Introduction

In the river ecosystem, the influence of vegetation cannot be neglected. Vegetation has a stabilizing effect on riverbeds, defends the hirsts and dikes, and protects and rests ecological environment, but at the same time, vegetation can increase the roughness of banks and change the flow regime, thus affecting the flood diversion capacity of the river (Carroll et al. 1997; Cerdà 1997; Fattet et al. 2011; Zhao et al. 2017; Liu et al. 2018). Aquatic vegetation is prone to lodging under the flow of water, and there are many factors that induce the lodging of vegetation. In addition to intrinsic factors, such as the flexibility and overall structure of the vegetation, the slope is also one of the most important factors affecting the lodging of vegetation. Studying the relationship between slope and lodged vegetation roughness not only provides theoretical support for flood control, but also has practical implications in river ecological environment management.

While there have been many studies on the vegetation roughness on slopes (e.g., Abrahams and Parsons 1994; Atkinson et al. 2000), there have been few on vegetation lodging. Among them, Ferro et al. (2005) showed that vegetation flow resistance decreases with an increase in slope, although the relationship is complex, and cannot be expressed by a simple function. Han et al. (2016) examined the non-uniform distribution of flexible, submerged vegetation in a rectangular channel and concluded that the mean velocity decreased with increasing flow resistance. Meanwhile, Velasco et al. (2003) used simulated plastic plants instead of real plants in flume experiments, and the relationship between the deflected height of flexible plants and the velocity field was measured. They found that plant roughness correlated directly with the lodging deformation of plants. Busari and Li (2015) estimated the uncertainty of a hydraulic roughness model of submerged flexible vegetation, and suggested that the hydraulic resistance produced by submerged flexible vegetation depends on many factors, including plant stem size, height, number, and density, as well as flow depth.

The research on the flow characteristics of vegetation accounted for a high percentage in the past (Cerdà 1997; Järvelä 2002; Yagci et al. 2010; Guo et al. 2016). According to vegetation characteristics, it can be divided into coverage area, flexibility, diameter, and leaf number on the basis of the prevailing research (Wilson et al. 2003; Kothyari et al. 2009; Hu et al. 2012). At present, the studies on the effects of slope on the hydrodynamic characteristics of overland runoff are becoming more advanced. However, studies on the hydrodynamic characteristics of lodged vegetation remain limited (Ferro et al. (2005)), especially with respect to the effect of slope on the flow roughness of lodged vegetation. Therefore, it is necessary to experimentally investigate the effects of changes in slope on the surface roughness of lodged vegetation. This provides a theoretical basis for further exploring the river flow structure and movement characteristics, and has practical significance for river ecological restoration and flood control.

Experimental setup

According to previous studies, it is necessary to perform open channel flow simulation experiments (e.g., through indoor simulations), and the data processing and theoretical research on the experiments should be performed using the formulae and theory of open channel flows. Furthermore, there are many factors that affect the flow resistance of vegetation. To clearly study changes in flow resistance under different lodging states, it was necessary to simplify the simulations in this study. Before formal testing, a preliminary experiment was performed to select the experimental materials and to determine the slope and lodging angle. In the indoor open-channel flow simulation, a plexiglass plate was positioned on the bottom of the instrument as the reference plane, and the angle of the vegetation from the vertical direction of the reference plane was used as the lodging angle. In addition, a cylindrical aluminum column (Hsieh 1964; Huthoff et al. 2007; Luo et al. 2009; Yagci et al. 2010; Zhu et al. 2018) with a diameter of 4 mm and a fixed height of 10 cm was used to simulate natural vegetation. Seven classes of slope (indicated by i, where i = 0%, 0.5%, 1.0%, 1.5%, 2.0%, 2.5%, and 3.0%), and four categories of lodging angles (indicated by θ, where θ = 20°, 40°, 60°, and 80°) were also used to perform the experiment.

Due to the large volume of water used in the test, and to conserve water resources, a device for recirculating water flow within the closed system was used. The device consisted of an open-channel flume with a rectangular section, a water tank, water pump, and a tailgate. During simulations, water flow could be pumped from the water tank into the open-channel flume, and then returned to the water tank through the test section to recycle the water (Fig. 1). The rectangular flume was 5 m long, 0.4 m wide at the bottom, and the side walls were 0.3 m in height. A plexiglass plate was placed on the bottom, and the surface was drilled with small holes at a longitudinal and lateral spacing of 60 mm × 60 mm for the placement of simulated vegetation. The flume was divided into three sections: an upper equalizing section (1 m in length), a middle test section (3 m in length), and a tail-gate section (1 m in length). In the experimental section, two cross sections, 1 and 2, were put in place with a separation distance of 1.5 m, and both of them were equipped with piezometer tubes to observe water level. A steel beam was placed below the flume to adjust the slope, and the range in slope varied between 0 and 3%. A flow control valve was positioned at the connection point between the water tank and the open-channel flume, and the flow rate varied from 0 to 0.0125 m3/s.

Fig. 1
figure 1

Experimental setup for the monitoring of the effect of vegetation lodging

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Theory and data

The roughness coefficient is one of the most important hydrodynamic parameters to understand as it indicates the roughness of the surface and the obstructive effects of vegetation on the flow of water (Barros and Colello 2001; Wang et al. 2014; Zhang et al. 2018). The primary means of expressing the roughness coefficient are the Manning, Darcy-Weisbach, and Chezy flow resistance equations (Rouhipour et al. 1999; Hogarth et al. 2005; Smith et al. 2007). Moreover, according to the experimental data processing, the minimum Reynolds number (Re; Eq. 1) was ~ 1400, which is much larger than the critical value of 500, meaning that the flow was in a turbulent state throughout the experiment. Therefore, the Manning’s roughness coefficient (n; Eq. 2) was considered to be the most accurate parameter:

$$Re=\frac{vR}{\upsilon },$$
(1)

where \(v\) is the mean velocity (m/s) between cross sections 1 and 2, R is the hydraulic radius (m), \(\upsilon\) is kinematic viscosity (m2/s), and

$$n=\frac{1}{v}{R}^{2/3}{J}^{1/2},$$
(2)

where J is the hydraulic gradient (dimensionless), n is Manning’s roughness coefficient (s/m1/3) (Smith et al. 2007).

In the process of calculating the roughness coefficient, both the hydraulic radius and the hydraulic gradient are important parameters that affect the results. The hydraulic radius is the ratio of the area of flow passing through a water section to the boundary line (i.e., wet cycle) of the contact between the fluid and the solid wall (Eq. 3; Querner 1997; Cheng and Nguyen 2011; Vatankhah et al. 2015). Meanwhile, the hydraulic gradient is the head loss per unit distance along the water flow path (Eq. 4; Zheng et al. 2000; Heuperman 2007; Nouwakpo et al. 2010), such that

$$R=\frac{A}{\chi },$$
(3)

where A is the cross-sectional area of water flow (m2) and \(\chi\) is the wetted perimeter (m):

$$J=\frac{{h}_{f}}{l},$$
(4)

where \({h}_{f}\) is the frictional head loss (m) and \(l\) is the length of water along the course (m).

During the experiment, we measured the pressure with piezometer tubes in Sects. 1 and 2, and recorded the flow depths and flow velocities as h1, h2, \({v}_{1}\), and \({v}_{2}\), respectively. The flow depth (hc), current velocity (\(v\)), and the hydraulic radius (R) were calculated using the mean values of cross sections 1 and 2 (i.e., hc = (h1 + h2)/2; = \(v\)(\({v}_{1}\)+ \({v}_{2}\))/2; R = (R1 + R2)/2). The formulae for calculating the current velocities of each cross section are shown in Eq. 5:

$$v_{1} = \frac{Q}{{Bh_{1} }};\,v_{2} = \frac{Q}{{Bh_{2} }}$$
(5)

where \({v}_{1}\) is the current velocity and h1 is the flow depth for cross section 1, \({v}_{2}\) is the current velocity and h2 is the flow depth for cross section 2, B is the channel width (m), and Q is the flow rate (m3/s).

Four categories of lodging angles (20°, 40°, 60°, and 80°) were used in the experiment, and 7 classes of slopes were assigned to each angle (where i = 0% indicates horizontality; i = 0.5% and 1.0% indicate a shallow slope; i = 1.5% and 2.0% indicate a medium slope; i = 2.5% and 3.0% indicate a steep slope). During the experiment, the flow rate (Q) and the water depth (hc) corresponding to different slopes, (i), and different lodging angles, (θ), were measured, and then the corresponding Manning’s roughness coefficient (n)was calculated using Eq. 2; the results are shown in Table 1.

Table 1 Experimental data under different slopes (i = 0%, 0.5%, 1.0%, 1.5%, 2.0%, 2.5%, and 3.0%) and lodging angles (θ = 20°, 40°, 60°, and 80°)
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Results and discussion

The relationships between the Manning's roughness coefficient (n) and water depth (h) calculated under different experimental slopes are shown in (Fig. 2). Figure 2a shows the n–h relationship for a horizontal state (i.e., with i = 0%). When the vegetation is at the same lodging angle, the Manning’s roughness coefficient (n) increases gradually as the water depth (h) increases, before gradually decreasing. The reason for this behavior may be that with increasing water depth, the degree of submergence of the vegetation increases, causing the area of water blockage to increases. Under these circumstances, the Manning's roughness coefficient (n) exhibits an increasing trend. When the vegetation is completely submerged, as the water depth increases and the water blocking area does not change, the resistance generated by the vegetation does not change, but the water depth continues to increase. Compared to the unsubmerged state, the Manning's roughness coefficient (n) shows a decreasing trend. Under the same water depth, as the lodging angle (θ) increases, the Manning's roughness coefficient (n) gradually decreases in the order of: n20° > n40° > n60° > n80°. This may be because with an increasing degree of lodging, the vertical projection of the vegetation, which has a fixed height, decreases gradually, and the area of water blockage decreases accordingly; thus, the Manning's roughness coefficient (n) exhibits a decreasing trend.

Fig. 2
figure 2

Relationships between Manning's roughness coefficient (n) and flow depth (h) under different lodging angles and slopes (ai = 0.0%; bi = 0.5%; ci = 1.0%; di = 1.5%; ei = 2.0%; fi = 2.5%; gi = 3.0%), the hollow points stand for submersed and the filled points for unsubmersed vegetation

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Figure 2b, c shows the relationships of n–h under shallow slope conditions (i = 0.5% and 1.0%). It can be seen from the figures that, compared with the horizontal state, the n–h curves at shallow water depths and for shallow slopes just begin to converge with one other, and the n–h curves at greater water depths do not change significantly. From this pattern, we infer that shallow slopes can only affect the size of the Manning's roughness coefficient (n) for lodged vegetation under shallow water depths. Figure 2d, e shows the relationship of n–h for medium slopes (i = 1.5% and 2.0%). Compared to the horizontal and shallow slope states, the n–h curves with medium slopes are obviously closer; the n–h curves at shallow water depths remain converged, and the n–h curves at greater water depths begin to converge. Finally, Fig. 2f, g shows the relationship of n–h with steep slopes (i = 2.5% and 3.0%). It can be seen from these figures that whether at deep or shallow water depths, the n–h curves almost completely converge for all lodging angles, especially for the steepest slope (i = 3%), as shown in Fig. 2g.

The general trends shown in Fig. 2 are that of convergence in the relationship of n–h as the slope increases, and that the water depth that can be achieved under the same flow conditions decreases with increasing slope. The reason for this phenomenon may be that the Manning's roughness coefficient (n) of vegetation was mainly controlled by three factors during the experiment: the lodging angle (θ), slope (i), and water depth (h). In the horizontal state, the Manning's roughness coefficient (n) of the vegetation is not affected by the slope, and the lodging angle is the main controlling factor. As the slope gradually increases, the influence on Manning's roughness coefficient (n) increases, and gradually exhibits a greater influence than the lodging angle (θ) until the slope (i) becomes the dominant factor (Fig. 2g). In the process of the slope effect increasing and the influence of the lodging angle decreasing, water depth is an important criterion. Under the conditions of a shallow slope, only the hydraulic characteristics under shallow water depths are affected by the slope, and with an increase in slope, the affected water depth increases gradually.

Therefore, the lodging angle is fixed at 20° to observe the relationship between Manning's roughness coefficient (n) and water depth (h) on different slopes, as shown in Fig. 3. It can be seen from the figure that with increasing water depth, the Manning’s roughness coefficient (n) for different slopes generally increases. This is because for unsubmerged vegetation, with increasing water depth, the degree in vegetation submergence increases, and the area of water blockage also increases, thus increasing the Manning's roughness coefficient (n). In addition, under the same water depth, the Manning's roughness coefficient (n) is positively correlated with slope at shallow water depths (0 < h < 0.05 m), but negatively correlated with slope at greater water depths (0.05 < h < 0.11 m). Therefore, through comparative study, it can be concluded that the flow resistance generated by vegetation is closely related to the slope, the lodging angle, and the water depth.

Fig. 3
figure 3

Relationship between Manning's roughness coefficient (n) and flow depth (h) for different slopes when the lodging angle is 20°, the hollow points stand for submersed and the filled points for unsubmersed vegetation

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Conclusions

Vegetation is one of the important components of river ecosystems. To further study the effect of vegetation roughness on water flow, open-channel flow simulation experiments were carried out. The following conclusions were drawn:

  1. 1.

    When i = 0%, the Manning’s roughness coefficient (n) increases gradually as the water depth (h) increases at the same lodging angle (θ), and then gradually decreases. Under the same water depth, the Manning's roughness coefficient (n) decreases gradually with the increase in lodging angle (θ), such that n20º > n40º > n60º > n80º.

  2. 2.

    By laterally comparing the relationships shown in n–h curves under different slopes (Fig. 2b–g), it can be concluded that as the slope increases, the n–h curves appear to converge, and the degree of convergence gradually increases. In addition, the water depth can be reached under the same discharge range decreases, and the effect of the slope gradient on the roughness coefficient of lodged vegetation increases gradually. This process is mainly controlled by three factors: the lodging angle, slope, and water depth.

  3. 3.

    By longitudinally comparing the n–h relationship at a fixed lodging angle (Fig. 3), it can be concluded that with increasing water depth, the Manning's roughness coefficient (n) generally increases when the lodging angle is 20°. Under the same water depth, the Manning's roughness coefficient (n) increases as the slope increases at shallow water depths, but decreases with increases in slope at greater water depths.

It should be noted that our conclusions were derived by controlling many factors. To simplify the study, uniform vegetation heights and stem diameters were used, and four representative lodging angles and seven slope classes were selected. Therefore, the conclusions of this study are representative, but the reliability and adaptability of their application warrant further exploration.