# An Analytical Solution to Predict the Distribution of Streamwise Flow Velocity in an Ecological River with Submerged Vegetation

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## Abstract

**:**

_{d}, and porosity α were used for both the non-vegetated area and the vegetated area, and the range of the depth-averaged secondary flow coefficient was investigated. An analytical solution for predicting the transverse distribution of the water depth-averaged streamwise velocity was obtained in channels that were partially covered by submerged vegetation, which was experimentally verified in previous studies. Additionally, the improved ratio proposed here was compared to previous ratios from other studies. Our findings showed that the ratio in this study could perform velocity prediction more effectively in the partially covered vegetated channel, with a maximum average relative error of 4.77%. The improved ratio model reduced the number of parameters, which introduced the diameter of the vegetation, the amount of vegetation per unit area, and the flow depth. This theoretical ratio lays the foundation for analyzing the flow structure of submerged vegetation.

## 1. Introduction

## 2. Theoretical Analysis

_{0}represents the channel bed slope. Along with the constitutive relations for the stresses, the τ

_{xx}, τ

_{yx}, and τ

_{zx}represent forces acting on the surface of the control body in the derivation of the N–S equations [6]. F

_{v}represents the drag force caused by vegetation per unit volume of a fluid, and can be expressed as follows [10]:

_{d}represents the drag force coefficient of vegetation, β represents the shape factor of vegetation, and A

_{v}represents the projected area of vegetation per unit volume in the direction toward the downstream flow, A

_{v}= mD; m represents the amount of vegetation per unit area, and D represents the stem diameter of the vegetation. Equations (1) and (2) were integrated over the flow depth H. As the water flow was uniform, the water surface in the lateral direction was parallel to the channel bed, indicating that the water depth was constant over the entire cross section [13]. $\partial $(HU

^{2})/$\partial $x ≈ 0, $\partial {(H\tau}_{xx})$/$\partial $x ≈ 0. Assuming that W(H) = W(0) = 0, Equation (1) can be simplified as follows [6]:

_{b}=$-{\int}_{0}^{H}\frac{\partial {\tau}_{zx}}{\partial z}dz$. The integral depth H for the vegetation resistance term can be divided into two parts: 0~H

_{v}and H

_{v}~H. H

_{v}represents the height of the vegetation. No vegetation occurs in the range of H

_{v}~H, and thus, the resistance is 0. The integral term used in Equation (3) can be presented as follows:

_{v}and the water depth-averaged streamwise velocity U

_{d}is necessary. Previous studies obtained different results for the relationship between U

_{v}and U

_{d}.

_{v}, H]/H. Therefore, the value of k

_{v}is 1.0 for the emergent vegetation, and less than 1.0 for submerged vegetation.

^{2}/4, which represents the ratio of the vertical projected area of vegetation to the unit bed area; h

_{s}= H − H

_{v}. Regarding flow velocity ratios, in addition to the above-mentioned equations, another equation was presented by Huthoff et al. [27] as follows:

_{v}and the water depth-averaged streamwise velocity U

_{d}can be determined as follows:

_{v}/U

_{d}as φ, Equation (12) can be substituted into Equation (4) to obtain Equation (13), as follows:

_{0}. When vegetation blockage is considered, the gravity component and other terms in the equation can be obtained by multiplying the terms in Equation (15) by α. However, the drag force due to vegetation is still $\frac{1}{2}\rho \left({C}_{D}\beta {A}_{v}\right){H}_{v}{\phi}^{2}{U}_{d}{}^{2}$, indicating that the drag force term remains unchanged [13].

_{b}represents the comprehensive shear stress of the boundary. Equations (16)–(19) were substituted into (15) to obtain Equation (20), as follows:

- For the non-vegetated area, i.e., the area I in Figure 1b, the solutions of Equation (20) for the non-vegetated area and vegetated area are different. This is because, for the non-vegetated area, the drag force coefficient is 0 in Equation (20). Ignoring the vegetation resistance term “$\frac{1}{2}\rho \left({C}_{d}{\beta A}_{v}\right){H}_{v}{\phi}^{2}{U}_{d}{}^{2}$”, U
_{d}is expressed as follows:

- 2.
- For the vegetated area, i.e., area II in Figure 1b, U
_{d}is expressed as follows:

_{1}, C

_{1}, A

_{2}, and C

_{2}are unknown constants. The superscripts (1) and (2) indicate the non-vegetated area and vegetated area, respectively.

## 3. Boundary Conditions

_{1}, C

_{1}, A

_{2}, and C

_{2}in Equations (22) and (24), four boundary conditions are required, which are described as follows:

- (1)
- On the side wall, the no-slip boundary condition is present, and for the velocity near the side wall, i.e., when y = 0 and y = B, U
_{d}= 0 (two boundary conditions). - (2)
- The velocity continuity condition exists at the junction between the non-vegetated area and the vegetated area, i.e., when y = B, U
_{d}(i) = U_{d}(i + 1). - (3)
- The stress continuity condition is present when the water flow is uniform at the junction between the non-vegetated and vegetated areas. Thus, the flow depth transition is not abrupt at the junction between the areas. The stress continuity condition can be expressed as follows:

## 4. Parameter Determination

_{d}, $\overline{{K}_{1}}$, and $\overline{{K}_{2}}$) is necessary. The values of these parameters are generally different in the non-vegetated and vegetated areas. They can be calculated using the methods described below.

#### 4.1. Transverse Eddy Viscosity Coefficient ξ

_{r}is defined as the relative depth ratio, expressed as the ratio of vegetation depth to the water depth.

#### 4.2. Darcy–Weisbach Friction Coefficient f

^{−6}m

^{2}/s, and ϕ = {12.3, 1.2} in the non-vegetated and vegetated areas, respectively; ${k}_{s}$ represents the equivalent roughness height, and can be calculated using the equation developed by Ackers [31], as follows:

#### 4.3. Porosity α

#### 4.4. The Drag Force Coefficient C_{d}

_{d}, described by Liu et al. [13], is related to the Reynolds number (Re), vegetation shape, and vegetation density. The drag force coefficient C

_{d}decreases with an increase in the Reynolds number of simulated cylindrical vegetation, and increases with an increase in the volume fraction of vegetation [34]. James et al. [35] measured the drag force coefficient of cylindrical vegetation when 200 < Re < 10,000, and found that C

_{d}fluctuated around 1. The C

_{d}value increases significantly with an increase in the number of leaves of cylindrical vegetation. The Reynolds numbers calculated for each case in this study are detailed in Table 1. The drag force coefficient C

_{d}was considered to be 1 in this study.

#### 4.5. Secondary Flow Coefficients $\overline{{K}_{1}}$ and $\overline{{K}_{2}}$

## 5. Experimental Data

#### 5.1. Experimental Data Obtained by Naot et al. [32]

_{0}of 0.0064. The cylindrical diameter D of the simulated rigid vegetation was 0.0036 m. The height of vegetation H

_{v}was 0.03 m, and the flow depth H was 0.06 m. The non-dimensional vegetation density formula, defined as N = mHD, was proposed by Naot et al. [32] and used only in this case. The experiments were conducted with non-dimensional vegetation densities of N = 0.06, N = 0.24, and N = 0.96.

#### 5.2. Experimental Data Obtained by Shi and Huai [33]

## 6. Comparison of Theoretical and Experimental Data

## 7. Discussion

_{v}/U

_{d}. Table 1 shows that in four cases when the flow depth increased from 0.06 m to 0.31 m, the $\overline{{K}_{1}}$ values changed from −0.001 to −0.06, indicating that the absolute value of $\overline{{K}_{1}}$ increased with an increase in flow depth. In order to quantitatively describe the difference between the results of the model and the experimental data, we performed an error analysis from two perspectives: the average values of the absolute error $\overline{\epsilon}$, and the relative error $\overline{{\epsilon}^{\prime}}$. The absolute error ε is expressed as follows:

_{m}represents the number of experimental measurement points. The relative error ε’ is expressed as follows:

## 8. Conclusions

_{d}, were introduced to determine the analytical solution. Additionally, we discussed the methods for calculating the parameters in different zones. Close inspection of the range of the secondary flow coefficient showed that the secondary flow coefficient $\overline{K}$ could not be ignored in different regions. $\overline{K}$ could be determined by the flow depth and the depth of the vegetation layer. According to the error analysis for the velocity data from the analytical solution and experiments, the average relative error is smaller than those from previous studies, revealing that a relative satisfactory prediction of the transverse distribution of water depth-averaged streamwise velocity in the channel flow with submerged vegetation was obtained using the present model. Additionally, when the flow is not assumed to be uniform, the model parameters may be modified, although further experiments are required to verify this. Future studies may put emphasis on the secondary flow coefficient, and extend the model into flows with flexible vegetation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Notation

A_{v} | projected vegetated area per unit volume in the direction of downstream flow |

B | width of the flume |

b | width of the vegetation layer |

C_{d} | drag force coefficient for vegetation |

D | vegetation stem diameter |

D_{r} | relative depth ratio |

F_{v} | drag force |

f | Darcy–Weisbach friction coefficient |

g | gravitational acceleration |

H | flow depth |

H_{v} | height of vegetation |

$\overline{K}$ | secondary flow coefficient |

m | number of vegetation per unit of area |

N | non-dimensional vegetation density |

S_{0} | channel bed slope |

U_{v} | depth-averaged streamwise velocity along the vegetation height |

U_{d} | water depth-averaged streamwise velocity |

ρ | flow density |

α | porosity |

β | shape factor of the vegetation |

ξ | transverse eddy viscosity coefficient |

φ | U_{v}/U_{d} |

$\overline{\epsilon}$ | average value of absolute error |

$\overline{{\epsilon}^{\prime}}$ | average value of relative error |

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**Figure 1.**Layout of the vegetation in the flume. (

**a**) Top view arrangement; (

**b**) arrangement of the cross-section profile. Here, b represents the width of the vegetated area, B represents the width of the flume, H

_{v}represents the height of the vegetation, and H represents the flow depth. I and II represent the non-vegetated area and the vegetated area, respectively.

**Figure 2.**Water depth-averaged streamwise velocity between the experimental data and analytical solution in case 1. The green line occurs when φ

_{1}= 0.8279 (from Cheng [26]). The blue line occurs when φ

_{2}= 0.6914 (from Stone and Shen [25]). The red line occurs when φ

_{3}= 0.4845 (from the presented model (φ = U

_{v}/U

_{d})). The dotted line indicates the boundary between the non-vegetated and vegetated areas.

**Figure 3.**The water depth-averaged streamwise velocity between the experimental data and analytical solution in case 2. φ

_{1}= 0.6522, φ

_{2}= 0.662, and φ

_{3}= 0.4681.

**Figure 4.**Water depth-averaged streamwise velocity between the experimental data and analytical solution in case 3. φ

_{1}= 0.3044, φ

_{2}= 0.6107, and φ

_{3}= 0.4318.

**Figure 5.**Water depth-averaged streamwise velocity between the experimental data and analytical solution in case 4; φ

_{1}= 0.8013, φ

_{2}= 0.8685, and φ

_{3}= 0.7778.

**Figure 6.**Effect of the secondary flow coefficient, $\overline{K}$, on the results of the model for the prediction of the transverse distribution of water depth-averaged streamwise velocity with submerged vegetation. (

**a**) For case 2, in the presented model, $\overline{{K}_{1}}$ = −0.001 and $\overline{{K}_{2}}$ = 0.06. (

**b**) For case 4, in the presented model, $\overline{{K}_{1}}$ = −0.06 and $\overline{{K}_{2}}$ = 0.06.

Sources | Cases | H (m) | H_{v} (m) | D (m) | m (m^{−2}) | β | $\overline{{\mathit{K}}_{1}}$ | $\overline{{\mathit{K}}_{2}}$ | Re |
---|---|---|---|---|---|---|---|---|---|

Naot et al. [32] | 1 | 0.06 | 0.03 | 0.0036 | 278 | 0.51 | −0.0009 | 0.15 | 9000 |

2 | 0.06 | 0.03 | 0.0036 | 1111 | 0.43 | −0.001 | 0.06 | 9000 | |

3 | 0.06 | 0.03 | 0.0036 | 4444 | 0.26 | −0.001 | 0.06 | 9000 | |

Shi and Huai [33] | 4 | 0.31 | 0.25 | 0.008 | 400 | 1 | −0.06 | 0.06 | 8550 |

**Table 2.**Error statistics of the transverse distribution of the water depth-averaged streamwise velocity calculated by the models.

Sources | The Average Value of Error | Cases | |||
---|---|---|---|---|---|

Case 1 | Case 2 | Case 3 | Case 4 | ||

Cheng [26] | $\overline{\epsilon}\left(m/s\right)$ | 0.0072 | 0.0125 | 0.0191 | 0.0045 |

$\overline{{\epsilon}^{\prime}}$(%) | 2.8 | 6.15 | 10.3 | 5.47 | |

Stone and Shen [25] | $\overline{\epsilon}\left(m/s\right)$ | 0.0047 | 0.0131 | 0.02 | 0.0059 |

$\overline{{\epsilon}^{\prime}}$(%) | 1.81 | 6.47 | 11.7 | 8.04 | |

Present Model | $\overline{\epsilon}\left(m/s\right)$ | 0.0048 | 0.0041 | 0.0080 | 0.004 |

$\overline{{\epsilon}^{\prime}}$(%) | 1.85 | 1.84 | 4.34 | 4.77 |

Sources | $\overline{\mathit{K}}$ | |||
---|---|---|---|---|

Present Model | $\mathbf{Modify}\overline{{\mathit{K}}_{1}}$ | $\mathbf{Modify}\overline{{\mathit{K}}_{2}}$ | $\mathbf{Ignore}\overline{{\mathit{K}}_{1}}$$,\overline{{\mathit{K}}_{2}}$ | |

Case 2 | $\overline{{K}_{1}}$ = −0.001 $\overline{{K}_{2}}$ = 0.06 | $\overline{{K}_{1}}$ = 0.001 $\overline{{K}_{2}}$ = 0.06 | $\overline{{K}_{1}}$ = −0.001 $\overline{{K}_{2}}$ = −0.06 | $\overline{{K}_{1}}$$=\overline{{K}_{2}}$ = 0 |

Case 4 | $\overline{{K}_{1}}$ = −0.06 $\overline{{K}_{2}}$ = 0.06 | $\overline{{K}_{1}}$ = 0.06 $\overline{{K}_{2}}$ = 0.06 | $\overline{{K}_{1}}$ = −0.06 $\overline{{K}_{2}}$ = −0.06 | $\overline{{K}_{1}}$$=\overline{{K}_{2}}$ = 0 |

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**MDPI and ACS Style**

Zhang, J.; Mi, Z.; Wang, W.; Li, Z.; Wang, H.; Wang, Q.; Zhang, X.; Du, X.
An Analytical Solution to Predict the Distribution of Streamwise Flow Velocity in an Ecological River with Submerged Vegetation. *Water* **2022**, *14*, 3562.
https://doi.org/10.3390/w14213562

**AMA Style**

Zhang J, Mi Z, Wang W, Li Z, Wang H, Wang Q, Zhang X, Du X.
An Analytical Solution to Predict the Distribution of Streamwise Flow Velocity in an Ecological River with Submerged Vegetation. *Water*. 2022; 14(21):3562.
https://doi.org/10.3390/w14213562

**Chicago/Turabian Style**

Zhang, Jiao, Zhangyi Mi, Wen Wang, Zhanbin Li, Huilin Wang, Qingjing Wang, Xunle Zhang, and Xinchun Du.
2022. "An Analytical Solution to Predict the Distribution of Streamwise Flow Velocity in an Ecological River with Submerged Vegetation" *Water* 14, no. 21: 3562.
https://doi.org/10.3390/w14213562