The Deformation and Shear Vortex Width of Flexible Vegetation Roots in an Artificial Floating Bed Channel

Abstract

As an ecological measure to deal with river water quality problems, artificial floating islands have been widely used all over the world, but the research on root deformation and shear vortex width under the action of artificial floating islands is rare. In this paper, the relationship between the deformation of vegetation roots and parameters of vegetation roots under different hydrodynamic conditions is experimentally studied. The results show that the Cauchy number ($C_a$) value gradually increases with the increase of velocity, and that the smaller the diameter is, the greater the $C_a$ value is. The value of the buoyancy number ($B$) will increase with the increase of root length and will decrease with the increase of root diameter. The corresponding deformation formula of flexible root systems under hydrodynamic conditions is obtained, and has high simulation accuracy. Based on theoretical analysis and machine learning, a formula for the shear vortex width of flexible vegetation is established, $\delta =\frac{0.361+0.0738\frac{l_e}{l}}{\alpha C_{df}}$. The research results can provide a theoretical basis for hydrodynamic and solute transport in artificial floating island channels.

1. Introduction

As people pay more and more attention to river environments, the problem of river water pollution is a hot issue. Excessive pollutants such as nitrogen and phosphorus in the river will make the river eutrophic and affect the downstream water environment. Adding ecological measures into channels is one of the effective methods to solve river pollution [1,2]. An artificial floating island is one of the ecological measures used to effectively reduce river water pollution. They have been widely used all over the world because they do not need to occupy additional land, the floating bed can effectively slow down the velocity of water, and the vegetation on the floating bed can effectively absorb pollutants in the river [3,4].

Practice has proven that artificial floating islands can effectively purify water bodies and significantly improve the self-healing ability of rivers and lakes [5]. Subsequently, artificial floating islands were also introduced into farmland drainage channels, and achieved good results. Due to the fact that the artificial floating island can rise and fall with the change of water level in the channel, it provides favorable conditions for the artificial floating island to deal with the unstable flow in the drainage ditch [6]. At the same time, the artificial floating island system is cheap and easy to maintain [7]. A large number of studies on the water purification effect of different kinds of artificial floating islands have been carried out both in the laboratory and field, which provides a theoretical basis for wastewater treatment [8,9,10,11]. Moreover, artificial floating islands have been widely used in Europe [12], Japan [13,14], the United States [15], Australia [16], and China [17].

The interface between vegetation and water flow will create an obvious shear vortex. Ai et al. [18] studied the hydrodynamic conditions of river channels under the action of artificial floating islands with rigid vegetation roots based on the finite volume method, and described the generation of shear vortices in artificial floating islands with rigid vegetation. Zong and Nepf [19] gave the boundary width between the shear vortex and the root area in the submerged rigid vegetation channel. The formula proved that the shear vortex width $\delta$ and drag force coefficient $C_D$ are directly related to the vegetation characteristic diameter $d$:

$$\delta =0.5/\left(C_D\alpha \right)$$

(1)

where $\alpha =nd$, $d$ is the characteristic diameter of the vegetation, and $n$ is the amount of vegetation per unit area.

Artificial floating islands are a common method of water pollution treatment, and their roots often have a great impact on hydrodynamic forces. The change of hydrodynamic forces will affect the migration law of pollutants. In previous years, there have been many studies on the hydrodynamics of vegetated channels [20,21,22,23,24]. Bai and Duan [25] gave the turbulent diffusion coefficient forms of different velocity layers according to the stratification of flow velocity in the glacial channel. Nepf et al. [26] found that the formation of shear vortices in submerged rigid vegetation channels would affect the distribution of the turbulent diffusion coefficient, and combined with experiments to give the formula of turbulent diffusion coefficient in vegetation areas. Liu et al. [27] investigated the diffusion law of pollutants in a river under the action of artificial floating islands with rigid vegetation roots, and recommended the arrangement of artificial floating islands in series when assuming that the absorption of pollutants in the vegetation area conforms to the first-order dynamic adsorption model.

However, different from rigid vegetation, flexible vegetation should also consider its deformation under different hydrodynamic conditions and effective lengths of flexible root systems (Figure 1). This is also a necessary factor affecting the position of shear vortex. There have been some studies on the hydrodynamics of artificial floating islands, but the traditional research mainly focuses on the hydrodynamics of artificial floating island channels, which are biased towards rigid vegetation roots, and the deformation of the root system needs to be considered for the flexible root system. At the same time, the shear vortex width needs to be considered in a study of the migration of pollutants in the vegetation channel, and the variation law of the shear vortex width under the action of the flexible root system of artificial floating islands has not been given yet. The existence of these problems has seriously affected the theoretical research and promotion of artificial floating islands.

In response to the above problems, this paper, based on the experiment of the water tank in the laboratory, takes the flexible root system of the artificial floating island as the research object, analyzes the deformation of vegetation under different hydrodynamic conditions, and explores the parameters that affect the deformation of vegetation; through theoretical analysis and machine learning, the paper establishes the shear relationship between vortex width and root deformation, root drag force, and root density. It provides a theoretical basis for research on the influence of artificial floating islands on canal hydrodynamics and hydrodynamics.

2. Study Methods

2.1. Experimental Study

The experiments were performed in a long circulating flume 10 m long, 0.4 m wide, and 0.5 m deep. There was a tailgate at end of the flume that could adjust the water depth (Figure 2). The vegetation root system in artificial floating islands only needs to absorb nutrients without keeping shape as stem [28,29], and the elastic modulus is relatively small, which often produces a large deformation with the flow of water [30]. According to previous experimental experience [31,32], the flexible silicone foam cylindrical strip with elastic modulus $E=560\mathrm{kPa}$ was used to simulate the flexible vegetation root. Flexible cylinders with different diameters, lengths, and discharges were used in the experiment (

Table 1. Experimental parameters in different treatments.

Treatments	$\mathit{l}$ (cm)	Density (1/m2)	Discharge (L/s)	Slope	$\mathit{h}$ (cm)	${\mathit{u}}_{\mathit{m}}$ (cm/s)	$\mathit{d}$ (mm)
1	5	1600	8.17	0.0005	18.48	11.05	4
2	5	1600	11.88	0.0005	19.53	15.21	4
3	5	1600	15.68	0.0005	20.65	18.98	4
4	5	1600	18.60	0.0005	21.14	22.00	4
5	5	1600	22.91	0.0005	22.25	25.75	4
6	10	1600	8.22	0.0005	19.29	10.65	4
7	10	1600	11.99	0.0005	20.41	14.69	4
8	10	1600	15.82	0.0005	21.59	18.31	4
9	10	1600	18.62	0.0005	22.11	21.06	4
10	10	1600	22.51	0.0005	23.24	24.22	4
11	15	1600	8.38	0.0005	20.23	10.36	4
12	15	1600	11.77	0.0005	21.47	13.70	4
13	15	1600	15.98	0.0005	22.57	17.70	4
14	15	1600	18.73	0.0005	23.07	20.30	4
15	15	1600	22.82	0.0005	24.26	23.51	4
16	5	1600	8.30	0.0005	19.04	10.89	6
17	5	1600	11.77	0.0005	20.12	14.63	6
18	5	1600	15.63	0.0005	21.19	18.43	6
19	5	1600	18.74	0.0005	21.71	21.58	6
20	5	1600	22.66	0.0005	22.79	24.85	6
21	10	1600	8.48	0.0005	19.93	10.64	6
22	10	1600	11.96	0.0005	21.04	14.21	6
23	10	1600	15.39	0.0005	22.18	17.35	6
24	10	1600	18.80	0.0005	22.75	20.66	6
25	10	1600	22.33	0.0005	23.87	23.39	6
26	15	1600	8.39	0.0005	20.77	10.09	6
27	15	1600	11.61	0.0005	22.01	13.18	6
28	15	1600	15.44	0.0005	23.07	16.73	6
29	15	1600	19.00	0.0005	23.73	20.02	6
30	15	1600	22.24	0.0005	24.86	22.36	6
31	5	1600	8.38	0.0005	19.95	10.50	8
32	5	1600	11.70	0.0005	21.08	13.88	8
33	5	1600	15.48	0.0005	22.21	17.43	8
34	5	1600	18.57	0.0005	22.73	20.42	8
35	5	1600	22.52	0.0005	23.91	23.55	8
36	10	1600	8.49	0.0005	20.79	10.22	8
37	10	1600	11.87	0.0005	21.97	13.51	8
38	10	1600	15.58	0.0005	23.10	16.87	8
39	10	1600	18.68	0.0005	23.81	19.62	8
40	10	1600	22.18	0.0005	24.92	22.25	8
41	15	1600	8.30	0.0005	21.67	9.58	8
42	15	1600	11.57	0.0005	22.84	12.66	8
43	15	1600	15.74	0.0005	24.17	16.29	8
44	15	1600	18.60	0.0005	24.63	18.87	8
45	15	1600	22.72	0.0005	25.92	21.92	8

Note: $l$ is the length of the vegetation; $u_m$ is the average velocity; $h$ is the water depth; $d$ is the diameter of the vegetation.

). The plants were in a fixed-row arrangement in the artificial floating bed (Figure 3), and all experiments were performed under the steady uniform flow conditions. The deflection height of the root system is measured by a scale, and the area with stable reading is selected. The scale was in the water tank, and the water tank was covered with scale paper. The root deformation can be obtained by the alignment size of the scale paper on both sides (Figure 2). To characterize the flow field, three velocity components were measured simultaneously using ADV, with a sampling volume of 6 mm wide and 3 mm high. Longitudinal cross-sections starting 2 m upstream of the patch and extending to the end of the patch were made. The support structure across the trough prevented the carriage supporting the ADV from traveling further downstream. The vegetation density often ranges from 97 to 2857 m⁻², according to the previous literatures [33,34,35,36,37,38,39,40], and the vegetation density we adopted was 1600 m⁻².

2.2. The Effective Blade Length

The deflection height of the emergent flexible vegetation with uniform diameter in water can be solved by Luhar and Nepf [41].

$$\frac{l_e}{l}=1-\frac{1-0.9C_a^{-1/3}}{1+C_a^{-3/2}\left(8+B^{3/2}\right)}$$

(2)

where $C_a$ is the Cauchy number, which represents the ratio between resistance and vegetation stiffness-force, and $B$ is the buoyancy number, which represents the ratio between buoyancy and stiffness-force.

A schematic showing the coordinate system and force balance used to derive the mathematical model for the flow induced reconfiguration of root is shown in Figure 4. The Equation (3) was obtained by force-analyzing the flexible cylinder. The first term on the left of the equation is the blade-normal restoring force generated by the blade-normal restoring force due to stiffness, and ($V^{*}$) is the spatial derivative of the internal bending moment; the second item is the vertical buoyancy force; the drag force per unit blade length is on the right.

$$V^{*}\left(s^{*}\right)+{{\displaystyle \int }}_{s^{*}}^l{f}_{B}sin{\theta }^{*}\left(s^{*}\right)ds=\frac{1}{2}\rho C_{d}d{\overline{u}}^2{{\displaystyle \int }}_{s^{*}}^l\cos \left(\theta \left(s\right)-{\theta }^{*}\left(s^{*}\right)\right)f_D\left(\theta \left(s\right)\right)ds$$

(3)

The following equation can be obtained by assuming that the blades can be modeled as isolated, buoyant, inextensible elastic beams of constant diameter, density, and elastic modulus. The horizontal velocity is uniform over depth. The dominant hydrodynamic force is form drag.

$$\left\{\begin{array}{c}V=-EI\frac{d^2\theta }{d{s}^2}\\ f_D=\frac{1}{2}\rho C_{df}d{u_m}^{2}co{s}^2\theta \\ f_B=\left(\rho -{\rho }_v\right)g\pi d^2/4\end{array}\right.$$

(4)

where $d$ is the diameter of the root, ${\rho }_v$ is the vegetation density, $\rho$ is the water density, $EI$ is the flexural rigidity, and $I$ ($\pi d^4/64$) is the second moment of the area [42,43]. Based on this definition, the drag coefficient, $C_{df}$, for the flexible blades is identical to that for rigid, vertical blades.

By bringing Equation (4) into Equation (3)

$$-EI{\left.\frac{d^2\theta }{d{s}^2}\right|}_{s^{*}}+\left(\rho -{\rho }_v\right)g\pi d^2/4\left(l-s^{*}\right)sin{\theta }^{*}=\frac{1}{2}\rho C_{df}d{u_m}^2{{\displaystyle \int }}_{s^{*}}^l\cos \left(\theta -{\theta }^{*}\right)co{s}^2\theta ds$$

(5)

where $B$ and $C_a$ can be brought into Equation (5)

$$-{\left.\frac{d^2\theta }{d{s}^2}\right|}_{s^{*}}+B\left(l-s^{*}\right)sin{\theta }^{*}=C_a{{\displaystyle \int }}_{s^{*}}^l\cos \left(\theta -{\theta }^{*}\right)co{s}^2\theta ds$$

(6)

$$C_a=\frac{\rho C_{df}d{u_m}^2}{2EI}$$

(7)

$$B=\frac{\left(\rho -{\rho }_v\right)g{l}^3\pi d^2/4}{EI}$$

(8)

By introducing Equations (7) and (8) into Equation (2), the deformation of the flexible cylinder can be obtained.

2.3. The Formula of $\delta$

$\delta$ is a parameter correlated with the drag coefficient of rigid vegetation ($C_d$) and $\alpha$ in a rigid vegetated river [19], and it also related with deformation when in flexible a vegetation channel. $\delta$ can be established as follows:

$$\delta =f(C_{df}, \alpha , \frac{l_e}{l})$$

(9)

The drag coefficient of flexible vegetation is related to the drag coefficient of rigid vegetation and flow velocity, which can be deduced from the following formula [44]

$$C_{df}=C_d{\left(u_m/u_b\right)}^b$$

(10)

where $u_b$ and $b$ denote a reference velocity and a reference parameter, the value of $u_b$ and $b$ were chosen according to Bai and Xuan [30], $u_b=10$ cm/s, and $b=-0.33$. $C_d$ is the drag coefficient of the rigid vegetation, and $C_d=1$ when the Reynolds number ranges from 800 to 8000 [45].

To determine the form of the unknown $\delta$ equation, Eureqa [27], a method that can be effectively applied to the prediction of the delta equation, should be used. The prediction formula was built by Eureqa, a computer program that finds correlations between data (https://www.nutonian.com/download/eureqa-onprem-download/, accessed on 15 April 2021). Meanwhile, the data was divided into three groups: 40% of the data was used for training, 30% was used for validation, and 30% was used for testing. The choice of the formula should take into account the accuracy and complexity of the formula.

2.4. Error Analysis

Error analysis was conducted to determine the difference between the predicted and measured data. The mean absolute error (MAE), root mean square error (RMSE) and coefficient of determination (R²) were calculated by the following equations:

$$R^2=1-\frac{SSE}{SST},$$

(11)

$$SST={{\displaystyle \sum }}_{i=1}^N{\left(Y_i-meanY\right)}^2,$$

(12)

$$SSE={{\displaystyle \sum }}_{i=1}^N{\left(Y_i-X_i\right)}^2,$$

(13)

$$meanY=\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N{Y}_i,$$

(14)

$$MAE=\frac{{{\displaystyle \sum }}_1^N\left|Y_i-X_i\right|}{N},$$

(15)

$$RMSE=\sqrt{\frac{{{\displaystyle \sum }}_1^N{\left(X_i-Y_i\right)}^2}{N}},$$

(16)

where $N$ is the number of lateral measuring points and X and Y are the calculated and measured values.

3. Results

3.1. The Deformation of the Flexible Vegetation

The relationship between measured $\frac{l_e}{l}$ and flow velocity is shown in the Figure 5. When $l$ is 5 cm with diameter of 6 mm and 8 mm, there is almost no deformation under different velocities. At the initial stage of root growth, the root length is too short and is not easy to be affected by hydrodynamic forces. With the decrease of root diameter and the increase of root length, the root deformation with velocity becomes larger and larger. The maximum root length and diameter of deformation is a length of 15 cm and a diameter of 4 mm, and $\frac{l_e}{l}$ is ranged from 0.26 to 0.56.

The value of $C_a$ gradually increases with the increase of the discharge, and the smaller the diameter is, the greater $C_a$ is (Figure 6). This is because with the increase of flow velocity, the resistance of the vegetation also increases significantly, and the greater $C_a$ will be. The maximum is a diameter of 4 mm and a length of 15 cm, and its value ranged from 10 to 53. This is because the longer the vegetation, the greater the resistance, and the smaller the diameter of the vegetation, and the smaller the stiffness of the vegetation.

The value of $B$ is not affected by the velocity, but is only related to the root length and root diameter (Figure 7). The value of $B$ will increase with the increase of root length. The longer the vegetation, the greater the buoyancy of the vegetation is. The value of $B$ will decrease with the increase of root diameter. The larger the diameter, the greater the stiffness of vegetation is. The increase of stiffness caused by the increase of diameter is greater than the increase of buoyancy caused by the increase of diameter.

As shown in the Figure 8, the $\frac{l_e}{l}$ value calculated by the formula is very close to the actually measured $\frac{l_e}{l}$ value, with a $RMSE$ of 0.0307 and a $R^2$ of 0.9828. All points in the figure are evenly distributed around the straight line, and this indicates that the calculation effect of the formula is good.

3.2. The Vortex Width of the Flexible Vegetated Channel

The default value is because it is not measured, such as the condition with a diameter of 4 mm and a length of 5 cm. Three values can be measured at the condition with a diameter of 4 mm and a length of 10 cm. The overall trend of shear vortex width is to increase with increasing flow velocity, and the shear vortex width ranged from 2 cm to 6 cm (Figure 9).

In the formula interpretation, the precision and the complexity should be considered. At the complexity higher than 10, the formula was extremely complex, and the precision increased slowly (Figure 10). Accordingly, the present study took the formula with a complexity of 10, and the solution results are listed in

Table 2. The relationship between complexity and solution.

Complexity	MAE	Solution
1	0.00781	$\delta =0.0333$
4	0.00338	$\delta =\frac{0.511}{\alpha }$
6	0.00218	$\delta =\frac{0.426}{\alpha C_{df}}$
8	0.00200	$\delta =\frac{0.513}{2.45+\alpha C_{df}}$
10	0.00180	$\delta =\frac{0.361+0.0738\frac{l_e}{l}}{\alpha C_{df}}$
12	0.001795	$\delta =\frac{0.383+0.0741\frac{l_e}{l}}{\alpha C_{df}}-0.00142$
15	0.00177	$\delta =\frac{\frac{l_e}{l}\cos \left(C_{df}\right)}{0.632+1.22\alpha \frac{l_e}{l}}$
17	0.00173	$\delta =\frac{0.937\frac{l_e}{l}\cos \left(C_{df}\right)}{C_{df}+\alpha \frac{l_e}{l}}-0.00319$
19	0.00173	$\delta =\frac{0.942\frac{l_e}{l}\cos \left(C_{df}\right)}{C_{df}+1.01\alpha \frac{l_e}{l}}-0.00314$
25	0.00171	$\delta =0.152+0.00282\alpha \frac{l_e}{l}{C}_{df}-0.00425\alpha -0.0834C_{df}-0.0186 {\frac{l_e}{l}}^3$

. The testing and all data of simulated and measured $F_{dx}$ can be seen from Figure 11, and the simulation effect is quite good. In the process of testing, the R² was 0.934 and the MAE was 0.0018. In all data, the accuracy slightly decreased, R² was 0.926 and MAE reached 0.00191 (Figure 11). If R² can reach 0.6, it is considered acceptable. If R² reaches 0.9, it is considered that the simulation effect is very good. If MAE is less than 0.2, it is considered that the simulation result is good [30,46,47].

4. Discussion

Root diameter is one of the key factors that determine the size of the root drag force. With the increase of diameter, the size of the drag force increases significantly [48,49], and the same is true for flexible vegetation roots in this paper. Due to the fact that the increase of diameter increases the stress area of the vegetation, the deflection of vegetation is also significantly smaller [31,42,50]. The diameters and lengths of the vegetation roots need to be considered at the same time, because in the process of vegetation growth, the diameter of the roots changes from thick to crude, and the length of roots changes from short to long. These root changes should be considered in the impact of artificial floating islands on hydrodynamic. For the purification effect of water quality on roots, the model is an effective means. Liu et al. [27] assumed that the absorption of pollutants by artificial floating islands with rigid vegetation roots conforms to the first-order dynamic equation and supports the arrangement of crossed artificial floating beds. However, the adsorption of pollutants by artificial floating islands is not only linear, but also gradually increases with the growth of vegetation roots, which is also a problem that we need to consider. On the other hand, the diameter distribution of plant roots in nature is uneven.

The width of the shear vortex is a formula related to diameter and density [19]. However, the flexible root system will produce deformation, which will affect the formation of a shear nest to a certain extent. It is necessary to add the factor of flexible vegetation deformation into the formula. In the previous rigid vegetation study, the shear vortex width did not vary with flow velocity [19]. However, the shear vortex varies with the flow velocity in the flexible root channel, which may be because the drag coefficient of the flexible vegetation is related to the flow velocity [30,51]. When the flow velocity increases, the drag force increases, while the width of the shear vortex decreases.

The arrangement pattern of the elements can significantly change the drag on an element [52]. In this paper, only the parallel pattern of vegetation has been investigated, and for the staggered pattern the drag coefficient will be different from that of the parallel one, as discussed by Zhang et al. [53]. The drag coefficient of random vegetation forms which are closer to the natural distribution of vegetation was been studied by Tanino and Nepf [54].

The shear vortex formula of flexible vegetation is obtained as Equation (17), where the width of a shear vortex obviously decreases with the increase of drag force and vegetation density, and the formula form is basically the same as that of rigid vegetation shear vortex [55]. The previous studies are consistent. At the same time, in the flexible vegetation, the factors of vegetation deformation also need to be considered. When the vegetation deformation is larger ($\frac{l_e}{l}$ is smaller), the width of the shear vortex is also smaller.

$$\delta =\frac{0.361+0.0738\frac{l_e}{l}}{\alpha C_{df}}$$

(17)

For the uneven root diameter, we can try to find out the characteristic diameter and characteristic elastic modulus. Meanwhile, submerged rigid cylindrical vegetation will produce a shear vortex between the upper layer and the water flow. Previous studies on the distribution of shear vortices in horizontal partially vegetated channels show that there are more, while the distribution of vertical shear pits in relatively flexible floating island root channels is relatively less [56,57,58]. The distribution of shear dimples is conducive to the study of water dynamics and solute transport in channels. Turbulent diffusion caused by shear dimples and mechanical diffusion caused by flexible vegetation also need to be studied. This is the focus of our research in the next stage. At the same time, more outdoor experiments should be carried out to verify our research results.

The artificial floating island vegetation in different regions may be different, as the climatic conditions vary in different regions [59,60,61]. Therefore, if we want to popularize our test results to other regions, we need to test materials with similar characteristics to the artificial floating island vegetation in this region. With the deflection-height formula of the root zone of the floating island vegetation, we can more clearly determine the impact of the artificial floating island on the hydrodynamic force of the river channel, judge its impact on flood discharge or the interception of pollutants by vegetation, and effectively provide a theoretical reference for the selection of artificial floating islands and artificial floating island vegetation.

5. Conclusions

Based on the flume experimental data and theoretical analysis, the deformation law of flexible vegetation under hydrodynamic conditions was studied. The main conclusions are as follows:

(1)

The $C_a$ and $B$ values of the roots of the artificial floating island’s flexible vegetation under hydrodynamic conditions were obtained. The value of $C_a$ gradually increases with the increase of velocity, and the smaller the diameter is, the greater the $C_a$ value of the root with a longer length. The value of $B$ will increase with the increase of root length and will decrease with the increase of root diameter.

(2)

According to the stress analysis, the simulation formula of flexible root deformation is given. The simulated deformation is in good agreement with the actual deformation, which can effectively provide support for the application-design of artificial floating islands.

(3)

Based on the theoretical analysis and machine learning, a formula for the shear vortex width of flexible vegetation is established, $\delta =\frac{0.361+0.0738\frac{l_e}{l}}{\alpha C_{df}}$. The formula simulation has a high precision; R² was 0.926 and MAE reached 0.00191.