Study on the Characteristic Point Location of Depth Average Velocity in Smooth Open Channels: Applied to Channels with Flat or Concave Boundaries
Abstract
:1. Introduction
2. Methodology
2.1. CPL in Rectangular Channels
2.1.1. Existing Form of the Division Line in Rectangular Channels
- (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)
2.1.2. CPL of Lines in Rectangular Channel
2.1.3. CPL of Regions in Rectangular Channel
2.2. CPL in Semi-Circular Channel
2.3. Discharge of Rectangular and Semi-Circular Channel
2.3.1. Discharge in a Rectangular Channel
2.3.2. Discharge in a Semi-Circular Channel
3. Experimental Setup
4. Results and Discussion
4.1. Analysis with Rectangular Channels
4.2. Analysis with Semi-Circular Channels
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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Cross-Sectional Shape | Conditions | Channel Width: b (m) | Channel Radius: r (m) | Flow Discharge: Q (m3/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 |
Average Error Value along Normal Direction | Average Error Value along Vertical Direction | ||
---|---|---|---|
Normal Slope kn | 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 (m3/s) | Q Measurement (m3/s) | Relative Error (%) | ||
---|---|---|---|---|---|
θ1 (°) | θ2 (°) | θ3 (°) | |||
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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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
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 StyleLi, 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