Effects of Vegetation Density on Sediment Transport in Lateral Cavities ^†

Abstract

In rivers and canals, lateral cavities are regions of low velocities and recirculation, which have ecological importance, such as sediment retention. The presence of vegetation in cavities has the potential to modify the flow and alter the retention of sediments inside the cavity. In this study, the impact of vegetation on hydrodynamics and sediment transport was investigated with a numerical model. The vegetation density was distributed from 0 to 10.65% in four cases. Sediment transport was investigated through the Rouse number, Hjulström, and Shields diagrams. The increase in vegetation density did not change the predominant sediment transport type. Furthermore, the increase in vegetation favoured the deposition of sediments in the lateral cavity.

1. Introduction

Lateral cavities are semi-enclosed volumes adjacent to the main channel flowing along its only open side, being found in a single cavity (as in the present study) or a series of cavities, occurring naturally or man-made. The presence of cavities in rivers (a) increases the lateral macro-roughness in rivers [1], (b) drives mass exchange processes with the open channel [2,3,4,5], (c) acts as transient storage zones [6,7], and (d) enhances biodiversity in the system [8,9,10].

Cavities are characterized by the presence of a recirculation system with slow velocity magnitudes. Inside the cavity, the flow velocity can become zero at the centre of the circulation and have magnitudes up to 60% of the main channel velocity [11]. These slow velocity profiles contribute to the settling of particles as the flow traction force is lower. The retention of fine sediments and, consequently, nutrients can constitute a favourable substrate for vegetation establishment and growth [12,13]. The presence of vegetation decreases the flow velocity and turbulence [11,14], increasing the functionality of lateral cavities by giving refuge and sustaining fish communities [15,16,17], trapping suspended material [18], and protecting from bank erosion [19].

Little is known about the influence of vegetation on the processes that happen inside a cavity, especially those related to sediment transport. For example, to the best of the authors’ knowledge, nobody has inferred from the hydrodynamics regions in the cavity that the possibility of deposition of sediment is higher. Thus, in this study, we investigated the effect of vegetation density inside a cavity on sediment transport.

This paper is organized into five main sections. Following the introduction, the details of the numerical model are described, along with the grid independence test and solution quality. Third, the main results about flow hydrodynamics and sediment transport are presented. Fourth, the results are discussed and, finally, conclusive remarks about the influence of vegetation on sediment transport inside lateral cavities are presented.

2. Materials and Methods

2.1. Model Equations

The simulations were performed in a computational fluid dynamics (CFD) environment. The approach used to solve the flow and turbulence was the large Eddy simulation (LES), which uses spatial filtering of the Navier–Stokes equations [20]. In this model, the fluid was considered incompressible, which simplifies the conservation equations of mass and momentum into the following:

$$\partial \overline{{\mathrm{u}}_{\mathrm{i}}}/\partial {\mathrm{x}}_{\mathrm{i}}=0$$

(1)

$$\frac{\partial \overline{{\mathrm{u}}_{\mathrm{i}}}}{\partial \mathrm{t}}+\frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}\left(\overline{{\mathrm{u}}_{\mathrm{i}}}\overline{{\mathrm{u}}_{\mathrm{j}}}\right)=\frac{-1}{\mathsf{\rho }}\frac{\partial \overline{\mathrm{p}}}{\partial {\mathrm{x}}_{\mathrm{i}}}+\frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}\left[\mathsf{\vartheta }\left({2\overline{\mathrm{S}}}_{\mathrm{ij}}\right){-\mathsf{\tau }}_{\mathrm{ij}}\right]+{\mathrm{S}}_{\mathrm{M},\mathrm{i}}$$

(2)

where the overbar indicates resolved quantities; i and j = 1, 2, 3 correspond to x, y, and z directions, respectively; u_i (m/s) is the velocity component in the i direction; $\mathsf{\rho }$ (kg/m³) is the fluid density; p (N/m²) is the dynamic pressure; ϑ (m²/s) is the kinematic viscosity; $S_{\mathrm{ij}}=0.5(\partial u_i/\partial x_j+\partial u_j/\partial x_i)$ (1/s) is the strain-rate tensor; ${\tau }_{ij}={\overline{u}}_i{\overline{u}}_j-\overline{u_i{u}_j}$ (m²/s²) is the subgrid-scale stress; and S_M,i (m/s²) is the sink term related to vegetation drag (Equation (3)). τ_ij represents the effect of unresolved small-scale motion on the resolved flow and is based on the Eddy-viscosity assumption ${\mathsf{\tau }}_{\mathrm{ij}}-\frac{1}{3}{\mathsf{\tau }}_{\mathrm{kk}}{\mathsf{\delta }}_{\mathrm{ij}}=-{\mathsf{\vartheta }}_{\mathrm{t}}\left(2{\overline{\mathrm{S}}}_{\mathrm{ij}}\right)$, where ϑ_t (m²/s) is the eddy viscosity. The wall-adapting local Eddy-viscosity (WALE) model, proposed by [21], was chosen as the subgrid-scale model to calculate ϑ_t.

The vegetation was modelled as a porous medium following [22], in which the flow resistance induced by vegetation generates a momentum loss that is added as a sink term in Equation (2). In our study, momentum loss was computed with the Darcy–Forchheimer (DF) model:

$${\mathrm{S}}_{\mathrm{M},\mathrm{i}}=-\left(\mathsf{\vartheta }\mathrm{d}+0.5\mathsf{\rho }\left|{\mathrm{u}}_{\mathrm{jj}}\right|\mathrm{f}\right){\mathrm{u}}_{\mathrm{i}}$$

(3)

in which d (1/m²) is the viscosity drag coefficient and f (1/m) is the inertial coefficient. The coefficients d and f were calculated using the Ergun equation:

$$\mathrm{d}=\frac{150}{{\mathrm{D}}_{\mathrm{p}}^2}\frac{{\left(1-ϵ\right)}^2}{{ϵ}^3}$$

(4)

$$\mathrm{f}=\frac{3.5}{{\mathrm{D}}_{\mathrm{p}}}\frac{\left(1-ϵ\right)}{{ϵ}^3}$$

(5)

in which D_p (cm) is the mean particle diameter, ϵ (= 1 − a) is the void fraction, a = nS_v/S_cav is the vegetation density in the cavity, n is the number of vegetation cylinders, S_V (m²) is the horizontal cross-section area of the vegetation cylinders, and S_cav (m²) is the cavity area. In order to account the anisotropic resistance of vegetation, the coefficients were calculated differently in the horizontal (XY plane) and the vertical direction (z-axis) [22,23]. In the horizontal direction, D_p assumed the value of the vegetation diameter extracted from [14]. In the vertical direction, the coefficients were calculated with the equivalent hydraulic diameter (d_h), where D_p = d_h. The value of a = 100% corresponds to a wall with a no-slip condition.

2.2. Numerical Model and Boundary Conditions

The 3D geometry consisted of a lateral cavity adjacent to a rectangular open channel (Figure 1). The geometry and flow conditions were extracted from [14], in order to validate this model. The main channel was L_c = 1.25 m long, B = 0.30 m wide, and had a depth H = 0.10 m that was held constant throughout all the geometry. The lateral cavity was W = 0.15 m wide and L = 0.25 m long, resulting in the aspect ratio W/L = 0.6. In all cases, the flow in the main channel was turbulent (Re = 9000) and subcritical (Fr = 0.102), with an averaged velocity U = 0.101 m/s.

The rigid-lid approximation was applied at the free surface of the domain (z = 0.10 m), which is valid for subcritical flows within Fr < 0.36 [24]. The free-slip wall condition was used at y = 0.30 m. The width of the main channel covered by the numerical model was only 0.30 m (the full width of the experimental main channel was 0.85 m), because, at this location, the effect caused by the cavity in the main channel is negligible [25]. The inlet was calculated in two phases: (a) precalculated velocity fields that were fully developed in a periodic channel, under the same flow conditions and the same main channel geometry; and (b) the average incoming flow was used to generate turbulent fluctuations using the turbulence divergence-free synthetic Eddy method [26]. A convective outflow boundary condition was adopted at the outlet, in which the zero-gradient condition allows the flow to exit the domain without having any backflow. The bottom of the domain, the walls of the main channel, and the cavity were considered no-slip walls. The turbulent viscosity was modelled in all of the no-slip walls of the domain with the Spalding wall function.

The vegetation density varied between a = 0 (no vegetation) and a = 10.65% and was distributed in four scenarios (

Table 1. Vegetation densities and the calculated Darcy–Forchheimer coefficients.

Case	a (%)	Horizontal Direction (x- and y-axis)		Vertical Direction (z-axis)
		d (1/m2)	f (1/m)	dh (m)	d (1/m2)	f (1/m)
0	0	0.00	0.00	0.00	0.00	0.00
1	0.13	116.53	3.09	0.7624	0.0004	0.006
2	3.99	120,314.00	105.38	0.0210	613.12	7.52
3	10.65	1,061,150.94	348.58	0.0041	140,829.09	126.99

where a (%) is the vegetation density, d (1/m²) is the viscosity drag coefficient, f (1/m) is the inertial coefficient, and d_h (m) is the hydraulic diameter.

). The scenarios were chosen based on typical values of vegetation density found in the literature [27]. It was assumed that vegetation was uniformly distributed in the cavity throughout the entire water depth.

The computational points in the entire domain were displaced in a rectangular grid, composed of hexahedral elements. The main channel was divided into a grid of 120 × 120 × 40, for the streamwise (x-axis), spanwise (y-axis), and vertical directions (z-axis), respectively, while the cavity was divided into 80 × 80 × 40. Figure 2 shows the computational grid in the free-surface plane. The total number of hexahedral elements was 1,408,000 elements and a maximum y+ = 6.8 and z+ = 2.2. The discretization of the domain was defined with the grid uncertainty evaluation proposed by [28]. Three grids were employed in which the calculated uncertainty was 1.74% in the medium grid, which was chosen for the next simulations. The model was decomposed and solved on 48 cores each (Intel Xeon E5-2670v3 (Haswell) with clock frequencies of 2.3 GHz) and took an average of ≈18 h to simulate 100 s of flow.

The simulations were calculated with the open-source package OpenFOAM (version 1912). To discretize the governing equations and numerical schemes, the module pimpleFoam, which employs the finite volume method (FVM) in a transient formulation, was used. To solve the convection-diffusion equations, the implicit second-order backward time-stepping scheme and additional second-order schemes were used. The residual tolerance was set to 1 × 10⁻⁴ for all variables. The timestep size was set as adaptative with a maximum Courant number of 0.9, a maximum time step of 0.05 s, and an averaged timestep size of 0.001 s. The simulation ran for nearly 150 H/U to stabilize the solution and develop the flow; after this period, the time-averaging procedure started and lasted for another 3600 H/U.

3. Results and Discussion

Figure 3 shows the contours of bed shear stress and the flow streamlines at z/H = 0.2 (z = 20Φ for very coarse sand, see

Table 2. Typical values of particle diameter (Φ) and settling velocity (w_s) extracted from [30].

Sand Granulometry	Φ [m]	ws [m/s]
Very Fine	0.0000625	0.112
Fine	0.000125	0.0703
Medium	0.00025	0.036
Coarse	0.0005	0.0128
Very Coarse	0.001	0.00347

). For a < 0.13%, a main anti-clockwise motion was formed (Figure 3a,b). For a > 3.99%, the main circulation started to develop a secondary circulation at x/L ≈ 0.8 and y/W ≈ 0.2 and the main circulation’s centre was translated to x/L ≈ 0.2 (Figure 3c). The further increase in density increased the size of the right circulation and started to suppress the left one (Figure 3d). Both circulations had contact with the main channel. Unlike the pattern for non-vegetated lateral cavities [29], the flow entered from the most upstream (x/L ≈ 0.3) part of the cavity and exited through the downstream (right portion) for a ≥ 3.99% (Figure 3c,d).

The maximum bed shear stress was located at the downstream wall of the cavity and at the interfacial region between the main channel and the cavity (Figure 3). Higher bed shear stresses were observed in the outer parts of the recirculation regions. Very low shear stresses were found in the central areas of the recirculation regions. As the vegetation density increased, in general, the bed shear stresses’ intensity diminished. The shear stress field at the cavity bed was used to estimate the sediment transport mode and the sediment motion by employing well-known criteria from the literature.

The cavity’s downstream wall and the interfacial region between the main channel and the cavity had the highest bed shear stress (Figure 3). The outlying parts of the recirculation regions have higher bed shear stresses, while the core parts of the recirculation regions have very low shear stresses. The intensity of bed shear stresses decreased as vegetation density increased. Using well-known criteria from the literature, the shear stress field at the cavity bed was used to estimate the sediment transport mode and sediment motion. For this analysis, the Shields number, $Sh=\rho {\mathrm{u}}_{*} /\left({\rho }_s-\rho \right)g\varphi$, and the Rouse number, $Ro=w_s/k{\mathrm{u}}_{*}$, were calculated. In those numbers, $\rho$ (kg/m³) is the density of water, $u_{*}=\sqrt{{\tau }_b/\rho }$ (m/s) is the shear velocity, ${\tau }_b$ (kg/ms²) is the bed shear stress, ${\rho }_s$ = 2650 (kg/m³) is the density of sediment (typical value for quartz and clay minerals), g = 9.802 (m/s²), Φ (m) is the particle diameter, w_s (m/s) is the particle settling velocity, and k = 0.41 is the von Karmán constant. The tests were conducted for all ranges of sands. Typical values of particle diameter and settling velocity (Table 2) were extracted from [30].

Concerning the sediment transport mode,

Table 3. Estimated results of Rouse number (Ro) inside the lateral cavity at z = 20Φ. Values from left to right represent cases 0, 1, 2, and 3, respectively. The mean values represent a spatial averaging of the Rouse values in the XY plane.

Sediment	Max (102)	Mean	Median
Very Fine	2.00/10.31/38.45/55.20	4.26/8.02/63.04/84.82	2.45/3.10/20.08/31.17
Fine	7.63/63.03/134.87/255.42	10.91/20.69/200.28/294.80	7.02/8.63/68.57/111.48
Medium	15.61/46.07/379.28/411.36	24.61/39.43/559.63/820.10	17.59/21.15/188.95/311.26
Coarse	54.77/68.02/814.16/1467.31	48.54/78.04/1092.73/1616.80	33.09/40.67/373.46/612.30
Very Coarse	95.00/129.07/1562.37/1316.96	78.11/125.22/1762.61/2570.15	52.80/71.68/608.14/954.70

summarizes the statistical values of the Rouse number Ro. The minimum values were omitted from Table 3 because they represent the null value at the wall. Figure 4 presents the contours of the Rouse number in the cavity at the depth z = 20Φ for a = 0 and 3.99%. When Ro < 0.8, the transport is wash-load; when 0.8 < Ro < 1.2, 100% of the transport is suspended; when 1.2 < Ro < 2.5, 50% of the transport is suspended; and when Ro > 2.5, the transport is bedload [31]. For the non-vegetated case, a = 0%, with very fine sand, the main mode of movement was bedload transport. Further, in the non-vegetated cavity with very fine sand, there was a spatial variation in the values of Ro in the mixing layer region (−0.1 < y/W < 0.2) and outer part of the recirculation (Figure 4a), in which the predominant expected behaviour was 50% and 100% suspended transport. With the increase in vegetation density, the shear velocity reduced, which gradually increased the values of Ro (Figure 4b). In all vegetated cases, the main class of movement was bedload transport, as Ro > 2.5. As the velocity became more homogenous inside the cavity, with the increase in a (Figure 3), so did the distributions of Ro (Figure 4b) and Sh (not included in this paper).

Sediment motion was analysed with Hjulström and Shields curves. Concerning the Hjulström curve, Figure 5a shows that, in the zones with high velocity (e.g., downstream wall of the cavity), and for sediments with Φ < 0.00025 m, transport can occur. For low and average velocities, sediment was deposited. In relation to the Shields curve in Figure 5b, sediment movement also only occurred in regions with high velocity and when the granulometry was larger than very fine. The increase in vegetation density reduced the bed shear stress, which secures that the “no sediment motion” status continued for denser vegetations. Thus, the increase in vegetation density tends to preserve trapped sediments inside the region. Furthermore, vegetation impacted sediment transport at different rates. In both diagrams in Figure 5, the maximum values of u and ${\tau }^{*}$ were not largely influenced by the increase in vegetation density. Nevertheless, the enhancement in vegetation density impacted the spatial average and minimum values of u and ${\tau }^{*}$. Moreover, at the minimum series of the Hjulström diagram (Figure 5a), the velocity, for a = 3.99%, was higher than the higher densities (a = 10.65%), which did not occur for the shear stresses in the Shields diagram (Figure 5b). This behaviour indicates that a threshold not only occurs for the velocity, but also for sediment transport.

4. Conclusions

This study investigated the influence of vegetation density on sediment transport and velocity in lateral cavities. The increase in vegetation density reduced flow velocity inside the lateral cavity. For the vegetation density a > 3.99%, a secondary circulation appeared that was not expected with the chosen aspect ratio (width/length = 0.6). The presence of vegetation uniformised the sediment transport type and motion status along the entire cavity. The increase in vegetation density overall promoted the deposition/no sediment motion inside the cavity (Figure 5), which was not observed for all the sediment granulometries in a non-vegetated cavity. Thus, the vegetation enhances the protection mechanism to sediment movement, which might be beneficial for conserving the main channel depth and trapping contaminants attached to sediment particles.