# Large Eddy Simulation of Compound Open Channel Flows with Floodplain Vegetation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{rk}= 0, 0.28 m

^{−1}, 1.13 m

^{−1}and 2.26 m

^{−1}). The main flow velocity, secondary flow, bed shear stress and vortex coherent structure, based on the Q criterion, were obtained and analyzed. Based on the numerical results, the influences of floodplain vegetation density on the flow field and turbulent structure of compound open channel flows were summarized and discussed. Compared to the case without floodplain vegetation, the streamwise velocity in the main channel increased by 10.8%, 19.9% and 24.4% with the f

_{rk}= 0.28 m

^{−1}, 1.13 m

^{−1}and 2.26 m

^{−1}, respectively. The results also indicated that, when the floodplain vegetation density increased, the following occurred: the velocity increased in the main channel, while the velocity decreased in the floodplain; the transverse momentum exchange was enhanced; and the strip structures were more concentrated near the junction area of compound open channel flows.

## 1. Introduction

_{r}, defined as the ratio of the water depth in the floodplain and the water depth in the main channel) on the flow field and turbulent structure [25], and the distributions of energy and momentum correction coefficients in compound open channel flows [26]. In the present study, the WMLES model was applied to simulate the compound open channel flows with floodplain vegetation. The drag force method was adopted to model the resistance effect of vegetation. The WMLES model, incorporating the drag force method, was verified against flume measurements and an analytical solution of vegetated open channel flows. Numerical simulations were conducted with four different vegetation densities (f

_{rk}= 0, 0.28 m

^{−1}, 1.13 m

^{−1}and 2.26 m

^{−1}). Based on the numerical results, the influence of floodplain vegetation density on the flow field and turbulent structure of compound open channel flows was studied and summarized.

## 2. Mathematical Model

#### 2.1. Governing Equations

_{i}represents the Cartesian coordinates (i, j = 1, 2, 3 corresponding to x, y and z, meaning the streamwise, vertical and spanwise directions, respectively); ${\overline{u}}_{i}$ and ${\overline{u}}_{j}$ (i, j = 1, 2, 3) are the filtered velocity components; t is the time; $\overline{p}$ is the filtered pressure; ${g}_{i}=(g\mathrm{sin}\theta ,-g\mathrm{cos}\theta ,0)$ is the gravitational acceleration component in the x

_{i}direction, g is the gravitational acceleration and θ is the angle of the channel to the horizontal. The term τ

_{ij}in Equation (2) is the SGS stress tensor. In the present study, the Boussinesq’s Hypothesis was employed to calculate the SGS stress tensor as:

_{ij}is the strain rate tensor and μ

_{t}is the sub grid scale turbulent viscosity which can be modeled as:

_{w}is the nearest wall distance, Ω

_{ij}is the rotation rate tensor, κ = 0.41, C

_{s}= 0.2, and y

^{+}is the normal to the wall inner scaling. Based on a modified grid scale, the WMLES model considers the grid anisotropy in wall-modeled flows:

_{max}is the maximum length of the cell edge; h

_{wn}is the normal grid spacing and C

_{w}= 0.15 is a constant.

_{d}is the drag coefficient of stem; b

_{v}is the width of stem. The average force per unit volume within the vegetated floodplain could be calculated as:

_{i}(=F

_{x}, F

_{y}, F

_{z}= 0) are the resistance force components per unit volume induced by vegetation in x, y, z directions, N = number density and the resistance parameter f

_{rk}= C

_{d}b

_{v}N. The resistance parameter f

_{rk}can be used to characterize the influence of vegetation density on water flow. The value of C

_{d}of circular cylinder rods was found to be inconstant. Previous studies [27] suggested that the C

_{d}-value was in the range of 1.13 ± 0.15. In the present simulations, the C

_{d}-value was taken as 1.13.

#### 2.2. Boundary Condition and Numerical Algorithm

## 3. Verification of Drag Force Method

#### 3.1. Simulation Implementation

_{0}is the bed slope. The simulation area of the verification case was of length 2 m, width 0.91 m and height 0.61 m. The vegetation height h

_{v}was 0.1175 m. The schematic diagram of the longitudinal section of the simulation area is shown in Figure 2. A hexahedral structured grid was used, and the total number of grids was 384,000.

_{m}, in which H was the water depth and U

_{m}was the bulk velocity).

#### 3.2. Case Verification

_{m}. Figure 3 shows that the vertical profile of time-averaged streamwise velocity. The experimental measurements and the analytical results are plotted in the figure. For the empirical and quasi-theoretical analysis, the vegetation resistance was usually parameterized with the Darcy–Weisbach friction factor [32]. Kouwen [33] proposed a rational method to estimate the roughness coefficient for flow over submerged vegetation. In the present study, the analytical model of Huai et al. [34] was employed for comparison. For the flow layer within vegetation, the velocity could be calculated as:

_{0}is the mixing length at the interface between vegetation layer and the upper non-vegetated flow layer and can be obtained by referring to the region suggested by Righetti and Armanini [35], U* is the shear velocity and equal to:

## 4. Computational Cases

_{r}= 0.5. The computational domain is shown in Figure 6. The width of compound channel B was 0.4 m and the width of the main channel b was 0.2 m. The depth of the main channel H was 0.08 m and the depth of the floodplain h was 0.04 m. A hexahedral structured grid was used in the present simulation, and the grid size was uniform in the streamwise, spanwise and vertical directions. The total number of grids was 1,920,000.

_{v}was set to the water depth within the floodplain. Three cases, with different floodplain vegetation densities, were set. The resistance parameters f

_{rk}in these three cases were 0.28 m

^{−1}, 1.13 m

^{−1}and 2.26 m

^{−1}, respectively. Boundary conditions adopted in the present simulations were the same as previous simulations without floodplain vegetation [25]. To facilitate the comparative analysis in the present study, the computational results of the floodplain without floodplain vegetation were also included for comparison. Thus, there were four cases, with vegetation densities of 0, 0.28 m

^{−1}, 1.13 m

^{−1}and 2.26 m

^{−1}, considered in the present study. The impact of floodplain vegetation on the sectional velocity distribution, secondary currents, bed shear stress, Reynolds stress and turbulent structure was analyzed and summarized.

## 5. Results

#### 5.1. Flow Field and Secondary Flow

_{rk}= 0.28 m

^{−1}, 1.13 m

^{−1}and 2.26 m

^{−1}, respectively. Within the four cases of different floodplain vegetation densities, the location of the maximum streamwise velocity appeared in the main channel, and the smallest streamwise velocity was observed within the floodplain. With the increases of floodplain vegetation density, the high velocity area approached the side wall of the main channel, and the maximum velocity position moved toward the corner of the side wall and the water surface of the main channel. Although the computational flow field and secondary flow in this paper were generally consistent with the numerical simulation results of ASM [5], the present numerical model could faithfully replicate the velocity dip phenomena, while the phenomena were not observed in the results of ASM. It is worth noting that when the resistance parameter f

_{rk}was 1.13 m

^{−1}, the characteristics of the flow field were similar to those when f

_{rk}was 2.26 m

^{−1}, indicating that, when the vegetation density increased to a certain value, the floodplain vegetation tended to completely obstruct the water flow, and the flow field characteristics were less affected by the vegetation density.

_{rk}was 0.28 m

^{−1}, 1.13 m

^{−1}and 2.26 m

^{−1}, the maximum value of the secondary flow velocity was 4.4%, 4.8% and 5.8% of the maximum streamwise velocity, respectively. When the vegetation density increased, the intensity of the secondary flow near the junction area gradually increased, which also indicated that the presence of the floodplain vegetation enhanced the intensity of the secondary flow. In the main channel, the vortex close to the junction gradually approached the side wall due to the secondary flow, and the range of vortex scale reduced. When the floodplain vegetation density increased, the vortex near the bottom of the main channel became more obvious. The vortex at the water surface of the main channel evolved from two vortices in opposite directions to one vortex. The intensity of the floodplain vortex which was close to the interface became larger.

#### 5.2. Bed Shear Stress

#### 5.3. Reynolds Stress

_{rk}was 0.28 m

^{−1}, the Reynolds stress along the vertical direction was 0 at y/H = 1. With an increase in vegetation density, the zero-contour approached the side wall of main channel, while the absolute value of the Reynolds stress reached its maximum value near the interface area of the compound open channel flows. When the vegetation density increased, the Reynolds stress gradually increased near the interface area of compound open channel flows.

_{rk}was 1.13 m

^{−1}and 2.26 m

^{−1}, the negative area reached the water surface, while the value of the Reynolds stress near the bed gradually increased in the main channel. The computational results indicated that, when the floodplain roughened with vegetation, the exchange of mass and momentum was still the strongest near the interface of the main channel and floodplain. The Reynolds stress and the turbulence increased with the increase of floodplain vegetation density. When the floodplain was roughened with vegetation, increase of vegetation density did not result in any significant change in the Reynolds stress value near the side wall of the floodplain.

#### 5.4. Lateral Momentum Exchange

_{b}denotes the bed shear stress, its value corresponds to the total shear stress at z = 0; I

_{e}denotes the energy gradient; h′ denotes the water depth; T denotes the shear stress induced by the Reynolds stress $-\langle {u}^{\prime}{v}^{\prime}\rangle $; J denotes the shear stress induced by the secondary flow; (T-J) is the apparent shear stress, which quantitatively estimates the magnitude of the lateral transport of momentum.

_{rk}was 1.13 m

^{−1}and 2.26 m

^{−1}, two peak values appeared in the secondary current component J.

#### 5.5. Vertical Structures

_{rk}was 1.13 m

^{−1}and 2.26 m

^{−1}, the strip structure in the floodplain significantly reduced. It was found that when the Q value was constant, more strip structures were captured in the main channel and less strip structures were observed in the floodplain while the floodplain was roughened with vegetation.

## 6. Conclusions

_{rk}= 0, 0.28 m

^{−1}, 1.13 m

^{−1}and 2.26 m

^{−1}). The LES results confirmed the findings of existing studies. The impact of vegetation density on flow field and turbulent structure was investigated and summarized. The following conclusions could be drawn.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Sectional distribution of streamwise velocity U and secondary currents (V, W) with different floodplain vegetation densities. (

**a**) Case with f

_{rk}= 0. (

**b**) Case with f

_{rk}= 0.28 m

^{−1}. (

**c**) Case with f

_{rk}= 1.13 m

^{−1}. (

**d**) Case with f

_{rk}= 2.26 m

^{−1}.

**Figure 9.**Reynolds stress $-\langle {u}^{\prime}{v}^{\prime}\rangle /{u}_{\tau}^{2}$ with different floodplain vegetation densities. (

**a**) Case with f

_{rk}= 0. (

**b**) Case with f

_{rk}= 0.28 m

^{−1}. (

**c**) Case with f

_{rk}= 1.13 m

^{−1}. (

**d**) Case with f

_{rk}= 2.26 m

^{−1}.

**Figure 10.**Reynolds stress $-\langle {u}^{\prime}{w}^{\prime}\rangle /{u}_{\tau}^{2}$ with different floodplain vegetation densities. (

**a**) Case with f

_{rk}= 0. (

**b**) Case with f

_{rk}= 0.28 m

^{−1}. (

**c**) Case with f

_{rk}= 1.13 m

^{−1}. (

**d**) Case with f

_{rk}= 2.26 m

^{−1}.

**Figure 11.**The apparent shear stress distribution with different floodplain vegetation densities. (

**a**) Case with f

_{rk}= 0. (

**b**) Case with f

_{rk}= 0.28 m

^{−1}. (

**c**) Case with f

_{rk}= 1.13 m

^{−1}. (

**d**) Case with f

_{rk}= 2.26 m

^{−1}.

**Figure 12.**Instantaneous turbulent structure with different floodplain vegetation densities. (

**a**) Case with f

_{rk}= 0. (

**b**) Case with f

_{rk}= 0.28 m

^{−1}. (

**c**) Case with f

_{rk}= 1.13 m

^{−1}. (

**d**) Case with f

_{rk}= 2.26 m

^{−1}.

Q (m^{3}/s) | H (m) | h_{v} (m) | b_{v} (m) | S_{0} | f_{rk} (m^{−1}) |
---|---|---|---|---|---|

0.179 | 0.335 | 0.1175 | 0.0064 | 0.0036 | 1.23 |

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## Share and Cite

**MDPI and ACS Style**

Zeng, C.; Bai, Y.; Zhou, J.; Qiu, F.; Ding, S.; Hu, Y.; Wang, L. Large Eddy Simulation of Compound Open Channel Flows with Floodplain Vegetation. *Water* **2022**, *14*, 3951.
https://doi.org/10.3390/w14233951

**AMA Style**

Zeng C, Bai Y, Zhou J, Qiu F, Ding S, Hu Y, Wang L. Large Eddy Simulation of Compound Open Channel Flows with Floodplain Vegetation. *Water*. 2022; 14(23):3951.
https://doi.org/10.3390/w14233951

**Chicago/Turabian Style**

Zeng, Cheng, Yimo Bai, Jie Zhou, Fei Qiu, Shaowei Ding, Yudie Hu, and Lingling Wang. 2022. "Large Eddy Simulation of Compound Open Channel Flows with Floodplain Vegetation" *Water* 14, no. 23: 3951.
https://doi.org/10.3390/w14233951