# Numerical Simulation Research on the Diversion Characteristics of a Trapezoidal Channel

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Experimental Setup

_{1}; the downstream of the open-channel bifurcation was called the trapezoidal branch channel, and the flow discharge was Q

_{3}. The elevation of the bottom of the trapezoidal branch channel was consistent with that of the bottom of the trapezoidal main channel. The rectangular branch channel was 8 m long, with a width of 0.36 m, and a channel height of 0.6 m, and the flow discharge was Q

_{2}. The elevation of the bottom of the rectangular branch channel was higher than the elevation of the bottom of the trapezoidal main channel, resulting in a bottom sill of 7 cm. This bottom sill was set in the rectangular branch channel 60 cm downstream at the rectangular branch channel inlet. In this channel system, the open-channel bifurcation was 20 m away from the trapezoidal main channel head, ensuring uniform flow conditions at the branch channel. The angle between the central axis of the rectangular branch channel and the central axis of the trapezoidal channel was 90°. The roughness of both channels was 0.011. Measurements were collected in the experiments at five different trapezoidal main channel flow discharges. The water depth was measured using SCM 60 needle water-level gauges with an accuracy of ±0.1 mm (manufactured by Weifang Jinshui Huayu Information Technology CO., Ltd., Weifang City, China.).

#### 2.2. Numerical Simulation Model

#### 2.3. Governing Equations

_{t}is the fluid turbulent viscosity; G

_{k}is the turbulent kinetic energy k generation term due to the average velocity gradient; with α

_{k}= α

_{ε}= 1.39, C

_{1ε}* = 1.42, and C

_{2ε}= 1.68, all being empirical constants.

#### 2.4. Open-Channel Bifurcation Structure and Boundary Conditions

#### 2.5. Meshing and Simulation Conditions

## 3. Results and Discussion

#### 3.1. Water Depth

#### 3.2. Froude Number

#### 3.3. Width of the Recirculation Zone

_{1}, whereas the width of the subsurface flow acceleration zone slowly increases. Their change ranges are small, and the narrowing effect of the recirculation zone does not change significantly with the change in Q

_{1}. When Q

_{1}is fixed, changing the rectangular branch widths return the widths of recirculation zone varies substantially. However, nearly no difference is observed in the width of the submerged acceleration zone. Therefore, the flow-narrowing effect of the recirculation zone does not change significantly with the changing rectangular branch channel width. As a result, in an open-channel bifurcation with a bottom sill fixed 60 cm from the branch inlet, the width of the submerged acceleration zone and the flow narrowing effect are almost constant under different working conditions.

#### 3.4. Velocity Distribution in the Main Channel

#### 3.5. Turbulent Kinetic Energy

#### 3.6. Flow Diversion Width

_{s}and B

_{b}are the surface and bottom flow diversion widths; K

_{1}= 4.90Q

_{1}+ 0.11 and K

_{2}= 4.60Q

_{1}+ 0.06 in which Q

_{1}is the flow discharge in the trapezoidal main channel in m

^{3}/s; and b is the width of the rectangular branch channel in units consistent with B. It can be seen from Figure 10b, c that the equations are in good agreement with the simulation data.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Experimental plan and section measurement layout. Note: Red points in the figure represent the measurement point arrangement, and Roman numerals represent measurement section numbers.

**Figure 3.**Grid independence test. Note: b is the branch channel width and Q

_{1}is flow discharge in the main channel.

**Figure 4.**Water depth change along the direction of water flow in trapezoidal open channel for (

**a**) variation in flow discharge and (

**b**) variation in branch channel width. Note: Red lines indicate the water depth on the line y = 64 cm, blue lines indicate the water depth on the line y = 94 cm.

**Figure 5.**Froude number (F

_{r}) contour map at different water depths. Note: Q

_{1}= 40 L/s; b = 30 cm. X* and Y* are obtained by dimensionless processing of X-axis and Y-axis coordinates. (

**a**) depth of water below the sill height; (

**b**) depth of water above the sill height.

**Figure 6.**Relationships between the width of the recirculation zone, flow discharge, and rectangular channel width.

**Figure 7.**Velocity distribution at different water depths. Note: Q

_{1}= 40 L/s; b = 30 cm; P = 7 cm, which is the bottom sill height. (

**a**) The x-direction with velocity u; (

**b**) the y-direction with velocity v; and (

**c**) the z-direction with velocity w.

**Figure 8.**Velocity distribution at different sections. Note: Water depth of 1.5P; Q

_{1}= 40 L/s; b = 30 cm. (

**a**) The x-direction with velocity u; (

**b**) the y-direction with velocity v; and (

**c**) the z-direction with velocity w.

**Figure 10.**(

**a**) Comparison of flow diversion widths in the trapezoidal and rectangular channels and comparisons of the calculated and simulated values of the flow diversion width for (

**b**) surface flow and (

**c**) bottom flow.

**Table 1.**Analysis of the relative error of water depth in different sections at the flow rate of 40 L/s and the branch channel width of 36 cm.

Section | Analog Value/cm | Measured Value/cm | Relative Error/% |
---|---|---|---|

Ⅱ | 13.83 | 13.46 | 2.75 |

Ⅲ | 13.8 | 13.78 | 0.18 |

Ⅳ | 13.92 | 13.93 | −0.08 |

Ⅴ | 14.25 | 14.34 | −0.62 |

Ⅵ | 14.63 | 14.88 | −1.66 |

Ⅶ | 14.69 | 14.92 | −1.56 |

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**MDPI and ACS Style**

Cheng, Y.; Song, Y.; Liu, C.; Wang, W.; Hu, X.
Numerical Simulation Research on the Diversion Characteristics of a Trapezoidal Channel. *Water* **2022**, *14*, 2706.
https://doi.org/10.3390/w14172706

**AMA Style**

Cheng Y, Song Y, Liu C, Wang W, Hu X.
Numerical Simulation Research on the Diversion Characteristics of a Trapezoidal Channel. *Water*. 2022; 14(17):2706.
https://doi.org/10.3390/w14172706

**Chicago/Turabian Style**

Cheng, Yong, Yude Song, Chunye Liu, Wene Wang, and Xiaotao Hu.
2022. "Numerical Simulation Research on the Diversion Characteristics of a Trapezoidal Channel" *Water* 14, no. 17: 2706.
https://doi.org/10.3390/w14172706