# Case Study on Application of the Step with Non-Uniform Heights at the Bottom Using a Numerical and Experimental Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Physical Model and Problem Description

#### 2.1. Physical Model

^{3}/s, the maximum flow of the unit width was 44.8 m

^{3}/s, and the maximum head was 518.36 m.

^{3}/s of the prototype is presented in this study. The velocity was measured using a propeller-type current meter and pitot tube, and the time-averaged pressure was measured using a piezometer tube. The accuracy of the gauge used for measuring water depth was ± 1 mm. The flow visualizations were conducted with a digital video camera (Canon D800).

#### 2.2. Problem Description

#### 2.3. Effects of the Non-Uniform Steps

^{3}/s of the prototype are given in Table 2. For the concave bank of the smooth spillway, the water level was obviously higher than that of the convex bank, and the water level was extremely low (0.08 m) at an axial distance of 161.35 m, which was close to the bad flow state without water covering the base plate. The non-uniform-height steps proposed in this study were placed in the smooth spillway, which caused the difference in the transverse water level on the cross-section to decrease obviously. The flow velocity distribution was more uniform, which significantly improved the flow regime in the curved section. The cross-section velocity distribution of the curved stepped spillway is given in Table 3.

## 3. Numerical Methodology and Model Validation

#### 3.1. Numerical Methodology

^{3}/s are shown in Table 4 and Table 5. The simulated results at a flow rate of 272.0 m

^{3}/s that was converted to the prototype based on the similitude principle are presented. The simulation of air–water flow in the curved spillway with non-uniform-height steps was carried out using the ANSYS-FLUENT platform.

#### 3.2. Mesh Tests and Model Validation

^{+}, the values of which were obtained by Equation (10). The value of this parameter is related to whether the grid density near the wall is appropriate. If the value is too large or too small, the computational accuracy of the flow field near the wall will be affected. Due to the complexity of the model and the flow, this value was maintained at between 30 and 300 in this study [25]. The over-dense grid is not conducive to the calculation in this project.

## 4. Results and Discussion

#### 4.1. Main Flow Region

_{r}) on the maximum standing wave heights. For moderate and large Froude numbers (F

_{r}= 4 and 6), the uneven distribution separate bend flow on the cross-section resulted in extremely high-water levels. These hydraulic phenomena were only observed in Figure 14c, in which the severe situation of the base plate of the convex bank not having water cover occurred, while the water flow was completely concentrated on the concave bank, making the water level extremely high. When the non-uniform-height steps were arranged on the smooth spillway, the flow velocity in the main flow region and the water level on both sides were more uniform, which improved the bad flow pattern in the smooth spillway, as seen in Figure 14d.

#### 4.2. Flow Pattern on a Step

#### 4.3. Cavitation Characteristics

#### 4.4. Energy Loss

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Diagram of a reservoir spillway. (

**a**) Top view of smooth spillway; (

**b**) top view of stepped spillway; (

**c**) side view of stepped spillway. “A” denotes step No. 9.

**Figure 8.**Schematic of the grid: (

**a**) in the weir crest; and (

**b**) in the curved section of the smooth spillway.

**Figure 9.**The vertical velocity at position [x = 1.4, y = 0, z = −0.3] of the smooth spillway for ${\mathrm{z}}_{0}=3.5$, ${\text{}\mathrm{z}}_{0}=5.0$, and ${\mathrm{z}}_{0}=7.5$.

**Figure 10.**The experimentally measured (

**left**) and numerically simulated (

**right**) flow regime. (

**a**) and (

**b**) are located on the weir crest and (

**c**,

**d**) are located in the curved section.

**Figure 11.**Flow pattern on the steps. (

**a**) The experimental measurement results; and (

**b**) numerical results. VOF: volume-of-fluid.

**Figure 14.**The cross-sectional velocity distributions at the ends of steps No. 1, 17, 24, and 28 of the smooth spillway and step spillway. (

**a**,

**c**,

**e**,

**g**) are smooth spillways; and (

**b**,

**d**,

**f**,

**h**) are stepped spillways.

Prototype | Physical Model | |||
---|---|---|---|---|

Index | Water Level in Upstream (m) | Discharge (m^{3}/s) | Water Level in Upstream (m) | Discharge (L/s) |

Case 1 | 516.09 | 272.00 | 12.90 | 26.87 |

Case 2 | 517.00 | 348.08 | 12.92 | 34.40 |

Case 3 | 518.36 | 477.64 | 12.96 | 47.20 |

**Table 2.**The average water depths of each measuring point at a flow discharge rate of 272.0 m

^{3}/s of the prototype.

Smooth Spillway | Curved Stepped Spillway | ||||
---|---|---|---|---|---|

Axial Distance (m) | Water Level (m) | Axial Distance (m) | Water Level (m) | ||

Concave | Convex | Concave | Convex | ||

103.66 | 1.44 | 1.28 | 103.16 | 1.12 | 1.80 |

126.73 | 2.40 | 0.52 | 124.16 | 1.60 | 2.60 |

149.81 | 5.32 | 0.16 | 145.16 | 1.80 | 2.40 |

161.35 | 5.92 | 0.08 | 166.16 | 2.60 | 2.20 |

194.97 | 3.60 | 0.40 | 192.41 | 2.20 | 2.60 |

206.51 | 1.60 | 0.92 | 208.16 | 2.00 | 2.80 |

**Table 3.**The cross-sectional velocity distribution of the curved stepped spillway at a flow discharge rate of 272.0 m

^{3}/s of the prototype.

Axial Distance (m) | Velocity Distribution (m/s) | ||
---|---|---|---|

Concave | Middle | Convex | |

103.16 | 16.57 | 16.63 | 16.70 |

124.16 | 16.76 | 18.97 | 16.95 |

145.16 | 17.14 | 17.77 | 15.68 |

166.16 | 19.04 | 18.09 | 15.18 |

192.41 | 19.61 | 15.81 | 14.29 |

208.16 | 16.57 | 16.63 | 16.70 |

Index | Velocity on the Weir Crest (m/s) | Water depth on the Weir Crest (cm) | F_{r} |
---|---|---|---|

Smooth | 1.15 | 8.78 | 1.16 |

Stepped | 1.15 | 8.78 | 1.16 |

Index | Velocity of Flow into the Curved Section (m/s) | Water Depth of Flow into the Curved Section (cm) | F_{r} |
---|---|---|---|

Smooth | 2.00 | 5.00 | 2.86 |

Stepped | 2.00 | 5.00 | 2.86 |

Case | Water Depth (h_{0}) (m) | Mean Velocity (V_{0}) (m/s) | Energy (E) (m) | Relative Energy Loss (ΔE/E^{′}) |
---|---|---|---|---|

Smooth curved spillway | 1.25 | 27.4 | 39.51 | 0.66 |

Stepped curved spillway | 2.0 | 15.0 | 13.47 | 1.93 |

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**MDPI and ACS Style**

Li, D.; Yang, Q.; Ma, X.; Dai, G.
Case Study on Application of the Step with Non-Uniform Heights at the Bottom Using a Numerical and Experimental Model. *Water* **2018**, *10*, 1762.
https://doi.org/10.3390/w10121762

**AMA Style**

Li D, Yang Q, Ma X, Dai G.
Case Study on Application of the Step with Non-Uniform Heights at the Bottom Using a Numerical and Experimental Model. *Water*. 2018; 10(12):1762.
https://doi.org/10.3390/w10121762

**Chicago/Turabian Style**

Li, Dengsong, Qing Yang, Xudong Ma, and Guangqing Dai.
2018. "Case Study on Application of the Step with Non-Uniform Heights at the Bottom Using a Numerical and Experimental Model" *Water* 10, no. 12: 1762.
https://doi.org/10.3390/w10121762