# Impact of Local Scour around a Bridge Pier on Migration of Waved-Shape Accumulation of Ice Particles under an Ice Cover

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Materials

_{50}) of 0.713. Before each experimental run, the entire sand bed was leveled with the slope of 0. To initiate the formation of an ice jam, a Styrofoam panel, which was 1.0 m long and 0.39 m wide, was placed on the water surface at CS-20. The CS-4 was chosen as the control cross section. At the beginning of each experimental run, the average approaching flow velocity (V

_{0}) and flow depth (H

_{0}) at CS-4 were used as the initial hydraulic condition. In this experimental study, the model ice particles were made of polyethylene material with a mass density of 0.918 g/cm

^{3}, which was nearly identical to that of natural ice of 0.917 g/cm

^{3}. The shape of the model ice particles was flat-ellipsoid with the longest diameter of 3.5 mm. Model ice particles were discharged from an ice hopper that was located over the flume between CS-2 and CS-3. The discharge rate of ice particles from the ice hopper could be adjusted as required. A cylinder with a diameter of D = 2 cm was used to model a bridge pier and installed in the center of the flume at CS-16.

## 3. Interaction of Local Scour and Ice Wave Migration

_{w}

_{0}(x) is the depth-average flow velocity in front of the ice wave at a distance of x from the left flume wall; H

_{w1}(x)is the flow depth above the initial sand bed at the pier at a distance of x from the left flume wall; v

_{w}

_{1}(x) is the depth-average flow velocity at the pier under the ice jam at a distance of x from the left flume wall; H

_{s1}(x) is the depth of the scour hole at a distance of x from the left flume wall; and H

_{s1}(x) = 0 in a sand bed without a scour, as shown in Figure 4.

_{m}

_{0}is the depth-average flow velocity in the center of the flume in front of the ice wave, v

_{m}

_{1}is the depth-average flow velocity in the center of the flume at the pier (note, the bridge pier is located at the center of the flume); K is the transverse distribution parameter of flow velocity.

_{w}

_{1}, H

_{s}

_{1}are flow depth under an ice jam (above the initial sand bed) and the scour hole depth in this transverse unit, respectively, as shown in Figure 5.

_{w}is the mass density of water; Q

_{w}is the discharge of flow.

_{P}is the difference between the pressure force at upstream control surface (at the ice wave front) and that at downstream cross section (at the pier); H

_{i}is the thickness of ice wave crest; g is gravitational acceleration.

_{1}is the drag force caused by an ice wave; C

_{i}is the drag coefficient, which ranges from 5.5 to 12.5 and increases with the increase of the ice discharge rate Q

_{i}.

_{2}is the resistance force caused by the pier, and other transverse units without the pier are F

_{2}= 0; C

_{D}is the resistance coefficient, which is related to the Reynolds number; D is the pier diameter.

_{3}is the resistance caused by the channel bed and side wall of the flume; C

_{d}is the drag coefficient; n

_{c}is composite roughness coefficient of the flume; L is the length of the control volume between the upstream and downstream control surfaces.

_{i}. Because ice wave affects the local scour depth and the mechanical load on the pier, an ice wave with a thicker crest should be vulnerable to failure of the bridge. By substituting the experimental data into the above formula, the thickness of an ice wave can be obtained. At the cross section where the pier is located, the origin of the transverse coordinates is located at the center of the flume (the pier center) is set. The transverse distribution of thickness of ice waves at this cross section is shown in Figure 6. The calculated results using the above formula agree well with results of experiments.

- (1)
- As shown in Figure 6, the closer to the central axis of the flume, the smaller the thickness of an ice wave and the deeper the scour hole. Thus, the cross-sectional area for flow at the pier is larger. Around the pier, it is difficult for ice particles to accumulate here because of the existence of the downflow and vortices around the pier. Results of experiments showed that, similar to a scour hole in a sand bed, an “ice scour hole” appeared at the bottom of the ice jam around the pier. The appearance of the “ice scour hole” is resulted from the presence of the pier. The existence of the “ice scour hole” affects the development of ice waves. Interestingly, the thickness of the wave crest at this cross section (where the pier is located) is less than that at other cross sections without the presence of a pier.
- (2)
- With the increase in the distance from the flume center toward the flume wall, the thickness of the wave crest increases first and then decreases. In the zone around the pier, a scour hole in the sand bed developed, and the wave crest thickness increases slightly because the presence of a scour hole leads to the increase in the cross-sectional area for flow. Further away from the flume center (or the pier), for zones without the presence of scour holes in the sand bed, the wave crest thickness reaches a constant toward the flume side wall.

_{i}is the migration speed of an ice wave; T is the duration of the selected period for certain time for collecting data.

_{i}is the roughness coefficient of the ice jam; n

_{b}is the composite roughness coefficient of the channel bed and flume wall; ρ

_{i}is the density of ice; d

_{i}is the particle grain size of ice; T is the temperature; J is the water surface slope.

_{T}is the ice transport capacity; p is the porosity of an ice jam in this study, p = 0.4; y

_{i}is the thickness of an ice jam at any cross section; and t is time.

_{0}and at any instant time of t

_{0}, Equation (22) can be expressed as:

_{0}is the ice jam thickness at the cross section of the ice wave trough.

## 4. Conclusions

- By combining the continuity equation and momentum equation, the equation for calculating the thickness of ice wave crest under the influence of a local scour was derived. Results showed that the ice wave thickness near the central axis of the flume is smaller. This “ice scour hole” phenomenon at the bottom of an ice jam near the pier is similar to the scour hole at the pier in the sand bed. With the increase in the distance from the flume center, the thickness of the wave crest increases first and then decreases. Further away from the central axis of the flume, for zones without the presence of scour holes in the sand bed, the wave crest thickness reaches a constant toward the flume walls. The calculated average ice wave thickness of the whole section is in good agreement with result of experiments regardless of whether or not a local scour is present in channel bed. Under the same hydraulic condition, the presence of a local scour at a pier lead to an increase in the thickness of the wave crest.
- Under the same hydraulic condition, the appearance of a scour hole slows down the migration speed of an ice wave at the pier. With the presence of a local scour at the pier, the dimensionless relation for determining the migration speed of an ice wave was obtained based on experimental data. Results showed that the larger the ice discharge rate and flow Froude number is, the higher the migration speed of an ice wave is. The greater the ratio of the water depth under an ice jam, including the scour hole depth to total water depth, the smaller the migration speed of an ice wave.
- The ice transport capacity with the presence of a scour hole was analyzed. The ice transport capacity at the pier decreases with the increase in both the scour hole depth and the flow depth under an ice jam. The developed equation can be used to determine the ice transport capacity by means of the migration speed of ice waves considering a local scour process at a bridge pier. The calculated results agree well with those of experiments in the laboratory.
- The interaction between an ice jam and local scour at a pier is very complicated and has hardly been conducted. This study, based on laboratory experiments, belongs to conceptual research instead of research based on field prototype data from a natural river. More field observation data are needed to verify results obtained from laboratory experiments. Considering the influence of different pier types, river bends, and side wall effects, relevant experiments need to be further carried out.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Forces acting on the control volume of flowing water under an ice wave around the bridge pier.

**Figure 5.**Channel cross-section at the pier (where, b is the width of each unit, and B is the sum of width of all units).

**Figure 6.**Transverse distribution of ice wave thickness at the cross section where the pier is located.

Experimental Run # | v_{0} (m/s) | H_{0} (m) | D (cm) | Q_{i} (L/s) | d_{50} (mm) | H_{S} (m) |
---|---|---|---|---|---|---|

A1 | 0.17 | 0.25 | 2 | 0.0205 | 0.713 | 0.0319 |

A2 | 0.15 | 0.25 | 2 | 0.0205 | 0.713 | 0.0286 |

A3 | 0.17 | 0.20 | 2 | 0.0205 | 0.713 | 0.0343 |

A4 | 0.15 | 0.20 | 2 | 0.0205 | 0.713 | 0.0323 |

A5 | 0.18 | 0.25 | 2 | 0.018 | 0.713 | 0.0288 |

A6 | 0.18 | 0.25 | 2 | 0.022 | 0.713 | 0.0327 |

A7 | 0.18 | 0.25 | 2 | 0.026 | 0.713 | 0.0350 |

A8 | 0.18 | 0.25 | 2 | 0.030 | 0.713 | 0.0365 |

A9 | 0.18 | 0.20 | 2 | 0.018 | 0.713 | 0.0308 |

A10 | 0.18 | 0.20 | 2 | 0.022 | 0.713 | 0.0339 |

A11 | 0.18 | 0.20 | 2 | 0.026 | 0.713 | 0.0384 |

A12 | 0.18 | 0.20 | 2 | 0.030 | 0.713 | 0.0391 |

B1 | 0.17 | 0.25 | 2 | 0.0205 | / | 0 |

B2 | 0.15 | 0.25 | 2 | 0.0205 | / | 0 |

B3 | 0.17 | 0.20 | 2 | 0.0205 | / | 0 |

B4 | 0.15 | 0.20 | 2 | 0.0205 | / | 0 |

_{0}is the average approaching velocity at CS-4; H

_{0}is the approaching flow depth at CS-4; D is the pier diameter; Q

_{i}is the discharge rate of ice particles released from the ice hopper; d

_{50}is the median grain size of sand particles; H

_{S}is the maximum scour depths. B1–B4 represent experiments without local scour (without bed sand on the flume bottom).

Experiment Run # | Calculated Value (m) | Experimental Value (m) | Experiment Run # | Calculated Value (m) | Experimental Value (m) |
---|---|---|---|---|---|

A1 | 0.082 | 0.073 | B1 | 0.068 | 0.070 |

A2 | 0.082 | 0.079 | B2 | 0.053 | 0.077 |

A3 | 0.065 | 0.056 | B3 | 0.059 | 0.051 |

A4 | 0.065 | 0.068 | B4 | 0.053 | 0.065 |

Number | H_{s} (m) | v_{i} (m/s) | Number | H_{s} (m) | v_{i} (m/s) |
---|---|---|---|---|---|

A1 | 0.0319 | 0.00142 | B1 | 0 | 0.00147 |

A2 | 0.0286 | 0.00128 | B2 | 0 | 0.00134 |

A3 | 0.0343 | 0.00184 | B3 | 0 | 0.00200 |

A4 | 0.0323 | 0.00152 | B4 | 0 | 0.00159 |

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**MDPI and ACS Style**

Hou, Z.; Wang, J.; Sui, J.; Song, F.; Li, Z. Impact of Local Scour around a Bridge Pier on Migration of Waved-Shape Accumulation of Ice Particles under an Ice Cover. *Water* **2022**, *14*, 2193.
https://doi.org/10.3390/w14142193

**AMA Style**

Hou Z, Wang J, Sui J, Song F, Li Z. Impact of Local Scour around a Bridge Pier on Migration of Waved-Shape Accumulation of Ice Particles under an Ice Cover. *Water*. 2022; 14(14):2193.
https://doi.org/10.3390/w14142193

**Chicago/Turabian Style**

Hou, Zhixing, Jun Wang, Jueyi Sui, Feihu Song, and Zhicong Li. 2022. "Impact of Local Scour around a Bridge Pier on Migration of Waved-Shape Accumulation of Ice Particles under an Ice Cover" *Water* 14, no. 14: 2193.
https://doi.org/10.3390/w14142193