# Double Parameters Generalization of Water-Blocking Effect of Submerged Vegetation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Simulation of Open Channel Flow with Submerged Vegetation

#### 2.1.1. Mathematical Model

^{2}/s);

^{2}/s).

_{s}, k

_{w}and ε

_{s}are as follows.

_{w}is directly taken to be:

_{k}, σ

_{ε}, C

_{ε1}, and C

_{ε2}in Equations (4)–(6) are consistent with those in the standard k-ε model. W

_{D}is a term reflecting the transformation of shear turbulent kinetic energy to stem-scale wake kinetic energy. The formula is

_{μ}= 0.09.

#### 2.1.2. Verification

#### 2.2. Study of the Water-Blocking Effect Generalization of Submerged Vegetation

^{2});

^{3});

#### 2.2.1. Generalization Model and Principle

_{s}is the height of the theoretical zero point, and the flow velocity here is zero after generalization; Δ

_{p}is the height of the equivalent virtual obstacle after generalization, and it’s called virtual channel height; Δ

_{h}is the thickness of the affected region in which submerged vegetation changes the longitudinal velocity above the top of vegetation; and Δ

_{s}+ Δ

_{h}is defined as the influence depth. Before generalization, the real flow is the pore flow in the vegetation layer and the free flow above the vegetation layer, both of which are described uniformly by means of double averaging. However, flow after generalization (called generalized flow) is constant uniform flow in the open channel described by the Reynolds average method. The latter’s effective water depth H

_{m}can be defined as H

_{m}= H − h + Δ

_{s}.

_{s}is the height discrepancy from the theoretical zero point of velocity (i.e., the top of virtual channel after generalization) to the top of the submerged vegetation layer.

_{h}+ Δ

_{s}), the double-average velocity before generalization equals the velocity after generalization.

#### 2.2.2. Handling of Key Issues

_{s}and Δ

_{h}, the following problems need to be solved. One is the vertical distribution of the longitudinal velocity of the constant uniform flow, which is generated after generalization. In this paper, we choose the logarithmic velocity distribution formula (the constant shear stress hypothesis is also used, i.e., ${\tau}_{b}=\rho {u}_{*b}^{2}$). The other is that many flow characteristic variables after generalization are interrelated and need a coupling solution. This paper uses the following methods to obtain simultaneous equations.

_{3}= Δ

_{p}= h − Δ

_{s}, $\overline{{u}_{1}}=0$, i.e., the theoretical zero point of velocity is at the place of x

_{3}= Δ

_{p}. Let z = x

_{3}− Δ

_{p}, and take the theoretical zero point as the origin to establish a new coordinate axis z. Then, the flow velocity distribution after generalization is:

_{h}+ Δ

_{s}, the flow velocity after generalization equals that before generalization. In the calculation process, the absolute value of the difference between the two is set to be less than a minimal value ε

_{m}. That is, 0

_{h}and τ

_{b}, respectively, represent the shear stress at the top of the vegetation layer before generalization and the shear stress at the top of the virtual channel after generalization. We take β = 0.75 (determined after trial). By combining Equations (15)–(17), Δ

_{s}, Δ

_{h}and u

_{*b}can be solved. The roughness coefficient n

_{p}is determined by the Chezy formula:

_{m}is the effective water depth (the definition is in Section 2.2.1), $U=\frac{Q}{B{H}_{m}}$, Q is the flow discharge quantity, B is the channel width, and g is the acceleration of gravity.

#### 2.2.3. Data for Generalization

## 3. Results

#### 3.1. Numerical Simulation of Open Channel Flow with Submerged Vegetation

#### 3.1.1. Verification 1

#### 3.1.2. Verification 2

#### 3.2. Water-Blocking Effect Generalization of Submerged Vegetation

#### 3.2.1. Preliminary Result

_{p}is the height of the generalized obstacle—the virtual channel after generalization. At this height, the generalized flow velocity is zero. From here up, the generalized flow velocity increases from zero. When it rises to the height of vegetation, h, the generalized flow velocity still does not recover to the original flow velocity due to the effect of vegetation obstruction. The height difference in this section is the theoretical zero point’s height of the velocity, Δ

_{s}. Then continue to rise to the highest dotted line, generalized flow velocity adjustment is completed, and it returns to the original flow velocity. That is, these two kinds of velocity overlap. The height difference of this section is the influence thickness, Δ

_{h}. Then these two kinds of flow velocity remain consistently overlap, basically restore the original flow.

#### 3.2.2. Height of the Theoretical Zero Point Δ_{s}

_{s}is nondimensionalized by vegetation height h, and a dimensionless parameter Δ

_{s}/h is obtained. Figure 5 shows the relationship between Δ

_{s}/h and ah/H. The fitting formula is

_{s}/h with increasing ah/H. That is, when the water depth H and vegetation height h remain unchanged, the denser the vegetation arrangement is, the higher the theoretical zero point (in other words, the top of virtual channel rises higher). Analysis for the reason: increasing ah/H means a denser vegetation arrangement. Then, there is more resistance to the flow, and the water-blocking effect of vegetation is strengthened. After the generalization, the relative height of virtual channel (Δ

_{p}/h) will increase. Accordingly, the relative height of the theoretical zero point (Δ

_{s}/h) goes down. When ah/H is zero, there is no vegetation holding back water, so there is no virtual channel after generalization. That is, the virtual channel height Δ

_{p}is zero. Then, the theoretical zero point’s height Δ

_{s}is equal to the height of vegetation, h. Therefore, the y-coordinate value is zero. The extension prediction of fitting line accords with the actual law. The fitting line is suitable to express the relation between vegetation density and generalization results.

_{s}. Here, Δ

_{p}= h − Δ

_{s}, so we can obtain the variation trend of Δ

_{p}. Virtual channel elevation Δ

_{p}is an important parameter obtained after generalization of water-blocking effect. Because it can be directly put into the plane two-dimensional model for engineering calculation.

#### 3.2.3. Influence Depth Δ_{s} + Δ_{h}

_{s}+ Δ

_{h}is nondimensionalized by vegetation height h. Figure 6 shows the (Δ

_{s}+ Δ

_{h})/h~ah/H graph. The fitting formula is

_{s}+ Δ

_{h}is. Further analysis: similarly, the greater ah/H, the stronger the water-blocking effect of vegetation. Then, the water needs a higher depth in the vertical direction to restore its original flow. Therefore, the greater the range of influence thickness zone is, i.e., the greater Δ

_{h}is. However, the actual increase amplitude of influence thickness, Δ

_{h}, is small. This is related to the change in the influence depth, Δ

_{h}+ Δ

_{s}, subsequently. According to Section 3.2.2, the smaller the theoretical zero point’s height Δ

_{s}is, and the reduction is greater than the increase in Δ

_{h}. Therefore, the influence depth Δ

_{s}+ Δ

_{h}shows a decreasing trend as a whole. The height of vegetation h is fixed, so the ordinate value in Figure 6 shows a decreasing trend. When the abscissa value in the Figure 6 is −4, the extension value of the fitting line is exactly about 1. In addition, the relative value of the influence depth (Δ

_{s}+ Δ

_{h})/h is about e. In other words, in the condition of this vegetation arrangement, the influence depth Δ

_{s}+ Δ

_{h}is about e times of the height of vegetation h (about 2.72 times).

#### 3.2.4. Generalized Roughness Coefficient n_{p}

_{p}. Here, n

_{p}calculated from the experimental data in Dunn et al. [1] basically remains at approximately 0.029. However, there is also a trend of a slight increase in n

_{p}with increasing ah/H. The denser the vegetation arrangement is, the higher generalized roughness coefficient n

_{p}is. The fitting formula is

_{p}is. Because of the dense vegetation arrangement, the water-blocking effect of vegetation will be enhanced. Then the equivalent roughness coefficient after generalization, n

_{p}, will increase.

_{p}is another important parameter obtained after generalization of water-blocking effect. Because it can also be directly put into plane two-dimensional model for engineering calculation.

#### 3.2.5. Ratio of Single-Width-Discharges q/q_{0}

_{0}represent the single-width discharge of open channel flow with or without submerged vegetation, respectively. The following formula can be obtained.

_{p}and Δ

_{s}(from Figure 5 and Figure 7) into Equation (25) to obtain the calculated values of q/q

_{0}. Figure 8 shows the variation in q/q

_{0}with the changes in ah/H. The corresponding experimental values of q/q

_{0}(data from the experiment performed by Dunn et al. [1]) are also given in Figure 8. We also extend the calculated values of q/q

_{0}. Figure 8 shows the following: (1) The calculated values are basically consistent with the experimental values of Dunn et al. [1]. This shows that the calculation method of the water-blocking effect proposed in this paper has high precision; (2) q/q

_{0}decreases monotonically with increasing ah/H. This result shows that the denser the submerged vegetation is or the higher the submerged vegetation is, the greater the discharge capacity decreases. This is in line with the objective law. Because of the dense vegetation arrangement, the water-blocking effect of vegetation will be enhanced. That is, the flow of water becomes more obstructed. With the decrease in flow velocity and flow space, the single-width-discharge, q, will become smaller. Then, divided by the single-width-discharge without vegetation obstruction, i.e., q

_{0}, the single-width-discharges ratio q/q

_{0}will decrease. That is, the flow loss will increase, so Figure 8 shows a downward trend. It can be seen from Figure 8 that the cut-off point is the single-width-discharges ratio q/q

_{0}equals to 0.4 (ah/H equals to about 0.75). Before and after this cut-off point, the single-width-discharges ratio q/q

_{0}changes from fast to slow. When ah/H is zero, that is, there is no vegetation obstruction, the single-width-discharges ratio q/q

_{0}is equal to 1. There is no flow loss, in line with the objective law. When ah/H is between 0 and 0.75, the single-width-discharge loss is rapid. (q/q

_{0}rapidly decreases from 1 to 0.4. ah/H as short as 0.75 range, the single-width-discharge loses 60%.) It can be seen that in the early stage of vegetation density increase, a little change in vegetation density will cause a large loss of single-width-discharge. When ah/H is greater than 0.75, the single-width-discharge loss slows down. Over a long ah/H interval, the single-width-discharge ratio q/q

_{0}only decreases from 0.4 to 0.2. That is, the single-width-discharge loss only increases by 20%. This is a stable interval of high loss rate of single-width-discharge. It can be seen that after the early stage of vegetation density increase, even if vegetation density increases again, it has little impact on the loss of single-width-discharge. Figure 8 can be used as a reference for estimating single-width-discharge variation and loss under the action of vegetation arrangement parameter, ah/H.

#### 3.2.6. Comparison Results Related to Velocity Integration

_{s}, Δ

_{s}+ Δ

_{h}, n

_{p}, q/q

_{0}. The velocity of four sections is different, so the single-width discharge of each section is calculated (i.e., the integral area of the velocity distribution diagram). These four sections are then sorted by the size of the single-width discharge in every group, labeled in descending order as Max1, Max2, Max3, and Max4. Then the influence of velocity integral areas in four sections on the four parameters is explored.

_{s}

_{s}, which is calculated after generalization using original flow velocity data. Then, ∆

_{s}is also nondimensionalized in the same way. Figure 9 shows that: (1) When using original velocity to be generalized, the overall trend of the parameter ∆

_{s}is consistent with that of the numerical calculation velocity in Section 3.2.2, i.e., both show a decreasing trend of ∆

_{s}/h with the increase in ah/H. That is, when water depth H and vegetation height h remain unchanged, the denser vegetation layout is, the higher the theoretical zero point is (the higher virtual channel upper boundary is lifted). (2) For the same ah/H, the larger the integrated area of the flow velocity is, or say the single width flow discharge increases, (Max1 blue line is the largest, Max4 purple line is the smallest), the larger parameter Δ

_{s}is, and the smaller it is otherwise. Further analysis for (2): for the same ah/H (i.e., the same group), although water depth H is equal, the greater integrated area of flow velocity in four cross sections is, the larger velocity value is, meaning flow moves faster. Therefore, the generalized virtual channel will be lower, i.e., Δ

_{p}will be smaller, and the corresponding proportion Δ

_{p}/h will be smaller too. Thus, faster flow can be allowed. Accordingly, the proportion of theoretical zero point height, Δ

_{s}/h, is larger.

_{p}= h − Δ

_{s}, the relationship between velocity integration and virtual channel elevation is negative.

_{s}+ Δ

_{h}

_{s}+ Δ

_{h}, which is calculated after generalization using original flow velocity data. Then, Δ

_{s}+ Δ

_{h}is also nondimensionalized by the same way. Figure 10 shows that: (1) when using original velocity to be generalized, the overall trend of the parameter Δ

_{s}+ Δ

_{h}is consistent with that of the numerical calculation velocity in Section 3.2.3, i.e., both show a decreasing trend of (Δ

_{s}+ Δ

_{h})/h with the increase in ah/H. That is, when vegetation height h remains unchanged, the denser the vegetation layout is, the smaller the influence depth Δ

_{s}+ Δ

_{h}is. (2) The greater the integrated area of flow velocity is, the larger the influence depth Δ

_{s}+ Δ

_{h}is. Further analysis for (2): for the same ah/H (i.e., the same group), although the water depth H is equal, the greater the integrated area of flow velocity in four cross sections is, the larger the velocity value is, meaning the flow moves faster. Therefore, at the same h (i.e., at the same water-blocking height), the larger the generalized influence depth Δ

_{s}+ Δ

_{h}will be, meaning that the water needs a higher depth in the vertical direction to restore its original flow. In other words, the water-blocking-effect is stronger. A stronger water-blocking-effect reflects a stronger force between the faster flow and the obstruction.

_{p}

_{p}, which is calculated after generalization using original flow velocity data. Figure 11 shows that: (1) The magnitude of the parameter n

_{p}in this section is basically the same as that of the numerical flow velocity in Section 3.2.4, but the variation with ah/H is smaller than that in Section 3.2.4, i.e., when ah/H increases, n

_{p}increases weakly. (2) The greater integrated area of flow velocity is, the larger generalized roughness coefficient n

_{p}is. Further analysis for (2): for the same ah/H (i.e., the same group), although the water depth H is equal, the greater the integrated area of flow velocity in four cross sections is, the larger the velocity value is, meaning the flow moves faster. Therefore, at the same h (i.e., at the same water-blocking height), the larger the generalized roughness coefficient n

_{p}will be, meaning the water-blocking-effect will be stronger. As mentioned above, a stronger water-blocking-effect reflects a stronger force between the faster flow and the obstruction.

_{p}is positive.

_{0}

_{0}, which is calculated after generalization using original flow velocity data. Figure 12 shows that: (1) when using original velocity to be generalized, the overall trend of q/q

_{0}is consistent with that of the numerical calculation velocity in Section 3.2.5, i.e., both show a decreasing trend of q/q

_{0}with the increase in ah/H. That is, the denser the vegetation layout is, or the higher the vegetation is, the loss of discharge capacity is more significant. (2) The larger integrated area of the flow velocity is, the smaller generalized q/q

_{0}is. Further analysis for (2): for the same ah/H (i.e., the same group), although the water depth H is equal, the greater the integrated area of flow velocity in four cross sections is, the larger the velocity value is, meaning the flow moves faster. Therefore, the force between faster flow and obstruction will be stronger. Then, energy loss will be greater as well. This is the reason why generalized q/q

_{0}is smaller with the increase in the integrated area of the flow velocity.

## 4. Discussion

_{s}, Δ

_{s}+ Δ

_{h}, n

_{p}and q/q

_{0}) from three cases mentioned above.

#### 4.1. Height of the Theoretical Zero Point Δ_{s} in Three Cases

_{s}processed in three cases, and the fitting line obtained in Section 3.2.2 (here called the central line) are drawn into the coordinates together, as shown in Figure 13 below.

_{s}/h is. That is, with the dense vegetation layout, the theoretical zero height Δ

_{s}is smaller, and the virtual channel after generalization is elevated (i.e., Δ

_{p}is larger). (2) Comparing three cases’ data, those of numerical calculation (case 1) are scattered and deviate most from the central line. Data of other two cases (case 2 and case 3) are concentrated relatively. It shows that the two mean-methods can better reflect the central trend. (3) There is little difference between data of case 2 and case 3, indicating that there is little difference between cases of first-generalization and first-averaging for this parameter Δ

_{s}/h.

#### 4.2. Influence Depth Δ_{s} + Δ_{h} in Three Cases

_{s}+ Δ

_{h}processed in three cases, and the fitting line obtained in Section 3.2.3 (here called the central line) are drawn into the coordinates together, as shown in Figure 14 below.

_{s}+ Δ

_{h})/h is. The h in the figure is 0.1175 m, so (Δ

_{s}+ Δ

_{h}) decreases with the increase in ah/H. (2) Comparing data in three cases, the difference is small, and they all concentrate around the central line. It shows that there is little difference among the three cases for reflecting the central trend.

#### 4.3. Generalized Roughness Coefficient n_{p} in Three Cases

_{p}processed in three cases, and the fitting lines obtained in Section 3.2.4 (here called the central line) are drawn into the coordinates together, as shown in Figure 15 below.

_{p}is. That is, with the dense vegetation layout, the generalized roughness coefficient n

_{p}is larger, and all data generally distribute around the center line. (2) Comparing data in three cases, data in case 1 (i.e., numerical calculation data) are relatively larger. Data in case 2 and case 3 are relatively smaller and there is little difference between them. It shows that data processed by numerical calculation are larger than those processed by mean calculation, and there is little difference between the two latter cases.

#### 4.4. Ratio of Single-Width-Discharges q/q_{0} in Three Cases

_{0}processed in three cases, and the calculated line obtained in Section 3.2.5 (here called the central line) are drawn into the coordinates together, as shown in Figure 16 below.

_{0}is. That is, with the sparse vegetation layout, smaller flow discharge loss is, and all data generally distribute around the center line. (2) Comparing data in three cases, data in case 2 and case 3 (i.e., mean calculation cases) are generally equal. Data in case 1 are also generally equal to that of case 2 and case 3, except two are relatively smaller. Therefore, we can basically say that there is little difference in q/q

_{0}of three cases, and they can all reflect the same center trend.

#### 4.5. Influences of Arrangement of Vegetation

#### 4.6. Discussion of Implications for Environmental Flows and Hydraulic Engineering Problems

_{s}, k

_{w}, ε

_{s}, u

_{1}) are solved together. The numerical method is to discretize the equations using an unsteady finite volume method. Each equation is discretized as the governing equation of unsteady flow. Add the time partial derivative of the variable to the left-hand side of each governing equation. The central difference scheme is used to differentiate the time term. The flow is iteratively calculated with the number of steps in a certain time. In addition, then we put in the boundary conditions. In the boundary conditions, k

_{s}and ε

_{s}are the same as the standard k-ε equation, and the gradient of the newly introduced k

_{w}is zero at the bottom of the open channel and on the free surface. Finally, Gauss–Seidel iterative method is used to solve discrete algebraic equations. After that, the longitudinal velocity profile of open channel flow with submerged vegetation is obtained. These velocity values can then be used to calculate shear stress and determine the sediment transport rates. There are other implications as well, for example, a lot of aquatic vegetation is morphologically distinct from rigid cylinders (they have leaves, branches, etc.), so this would probably modify the velocity and shear stress profiles from the examples provided in this manuscript. The distribution pattern of velocity and shear stress is greatly affected by vegetation form and water flow condition. It is suggested that readers should first find experiments of submerged vegetation with leaves and branches for flow velocity distribution if they are interested. In addition, then you can think about whether to decompose k and ε into two or more parts. Accordingly, the governing Equations (4)–(6) may introduce new parts of k and ε. The variable expression in the equation may need to change some coefficients’ values. Vegetation porosity is the main parameter.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol Key | |

$\psi $ | instantaneous variable |

$\overline{\psi}$ | time mean variable |

${\psi}^{\prime}$ | fluctuation variable |

$\langle \overline{\psi}\rangle $ | local spatial mean |

${\psi}^{\u2033}$ | local spatial pulsation |

$\langle \overline{{u}_{1}}\rangle $ | longitudinal flow velocity after double averaging (m/s) |

$\overline{{u}_{1}}$ | longitudinal flow velocity after generalization (m/s) |

$\langle \overline{{\tau}_{1}}\rangle $ | shear stress after double averaging (Pa) |

$\overline{{\tau}_{1}}$ | shear stress after generalization (Pa) |

${i}_{b}$ | bottom slope of the open channel |

$D$ | diameter of submerged vegetation (m) |

$\theta $ | porosity of the flora |

${C}_{D}$ | drag coefficient |

$\nu $ | molecular viscosity coefficient (m^{2}/s) |

${\nu}_{t}$ | turbulence viscosity coefficient (m^{2}/s) |

k | turbulent kinetic energy |

k_{s} | turbulent kinetic energy of the large-scale shear turbulence |

k_{w} | turbulent kinetic energy of the small-scale stem turbulence |

ε | turbulent kinetic energy dissipation rate |

ε_{s} | turbulent kinetic energy dissipation rate of the large-scale shear turbulence |

ε_{w} | turbulent kinetic energy dissipation rate of the small-scale stem turbulence |

W_{D} | a term reflecting the transformation of shear turbulent kinetic energy to stem-scale wake kinetic energy |

ν_{t} | turbulent viscosity coefficient |

$a$ | vegetation density (1/m) |

$A$ | inflow area of a single cylinder (m^{2}) |

$V$ | volume affected by a single cylinder (m^{3}) |

$D$ | diameter of the cylinder (m) |

$h$ | height of submerged vegetation (m) |

$\Delta {x}_{1}$ | length of the volume $V$ in direction ${x}_{1}$ when averaged locally (m), ${x}_{1}$ and ${x}_{2}$ can be streamwise direction and spanwise direction, respectively. |

$\Delta {x}_{2}$ | length of the volume $V$ in direction ${x}_{2}$ when averaged locally (m). |

ah/H | a comprehensive parameter to describe the water-blocking effect of submerged vegetation on water (1/m) |

Δ_{s} | the height of the theoretical zero point, and the flow velocity here is zero after generalization (m) |

Δ_{p} | the height of the equivalent virtual obstacle after generalization, and it’s called virtual channel height (m) |

Δ_{h} | the thickness of the affected region in which submerged vegetation changes the longitudinal velocity above the top of vegetation (m) |

Δ_{s} + Δ_{h} | the influence depth (m) |

H_{m} | effective water depth (m), defined as H_{m} = H − h + Δ_{s} |

τ_{h} | shear stress at the top of the vegetation layer before generalization (Pa) |

τ_{b} | shear stress at the top of the virtual channel after generalization (Pa) |

n_{p} | roughness coefficient |

Q | flow discharge quantity (m^{3}/s) |

B | channel width (m) |

g | the acceleration of gravity (m^{2}/s) |

${u}_{*}^{}$ | frictional velocity (m/s), defined as ${u}_{*}^{2}={-\langle \overline{{u}_{1}^{\prime}{u}_{3}^{\prime}}\rangle |}_{{x}_{3}=h}$ |

q | single-width discharge of open channel flow with submerged vegetation (m^{3}/s) |

q_{0} | single-width discharge of open channel flow without submerged vegetation (m^{3}/s) |

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**Figure 2.**Verification calculation results of the double-mean longitudinal velocity and Reynolds stress (experimental data source: Nezu et al. [4]): (

**a**) verification calculation results of the double mean longitudinal velocity; (

**b**) verification calculation results of the double-mean Reynolds stress.

**Figure 3.**Verification calculation results of the double-mean longitudinal velocity and Reynolds stress (experimental data source: Dunn et al. [1]): (

**a**) verification calculation results of the double mean longitudinal velocity; (

**b**) verification calculation results of the double-mean Reynolds stress.

**Figure 4.**Comparison of the double-mean velocity $\langle \overline{{u}_{1}}\rangle $ and generalized velocity $\overline{{u}_{1}}$. (The upper dotted line is the depth of influence, the middle dotted line is the height of vegetation and the lower dotted line is the top of the virtual canal).

**Figure 9.**Height of the theoretical zero point Δ

_{s}varies with the comprehensive parameter ah/H. (calculated from original velocity data with different velocity integration in experiments of Dunn et al. [1]).

**Figure 10.**Influence depth Δ

_{s}+ Δ

_{h}varies with the comprehensive parameter ah/H. (calculated from original velocity data with different velocity integration in experiments of Dunn et al. [1]).

**Figure 11.**Generalized roughness coefficient n

_{p}varies with the comprehensive parameter ah/H. (calculated from original velocity data with different velocity integration in experiments of Dunn et al. [1]).

**Figure 12.**Ratio of single-width-discharges q/q

_{0}varies with the comprehensive parameter ah/H. (calculated from original velocity data with different velocity integration in experiments of Dunn et al. [1]).

**Figure 13.**Height of the theoretical zero point Δ

_{s}varies with the comprehensive parameter ah/H (3 cases).

**Figure 15.**Generalized roughness coefficient n

_{p}varies with the comprehensive parameter ah/H. (3 cases).

**Figure 16.**Ratio of single-width-discharges q/q

_{0}varies with the comprehensive parameter ah/H. (3 cases).

**Table 1.**Main flow features of the experiments in Nezu et al. [4].

Case | Water Depth H (cm) | Height of Vegetation Element h (cm) | Mean Bulk Velocity U_{m} (cm/s) | U_{h} ^{1} (cm/s) | Friction Velocity U* ^{2} (cm/s) | Fr | Reynolds Number Re |
---|---|---|---|---|---|---|---|

A-10 | 15.0 | 5.0 | 12.0 | 5.83 | 2.76 | 0.10 | 1.8 × 10^{4} |

B-10 | 15.0 | 5.0 | 12.0 | 5.25 | 2.53 | 0.10 | 1.8 × 10^{4} |

C-10 | 15.0 | 5.0 | 12.0 | 5.77 | 2.31 | 0.10 | 1.8 × 10^{4} |

^{1}U

_{h}is the space-averaged (horizontally averaged) mean velocity at the vegetation top edge.

^{2}U* here is defined as the value of Reynolds stress at the vegetation top edge.

**Table 2.**Main flow features of the experiments in Dunn et al. [1].

Experiment | Q (L/s) | Bed Slop S (%) | Bulk FlowVelocity (m/s) | Water Depth (m) | Reynolds Number Re | Manning’s n (m^{1/6}) | Fr |
---|---|---|---|---|---|---|---|

1 | 179 | 0.36 | 0.587 | 0.335 | 2.24 × 10^{5} | 0.034 | 0.33 |

2 | 88 | 0.36 | 0.422 | 0.229 | 1.13 × 10^{5} | 0.041 | 0.29 |

3 | 46 | 0.36 | 0.308 | 0.164 | 0.57 × 10^{5} | 0.048 | 0.24 |

4 | 178 | 0.76 | 0.709 | 0.276 | 1.91 × 10^{5} | 0.038 | 0.36 |

5 | 98 | 0.76 | 0.531 | 0.203 | 1.25 × 10^{5} | 0.045 | 0.37 |

6 | 178 | 0.36 | 0.733 | 0.267 | 1.96 × 10^{5} | 0.025 | 0.39 |

7 | 95 | 0.36 | 0.570 | 0.183 | 1.20 × 10^{5} | 0.027 | 0.42 |

8 | 180 | 0.36 | 0.506 | 0.391 | 2.58 × 10^{5} | 0.042 | 0.29 |

9 | 58 | 0.36 | 0.298 | 0.214 | 0.70 × 10^{5} | 0.056 | 0.19 |

10 | 180 | 1.61 | 0.746 | 0.265 | 2.03 × 10^{5} | 0.052 | 0.40 |

11 | 177 | 0.36 | 0.625 | 0.311 | 2.22 × 10^{5} | 0.031 | 0.35 |

12 | 181 | 1.08 | 0.854 | 0.233 | 2.38 × 10^{5} | 0.036 | 0.58 |

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## Share and Cite

**MDPI and ACS Style**

Qiu, C.; Huang, J.; Liu, S.; Pan, W.
Double Parameters Generalization of Water-Blocking Effect of Submerged Vegetation. *Water* **2023**, *15*, 764.
https://doi.org/10.3390/w15040764

**AMA Style**

Qiu C, Huang J, Liu S, Pan W.
Double Parameters Generalization of Water-Blocking Effect of Submerged Vegetation. *Water*. 2023; 15(4):764.
https://doi.org/10.3390/w15040764

**Chicago/Turabian Style**

Qiu, Chunlin, Jiesheng Huang, Shihe Liu, and Wenhao Pan.
2023. "Double Parameters Generalization of Water-Blocking Effect of Submerged Vegetation" *Water* 15, no. 4: 764.
https://doi.org/10.3390/w15040764