## Abstract

Vegetation flow is more and more widely studied by scholars at home and abroad because it is an important condition affecting river water quality. However, most of the studies were carried out based on the data of indoor experimental flumes, because the vegetation conditions in nature are more complex. The analytical solution of the flow velocity based on indoor conditions often has some problems when applied to practical projects. Therefore, we propose a numerical method based on the lattice Boltzmann method to simulate the vertical velocity distribution in an open channel with double-layered rigid vegetation. This method has high simulation accuracy in different vegetation conditions. At the same time, because the lattice Boltzmann method is more conducive to simulating complex boundary conditions, it is easier to combine with a multi-layered rigid vegetation flow and a flexible vegetation flow in nature after improvement, providing a basis for the application of indoor theoretical results to the outdoor.

## HIGHLIGHTS

This method has a high simulation accuracy in different vegetation conditions.

It is helpful to study hydrological processes in the vegetation river.

## INTRODUCTION

In recent years, the international river pollution problem has become more and more serious, which has a relatively important impact on human production and life (Li *et al.* 2021; Zhang *et al.* 2021; Alam *et al.* 2023). Therefore, more and more measures have been applied to the study of river water quality, among which plating vegetation is an effective method of improving water quality (Devi *et al.* 2019; Unigarro Vilotta *et al.* 2022; Cui *et al.* 2023). Vegetation in the river can effectively slow down the velocity of the river, prolong the retention time of the river water body, and increase the absorption of excessive nutrients in the water body by vegetation and bed sediment (Bundschuh *et al.* 2016; Soana *et al.* 2017). Meanwhile, the vegetation may be associated with conditions of excessive eutrophication of the river waters. There is a remarkable congruence between the results of our floodplain vegetation analysis and the longitudinal river eutrophication patterns (Mölder & Schneider 2011).

*et al.*2022; Rao

*et al.*2022). Rigid vegetation is a kind of river vegetation, which has also been widely studied by previous scholars (Ratha

*et al.*2018; Caroppi

*et al.*2021). The most common forms of rigid vegetation flow are submerged vegetation flow and emergent vegetation flow. As shown in Figure 1, the velocity in rivers with rigid emergent vegetation is relatively stable (D'Ippolito

*et al.*2019). In the submerged vegetation flow, the velocity in the vegetation layer is relatively stable (Ren

*et al.*2021). However, the riverway vegetation in nature is not uniform in height, and there may be certain height differences between different types of vegetation, thus forming a double-layer vegetation flow (Rao

*et al.*2022). Following the previous research, it can be divided into double-layer submerged vegetation flow and double-layer emergent vegetation flow (Huai

*et al.*2014). Double-layer emergent vegetation flow has a relatively steep velocity transition at the intersection of different vegetation areas, and the velocity in the vegetation area tends to be uniform (Wang

*et al.*2023). The double-layer vegetation submerged flow has the same law as the double-layer vegetation emergent flow in the vegetation area, and the velocity in the non-vegetation layer is basically the same as that in the conventional non-vegetation channel (Tang

*et al.*2021).

The drag force of vegetation has a significant impact on the velocity distribution in a rigid vegetation channel, and the velocity in vegetation area will be uniform (Liu & Zeng 2017). Kumar & Sharma (2022) evaluated and compared the turbulent kinetic energy, skewness, and kurtosis of the vegetation area in the emergent vegetation channel, and proved that the presence of vegetation reduced the longitudinal distribution of Reynolds flow, shear stress, and turbulence intensity. The submerged vegetation will form a certain shear vortex at the interface between vegetation and non-vegetation layers, which will cause rapid changes in the velocity at the interface. On this basis, Huai *et al.* (2009) gave the analytical solution of the vertical distribution of velocity in the submerged vegetation channel in combination with the flume experimental study. For a double vegetation flow, Huai *et al.* (2014) analyzed the momentum equation in vegetation and non-vegetation layers, and finally gave the analytical solution of velocity in different layers.

Although the analytical solution can provide accurate velocity distribution, the solution process is relatively complex. At the same time, if the vegetation layout conditions in the river are more complex, such as there are more than two layers of vegetation distribution, or the vegetation is flexible or uneven, the analytical solution is difficult to apply. In order to better apply the theoretical results to practice, it is necessary to establish a numerical model. In view of this problem, this paper establishes a two-layer vegetation velocity distribution model based on the lattice Boltzmann method. This model can better simulate complex boundary conditions, and can easily add a water quality transport module, which provides help for simulating the velocity and pollutant transport of vegetated river under natural conditions.

## NUMERICAL MODEL

### Governing equations

*et al.*2014):where the Einstein summation convention over Latin indices is adopted;

*t*denotes the time; is the velocity; and represent kinematic and eddy viscosity, respectively; is the bed elevation.

*et al.*(2012):where is the coefficient of the rigid vegetation which can be assumed as 1 (Wu

*et al.*1999),

*a*is the vegetation density coefficient and , is the vegetation density,

*d*is the diameter of the vegetation.

### Lattice Boltzmann method

*et al.*2019; Xu & Yan 2023). The lattice Boltzmann method is mainly divided into two processes: collision process and migration process. These two processes can be expressed by the following equation:where represents the distribution function of particles; is the local equilibrium distribution function;

*x*is the space vector in Cartesian coordinates; ; is the lattice size; is the time step; is the total relaxation time parameter; denotes the external forces.

*et al.*(2010):where denotes the single-relaxation time and is the Smagorinsky constant.

### Boundary conditions

*et al.*(2012):

*r*refers to a constant (), .

The simulation steps are as follows:

- 1.
Establish initial conditions.

- 2.
Simulation of particle migration.

- 3.
Redistributing particles.

- 4.
Judge whether it is over, and output the results when it is over; otherwise, continue the particle migration.

## DATA SOURCE

Source . | Number . | Flume type . | Diameter of vegetation (mm) . | Density of vegetation (1/m^{2})
. | Water depth (cm) . | Note . |
---|---|---|---|---|---|---|

Kumar & Sharma (2022) | 1 | 13-m long, 0.9-m wide and 0.7-m re-circulating straight rectangular channel | 8 | 30.86 | 20 | Emergent vegetation, select the average value at S1 and S2 |

Huai et al. (2009) | 2 | 20-m long, 0.5-m wide, and 0.44-m deep glass flume | 1.5 | 2,500 | 9.36 | Submerged vegetation, select treatment group A31 |

Huai et al. (2014) | 3 | 20-m long, 1-m wide, and 0.5-m deep glass flume | 6 | 171.5 | 20.7 | Emergent double-layer vegetation, select treatment group X1 |

Huai et al. (2014) | 4 | 20-m long, 1-m wide, and 0.5-m deep glass flume | 6 | 171.5 | 28.7 | Submerged double-layer vegetation, select treatment group X3 |

Source . | Number . | Flume type . | Diameter of vegetation (mm) . | Density of vegetation (1/m^{2})
. | Water depth (cm) . | Note . |
---|---|---|---|---|---|---|

Kumar & Sharma (2022) | 1 | 13-m long, 0.9-m wide and 0.7-m re-circulating straight rectangular channel | 8 | 30.86 | 20 | Emergent vegetation, select the average value at S1 and S2 |

Huai et al. (2009) | 2 | 20-m long, 0.5-m wide, and 0.44-m deep glass flume | 1.5 | 2,500 | 9.36 | Submerged vegetation, select treatment group A31 |

Huai et al. (2014) | 3 | 20-m long, 1-m wide, and 0.5-m deep glass flume | 6 | 171.5 | 20.7 | Emergent double-layer vegetation, select treatment group X1 |

Huai et al. (2014) | 4 | 20-m long, 1-m wide, and 0.5-m deep glass flume | 6 | 171.5 | 28.7 | Submerged double-layer vegetation, select treatment group X3 |

## MODEL APPLICATION

## DISCUSSION

There are many factors affecting the accuracy of model simulation. The first is the drag coefficient of vegetation (Wu 2008; Kothyari *et al.* 2009; Liu *et al.* 2020). The research on the drag force coefficient of rigid vegetation has been relatively common. It is generally believed that with the characteristic Reynolds number between 800 and 8,000 is 1. In this paper, the Reynolds number ranged from 1,850 to 7,822. If the characteristic Reynolds number is large or small, the value of the drag force coefficient will change. If the Reynolds number is smaller than 800, then . If the Reynolds number is larger than 8,000, then (Schlichting 1979). At the same time, except for the river where the vegetation is planted artificially, the vegetation in the river is not necessarily arranged in a parallel way. The drag force coefficient of rigid vegetation in the cross the arrangement may be different (Zhang *et al.* 2018). The most common distribution in nature is probably random distribution, whose drag force coefficient is also close to 1 (Li & Shen 1973; Tanino & Nepf 2008).

At the bottom of the vegetated channel, the shear nest effect between the riverbed and the root system is very strong, and the flow velocity measured by the velocity meter is relatively low (Huai *et al.* 2014). Previous simulations of the flow velocity at the bottom of the channel have also exceeded the measured values (Huai *et al.* 2009). Because the stratification of submerged vegetation is more complex than the velocity stratification of emerged vegetation, the shear dimples between the vegetation top and the water flow can also significantly affect the vertical distribution of the flow velocity in the river channel. Therefore, simulating the velocity distribution in submerged vegetation is more difficult, and the simulation accuracy is relatively lower than that of submerged vegetation channels. Compared with the analytical expression model of Huai *et al.* (2014), the flow velocity between two vegetation layers is connected in a straight line, but the flow velocity is closer to the smooth curve distribution at this time, and the model by Huai *et al.* (2014) cannot simulate the flow velocity distribution between vegetation layers well. The steps of solving analytical expression are complex, and difficult to combine with the pollution transport model.

This article mainly discusses the method of simulating the vertical distribution of flow velocity in layered rigid vegetation channels through numerical models, which divides the drag force of vegetation on water flow based on the density of vegetation in different layers. Studying the hydrodynamic laws of layered rigid vegetation channels helps to explore the impact of water flow on vegetation and riverbed, and can provide assistance in studying the transport laws of pollutants in such channels. The limitation is to study uniform rigid vegetation, and further discussion is needed on the study of uneven rigid vegetation or flexible vegetation. At the same time, the distribution of vegetation is relatively uniform, and research on uneven vegetation distribution should also be carried out in the future.

The numerical model can also simulate multi-layer vegetation such as more complex, uneven vegetation layouts by dividing the river into different layers of vegetation density. Flexible vegetation often exists in the channel, and the deformation of flexible vegetation with the change in the water flow should be considered in velocity simulation (Järvelä 2004). There have been some studies on the drag force of flexible vegetation, and it is believed that the drag force of flexible vegetation is generally less than that of rigid vegetation (Aberle & Järvelä 2013). Chapman *et al.* (2015) gave the drag force coefficient formula of flexible vegetation. There are also some studies on the deformation of flexible vegetation. It is determined that the deformation of flexible vegetation is related to the elastic modulus and the leaf area index of vegetation (Luhar & Nepf 2011). This should be considered in the simulation of multi-layer flexible vegetation flow.

This model can be effectively applied to the prediction of vertical velocity distribution in rivers covered by rigid vegetation. At the same time, the lattice Boltzmann method, as a method that can adapt to complex boundary conditions, is suitable for combining with the pollutant transport equation, so that the results of this paper can be more widely used in the simulation of river water quality environment in the future (Prestininzi *et al.* 2016; Chen *et al.* 2018; Wang *et al.* 2018).

## CONCLUSIONS

A numerical model based on the lattice Boltzmann method is proposed to predict the vertical distribution velocity in two-layer rigid vegetation. Compared with the analytical solution, laboratory measurement and field investigation, the numerical model is more suitable for the prediction of velocity under complex and multi-layer conditions of rivers. The numerical prediction of the average velocity is compared with the experimental data, including emergent vegetation, submerged vegetation, emergent double-layer vegetation, and submerged double-layer vegetation. The good consistency shows that the numerical model is effective for open-channel flows with different rigid vegetation conditions. At the same time, more outdoor experiments should be carried out to verify the model, and the pollutant transport module should be added to solve the problem of predicting river pollution.

## ACKNOWLEDGEMENTS

National Science Foundation for Young Scientists of China (Grant No. 208 42207099) and Zhejiang Natural Science Foundation (LQ21E090003).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.