# Streamflow Measurement Using Mean Surface Velocity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Relation between Mean Velocity and Mean Surface Velocity

_{n-1}in the Figure 1 is the distance from initial point to the n-1

^{th}vertical; d

_{n-1}is the water depth at vertical n-1; ${\overline{u}}_{n-1}$ is the mean velocity of the n-1

^{th}vertical. Therefore, each subsection is rectangular. The subsectional discharge ${q}_{i}$ and the subsection area ${a}_{i}$ were calculated using (1) and (2), respectively:

_{i}is the distance from the initial point to vertical i; ${d}_{i}$ is the depth of flow at vertical i; and ${\overline{u}}_{i}$ is mean velocity at vertical i. The observed discharge (Q

_{obs}) and cross-sectional area (A

_{obs}) can be represented as (3) and (4), respectively:

_{si}is the surface velocity on vertical i. If the cross-section of a stream is a rectangle or close to a rectangle, and the intervals between the verticals are equal; then the area of each subsection will be equal as shown in (7):

_{s}is discharge estimated by surface velocity; and ${\overline{u}}_{s}$ is the mean surface velocity. However, (8) is not valid for estimating stream discharge. A surface-velocity coefficient must be applied to (8) to relate the data to the actual discharge amount as shown in (9):

#### 2.2. Estimation of Surface Velocity with Velocity Distribution Based on Probability

_{max}is the max velocity; M is a parameter; $\xi $ is the isovel in Figure 2 [25]; u is the velocity at $\xi $; ${\xi}_{max}$ and ${\xi}_{0}$ are the values of $\xi $ at which u = ${u}_{max}$ and u = 0, respectively. In addition, a $\eta -\xi $ coordinate system can be used to describe the velocity field with a set of isovels, in which $\xi $ and u has a one-to-one relationship, meaning that the velocities are the same on $\xi $, unlike the Cartesian coordinate system where the same velocity values can occur in difference locations. The $\xi $ on the vertical line is shown in (12):

#### 2.3. Estimation of Cross Section Area and Discharge

_{est}is the cross-sectional area estimated by water depth; d is the water depth of a vertical; and b, c, and e are coefficients.

_{est}are obtained, then the streamflow can be promptly evaluated from the surface velocities of the verticals.

_{est}is the streamflow estimated by mean surface velocity.

## 3. Case Study: Study Sites and Data Collection Methods

^{2}. It originates from the western foothills of the Hehuan Mountain at an elevation of 3417 m. Wu River is one of the most important rivers in Taiwan, providing vast amounts of water for industrial, agricultural and domestic uses. Thus, the Third River Management Office set up a gauge station at the Guanyin Bridge for the purposes of water resources and flood management.

## 4. Results and Discussion

^{3}/s. The flow pattern of the Longen Channel was quite similar to a large-scale hydraulic flume in a laboratory. In Figure 7, it is obvious that the flow patterns of Longen Channel were very different from those of natural rivers. Most of the maximum velocities on the verticals occurred at a depth of about 1/4 water depth from the water surface, while the surface velocity was relatively small. It also shows that the maximum velocities of the verticals excluding verticals (e) and (f) did not occur on the water surface. Experimental studies have been shown from considerations of momentum transfer that the velocity in an open channel should decrease toward the channel bed. In a very wide channel the velocity decreases toward the bed and walls, and theoretically the maximum occurs at the water surface. The Longen Channel is a small artificial flume; therefore, depression of the maximum velocities below the water surface was observed. The flow pattern cannot be described by a logarithmic distribution. The circle in Figure 7 is the actual velocity measurement on each vertical, and the line is the velocity distributions based on (11) indicating that vertical maximum velocity does not always occur on the water surface. It also shows that the velocity profile data of the Longen Channel is difficult to describe using conventional velocity distribution theories, such as logarithm velocity distribution. However, (11) can simulate velocity profiles effectively, regardless of whether the maximal velocity occurs on or below the water surface. Therefore, the surface velocities on the verticals could also obtained precisely by using (11). In addition, using the nonlinear regression method, M, h and u

_{max}can also be obtained from (11) with the vertical velocity and water depth. Thus, the mean vertical velocity ($\overline{{u}_{i}}in\left[1\right]$) on each vertical can be estimated. Therefore (11) can be used to accurately estimating the mean velocity of the vertical for obtaining reliable discharge.

^{2}> 0.97) between the above two methods demonstrates the accuracy and reliability of using the mean surface velocity method for discharge measurement in both the natural rivers and artificial channels. Therefore, the authors concluded the measurement of river discharge can be obtained promptly and accurately using the proposed approach, which only requires one to measure the surface velocity to obtain the mean surface velocity, and estimate the cross-sectional area based on the water depth (or water stage).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Velocity field in the $\eta -\xi $ coordinate system with a set of isovels. (

**a**) $h\le 0$; (

**b**) $h>0$.

**Figure 4.**Channel cross sections of the Nankang River at the Guanyin Bridge; (

**a**) left channel; (

**b**) right channel.

**Figure 5.**Instruments for measuring discharge. (

**a**) A magnetic-inductive current meter and a SVR are used at the Guanyin Bridge; (

**b**) A down-looking SW integrated with a sounding weight is used to measure the velocity profiles in the Longen Channel.

**Figure 7.**Subsectional velocity profile of the Run 4 in the Longen Channel. (

**a**) vertical I; (

**b**) vertical II; (

**c**) vertical III; (

**d**) vertical IV; (

**e**) vertical V; (

**f**) vertical VI; (

**g**) vertical VII; (

**h**) vertical VIII of Figure 6.

**Figure 8.**Relation between mean velocities of cross section and surface. (

**a**) the Nankang River at the Guanyin Bridge; (

**b**) the Longen Channel.

**Figure 10.**Accuracy of discharge measurement by surface mean velocity; (

**a**) the Nankang River at the Guanyin Bridge; (

**b**) the Longen Channel.

Run | Date | Depth (m) | Q_{obs} (Observed Discharge m^{3}/s) |
---|---|---|---|

1 | 22 February 2019 | 0.87 | 0.43 |

2 | 6 March 2019 | 0.98 | 0.92 |

3 | 8 March 2019 | 1.47 | 3.80 |

4 | 14 March 2019 | 1.43 | 4.24 |

5 | 20 March 2019 | 1.15 | 2.04 |

6 | 25 March 2019 | 1.31 | 3.16 |

7 | 3 May 2019 | 1.52 | 3.79 |

8 | 23 May 2019 | 1.48 | 4.35 |

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**MDPI and ACS Style**

Chen, Y.-C.; Hsu, Y.-C.; Zai, E.O.
Streamflow Measurement Using Mean Surface Velocity. *Water* **2022**, *14*, 2370.
https://doi.org/10.3390/w14152370

**AMA Style**

Chen Y-C, Hsu Y-C, Zai EO.
Streamflow Measurement Using Mean Surface Velocity. *Water*. 2022; 14(15):2370.
https://doi.org/10.3390/w14152370

**Chicago/Turabian Style**

Chen, Yen-Chang, Yung-Chia Hsu, and Eben Oktavianus Zai.
2022. "Streamflow Measurement Using Mean Surface Velocity" *Water* 14, no. 15: 2370.
https://doi.org/10.3390/w14152370