# Study on the Characteristic Point Location of Depth Average Velocity in Smooth Open Channels: Applied to Channels with Flat or Concave Boundaries

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{max}, which depends on the entropic parameter M. Maghrebi [14] used the similarity between the magnetic field of a current wire and the isovel contours in a channel cross-section to derive the isovel patterns. Then, one can easily obtain the discharge using a single point on the isovel patterns of velocity measurement. However, the measured points are only selected from the upper half of the water depth and away from the boundaries, which would be much closer to the measured discharge. Bretheim et al. [15] used the restricted nonlinear (RNL) model to simulate wall turbulence and obtained the real mean velocity distribution in the channel at a low Reynolds number. Hong et al. [16] employed a systematic measuring technology combining ground-penetrating radar and surface velocity radar and established the rating curves representing the relation of water surface velocity to the channel cross-sectional mean velocity and flow area. Then, stream discharge was deduced from the resulting mean velocity and flow area. Moramarco et al. [17] proposed a new method for estimating discharge from surface velocity monitoring (u

_{surf}). Based on entropy theory and sampling u

_{surf}, it can identify the two-dimensional velocity distribution in the cross-sectional flow area. This method is more accurate for rivers with a lower aspect ratio where secondary currents are expected. Johnson and Cowen [18,19] predicted the mean streamwise velocity and the depth-averaged velocity by permitting remote determination of the velocity power-law exponent. Then, the volumetric discharge from surface measurements of currents can be determined. Khuntia et al. [20] presented a new methodology to predict the depth-averaged velocity. They used multi-variable regression analysis to develop five models to predict the point velocities in terms of non-dimensional geometric and flow parameters at any desired location. Through these attempts, researchers have discussed new thoughts of quantifying channel discharge. But few people determine the average velocity of a channel by looking for the location of characteristic points.

## 2. Methodology

#### 2.1. CPL in Rectangular Channels

#### 2.1.1. Existing Form of the Division Line in Rectangular Channels

_{e}= energy slope.

_{b}= flow area corresponding to channel bed; A

_{w}= flow area corresponding to channel side wall; R

_{w}= hydraulic radius corresponding to side wall; and R

_{b}= hydraulic radius corresponding to bed.

_{s}).

_{1}in Figure 1b:

_{1}represents the slope of the division line OD in Figure 1b.

- (a)
- Determination of the location of division line in wide–shallow channel (b/h ≥ 2)

- (b)
- Determination of the location of division line in narrow–deep channel (b/h ≤ 2)

_{1}can be obtained by solving Equation (11).

_{1}, a comparison relationship between k

_{1}and $2h/b$ is as follows:

#### 2.1.2. CPL of Lines in Rectangular Channel

#### 2.1.3. CPL of Regions in Rectangular Channel

_{I}, P

_{II}, and P

_{III}are the characteristic points that represent the mean velocities in regions I, II, and III, respectively.

_{0}is shown in Figure 3.

_{II}can be calculated using Equation (33):

_{II}.

_{III}, which can be used to calculate the value of Q of region III, can be expressed using Equation (34):

_{III}.

#### 2.2. CPL in Semi-Circular Channel

_{n}in length and dr in width, where Ln is the length of the underwater part of the radius. With the rectangle FGJK, the area is ${A}_{FGJK}={L}_{n}\cdot dr$ and the discharge through FGJK can be expressed as:

#### 2.3. Discharge of Rectangular and Semi-Circular Channel

#### 2.3.1. Discharge in a Rectangular Channel

_{I}, A

_{II}and A

_{III}represent the areas of three regions, and ${u}_{{P}_{I}}$, ${u}_{{P}_{II}}$ and ${u}_{{P}_{III}}$ represent the velocities of points P

_{I}, P

_{II}, and P

_{III}.

#### 2.3.2. Discharge in a Semi-Circular Channel

_{1}, A

_{2}and A

_{3}represent sub-areas, ${u}_{CPL}$ is the velocity of the normal line and α

_{c}is the velocity coefficient along the bank. α

_{c}can be determined according to the test conditions.

## 3. Experimental Setup

## 4. Results and Discussion

#### 4.1. Analysis with Rectangular Channels

#### 4.2. Analysis with Semi-Circular Channels

_{m}denotes the measured value, and E

_{c}denotes the calculated value.

_{t}is bigger than U

_{m}. This is because the log-law appears to deviate near the water surface. Based on this, the log-wake law was proposed by Coles [35], which appears to be the most reasonable extension of the log-law. However, the value of Π in the log-wake law seems not to be universal [36]. So, more research is needed to correct the velocity near the water surface.

_{c}= 0.8. Table 3 shows the calculated discharge of the channel with the combination of six kinds of angles. Only the first is a uniform partition; the others are all non-uniform. Relative error refers to the error value of the calculated discharge relative to the measured discharge. It can be found that the relative error value is the smallest only when the angle is divided uniformly, about 5%. Therefore, we can reasonably conclude that the accuracy of the discharge calculation is the highest only if the angle is divided uniformly while using the two-line method.

## 5. Summary

- (1)
- Based on Yang et al.’ s partitioning theory [21,22,27], this paper gives a re-description of the existing form of the division line of a rectangular cross-section channel. That is, whether the channel cross-section is wide–shallow or narrow–deep with the center line of the cross-section as the symmetrical axis, and whether the intersection points of the left and right division lines intersect on or above the water surface.
- (2)
- This paper analyzes characteristic points in flat channels (e.g., rectangular channel) and concave boundary channels (e.g., semi-circular channel). In the rectangular channel, the division line divides the section into three regions. In each region, the analysis is conducted in the direction perpendicular to the bottom or side wall of the channel. In the semi-circular channel, the analysis is conducted along the normal direction. Based on the log-law, the theoretical expressions for calculating the location of the average velocity characteristic points in flat and concave boundary channels are derived through the formula transformation.
- (3)
- The velocity data in different experimental sites are used to verify the validity of the CPL formulas applied to flat and concave boundary channels. Moreover, the discharge calculation formulas of channels are given through discussion with CPL.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Comparison of the CPL between the theoretical and measured values in a rectangular channel.

**Figure 10.**Comparison of measured and theoretical values of mean radial velocity (

**a**) S1, S2; (

**b**) S3, S4, S5.

Cross-Sectional Shape | Conditions | Channel Width: b (m) | Channel Radius: r (m) | Flow Discharge: Q (m^{3}/s) | Water Depth: h (m) |
---|---|---|---|---|---|

Rectangle | R1 ^{a} | 0.3 | - | 0.004 | 0.065 |

R2 ^{a} | 0.004 | 0.091 | |||

R3 ^{a} | 0.004 | 0.110 | |||

R4 ^{b} | 0.8 | - | 0.031 | 0.120 | |

R5 ^{b} | 0.033 | 0.128 | |||

R6 ^{b} | 0.039 | 0.137 | |||

R7 ^{b} | 0.044 | 0.153 | |||

Semi-circle | S1 ^{c} | - | 0.120 | 0.005 | 0.0813 |

S2 ^{c} | 0.012 | 0.1200 | |||

S3 ^{a} | - | 0.150 | 0.003 | 0.075 | |

S4 ^{a} | 0.005 | 0.085 | |||

S5 ^{a} | 0.008 | 0.100 |

^{a}Experiment data are from the Fluid Laboratory of the University of Wollongong(UOW).

^{b}Experiment data are from the Hydraulic Experiment Hall of China Agricultural University(CAU).

^{c}Experiment data are from Knight [29].

**Table 2.**Accuracy comparison of velocity distribution along the vertical wall and normal directions with S2: h = 0.1200.

Average Error Value along Normal Direction | Average Error Value along Vertical Direction | ||
---|---|---|---|

Normal Slope k_{n} | Average Error Value E (%) | The Distance from the Vertical Line to the Central Line z/b | Average Error Value E (%) |

11.9 | 2.156661 | 0.083 | 3.528655 |

5.91 | 3.117373 | 0.167 | 3.441878 |

3.87 | 2.578154 | 0.250 | 2.827718 |

2.82 | 3.073467 | 0.333 | 7.846099 |

2.18 | 3.217222 | 0.417 | 6.557529 |

1.73 | 2.584048 | 0.500 | 10.55338 |

1.39 | 2.047317 | 0.583 | 12.39980 |

1.11 | 2.207014 | 0.667 | 19.05022 |

0.88 | 2.758453 | 0.750 | 21.84237 |

0.66 | 4.707039 | 0.833 | 30.18244 |

Degree of Angle with the Two-Line Method | Q Calculation (m ^{3}/s) | Q Measurement (m ^{3}/s) | Relative Error (%) | ||
---|---|---|---|---|---|

θ_{1} (°) | θ_{2} (°) | θ_{3} (°) | |||

71/3 | 71/3 | 71/3 | 0.00526 | 0.005 | 5.10 |

18 | 34 | 19 | 0.00565 | 12.90 | |

18 | 23 | 30 | 0.00531 | 6.27 | |

29 | 32 | 10 | 0.00551 | 10.09 | |

35 | 26 | 10 | 0.00542 | 8.48 | |

36 | 21 | 14 | 0.00533 | 6.61 |

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**MDPI and ACS Style**

Li, T.; Chen, J.; Han, Y.; Ma, Z.; Wu, J. Study on the Characteristic Point Location of Depth Average Velocity in Smooth Open Channels: Applied to Channels with Flat or Concave Boundaries. *Water* **2020**, *12*, 430.
https://doi.org/10.3390/w12020430

**AMA Style**

Li T, Chen J, Han Y, Ma Z, Wu J. Study on the Characteristic Point Location of Depth Average Velocity in Smooth Open Channels: Applied to Channels with Flat or Concave Boundaries. *Water*. 2020; 12(2):430.
https://doi.org/10.3390/w12020430

**Chicago/Turabian Style**

Li, Tongshu, Jian Chen, Yu Han, Zhuangzhuang Ma, and Jingjing Wu. 2020. "Study on the Characteristic Point Location of Depth Average Velocity in Smooth Open Channels: Applied to Channels with Flat or Concave Boundaries" *Water* 12, no. 2: 430.
https://doi.org/10.3390/w12020430