# Study on the Head Loss of the Inlet Gradient Section of the Aqueduct

^{*}

## Abstract

**:**

_{up}/W

_{down}; the local head loss coefficient has a good exponential function with W

_{up}/W

_{down}. The research results can provide a reference for the design of the inlet gradient section and the solution of the head loss coefficient.

## 1. Introduction

## 2. Establishment of Mathematical Model

#### 2.1. Model Layout

^{3}/s, the designed water level at the inlet is 146.801 m, and the designed water level at the outlet is 146.491 m. The increased flow is 420 m

^{3}/s, the increased water level at the inlet is 147.561 m, and the increased water level at the outlet is 147.211 m. The available head for design flow is 0.31 m. The parameters connecting the upstream channel of the aqueduct are the top elevation of the bottom plate (138.801 m), the bottom width (19 m), the internal slope (1:2), and the longitudinal slope (i = 1/25,000). The parameters of the downstream channel are the top elevation of the bottom plate (138.491 m), the bottom width (19 m), the internal slope (1:2), and the longitudinal slope (i = 1/25,000).

#### 2.2. Governing Equation

#### 2.3. Model Meshing

- (1)
- Grid irrelevance test

- (2)
- Grid division

#### 2.4. Setting of Model Boundary Conditions and Initial Conditions

## 3. Model Validation

#### 3.1. Comparison of Measured Data of Model Calculation Results

^{3}/s. The water level on the left bank in front of the gate was 147.00 m and the water depth was 8.20 m; The water level on the right bank in front of the gate was 146.90 m and the water depth was 8.18 m. When measuring the velocity of the aqueduct on site, the first cross section of the aqueduct along the water flow direction was section A and the second cross section was section B. Measuring points were arranged along the right side of section A; 1 measuring point was arranged every 1 m, 12 measuring points were arranged in each channel body, and a total of 24 measuring points were arranged in the two tanks. The arrangement of the measuring points is shown in Figure 4.

^{3}/s. The inlet flow boundary was set with a flow of 227.68 m

^{3}/s. The water level at the outlet pressure boundary was 146.33 m. The initial water level of the model was 146.33 m. The initial velocity was 0.8 m/s. The channel roughness was set at 0.014. The calculation time was set to 5000 s. The initial time step was set to 0.002 s, and the minimum time step was set to 1 × 10

^{−8}s. This paper selected the RNG k-ε turbulence model.

#### 3.2. Setting of Simulated Working Conditions

_{up}/W

_{down}< 1, model number 1–4; W

_{up}/W

_{down}= 1, model number 5; and W

_{up}/W

_{down}> 1, model number 6. The model conditions were selected to reflect the influence of W

_{up}/W

_{down}on the overflow capacity. The water level and flow rate of each model condition were set as shown in Table 5, and eight water level and flow rates are set for each model condition.

## 4. Analysis of Simulation Results

_{up}= 19 m and W

_{down}= 31 m, and the simulation working condition 1 results are shown in Table 6.

_{down}/A

_{up}) was obtained, and A

_{up}and A

_{down}were the upstream and downstream cross-sectional over-water areas, respectively. Tokyay fitted the local head loss expression $\mathsf{\zeta}=0.74{\left(\mathsf{\Delta}\mathrm{z}/{\mathrm{h}}_{1}\right)}^{0.5}$ for a 45° inclined negative step flow via an experimental study; h

_{1}was the downstream water depth and $\mathsf{\Delta}\mathrm{z}$ was the difference in the elevation of the bottom surface of the inlet gradient section. In this paper, through the analysis of the simulation results and taking into account the effects of the water surface contraction angle and the upstream and downstream cross-sectional areas of the gradient section, the expression of the head loss coefficient of the gradient section can be given on the basis of existing research [5].

_{up}/W

_{down}= 19/31 simulated working conditions into Equation (5), we get: coefficients x

_{1}= 20.755, x

_{2}= 0.837, and x

_{3}= 0.891, and the correlation coefficient is 0.995.

_{up}/W

_{down}. The water surface contraction angle θ (°) and the local head loss coefficient had a good exponential function relationship that met $\mathsf{\zeta}={\mathsf{\alpha}\mathrm{e}}^{\mathsf{\beta}\left({\mathrm{W}}_{\mathrm{up}}/{\mathrm{W}}_{\mathrm{down}}\right)}$, where α and β are the formula coefficient terms and the correlation coefficients R

^{2}of each model condition fit were above 0.98.

_{up}/W

_{down}with exponential function $\mathsf{\zeta}={\mathsf{\alpha}\mathrm{e}}^{\mathsf{\beta}\left({\mathrm{W}}_{\mathrm{up}}/{\mathrm{W}}_{\mathrm{down}}\right)}$ was performed for each model working condition. W

_{up}/W

_{down}had a good one-to-one quadratic function relationship with coefficient α, and the correlation coefficient R

^{2}was 0.9996; W

_{up}/W

_{down}had a good one-to-one quadratic function relationship with coefficient β, and the correlation coefficient R

^{2}was 0.9577.

_{up}/W

_{down}had a one-dimensional quadratic function relationship. In order to verify the accuracy and applicability of the formula, the formula was used to solve the local head loss coefficient when W

_{up}/W

_{down}was 19/31 and the water surface contraction angle was 10.76°. The formula calculated the local head loss coefficient to be 0.3796, the simulated value was 0.383, the error was 3.4 × 10

^{−3}, and the error percentage was 0.89%.

## 5. Conclusions

_{1}, x

_{2}, and x

_{3}are the formula’s coefficients.

_{up}/W

_{down}, the local head loss coefficient corresponding to the same water surface contraction angle also increased; the local head loss coefficient had a good exponential function relationship with W

_{up}/W

_{down}, which satisfied the functional form $\mathsf{\zeta}={\mathsf{\alpha}\mathrm{e}}^{\mathsf{\beta}\left({\mathrm{W}}_{\mathrm{up}}/{\mathrm{W}}_{\mathrm{down}}\right)}$, where α and β are the formula’s coefficients.

## 6. Patents

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Each model working condition water surface contraction angle as a function of the local head loss coefficient. (

**a**) θ as a function of ζ at W

_{up}/W

_{down}19/31; (

**b**) θ as a function of ζ at W

_{up}/W

_{down}22/31; (

**c**) θ as a function of ζ at W

_{up}/W

_{down}25/31; (

**d**) θ as a function of ζ at W

_{up}/W

_{down}28/31; (

**e**) θ as a function of ζ at W

_{up}/W

_{down}31/31; (

**f**) θ as a function of ζ at W

_{up}/W

_{down}34/31.

**Figure 7.**W

_{up}/W

_{down}as a function of coefficient. (

**a**) W

_{up}/W

_{down}as a function of coefficient α; (

**b**) W

_{up}/W

_{down}as a function of coefficient β.

Grid Width/m | Total Number of Grids | Number of Fluid Grids | Ending Time/s | Approximate Time of Stabilization/s | Simulated Water Depth at Point A/m |
---|---|---|---|---|---|

1.0 | 765,000 | 426,400 | 5000 | 3000 | 8.236 |

0.9 | 987,400 | 545,700 | 5000 | 3000 | 8.203 |

0.8 | 1,395,000 | 771,000 | 5000 | 3000 | 8.185 |

0.7 | 1,915,500 | 1,045,700 | 5000 | 3000 | 8.183 |

0.6 | 2,729,400 | 1,440,900 | 5000 | 3000 | 8.181 |

**Table 2.**Comparison between measured water depth and simulated water depth on the left bank of the aqueduct.

Left Bank Section of Aqueduct Body Section | ||||||
---|---|---|---|---|---|---|

Measuring point | L1 | L2 | L3 | L4 | L5 | L6 |

Measured water depth | 5.63 | 5.7 | 5.78 | 5.71 | 5.62 | 5.77 |

Simulated water depth | 5.58 | 5.68 | 5.75 | 5.75 | 5.67 | 5.71 |

Measuring point | L7 | L8 | L9 | L10 | L11 | L12 |

Measured water depth | 5.64 | 5.78 | 5.74 | 5.77 | 5.63 | 5.70 |

Simulated water depth | 5.58 | 5.73 | 5.68 | 5.71 | 5.72 | 5.76 |

**Table 3.**Comparison between measured water depth and simulated water depth on the right bank of the aqueduct.

Right Bank Section of Aqueduct Body Section | ||||||
---|---|---|---|---|---|---|

Measuring point | R1 | R2 | R3 | R4 | R5 | R6 |

Measured water depth | 5.65 | 5.77 | 5.80 | 5.71 | 5.74 | 5.65 |

Simulated water depth | 5.60 | 5.70 | 5.74 | 5.76 | 5.69 | 5.61 |

Measuring point | R7 | R8 | R9 | R10 | R11 | R12 |

Measured water depth | 5.79 | 5.66 | 5.74 | 5.80 | 5.64 | 5.77 |

Simulated water depth | 5.73 | 5.62 | 5.68 | 5.75 | 5.7 | 5.69 |

Model Number | W_{up} (m) | W_{down} (m) | Model Number | W_{up} (m) | W_{down} (m) |
---|---|---|---|---|---|

1 | 19 | 31 | 4 | 28 | 31 |

2 | 22 | 31 | 5 | 31 | 31 |

3 | 25 | 31 | 6 | 34 | 31 |

Work Conditions | Inlet Pressure P_{in} (m) | Outlet Pressure P_{out} (m) | Work Conditions | Inlet Pressure P_{in} (m) | Outlet Pressure P_{out} (m) |
---|---|---|---|---|---|

1 | 4.8 | 4.5 | 5 | 6.8 | 6.5 |

2 | 5.3 | 5.0 | 6 | 7.3 | 7.0 |

3 | 5.8 | 5.5 | 7 | 7.8 | 7.5 |

4 | 6.3 | 6.0 | 8 | 8.3 | 8.0 |

Water Level (m) | Angle (°) | Water Depth of Tank (m) | Downstream Area (m^{2}) | Upstream Water Depth (m) | Upstream Area (m^{2}) | Head Loss Coefficient |
---|---|---|---|---|---|---|

8.3 | 14.84 | 6.184 | 191.704 | 8.220 | 291.291 | 0.280 |

7.8 | 13.5 | 5.695 | 176.542 | 7.724 | 266.081 | 0.311 |

7.3 | 12.13 | 5.194 | 161.002 | 7.221 | 241.485 | 0.339 |

6.8 | 10.76 | 4.705 | 145.867 | 6.729 | 218.392 | 0.383 |

6.3 | 9.37 | 4.206 | 130.374 | 6.225 | 195.789 | 0.420 |

5.8 | 7.97 | 3.715 | 115.168 | 5.732 | 174.615 | 0.464 |

5.3 | 6.56 | 3.225 | 99.963 | 5.236 | 154.331 | 0.527 |

4.8 | 5.14 | 2.750 | 85.241 | 4.745 | 135.193 | 0.593 |

W_{up}/W_{down} | Angles (°) | Local Head Loss Coefficient | W_{up}/W_{down} | Angles (°) | Local Head Loss Coefficient | W_{up}/W_{down} | Angles (°) | Local Head Loss Coefficient |
---|---|---|---|---|---|---|---|---|

19/31 | 14.84 | 0.280 | 22/31 | 16.83 | 0.249 | 25/31 | 18.78 | 0.222 |

13.5 | 0.311 | 15.51 | 0.273 | 17.48 | 0.247 | |||

12.13 | 0.339 | 14.17 | 0.292 | 16.17 | 0.263 | |||

10.76 | 0.383 | 12.82 | 0.323 | 14.84 | 0.292 | |||

9.37 | 0.420 | 11.45 | 0.357 | 13.5 | 0.319 | |||

7.97 | 0.464 | 10.07 | 0.400 | 12.13 | 0.358 | |||

6.56 | 0.527 | 8.67 | 0.451 | 10.76 | 0.406 | |||

5.14 | 0.593 | 7.27 | 0.513 | 9.37 | 0.468 | |||

28/31 | 20.68 | 0.211 | 31/31 | 22.54 | 0.201 | 34/31 | 24.35 | 0.191 |

19.42 | 0.232 | 21.31 | 0.223 | 23.15 | 0.217 | |||

18.13 | 0.246 | 20.05 | 0.236 | 21.92 | 0.227 | |||

16.83 | 0.281 | 18.78 | 0.262 | 20.68 | 0.255 | |||

15.51 | 0.298 | 17.48 | 0.284 | 19.42 | 0.272 | |||

14.17 | 0.335 | 16.17 | 0.323 | 18.13 | 0.310 | |||

12.82 | 0.380 | 14.84 | 0.364 | 16.83 | 0.350 | |||

11.45 | 0.440 | 13.5 | 0.424 | 15.51 | 0.410 |

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## Share and Cite

**MDPI and ACS Style**

Chen, J.; Tian, Y.; Zhang, H.; Zhang, S.
Study on the Head Loss of the Inlet Gradient Section of the Aqueduct. *Water* **2023**, *15*, 1633.
https://doi.org/10.3390/w15081633

**AMA Style**

Chen J, Tian Y, Zhang H, Zhang S.
Study on the Head Loss of the Inlet Gradient Section of the Aqueduct. *Water*. 2023; 15(8):1633.
https://doi.org/10.3390/w15081633

**Chicago/Turabian Style**

Chen, Jian, Yangyang Tian, Huijie Zhang, and Shanju Zhang.
2023. "Study on the Head Loss of the Inlet Gradient Section of the Aqueduct" *Water* 15, no. 8: 1633.
https://doi.org/10.3390/w15081633