backflow appear, and the flow velocity decreases. The posterior lateral area of the rectangular bridge pier is affected by the scouring of the surrounding current and high velocity, which has a great impact on the river regime. The round pier is the second. The axis of the oval pier is parallel to the flow line, which greatly reduces the water blocking area.

### Data analysis

In this study, the 7 detection points of the bridge site section were used to verify the influence of different piers on the backwater of the bridge and the change in flow velocity at different flow rates. The stagnant water and flow velocity changes in the front of the bridge pier in the three kinds of working conditions are shown in Figures 9 and 10 and Tables 5 and 6.

Working conditions . | Detection points . | The rectangular piers . | The round piers . | The oval piers . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Bridge section elevation after bridge construction /m . | Bridge section elevation before bridge construction /m . | Backwater value /m . | Bridge section elevation after bridge construction /m . | Bridge section elevation before bridge construction /m . | Backwater value /m . | Bridge section elevation after bridge construction /m . | Bridge section elevation before bridge construction /m . | Backwater value /m . | ||

The 10-year cycle | 1 | 27.22 | 26.91 | 0.31 | 27.08 | 26.91 | 0.17 | 27.06 | 26.91 | 0.15 |

2 | 27.19 | 26.89 | 0.30 | 27.07 | 26.89 | 0.18 | 27.03 | 26.89 | 0.14 | |

3 | 27.24 | 26.85 | 0.39 | 27.16 | 26.85 | 0.31 | 27.11 | 26.85 | 0.26 | |

4 | 27.15 | 26.89 | 0.26 | 27.05 | 26.89 | 0.16 | 27.02 | 26.89 | 0.13 | |

5 | 27.22 | 26.82 | 0.40 | 27.13 | 26.82 | 0.31 | 27.09 | 26.82 | 0.27 | |

6 | 27.14 | 26.86 | 0.28 | 27.05 | 26.86 | 0.19 | 27.03 | 26.86 | 0.17 | |

7 | 27.16 | 26.88 | 0.28 | 27.04 | 26.88 | 0.16 | 27.02 | 26.88 | 0.14 | |

Annual average | 1 | 18.69 | 18.58 | 0.11 | 18.66 | 18.58 | 0.08 | 18.61 | 18.58 | 0.03 |

2 | 18.68 | 18.55 | 0.13 | 18.64 | 18.55 | 0.09 | 18.62 | 18.55 | 0.07 | |

3 | 18.71 | 18.54 | 0.17 | 18.68 | 18.54 | 0.14 | 18.65 | 18.54 | 0.11 | |

4 | 18.68 | 18.57 | 0.11 | 18.63 | 18.57 | 0.06 | 18.63 | 18.57 | 0.06 | |

5 | 18.70 | 18.59 | 0.11 | 18.67 | 18.59 | 0.08 | 18.64 | 18.59 | 0.05 | |

6 | 18.69 | 18.60 | 0.09 | 18.68 | 18.60 | 0.08 | 18.62 | 18.60 | 0.02 | |

7 | 18.73 | 18.59 | 0.14 | 18.71 | 18.59 | 0.12 | 18.60 | 18.59 | 0.01 | |

Lowest water level | 1 | 16.883 | 16.86 | 0.018 | 16.871 | 16.86 | 0.006 | 16.862 | 16.86 | 0.002 |

2 | 16.861 | 16.841 | 0.017 | 16.853 | 16.841 | 0.009 | 16.843 | 16.841 | 0.002 | |

3 | 16.871 | 16.833 | 0.027 | 16.858 | 16.833 | 0.014 | 16.847 | 16.833 | 0.014 | |

4 | 16.872 | 16.842 | 0.025 | 16.856 | 16.842 | 0.009 | 16.845 | 16.842 | 0.003 | |

5 | 16.875 | 16.826 | 0.049 | 16.857 | 16.826 | 0.031 | 16.848 | 16.826 | 0.022 | |

6 | 16.873 | 16.853 | 0.015 | 16.867 | 16.853 | 0.009 | 16.856 | 16.853 | 0.003 | |

7 | 16.876 | 16.856 | 0.010 | 16.872 | 16.856 | 0.006 | 16.859 | 16.856 | 0.003 |

Working conditions . | Detection points . | The rectangular piers . | The round piers . | The oval piers . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Bridge section elevation after bridge construction /m . | Bridge section elevation before bridge construction /m . | Backwater value /m . | Bridge section elevation after bridge construction /m . | Bridge section elevation before bridge construction /m . | Backwater value /m . | Bridge section elevation after bridge construction /m . | Bridge section elevation before bridge construction /m . | Backwater value /m . | ||

The 10-year cycle | 1 | 27.22 | 26.91 | 0.31 | 27.08 | 26.91 | 0.17 | 27.06 | 26.91 | 0.15 |

2 | 27.19 | 26.89 | 0.30 | 27.07 | 26.89 | 0.18 | 27.03 | 26.89 | 0.14 | |

3 | 27.24 | 26.85 | 0.39 | 27.16 | 26.85 | 0.31 | 27.11 | 26.85 | 0.26 | |

4 | 27.15 | 26.89 | 0.26 | 27.05 | 26.89 | 0.16 | 27.02 | 26.89 | 0.13 | |

5 | 27.22 | 26.82 | 0.40 | 27.13 | 26.82 | 0.31 | 27.09 | 26.82 | 0.27 | |

6 | 27.14 | 26.86 | 0.28 | 27.05 | 26.86 | 0.19 | 27.03 | 26.86 | 0.17 | |

7 | 27.16 | 26.88 | 0.28 | 27.04 | 26.88 | 0.16 | 27.02 | 26.88 | 0.14 | |

Annual average | 1 | 18.69 | 18.58 | 0.11 | 18.66 | 18.58 | 0.08 | 18.61 | 18.58 | 0.03 |

2 | 18.68 | 18.55 | 0.13 | 18.64 | 18.55 | 0.09 | 18.62 | 18.55 | 0.07 | |

3 | 18.71 | 18.54 | 0.17 | 18.68 | 18.54 | 0.14 | 18.65 | 18.54 | 0.11 | |

4 | 18.68 | 18.57 | 0.11 | 18.63 | 18.57 | 0.06 | 18.63 | 18.57 | 0.06 | |

5 | 18.70 | 18.59 | 0.11 | 18.67 | 18.59 | 0.08 | 18.64 | 18.59 | 0.05 | |

6 | 18.69 | 18.60 | 0.09 | 18.68 | 18.60 | 0.08 | 18.62 | 18.60 | 0.02 | |

7 | 18.73 | 18.59 | 0.14 | 18.71 | 18.59 | 0.12 | 18.60 | 18.59 | 0.01 | |

Lowest water level | 1 | 16.883 | 16.86 | 0.018 | 16.871 | 16.86 | 0.006 | 16.862 | 16.86 | 0.002 |

2 | 16.861 | 16.841 | 0.017 | 16.853 | 16.841 | 0.009 | 16.843 | 16.841 | 0.002 | |

3 | 16.871 | 16.833 | 0.027 | 16.858 | 16.833 | 0.014 | 16.847 | 16.833 | 0.014 | |

4 | 16.872 | 16.842 | 0.025 | 16.856 | 16.842 | 0.009 | 16.845 | 16.842 | 0.003 | |

5 | 16.875 | 16.826 | 0.049 | 16.857 | 16.826 | 0.031 | 16.848 | 16.826 | 0.022 | |

6 | 16.873 | 16.853 | 0.015 | 16.867 | 16.853 | 0.009 | 16.856 | 16.853 | 0.003 | |

7 | 16.876 | 16.856 | 0.010 | 16.872 | 16.856 | 0.006 | 16.859 | 16.856 | 0.003 |

Working conditions . | Detection points . | The rectangular piers . | The round piers . | The oval piers . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Flow velocity of bridge section after bridge construction /m . | Velocity of bridge section before bridge construction /m . | Backwater value /m . | Flow velocity of bridge section after bridge construction /m . | Velocity of bridge section before bridge construction /m . | Backwater value /m . | Flow velocity of bridge section after bridge construction /m . | Velocity of bridge section before bridge construction /m . | Backwater value /m . | ||

The 10-year cycle | 1 | 1.52 | 1.69 | 0.17 | 1.54 | 1.69 | 0.15 | 1.58 | 1.69 | 0.11 |

2 | 1.66 | 1.76 | 0.1 | 1.67 | 1.76 | 0.09 | 1.72 | 1.76 | 0.04 | |

3 | 1.05 | 1.87 | 0.82 | 1.21 | 1.87 | 0.66 | 1.25 | 1.87 | 0.62 | |

4 | 1.90 | 2.10 | 0.2 | 1.94 | 2.10 | 0.16 | 1.97 | 2.10 | 0.13 | |

5 | 1.35 | 2.12 | 0.77 | 1.42 | 2.12 | 0.7 | 1.48 | 2.12 | 0.64 | |

6 | 1.93 | 2.21 | 0.28 | 2.04 | 2.21 | 0.17 | 2.07 | 2.21 | 0.14 | |

7 | 1.76 | 1.99 | 0.23 | 1.78 | 1.99 | 0.21 | 1.82 | 1.99 | 0.17 | |

Annual average | 1 | 0.35 | 0.42 | 0.07 | 0.38 | 0.42 | 0.04 | 0.41 | 0.42 | 0.01 |

2 | 0.83 | 0.89 | 0.06 | 0.85 | 0.89 | 0.04 | 0.87 | 0.89 | 0.02 | |

3 | 0.34 | 0.90 | 0.56 | 0.42 | 0.90 | 0.48 | 0.47 | 0.90 | 0.43 | |

4 | 0.84 | 0.89 | 0.05 | 0.86 | 0.89 | 0.03 | 0.88 | 0.89 | 0.01 | |

5 | 0.43 | 0.92 | 0.49 | 0.57 | 0.92 | 0.35 | 0.68 | 0.92 | 0.24 | |

6 | 1.12 | 1.20 | 0.08 | 1.15 | 1.20 | 0.05 | 1.16 | 1.20 | 0.04 | |

7 | 0.71 | 0.87 | 0.16 | 0.79 | 0.87 | 0.08 | 0.82 | 0.87 | 0.05 | |

Lowest water level | 1 | 0.08 | 0.153 | 0.073 | 0.13 | 0.153 | 0.023 | 0.15 | 0.153 | 0.003 |

2 | 0.31 | 0.358 | 0.048 | 0.32 | 0.358 | 0.038 | 0.35 | 0.358 | 0.008 | |

3 | 0.15 | 0.361 | 0.211 | 0.20 | 0.361 | 0.161 | 0.24 | 0.361 | 0.121 | |

4 | 0.33 | 0.365 | 0.035 | 0.34 | 0.365 | 0.025 | 0.36 | 0.365 | 0.005 | |

5 | 0.26 | 0.368 | 0.108 | 0.27 | 0.368 | 0.098 | 0.30 | 0.368 | 0.068 | |

6 | 0.54 | 0.608 | 0.068 | 0.57 | 0.608 | 0.038 | 0.59 | 0.608 | 0.018 | |

7 | 0.16 | 0.221 | 0.057 | 0.19 | 0.221 | 0.027 | 0.21 | 0.217 | 0.007 |

Working conditions . | Detection points . | The rectangular piers . | The round piers . | The oval piers . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Flow velocity of bridge section after bridge construction /m . | Velocity of bridge section before bridge construction /m . | Backwater value /m . | Flow velocity of bridge section after bridge construction /m . | Velocity of bridge section before bridge construction /m . | Backwater value /m . | Flow velocity of bridge section after bridge construction /m . | Velocity of bridge section before bridge construction /m . | Backwater value /m . | ||

The 10-year cycle | 1 | 1.52 | 1.69 | 0.17 | 1.54 | 1.69 | 0.15 | 1.58 | 1.69 | 0.11 |

2 | 1.66 | 1.76 | 0.1 | 1.67 | 1.76 | 0.09 | 1.72 | 1.76 | 0.04 | |

3 | 1.05 | 1.87 | 0.82 | 1.21 | 1.87 | 0.66 | 1.25 | 1.87 | 0.62 | |

4 | 1.90 | 2.10 | 0.2 | 1.94 | 2.10 | 0.16 | 1.97 | 2.10 | 0.13 | |

5 | 1.35 | 2.12 | 0.77 | 1.42 | 2.12 | 0.7 | 1.48 | 2.12 | 0.64 | |

6 | 1.93 | 2.21 | 0.28 | 2.04 | 2.21 | 0.17 | 2.07 | 2.21 | 0.14 | |

7 | 1.76 | 1.99 | 0.23 | 1.78 | 1.99 | 0.21 | 1.82 | 1.99 | 0.17 | |

Annual average | 1 | 0.35 | 0.42 | 0.07 | 0.38 | 0.42 | 0.04 | 0.41 | 0.42 | 0.01 |

2 | 0.83 | 0.89 | 0.06 | 0.85 | 0.89 | 0.04 | 0.87 | 0.89 | 0.02 | |

3 | 0.34 | 0.90 | 0.56 | 0.42 | 0.90 | 0.48 | 0.47 | 0.90 | 0.43 | |

4 | 0.84 | 0.89 | 0.05 | 0.86 | 0.89 | 0.03 | 0.88 | 0.89 | 0.01 | |

5 | 0.43 | 0.92 | 0.49 | 0.57 | 0.92 | 0.35 | 0.68 | 0.92 | 0.24 | |

6 | 1.12 | 1.20 | 0.08 | 1.15 | 1.20 | 0.05 | 1.16 | 1.20 | 0.04 | |

7 | 0.71 | 0.87 | 0.16 | 0.79 | 0.87 | 0.08 | 0.82 | 0.87 | 0.05 | |

Lowest water level | 1 | 0.08 | 0.153 | 0.073 | 0.13 | 0.153 | 0.023 | 0.15 | 0.153 | 0.003 |

2 | 0.31 | 0.358 | 0.048 | 0.32 | 0.358 | 0.038 | 0.35 | 0.358 | 0.008 | |

3 | 0.15 | 0.361 | 0.211 | 0.20 | 0.361 | 0.161 | 0.24 | 0.361 | 0.121 | |

4 | 0.33 | 0.365 | 0.035 | 0.34 | 0.365 | 0.025 | 0.36 | 0.365 | 0.005 | |

5 | 0.26 | 0.368 | 0.108 | 0.27 | 0.368 | 0.098 | 0.30 | 0.368 | 0.068 | |

6 | 0.54 | 0.608 | 0.068 | 0.57 | 0.608 | 0.038 | 0.59 | 0.608 | 0.018 | |

7 | 0.16 | 0.221 | 0.057 | 0.19 | 0.221 | 0.027 | 0.21 | 0.217 | 0.007 |

The results show that the difference between the measured water surface elevation and the measured water surface elevation of rectangular piers at different frequencies is within 0.26 meters, while the difference between the measured water surface elevation and the measured water surface elevation of round piers at different frequencies is within 0.15 meters. The congestion in front of rectangular piers is the largest, followed by round piers and elliptical piers, and the congestion value of rectangular piers can reach 1.45 times that of round piers and 1.95 times that of elliptical piers, 1.45 times of the rectangular bridge pier, and can reach 1.95 times that of the oval bridge pier. Thus, it is proved that the oval pier has less influence on the pre-congestion of the bridge and is more conducive to the application in engineering practice.

The result shows that the flow velocity near the bridge pier is distributed along the longitudinal direction of the river channel. The water blocking effect results in local turbulence and higher velocity in some areas near the bridge pier in the main river. Therefore, velocity changes in the area near the bridge pier are obvious, and the velocity in the main river channel has a weak influence. The rectangular pier has the greatest influence on the flow velocity in front of the pier, followed by the round pier, and the oval pier has the least. As a result, it reviews that the elliptical pier has a little effect on the flow velocity in front of the bridge, which benefits the application in engineering practice.

### Analysis of results

In the study, the two-dimensional hydrodynamic model developed in the previous paper was used to numerically simulate the overall flow patterns in several groups with different flow levels and different combinations of piers. From the congestion relationship curves at each monitoring point, it can be seen that under the action of the three shapes of piers, the influence of congestion upstream of the piers is very large, and the congestion height increases significantly with the increase in flow. Under the working conditions, the congestion value of rectangular piers is generally higher than that of round and oval piers; under the same flow conditions, the congestion value of rectangular piers is the largest, followed by that of round piers, and oval piers are the smallest. This is mainly due to the large angle between the rectangular piers and the flow direction, especially in the main channel where the terrain is narrow and deep, which leads to an increase in the contact area between the simulated water flow and the piers, thus increasing the frictional resistance and the congestion height. Therefore, the use of rectangular piers should be avoided as much as possible in engineering practice.

Analysis of the diagram shows that the water flow within the range of the bridge piers is affected by the piers, the asymmetric velocity changes, the local range of near-shore velocity increase will cause the water to intensify the scouring of the river bank, the flow near the piers is significantly reduced; this is due to a certain distance in front of the piers, the congestion effect flow velocity decreases, part of the water flow directly to the front of the piers, the flow velocity is reduced to a minimum when it reaches the piers, another part of the flow is around the piers. And in the width of the pier at both ends of the maximum velocity; in the pier directly behind, due to the boundary layer separation on both sides of the pier a pair of wake vortices gradually formed, and the velocity decreases. These results are in agreement with the classical hydrodynamic theory. Under the conditions, the flow velocity difference of rectangular piers is generally higher than that of round piers and elliptical piers; under the same incoming flow conditions, the flow velocity difference of rectangular piers is the largest, followed by that of round piers, and elliptical piers is the smallest.

## CONCLUSION

This study illustrates a bridge across the Yellow River to analyze the influence of bridge pier shapes on the river flow based on the MIKE21 FM hydrodynamic model. The results are as follows:

- (1)
This study utilizes the MIKE21 FM model to analyze the numerical simulation and the backwater situations and velocity changes of different pier shapes in different working conditions respectively. The results reveal that: compared with the rectangular pier and the round pier, the oval pier has the smaller relative errors, the improved water flow structure, the increased water passing capacity of the bridge site, and significantly reduced flow velocity. The height of the backwater is significantly reduced, basically does not change the overall flow pattern of the river, and has little impact on the river regime. The backwater value of the rectangular pier is as 1.95 times as the ovel pier. The velocity of the rectangular pier is 1.61 times that of the oval pier.

- (2)
The MIKE21 numerical model can better simulate the complex terrain, and the model has a Nacy efficiency coefficient of 0.97 in the rate determination process. The MIKE21 mathematical model is used to calculate the congestion value, and the calculation results are reasonable.

- (3)
The model analysis should consider the unsteady flow factor for the bridge backwater problem of the river channel when the project is in complex terrain conditions. Since this calculation is performed in the assumption of a constant flow condition, the calculation results may have certain deviations, and further development will be planned in the future.

## ACKNOWLEDGEMENTS

This work is financially supported by Technology Research Center of Water Environment Governance and Ecological Restoration Academician Workstation of Henan Province, Program for Science & Technology Innovation Talents in Universities of Henan Province (No. 15HASTIT049).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

2020, 1–17