# Comparison of Empirical and Analytical Solutions for Open-Channel Flow Velocity with Common Grass Species in Taiwan

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flume Experiment Apparatus

#### 2.2. Experimental Procedure

#### 2.3. Measurements of Soil Layer Properties

_{p}

_{3}) can be calculated by the formula:

_{3}) and specific permeability (k

_{p}

_{3}) of the red soil layer were 0.602 and 1.392 × 10

^{−12}(${\mathrm{m}}^{2}$), respectively.

#### 2.4. Measurements of Grass Layer Properties

_{2}), depth (h

_{2}), and average grass stem diameter (${d}_{c}$) were measured. The grass layer porosity (n

_{2}) was measured based on the volume-replacing technique, and the steps included: (1) Selecting a control volume of the grass layer; (2) removing the roots below the soil surface to reduce measuring errors; (3) measuring and recording the length (l), width (w

_{2}), and height (h

_{2}) of the grass layer; (4) measuring and recording the average grass stem diameter (${d}_{c}$); (5) putting the grass into a 1000 mL volumetric cylinder and adding water of a known volume (V

_{1}); and (6) recording the graduation of the water surface in the cylinder (V

_{2}) and using the following equation to calculate the grass layer porosity:

_{p}

_{2}) can be calculated by the formula proposed by Kaviany (1991):

#### 2.5. Flow Velocity Measurements

#### 2.5.1. Ultrasonic Current Meter

#### 2.5.2. Electromagnetic Current Meter

#### 2.5.3. The Bucket Method

#### 2.5.4. Manning’s Equation

#### 2.5.5. Analytical Solution for Water Flow Passing a Grass Layer

_{1_anal}) and grass layer (V

_{2_anal}) are shown as follows:

## 3. Results

#### 3.1. Average Flow Velocities with Different Relative Heights

#### 3.1.1. Velocity Variation in Cases of Centipede Grass

#### 3.1.2. Velocity Variation in Cases of Bermuda Grass

#### 3.1.3. Velocity Variation in Cases of Carpet Grass

#### 3.2. Velocity Profiles with Different Grass Species

## 4. Discussion

#### 4.1. Comparison of Average Flow Velocity among the Three Grass Species

#### 4.2. Comparison between Measured Values and the Analytical Solution of Average Flow Velocity

#### 4.3. Comparison between Measured Values and Manning’s Estimations of Average Flow Velocity

#### 4.4. Application to Grassed Channel Design

- From Manning’s equation, ${\overline{V}}_{Manning}=\frac{1}{n}{R}^{\frac{2}{3}}{S}^{\frac{1}{2}}$,where $A=\frac{2}{3}bd=0.067$; $R=\frac{{b}^{2}d}{1.5{b}^{2}+4{d}^{2}}=0.093$.Take the suggested Manning’s n of Centipede grass, 0.055, in the Soil and Water Conservation Handbook [3], and thus, $Q=0.067\times \frac{1}{0.055}\times {0.093}^{\frac{2}{3}}\times {0.03}^{\frac{1}{2}}\cong 0.0433\text{}\left({m}^{3}/s\right)$.
- Given that the grass layer height (${h}_{2}$) is 0.05 m and the water depth ($d={h}_{1}+{h}_{2}$) is 0.182 m, $\frac{{h}_{2}}{{h}_{1}+{h}_{2}}\cong 0.275$.From Figure 7a, ${\overline{V}}_{anal}=0.62\text{}\left(m/s\right)$.
- Thus, ${Q}^{\prime}=A\times {\overline{V}}_{anal}=0.067\times 0.62\cong 0.0415\text{}\left({m}^{3}/s\right)$.
- Compare $Q$ and ${Q}^{\prime}$, and find $Q>{Q}^{\prime}$. This result implies that the flowrate calculated by Manning’s equation may be overestimated for grassed channel flows.
- Revise Manning’s n using the average velocity measured by the bucket method (Figure 3a), ${\overline{V}}_{bucket}$, $0.41=\frac{1}{{n}^{\prime}}\times {0.093}^{\frac{2}{3}}\times {0.035}^{\frac{1}{2}}$, and thus ${n}^{\prime}\cong 0.0867$.
- Revise Manning’s n using the average velocity calculated by the analytical solution, ${\overline{V}}_{anal}$, $0.62=\frac{1}{{n}^{\u2033}}\times {0.093}^{\frac{2}{3}}\times {0.03}^{\frac{1}{2}}$, and thus ${n}^{\u2033}\cong 0.0573$.
- Compare $n$, ${n}^{\prime}$, and $\text{}{n}^{\u2033}$, we find that both ${n}^{\prime}$ and $\text{}{n}^{\u2033}$ are larger than $n$. This result implies that Manning’s coefficient should be larger than the suggested value in the Soil and Water Conservation Handbook [3] in Taiwan.

## 5. Conclusions

- The comparison between the five evaluating methods suggested that the average velocity evaluated from the different methods showed a general trend of ${\overline{V}}_{Manning}>{\overline{V}}_{ultra}>{\overline{V}}_{anal}>{\overline{V}}_{elect}>{\overline{V}}_{bucket}$ in all the flow conditions. In the cases of a red soil bed (soil layer porosity of around 60%) and 3.5% slope, the values of ${\overline{V}}_{elect}/{\overline{V}}_{anal}$ varied from 0.92 to 1 and ${\overline{V}}_{elect}/{\overline{V}}_{Manning}$ varied from 0.51 to 0.63. When the slope was 6%, ${\overline{V}}_{elect}/{\overline{V}}_{anal}$ varied from 0.86 to 0.99 and ${\overline{V}}_{elect}/{\overline{V}}_{Manning}$ varied from 0.29 to 0.90. When the slope changed to 7%, ${\overline{V}}_{elect}/{\overline{V}}_{anal}$ varied from 0.91 to 0.99 and ${\overline{V}}_{elect}/{\overline{V}}_{Manning}$ varied from 0.33 to 0.93. Therefore, the experimental values of flow velocity (${\overline{V}}_{elect}$) fitted the analytical solution ($\text{}{\overline{V}}_{anal}$) very well, whereas ${\overline{V}}_{Manning}$ generally overestimated the grassed flow velocity.
- Based on the relationships between the average flow and relative height (${h}_{2}/\left({h}_{1}+{h}_{2}\right)$), Centipede grass showed the best flow decelerating effect when ${h}_{2}/\left({h}_{1}+{h}_{2}\right)<0.36$, and Carpet grass showed the best effect in cases of ${h}_{2}/\left({h}_{1}+{h}_{2}\right)>0.36$. Thus, Centipede grass or Carpet grass may be more effective when grassed channels are used to decelerate flow and enhance water infiltration for water conservancy, whereas Bermuda grass may be more suitable for grassed channels used to release floods for disaster prevention.
- The average velocity in grassed flow was found to be significantly affected by the morphological characteristics of grass, such as the height and porosity of the grass layer. Manning’s equation considered the roughness in channels using Manning’s coefficient, n, and thus was unable to properly reflect the grass layer characteristic effects on the flow velocity.
- The flow velocity profiles estimated using the analytical method matched well with the velocities observed at different water depths in grassed flow. Therefore, the experimental results will be beneficial for the verification of mathematical methods, including analytic solutions and numerical models of grassed flow. For application, we extended the analytical solution of flow velocity to grassed flow with three grass species and proposed curves of the average flow velocity against the relative height of the grass layer. When planning for a drainage system on hillslopes in Taiwan, the proposed curves can be used as references for grassed channel flowrate design in cases of red bed soil; 3% to 7% slopes; and grass species of Centipede grass, Bermuda grass, and Carpet grass.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Average flow velocities at different relative heights for Centipede grass with a bed slope of (

**a**) 3.5%, (

**b**) 6%, and (

**c**) 7%.

**Figure 4.**Average flow velocities at different relative heights for Bermuda grass with bed slopes of (

**a**) 3.5%, (

**b**) 6%, and (

**c**) 7%.

**Figure 5.**Average flow velocities at different relative heights for Carpet grass with bed slopes of (

**a**) 3.5%, (

**b**) 6%, and (

**c**) 7%.

**Figure 6.**Velocity profiles in the grassed flume with (

**a**) Centipede grass, (

**b**) Bermuda grass, and (

**c**) Carpet grass.

**Figure 7.**Relationships between the average flow velocity and the relative height in grassed channels covered with (

**a**) Centipede grass, (

**b**) Bermuda grass, and (

**c**) Carpet grass.

Grass Species | Parameters (Unit) | Values |
---|---|---|

Centipede grass (Eremochloa ophiuroides) | Height (m) | 0.031 |

Porosity (-) | 0.715 | |

Specific permeability (m^{2}) | 1.765 × 10^{−7} | |

Manning’s n ^{a} (-) | 0.055 | |

Bermuda grass (Cynodon dactylon) | Height (m) | 0.036 |

Porosity (-) | 0.947 | |

Specific permeability (m^{2}) | 2.941 × 10^{−7} | |

Manning’s n ^{b} (-) | 0.05 | |

Carpet grass (Axonopus) | Height (m) | 0.041 |

Porosity (-) | 0.781 | |

Specific permeability (m^{2}) | 1.987 × 10^{−6} | |

Manning’s n ^{a} (-) | 0.05 |

**Table 2.**Differences in the average velocities obtained from the different methods comparing with the bucket method.

Grass Species | Slope (%) | ${\overline{\mathit{V}}}_{\mathit{u}\mathit{l}\mathit{t}\mathit{r}\mathit{a}}-{\overline{\mathit{V}}}_{\mathit{b}\mathit{u}\mathit{c}\mathit{k}\mathit{e}\mathit{t}}\phantom{\rule{0ex}{0ex}}(\mathbf{m}/\mathbf{s})$ | ${\overline{\mathit{V}}}_{\mathit{e}\mathit{l}\mathit{e}\mathit{c}\mathit{t}}-{\overline{\mathit{V}}}_{\mathit{b}\mathit{u}\mathit{c}\mathit{k}\mathit{e}\mathit{t}}\phantom{\rule{0ex}{0ex}}(\mathbf{m}/\mathbf{s})$ | ${\overline{\mathit{V}}}_{\mathit{M}\mathit{a}\mathit{n}\mathit{n}\mathit{i}\mathit{n}\mathit{g}}-{\overline{\mathit{V}}}_{\mathit{b}\mathit{u}\mathit{c}\mathit{k}\mathit{e}\mathit{t}}\phantom{\rule{0ex}{0ex}}(\mathbf{m}/\mathbf{s})$ | ${\overline{\mathit{V}}}_{\mathit{a}\mathit{n}\mathit{a}\mathit{l}}-{\overline{\mathit{V}}}_{\mathit{b}\mathit{u}\mathit{c}\mathit{k}\mathit{e}\mathit{t}}\phantom{\rule{0ex}{0ex}}(\mathbf{m}/\mathbf{s})$ |
---|---|---|---|---|---|

Centipede grass | 3.5 | 0.44 | 0.08 | 0.79 | 0.14 |

6 | 0.52 | 0.12 | 1.04 | 0.20 | |

7 | 0.58 | 0.16 | 1.01 | 0.29 | |

Bermuda grass | 3.5 | 0.51 | 0.21 | 0.91 | 0.25 |

6 | 0.49 | 0.08 | 1.20 | 0.20 | |

7 | 0.44 | 0.12 | 1.04 | 0.19 | |

Carpet grass | 3.5 | 0.33 | 0.10 | 1.18 | 0.15 |

6 | 0.36 | 0.05 | 1.51 | 0.11 | |

7 | 0.25 | 0.08 | 1.38 | 0.11 | |

RMSE | 0.447 | 0.120 | 1.138 | 0.191 |

Grass Species | Slope | $\frac{{\mathit{h}}_{2}}{{\mathit{h}}_{1}+{\mathit{h}}_{2}}$ | $\frac{{\overline{\mathit{V}}}_{\mathit{b}\mathit{u}\mathit{c}\mathit{k}\mathit{e}\mathit{t}}}{{\overline{\mathit{V}}}_{\mathit{a}\mathit{n}\mathit{a}\mathit{l}}}$ | $\frac{\text{}{\overline{\mathit{V}}}_{\mathit{u}\mathit{l}\mathit{t}\mathit{r}\mathit{a}}}{{\overline{\mathit{V}}}_{\mathit{a}\mathit{n}\mathit{a}\mathit{l}}}$ | $\frac{{\overline{\mathit{V}}}_{\mathit{e}\mathit{l}\mathit{e}\mathit{c}\mathit{t}}}{{\overline{\mathit{V}}}_{\mathit{a}\mathit{n}\mathit{a}\mathit{l}}}$ |
---|---|---|---|---|---|

Centipede grass | 3.5% | 0.319 | 0.85 | 1.25 | 0.92 |

0.301 | 0.91 | 1.22 | 0.97 | ||

0.292 | 0.93 | 1.28 | 0.98 | ||

0.275 | 0.92 | 1.12 | 0.96 | ||

6% | 0.391 | 0.84 | 1.23 | 0.93 | |

0.337 | 0.80 | 1.26 | 0.90 | ||

0.326 | 0.93 | 1.23 | 0.97 | ||

0.293 | 0.93 | 1.04 | 0.98 | ||

7% | 0.423 | 0.83 | 1.20 | 0.91 | |

0.362 | 0.82 | 1.18 | 0.91 | ||

0.350 | 0.85 | 1.22 | 0.91 | ||

0.320 | 0.90 | 1.03 | 0.99 | ||

Bermuda grass | 3.5% | 0.365 | 0.86 | 1.36 | 0.96 |

0.352 | 0.79 | 1.12 | 0.94 | ||

0.342 | 0.82 | 1.22 | 0.98 | ||

0.324 | 0.91 | 1.02 | 1.00 | ||

6% | 0.429 | 0.81 | 1.14 | 0.86 | |

0.410 | 0.84 | 1.21 | 0.91 | ||

0.396 | 0.95 | 1.17 | 0.97 | ||

0.373 | 0.95 | 1.10 | 0.98 | ||

7% | 0.444 | 0.87 | 1.16 | 0.97 | |

0.425 | 0.85 | 1.14 | 0.95 | ||

0.416 | 0.91 | 1.15 | 0.94 | ||

0.398 | 0.97 | 1.03 | 0.99 | ||

Carpet grass | 3.5% | 0.359 | 0.84 | 1.39 | 0.94 |

0.337 | 0.82 | 1.15 | 0.93 | ||

0.331 | 0.92 | 1.09 | 0.97 | ||

0.313 | 0.93 | 1.04 | 0.98 | ||

6% | 0.364 | 0.84 | 1.24 | 0.92 | |

0.348 | 0.92 | 1.12 | 0.96 | ||

0.335 | 0.96 | 1.09 | 0.97 | ||

0.317 | 0.99 | 1.15 | 0.99 | ||

7% | 0.393 | 0.95 | 1.10 | 0.98 | |

0.374 | 0.96 | 1.04 | 0.98 | ||

0.368 | 0.93 | 1.08 | 0.98 | ||

0.346 | 0.94 | 1.05 | 0.99 |

Grass Species | Slope | $\frac{{\mathit{h}}_{2}}{{\mathit{h}}_{1}+{\mathit{h}}_{2}}$ | $\frac{{\overline{\mathit{V}}}_{\mathit{b}\mathit{u}\mathit{c}\mathit{k}\mathit{e}\mathit{t}}}{{\overline{\mathit{V}}}_{\mathit{M}\mathit{a}\mathit{n}\mathit{n}\mathit{i}\mathit{n}\mathit{g}}}$ | $\frac{\text{}{\overline{\mathit{V}}}_{\mathit{u}\mathit{l}\mathit{t}\mathit{r}\mathit{a}}}{{\overline{\mathit{V}}}_{\mathit{M}\mathit{a}\mathit{n}\mathit{n}\mathit{i}\mathit{n}\mathit{g}}}$ | $\frac{{\overline{\mathit{V}}}_{\mathit{e}\mathit{l}\mathit{e}\mathit{c}\mathit{t}}}{{\overline{\mathit{V}}}_{\mathit{M}\mathit{a}\mathit{n}\mathit{n}\mathit{i}\mathit{n}\mathit{g}}}$ |
---|---|---|---|---|---|

Centipede grass | 3.5% | 0.319 | 0.45 | 0.67 | 0.51 |

0.301 | 0.63 | 0.84 | 0.52 | ||

0.292 | 0.64 | 0.89 | 0.53 | ||

0.275 | 0.76 | 0.93 | 0.54 | ||

6% | 0.391 | 0.45 | 0.66 | 0.50 | |

0.337 | 0.47 | 0.74 | 0.53 | ||

0.326 | 0.63 | 0.84 | 0.66 | ||

0.293 | 0.86 | 0.96 | 0.90 | ||

7% | 0.423 | 0.48 | 0.70 | 0.53 | |

0.362 | 0.54 | 0.77 | 0.60 | ||

0.350 | 0.63 | 0.91 | 0.68 | ||

0.320 | 0.85 | 0.97 | 0.93 | ||

Bermuda grass | 3.5% | 0.365 | 0.46 | 0.72 | 0.57 |

0.352 | 0.54 | 0.76 | 0.58 | ||

0.342 | 0.60 | 0.89 | 0.59 | ||

0.324 | 0.85 | 0.95 | 0.60 | ||

6% | 0.429 | 0.44 | 0.62 | 0.47 | |

0.410 | 0.47 | 0.68 | 0.51 | ||

0.396 | 0.63 | 0.78 | 0.65 | ||

0.373 | 0.80 | 0.94 | 0.83 | ||

7% | 0.444 | 0.48 | 0.64 | 0.54 | |

0.425 | 0.54 | 0.73 | 0.61 | ||

0.416 | 0.69 | 0.88 | 0.72 | ||

0.398 | 0.91 | 0.96 | 0.92 | ||

Carpet grass | 3.5% | 0.359 | 0.28 | 0.47 | 0.60 |

0.337 | 0.36 | 0.51 | 0.61 | ||

0.331 | 0.58 | 0.69 | 0.62 | ||

0.313 | 0.84 | 0.94 | 0.63 | ||

6% | 0.364 | 0.27 | 0.39 | 0.29 | |

0.348 | 0.51 | 0.63 | 0.54 | ||

0.335 | 0.62 | 0.70 | 0.63 | ||

0.317 | 0.72 | 0.84 | 0.72 | ||

7% | 0.393 | 0.32 | 0.37 | 0.33 | |

0.374 | 0.53 | 0.58 | 0.54 | ||

0.368 | 0.68 | 0.79 | 0.71 | ||

0.346 | 0.81 | 0.91 | 0.85 |

Grass Species | Centipede Grass | Bermuda Grass | Carpet Grass | ||
---|---|---|---|---|---|

Porosity | |||||

Height (m) | |||||

0.02 | 0.710 | 0.944 | 0.765 | ||

0.03 | 0.715 | 0.947 | 0.770 | ||

0.04 | 0.718 | 0.949 | 0.779 | ||

0.05 | 0.720 | 0.951 | 0.783 | ||

0.06 | 0.725 | 0.955 | 0.792 | ||

0.07 | 0.729 | 0.957 | 0.796 |

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**MDPI and ACS Style**

Hsieh, P.-C.; Lin, Y.-C.; Wang, Y.-C.
Comparison of Empirical and Analytical Solutions for Open-Channel Flow Velocity with Common Grass Species in Taiwan. *Water* **2021**, *13*, 1839.
https://doi.org/10.3390/w13131839

**AMA Style**

Hsieh P-C, Lin Y-C, Wang Y-C.
Comparison of Empirical and Analytical Solutions for Open-Channel Flow Velocity with Common Grass Species in Taiwan. *Water*. 2021; 13(13):1839.
https://doi.org/10.3390/w13131839

**Chicago/Turabian Style**

Hsieh, Ping-Cheng, Yi-Cheng Lin, and Yung-Chieh Wang.
2021. "Comparison of Empirical and Analytical Solutions for Open-Channel Flow Velocity with Common Grass Species in Taiwan" *Water* 13, no. 13: 1839.
https://doi.org/10.3390/w13131839