Numerical Analysis of the Impact Factors on the Flow Fields in a Large Shallow Lake

Wetland acts as an important part of climatic regulation, water purification, and biodiversity maintenance. As an integral part of wetlands, large shallow lakes play an essential role in protecting ecosystem diversity and providing water sources. Baihe Lake in the Momoge Wetland is one such example, so it is necessary to study the flow pattern characteristics of this lake under different conditions. A new model, based on the lattice Boltzmann method, was used to investigate the effects of different impact factors on flow fields, such as water discharge from surrounding farmland, rainfall, wind speed, and aquatic vegetation. Importantly, this study provides a hydrodynamic basis for local ecological protection and restoration work.


Introduction
There are many wetlands in China that play an important role in water resource conservation and ecological diversity maintenance.Wetlands have irreplaceable ecological significance for human survival [1].Although China has emphasized wetland protection over the past few decades, there remains a sharp decline in the area of wetlands [2].According to official statistics of the China Forestry Administration, China's wetland area fell by 8.83% in 2013, compared with that of 2003.The shrinking of wetland area is closely related to the decrease of water resources in wetlands, and the atrophy of lake area is directly impacted by the shrinking of wetland area [3].Therefore, it is of vital importance to study the water flow in lakes to find out the feasible measures for wetland conservation [1,4].
A large number of microbes, animals, and plants live in lakes.Therefore, improving purification ability is essential to maintaining the wetland ecological environment [5].In China, the area of lakes in wetlands has reduced greatly, and as a result, finding ways to slow down the shrinkage of lakes has become an increasingly important topic [6].The reduction of lake area is bound to the flow of water, and there is no doubt that the characteristics of the water flow is of great importance [5][6][7].In general, gravity, bed friction, rainfall, wind speed, and topography may influence the hydrodynamic conditions of lakes of different levels [6].In addition, vegetation in lakes can also have an important impact on water flow with its drag effect [8,9].
Physical modelling and numerical simulation are the two main approaches to studying flow characteristics [10].With the development of computer science, numerical modeling has become prevalent [11].The MIKE series model, developed by the Danish Institute of Water Resources and Water Environment (DHI, Hørsholm, Denmark), is a general commercial package with a wide range of applications for water simulation [12].The Environmental Fluid Dynamics Code (EFDC) model, Water 2019, 11, 155 2 of 13 created by the Virginia Institute of Marine Science at the College of William and Mary (Williamsburg, VA, USA), has many functions for water quality modelling and its partially open-source code offers good flexibility [12][13][14].However, these commercial models are only based on the traditional finite difference method or the finite volume method, and further development is inhibited given that these models do not have a flexible external force term.
The lattice Boltzmann method (LBM) is a relatively new numerical method for fluid flows [15].The related theoretical basis of LBM was carried out in the 1980s [16].Salmon (1999) [17] developed a lattice Boltzmann model for shallow water.Compared with traditional computational methods based on the direct approach of the flow equation, it proposes a new solution for the flow equation [18].The method is characterized by simple calculation and easy handling of boundary conditions [19].
In recent years, it has become a promising approach in computational fluid dynamics [20].The use of the LBM to simulate different kinds of flow (e.g., open channel flows, tidal flows, and dam-break flows) has become popular.Ottolenghi et al. (2018) [21] used lattice Boltzmann method to investigate the properties of graphene oxide for environmental applications.O'Brien et al. ( 2002) [22] developed a lattice Boltzmann scheme to study reactive transport in porous media.Zhou (2007) [23] developed a lattice Boltzmann model for groundwater flow.Tubbs and Tsai (2009) [24] developed the parallel computation for multi-layer shallow water flows.Liu et al. (2012) [25] worked out a large eddy simulation of turbulent shallow water flows using the LBM.Prestininzi (2016) [26] presented a 2D multi-layer shallow water lattice Boltzmann model able to predict the salt wedge intrusion in river estuaries.Furthermore, Yang et al. (2017) [8] developed a rigid vegetation model with 2D shallow water equations using the LBM.
In this paper, the LBM is used to study the hydrodynamic characteristics of Baihe Lake.This is also the first numerical simulation study of Baihe Lake in China, which is helpful for local ecological and hydrological restoration.Through the numerical study of different scenarios, the model provides a quantitative investigation of different impact factors and the results could provide a theoretical basis for local wetland restoration and hydraulic engineering construction.

Study Area
The study area was Baihe Lake in the Momoge Wetland of Zhenlai County, Jilin Province, China.The Momoge Wetland (45 • 45'~46 • 10' N, 122 • 27'~124 • 04' E) is one of the most important wetlands in Northeastern China, as shown in Figure 1.It is the main migration path for white crane in China and the total area is 1440 km 2 .In recent years, the area of the Momoge Wetland has shrunk significantly.The local government has developed a series of wetland restoration projects.Baihe Lake is one of the largest lakes in the Momoge Wetland.The area of the lake is 15 km 2 and the eastern region of the lake is next to the Nenjiang River.The lake is also the main fishing area for local fishermen and its upstream is surrounded by local farmland.Every year a large amount of irrigation water recedes to Baihe Lake as a replenishment.At the same time, the lake also plays a role in purifying the water quality.The lake's outlet is next to the Taoer River.Therefore, Baihe Lake plays an extremely important role in local services such as economic activities, water quality purification, hydrological connectivity, and ecological protection.

Governing Equations
Baihe Lake is a shallow lake, where the horizontal scale is much larger than the vertical scale.According to field measurements, the largest water depth of Baihe Lake is 2.82 m.Therefore, the two-dimensional hydrodynamic model can be used to study its water flow.The depth-averaged hydrodynamic equations are derived from the incompressible Navier-Stokes equations, which are called the shallow water equations [25].
where h is the water depth; t is the time; the subscripts i and j are the space directions based on the Einstein summation convention; x j and u j are the distance and instantaneous velocity components in the j direction; g is the gravitational acceleration and equals 9.81 m 2 /s; ν is the kinematic viscosity; and F i is the external force term.
Water 2019, 11, x FOR PEER REVIEW 3 of 13 where h is the water depth; t is the time; the subscripts i and j are the space directions based on the Einstein summation convention; xj and uj are the distance and instantaneous velocity components in the j direction; g is the gravitational acceleration and equals 9.81 m 2 /s; ν is the kinematic viscosity; and Fi is the external force term.

External Force Term
Shallow water can be significantly influenced by external forces.Generally, the external forces caused by gravity, wind speed, and riverbed friction should be considered [27].In Equation (2), ignoring the Coriolis force, Fi is the force term and can be expressed as: ρ is the water density, which is equal to 1000 kg/m 3 .Zb is the bed elevation; τbi is the bed friction and can be expressed as: where cb = gn 2 /h 1/3 ; n is Manning's coefficient; and τwi is the wind shear stress that can be expressed as: Where ρa is the air density; cw is the resistance coefficient; and uwi and uwj are the wind velocities in the i and j directions.
In addition, if there is aquatic vegetation in lakes, the drag effect of vegetation on the water body should be considered as an external force [8,28].A large amount of vegetation is distributed in Baihe Lake, where reeds and bulrush are dominant [29].This type of aquatic vegetation shown in Figure 2

External Force Term
Shallow water can be significantly influenced by external forces.Generally, the external forces caused by gravity, wind speed, and riverbed friction should be considered [27].In Equation ( 2), ignoring the Coriolis force, F i is the force term and can be expressed as: ρ is the water density, which is equal to 1000 kg/m 3 .Z b is the bed elevation; τ bi is the bed friction and can be expressed as: where c b = gn 2 /h 1/3 ; n is Manning's coefficient; and τ wi is the wind shear stress that can be expressed as: where ρ a is the air density; c w is the resistance coefficient; and u wi and u wj are the wind velocities in the i and j directions.In addition, if there is aquatic vegetation in lakes, the drag effect of vegetation on the water body should be considered as an external force [8,28].A large amount of vegetation is distributed in Baihe Lake, where reeds and bulrush are dominant [29].This type of aquatic vegetation shown in Figure 2 with high toughness is generally higher than the water free surface and can be simplified as unsubmerged rigid vegetation [29,30].A rigid vegetation model based on two-dimensional shallow water equations is presented in Reference [8].The model treated the unsubmerged rigid vegetation as vertical cylinders and the drag force can be expressed as: where u vi is the average velocity on the vegetation elements in the i direction; C d is the drag force coefficient and is usually in the range of 1 and 1.5 [31]; and λ is the projected area (normal to the flow) of vegetation per unit volume of water and is calculated by: where α v represents the shape factor; c is the density of the vegetation zones and represents the projected area of vegetation per unit bed area; D v is the vegetation stems diameter; and u vi is equal to the average velocity u i .
Water 2019, 11, x FOR PEER REVIEW 4 of 13 with high toughness is generally higher than the water free surface and can be simplified as unsubmerged rigid vegetation [29,30].A rigid vegetation model based on two-dimensional shallow water equations is presented in Reference [8].The model treated the unsubmerged rigid vegetation as vertical cylinders and the drag force can be expressed as: where uvi is the average velocity on the vegetation elements in the i direction; Cd is the drag force coefficient and is usually in the range of 1 and 1.5 [31]; and λ is the projected area (normal to the flow) of vegetation per unit volume of water and is calculated by: where αv represents the shape factor; c is the density of the vegetation zones and represents the projected area of vegetation per unit bed area; Dv is the vegetation stems diameter; and uvi is equal to the average velocity ui.

Lattice Boltzmann Method (LBM)
The LBM is a modern numerical technique for computational fluid dynamics.The LBM for nonlinear two-dimensional shallow water equations has been widely used [10].It is a discrete computational method based on the lattice gas automata-a simplified, fictitious molecular model.The three main components in the LBM are lattice pattern, kinetic equation, and equilibrium distribution.The lattice Boltzmann equation can be expressed as: where fα is the particle distribution function; ∆x is the lattice size; ∆t is time step; the external force Fα is calculated by: where Fi is force term computed by Equation ( 3); e = Δx/Δt; ωα is the weight factor: ωα = 4/9 for α = 0; ωα = 1/9 for α = 1, 3, 5, 7; and ωα = 1/36 for α = 2, 4, 6, 8.
Lattice pattern in the LBM has two functions: indicating grid points and resolving particle motions.The former represents a similar role in the traditional numerical simulation methods.The latter shows a microscopic model for molecular dynamics.eαi is the particle velocity in the i direction.The nine-velocity square lattice is shown in Figure 3.Each particle moves one lattice unit at its

Lattice Boltzmann Method (LBM)
The LBM is a modern numerical technique for computational fluid dynamics.The LBM for non-linear two-dimensional shallow water equations has been widely used [10].It is a discrete computational method based on the lattice gas automata-a simplified, fictitious molecular model.The three main components in the LBM are lattice pattern, kinetic equation, and equilibrium distribution.The lattice Boltzmann equation can be expressed as: where f α is the particle distribution function; ∆x is the lattice size; ∆t is time step; the external force F α is calculated by: where F i is force term computed by Equation ( 3); e = ∆x/ ∆t; ω α is the weight factor: ω α = 4/9 for α = 0; ω α = 1/9 for α = 1, 3, 5, 7; and ω α = 1/36 for α = 2, 4, 6, 8. Lattice pattern in the LBM has two functions: indicating grid points and resolving particle motions.The former represents a similar role in the traditional numerical simulation methods.The latter shows a microscopic model for molecular dynamics.e αi is the particle velocity in the i direction.The nine-velocity square lattice is shown in Figure 3.Each particle moves one lattice unit at its velocity along the eight links represented by numbers 1-8, while 0 represents a particle at rest with zero speed.The velocity vector of the particles is defined by: , sin , sin A local equilibrium distribution function decides what flow equations are solved by means of the lattice Boltzmann equation.For 2D shallow water Equations ( 1) and ( 2), the local equilibrium distribution function f eq α is defined as: Then the remaining task is to determine the physical quantities.The macroscopic variables, water depth h and flow velocity u i can be expressed as: Water 2019, 11, x FOR PEER REVIEW 5 of 13 velocity along the eight links represented by numbers 1-8, while 0 represents a particle at rest with zero speed.The velocity vector of the particles is defined by: Then the remaining task is to determine the physical quantities.The macroscopic variables, water depth h and flow velocity ui can be expressed as:

Rainfall
Rainfall has an effect on the surface water and runoff, causes soil erosion and floods, and contaminants transport.A numerical model should be applied to investigate how rainfall influences water flow [32].In the LBM, the local equilibrium distribution function eq g α for shallow water equations with source term was developed.It can be expressed as follows: Where R is the rainfall intensity and the influence of dynamic pressure caused by precipitation is neglected.

Rainfall
Rainfall has an effect on the surface water and runoff, causes soil erosion and floods, and contaminants transport.A numerical model should be applied to investigate how rainfall influences water flow [32].In the LBM, the local equilibrium distribution function g eq α for shallow water equations with source term was developed.It can be expressed as follows: 6e 2 + h 3e 2 e αi u i + h 2e 2 e αi e αj u i u j − h 6e 2 u i u j α = 1, 3, 5, 7 gh 2 24e 2 + h 12e 2 e αi u i + h 8e 2 e αi e αj u i u j − h 24e 2 u i u j α= 2, 4, 6, 8 where R is the rainfall intensity and the influence of dynamic pressure caused by precipitation is neglected.

Boundary Conditions
A general treatment at the inlet is to set a constant velocity and a water depth, whereas a specific water depth is imposed at the outlet.In addition, a zero-gradient condition is used to obtain the velocity components u and v at the outlet.The standard bounce-back scheme, in which an incoming particle towards the boundary is bounced back into the fluid, is widely used.At the upper boundary: and at the lower boundary: The grids in the corner are involved with changes in boundary conditions, as shown in Figure 4.At the boundary corner point, there will be multiple directions without particle input due to the proximity of the model boundary.It cannot be calculated using the bounce-back scheme or macro variable boundary conditions.It is necessary to calculate the missing particle distribution according to the depth and velocity of the adjacent grids.The computational formula can be expressed as: A general treatment at the inlet is to set a constant velocity and a water depth, whereas a specific water depth is imposed at the outlet.In addition, a zero-gradient condition is used to obtain the velocity components u and v at the outlet.The standard bounce-back scheme, in which an incoming particle towards the boundary is bounced back into the fluid, is widely used.At the upper boundary: and at the lower boundary: The grids in the corner are involved with changes in boundary conditions, as shown in Figure 4.At the boundary corner point, there will be multiple directions without particle input due to the proximity of the model boundary.It cannot be calculated using the bounce-back scheme or macro variable boundary conditions.It is necessary to calculate the missing particle distribution according to the depth and velocity of the adjacent grids.The computational formula can be expressed as: ), 2 .
x y x y x y x y x y x y

Initial Conditions
Based on data from 2017, the drainage of farmland converged towards the lake inlet and discharged into Baihe Lake at an average rate of 26.5 m 3 /s from 11 May to 25 June, and at an average rate of 14.4 m 3 /s from 10 October to 21 November.The average wind speed was equal to 1.78 m/s in the northeast (132°) from 11 May to 25 June, and average wind speed was equal to 2.36 m/s in the northeast (110°) from 10 October to 21 November.
Figure 5 shows the vegetation distribution obtained by a geographic information system and a field investigation.There are a large number of reeds in Baihe Lake that are higher than the water surface and have high rigidity.From inspection of Figure 5, it can be seen that the vegetation is mainly

Initial Conditions
Based on data from 2017, the drainage of farmland converged towards the lake inlet and discharged into Baihe Lake at an average rate of 26.5 m 3 /s from 11 May to 25 June, and at an average rate of 14.4 m 3 /s from 10 October to 21 November.The average wind speed was equal to 1.78 m/s in the northeast (132 • ) from 11 May to 25 June, and average wind speed was equal to 2.36 m/s in the northeast (110 • ) from 10 October to 21 November.
Figure 5 shows the vegetation distribution obtained by a geographic information system and a field investigation.There are a large number of reeds in Baihe Lake that are higher than the water surface and have high rigidity.From inspection of Figure 5, it can be seen that the vegetation is mainly distributed in the northwest and southeast regions of the lake.Based on a field survey, it was found that the vegetation density varies significantly throughout the year.The density is highest from July to August and lowest from January to March.In order to confirm the effect of drag force caused by the vegetation, two situations involving high-and low-vegetation density were tested in the present model.In the computation, the vegetation element shape factor and drag coefficient C d are equal to 1.0 [28].
Water 2019, 11, x FOR PEER REVIEW 7 of 13 distributed in the northwest and southeast regions of the lake.Based on a field survey, it was found that the vegetation density varies significantly throughout the year.The density is highest from July to August and lowest from January to March.In order to confirm the effect of drag force caused by the vegetation, two situations involving high-and low-vegetation density were tested in the present model.In the computation, the vegetation element shape factor and drag coefficient Cd are equal to 1.0 [28].

Numerical Tests
A 2D hydrodynamic model was established using the LBM.Baihe Lake is covered by 340 × 161 grids with each lattice being 50 m × 50 m.The inflow velocity from the northeast inlet is given by 0.5 m/s, which is controlled by sluice gate.The initial water depth in June is shown in Figure 6.Overall, the water depth in the western and eastern regions was relatively shallow compared to the middle region.The average water depth was 1.4 m, and the deepest area was 2.14 m.
In order to discuss the hydrodynamic characteristics of the lake, the present study simulated the flow field with time from one day to five days.Figure 7 presents the flow field in Baihe Lake after five days for vegetation density c = 0.2.The western region of the lake is far away from the inlet, resulting in a very small flow velocity.The average velocity is about 0.0016 m/s, with an average water depth of 1.32 m.The middle region of the lake is the main area for local fishery.It has sparse vegetation and a larger water depth.The average velocity is about 0.36 cm/s, with an average water depth of 1.89 m.A large amount of vegetation is distributed in the southeast region of the lake.The velocity of the flow is small except for the areas near the outlet or the inlet where the velocities are about 26 cm/s and 0.063 m/s, respectively.The average velocity in the east area is about 0.32 cm/s.

Numerical Tests
A 2D hydrodynamic model was established using the LBM.Baihe Lake is covered by 340 × 161 grids with each lattice being 50 m × 50 m.The inflow velocity from the northeast inlet is given by 0.5 m/s, which is controlled by sluice gate.The initial water depth in June is shown in Figure 6.Overall, the water depth in the western and eastern regions was relatively shallow compared to the middle region.The average water depth was 1.4 m, and the deepest area was 2.14 m.In order to discuss the hydrodynamic characteristics of the lake, the present study simulated the flow field with time from one day to five days.Figure 7 presents the flow field in Baihe Lake after five Water 2019, 11, 155 8 of 13 days for vegetation density c = 0.2.The western region of the lake is far away from the inlet, resulting in a very small flow velocity.The average velocity is about 0.0016 m/s, with an average water depth of 1.32 m.The middle region of the lake is the main area for local fishery.It has sparse vegetation and a larger water depth.The average velocity is about 0.36 cm/s, with an average water depth of 1.89 m.A large amount of vegetation is distributed in the southeast region of the lake.The velocity of the flow is small except for the areas near the outlet or the inlet where the velocities are about 26 cm/s and 0.063 m/s, respectively.The average velocity in the east area is about 0.32 cm/s.

Wind Speed
The wind speed in the Momoge Wetland varies all year round.As a result, it is necessary to investigate the flow field under different wind conditions.The wind speed in the northeast direction was varied by 10%, 20%, 30%, −10%, −20%, and −30%.The flow field became steady after the running time reached five days.The simulation results revealed that the greater the wind speed, the greater the velocity.Wind speed fluctuated by 30%, the outflow velocity varied by 18.77%, and the average velocity of the whole lake varied by 22.16% (see Table 1).

Wind Speed
The wind speed in the Momoge Wetland varies all year round.As a result, it is necessary to investigate the flow field under different wind conditions.The wind speed in the northeast direction was varied by 10%, 20%, 30%, −10%, −20%, and −30%.The flow field became steady after the running time reached five days.The simulation results revealed that the greater the wind speed, the greater the velocity.Wind speed fluctuated by 30%, the outflow velocity varied by 18.77%, and the average velocity of the whole lake varied by 22.16% (see Table 1).

Scenario Simulation
Scenario simulations of different seasons were also run using data from July 2017 and September 2017.
In July 2017, the peak inflow discharge of the drainage from the surrounding farmland was 32 3 /s.The monthly rainfall was 184 mm; the average wind speed was 2.13 m/s in the northeast direction (132 • ); and the vegetation density was 0.26.When the simulation was stable and the result was steady, the outflow rate was 0.54 m/s.The change in velocity and depth is shown in Figure 8a.
In September 2017, the inflow discharge was 12.6 m 3 /s in the receding trough of the farmland drainage.The monthly rainfall was 38 mm; the average wind speed was 2.85 m/s in the northeast direction (110 • ); and the vegetation density was 0.12.The change in velocity and depth is shown in Figure 8b.

Discussion
In this paper, the LBM method is applied to Baihe Lake to reveal the temporal and spatial characteristics of water body.Overall, the water body of Baihe Lake suffered poor exchange, where the lowest flow rate based on simulation reached just 0.016 m/s.Simulations of different impact factors (e.g., wind speed, rainfall, vegetation density) were conducted here.The simulation results revealed that changes in inflow velocity, primarily as the result of water drainage from surrounding farmland, have the strongest effect.A 30% variation in the impact factors resulted in a 24.11% variation in the outflow rate at most.In addition, when the impact of wind speed was relatively small, the variation of the outflow rate reached just 18.87%.Relatedly, given the high distribution of aquatic vegetation in some areas of Baihe Lake, and that vegetation density varies with the season, the impact of vegetation on the flow field should be considered separately.The present model includes the change in vegetation density and investigates its effect on the flow fields.The simulation result revealed that vegetation density was negatively correlated with outflow rate.However, the influence of aquatic vegetation on flow distribution was relatively small, being only 9.59% when the vegetation

Discussion
In this paper, the LBM method is applied to Baihe Lake to reveal the temporal and spatial characteristics of water body.Overall, the water body of Baihe Lake suffered poor exchange, where the lowest flow rate based on simulation reached just 0.016 m/s.Simulations of different impact factors (e.g., wind speed, rainfall, vegetation density) were conducted here.The simulation results revealed that changes in inflow velocity, primarily as the result of water drainage from surrounding farmland, have the strongest effect.A 30% variation in the impact factors resulted in a 24.11% variation in the outflow rate at most.In addition, when the impact of wind speed was relatively small, the variation of the outflow rate reached just 18.87%.Relatedly, given the high distribution of aquatic vegetation in some areas of Baihe Lake, and that vegetation density varies with the season, the impact of vegetation on the flow field should be considered separately.The present model includes the change in vegetation density and investigates its effect on the flow fields.The simulation result revealed that vegetation density was negatively correlated with outflow rate.However, the influence of aquatic vegetation on flow distribution was relatively small, being only 9.59% when the vegetation density fluctuated 30%.On the whole, the outflow of Baihe Lake strongly depends on the drainage from upstream farmland, followed by the rainfall, and then by local wind speed.Aquatic vegetation can also have an impact.The inflow rate is the only factor that is easy to control.As a result, ecological protection and hydraulic engineering construction can be established upstream to control the water level and flow rate.
The monitored rainfall, wind speed, flowrate, and vegetation density data in different months was input into the present model in order to simulate and obtain the flow characteristics of Baihe Lake in July and September.Although the influence of wind speed and vegetation was relatively small in the single factor analysis, they can still have an impact on the flow as these two factors have potential uncertainty due to the change in climate.The monitored data revealed large drainage from the surrounding farmland in July, together with high rainfall.Consequently, based on the simulation results, the impact of wind speed and vegetation density was relatively small.In contrast, a reduction in rainfall with a similar farmland drainage happens in September.Importantly, this increases the influence of wind speed and vegetation density.The results indicated that the mean flow rate is much bigger in July than September, which also verifies the significance of different impact factors.Relatedly, the simulation results also demonstrate that the water exchange of Baihe Lake during the flood season from June to August is much better than it is in the dry season from September to November.Therefore, when developing an ecological restoration program, both the farmland drainage and the season should be emphasized.

Figure 7 .
Figure 7. Water depth and velocity vectors after 5 days.

Figure 7 .
Figure 7. Water depth and velocity vectors after 5 days.
A local equilibrium distribution function decides what flow equations are solved by means of the lattice Boltzmann equation.For 2D shallow water Equations (1) and (2), the local equilibrium

Table 1 .
Effect of wind speed on flow velocity.

Table 4 .
Effect of rainfall on flow velocity.