Experimental Tests of Lateral Bedload Transport Induced by a Yawed Submerged Vane Array in Open-Channel Flows

Bedform kinematics and morphology are identified using a bedform tracking algorithm developed by Lee et al. (2021) with high-resolution laser scanning bathymetry data. The algorithm first detects every concave downward shape from the time series of bathymetry data and finds the locations of the bedform crest and trough based on sign changes of the spatial gradient in longitudinal bed elevation profiles. Next, the corresponding bedform characteristics, such as bedform migration velocity, $u_b$, and bedform area, $A$, can be estimated by tracking the locations of the identified bedform crests and troughs in time series. More details on the bedform geometry extraction and tracking procedure are reported in Lee et al. (2021).

Fig. 2(a) shows an instantaneous two-dimensional field of bed elevations, and the transect-view elevation profile along the red dashed line is presented in Fig. 2(b), which clearly shows a wide range of coexisting migrating bedforms of different scales. To ensure capturing all small-scale bedform information, the representative bedform height, $h_b$, was estimated by dividing the bedform area, $A$, to the bedform length, $\lambda$, which is defined as the distance between the upstream and downstream troughs [Fig. 2(c)]. The information of the individual bedform migration velocity, $u_b$, and the bedform height, $h_b$, are set to be stored at the crest points of each individual bedform to represent the bedform migration [Fig. 2(d)]. The streamwise bedload transport rate, $q_{b,x}$, was calculated according to the equation (Simons et al. 1965):where $p$ = porosity of the sediment substrate. Note that we use a consistent porosity value to convert directly measured sediment weight by the weighing pans into volumetric flux. All the findings and validation discussed later are independent of $p$. In this study, we proposed a new Eulerian-average grid mapping algorithm to convert the discrete bedform characteristic data into regular grids. Following the Lagrangian approach that was applied to investigate individual bedform migrations [the bedform tracking method proposed by Lee et al. (2021, 2022a)], the Eulerian-average method allowed the calculation of time-averaged bedform transport characteristics on the 2D (X–Y) plane by monitoring bed surface evolution for a substantially long period (36 h), in which statistical convergence was achieved. In the Eulerian-average mapped grids, we calculated the Eulerian-averaged migration velocity, $U_b$, bedform height, $H_b$, and streamwise bedload transport rate, $q_{s,x}$, among all the bedforms inside each grid at each specific time instant (t):where $\mathfrak{E}\{\}$ = Eulerian-averaging operator. Note that bedforms that cover any part of the grid must be included when performing the Eulerian-average. We can simply follow the bedform length, $\lambda$, as the reference region to identify how many grids were occupied by each bedform and how many incoming or outgoing bedforms would need to be included in the calculation for each grid. As an example shown in Fig. 2(d), the center bedform ($j+1$) extends to both the left ($i$) and the right ($i+1$) grid. Therefore, both the left and the right grid should account for the contribution from the center bedform in the Eulerian-average. The quantities, $\mathbb{Q}=\{U_b,H_b,q_{s,x}{\}}_t$, obtained here are instantaneous, $\mathbb{Q}(X,Y,t)$, which can be used to calculate the time-averaged quantities, $\overline{\mathbb{Q}}(X,Y)=⟨\mathbb{Q}(X,Y,t)⟩$, where $⟨⟩$ represents the time-averaging operator. This operation accounts for bedform variability and scale-dependent migration velocity in the estimate of bedform transport (Lee et al. 2022b, 2023).

To design the Eulerian-average window size, we need to ensure a sufficient number of bedforms were captured in each grid to reach convergence in the quantities, such as $q_{s,x}$, while at the same time optimizing the window size to obtain the best resolution of the Eulerian-average values. Following this discussion, we set the Eulerian-average window to be 0.5 m long and 0.04 m wide and check the numbers of bedforms in each grid with and without the submerged vanes. Fig. 3(a) shows that in the baseline case, since no in-stream structure exists to interfere with the flow and redirect bedforms, bedform numbers are generally uniform across the whole domain, ranging from 10 to 15 in each grid. In Fig. 3(b), the presence of submerged vanes reduces the flow within the array and exerts additional spatial variability on both the flow field and the bed, leading to relatively fewer bedforms captured within the array. However, the selected size of the grid window from baseline data, still allowed to capture at least eight bedforms per grid, per time instant, within the array. A convergence analysis [Fig. 3(c)] indicates that $q_{s,x}$, locally averaged within the center grid in the baseline case (blue line) and the vane array case (red line), quickly converged when the grid included more than three bedforms at each time instant over 21 h because the substantially long period of the measurement time ensures convergence of the time-averaged values. In Fig. 3(d), lines representing one, four, and eight bedforms included in the Eulerian-average under the baseline case (solid lines) and the vane array case (dash lines) both converged when averaging over 20 h, which proves that the designed Eulerian-average window size is acceptable to ensure a converging average of local bedform transport rate. Note that this Eulerian-averaging provides a fairly robust measure of bedform transport in a specific location at a specific time.

A control volume (CV) theory in accordance with the mass conservation principle is applied to quantify the lateral bedload sediment transport so as to test the redirection efficiency by the vane array installment. In Fig. 4, two control volumes, ${\mathrm{C}\mathrm{V}}_{\mathrm{1}}$ and ${\mathrm{C}\mathrm{V}}_2$, were set to evenly separate the channel into the upper (unobstructed side) and lower (vane array) regions. The time-averaged streamwise bedload transport rate at the beginning of the two control volumes ($X=-0.5\mathrm{m}$) is shown to be uniform across the channel in the left panel of Fig. 4, suggesting that the volumetric sediment influx at ${\mathrm{C}\mathrm{V}}_1$ ($q_{\mathrm{i}\mathrm{n}(s,x1)}$) and ${\mathrm{C}\mathrm{V}}_2$ ($q_{\mathrm{i}\mathrm{n}(s,x1)}$) are nearly the same ($\approx q_{\mathrm{i}\mathrm{n}(s,x)}$), and the value is equal to the whole domain-averaged streamwise bedload transport rate, $q_{\mathrm{a}\mathrm{v}\mathrm{g}(s,x)}$, which can be used as a reference volumetric sediment flux per unit width:

We verified that the system reached a dynamic equilibrium in which statistical convergence was achieved by monitoring the time-averaged bed elevation to ensure that it remained constant within the two control volumes (not shown here). Therefore, the mass conservation of sediment in ${\mathrm{C}\mathrm{V}}_1$ and ${\mathrm{C}\mathrm{V}}_2$ can be expressed aswhere $q_{s,x1}$ and $q_{s,x2}$ are the width-averaged streamwise bedload transport rate at the end of ${\mathrm{C}\mathrm{V}}_1$ and ${\mathrm{C}\mathrm{V}}_2$, respectively. $B$ and $L$ are the width of the flume and the length of the control volume side. $q_{s,y}$ is the average lateral bedload transport rate, which can be obtained by subtracting the two equations [Eqs. (4) and (5)]:

However, because of the missing information within the blank gaps between the three laser scanning regions, the measured sediment outflow could deviate from the sediment inflow across the control volumes. Such unbalanced sediment mass could cause errors in calculating $q_{s,y}$ according to the mass conservation principle. To solve this potential issue, a mass balance correction coefficient, $C_M$, was introduced as a ratio between the reference domain-averaged volumetric sediment flux, $q_{\mathrm{a}\mathrm{v}\mathrm{g}(s,x)}$, and the mean of the computed outflow transport rate in the two CVs, $q_{s,x1}$ and $q_{s,x2}$, to account for the unresolved streamwise sediment flux and ensure sediment mass balance:

The value of $C_M$ varies in space and time, ranging from 0.85 to 1.18 for the baseline case and from 0.84 to 1.19 for the vane array case. Finally, Eq. (7) can be rewritten asand the total lateral bedload transport flux in the varying sizes of the control volumes can be obtained by multiplying the total lateral length of the control volume:

The approach provides an estimation of the lateral bedload transport while ensuring the streamwise sediment mass balance.

The physical mechanism governing the directionality of bedload transport is based on the cross-flow component of the drag force exerted by the vane on the incoming flow. This force steers the vane wake in the cross-stream direction and thus directly affects the flow near the sediment bed downstream of the vane and the local shear stress. In critical mobility experiments (Lee et al. 2022a), the sediments were locally mobilized near the base of the vane structure, leading to the formation of a scour region and then entrained in the wake. The performance of the vane(s) was evaluated by measuring how far the emerging sediment deposit was steered laterally with respect to the vane(s) center. In the design optimization, we focused on the lateral displacement of the deposit, compensating for the amount of sediment displaced from the scour region (not a desired feature). In live-bed conditions, bedforms migrate through the vane array, making boundaries of the scour and deposit regions less distinctive and unsteady, forcing us to change our methodology. The cross-flow components of the drag, the wake, and the shear stress persist in transitioning from critical to live-bed conditions. However, the overall effect on sediment transport is different. The shear stress spatial distribution and direction induced by the vane array primarily act on the incoming bedform field, not exclusively on the sediment in the scour hole. The macroscopic effect is a deflection and break up of incoming bedforms, resulting in the generation of sequences of yawed bedforms spatially evolving along the steered wake. This can be observed in Video S1 resulting from the laser scan measurements and in Fig. 5. These laterally redirected bedforms are, in fact, moving sediments in the cross-flow direction. Fig. 5 demonstrates the working principle of the vanes in live-bed conditions, showing instantaneous bedform fields at four selected times (9.01 h, 14.61 h, 27.20 h, and 30.35 h). Preferential lateral redirection of bedform fields is clearly shown by the chevron-shaped scour patterns highlighted by the magenta lines in the black solid boxes. The patterns are shifted in the cross-flow direction that aligns with the vane orientation and resembles those observed in the wake of MHK turbines (Hill et al. 2014; Musa et al. 2020) and vegetation patches (Rominger et al. 2010; Follett and Nepf 2012; Kim et al. 2015).

Fig. 6 presents comparisons of the Eulerian-averaged bedload transport rate, $q_{s,x}$, bedform migration velocity, $U_b$, and bedform height, $H_b$, between the baseline case and the vane array case. When there were no submerged vanes installed, the flow generated a uniform downstream bedform migration pattern, as Figs. 6(a, c, and e) present relatively uniform fields of $q_{s,x}$, $U_b$, and $H_b$, respectively. However, after installing the array, the additional drag exerted by the vanes slowed down the flow, thereby reducing both $U_b$ and $H_b$ within the array [Figs. 6(d and f)]. The reduced $U_b$ and $H_b$, in turn, decreased $q_{s,x}$ inside the array as indicated by the blue-colored region in Fig. 6(b). Nevertheless, all results suggest that the bedform field recovers back to normal flow conditions when leaving the vane array, as two cases show similar results of $q_{s,x}$, $U_b$, and $H_b$ downstream of the vanes when $X>5\mathrm{m}$.

The five weighing pans at the end of the channel provide a direct measurement of the streamwise bedload transport rate, $q_m$, far downstream from the vane array, whose time-averaged values are shown in Fig. 7(b) as a comparison to the estimated $q_{s,x}$ by the laser scanning bedform tracking method. The measured $q_m$ shows the expected bedload transport profile in open channel flows with higher transport at the center and lower transport near the two sides of the wall because of boundary effects similar to the baseline case result [Fig. 7(a)]. This confirms that bedform characteristics and transport have recovered back to normal flow conditions almost $12\mathrm{m}$ downstream (location of the weighing pans from the array), namely 3.7 times the array length ($L_a$). The time-averaged $q_m$ ranges roughly from 1 to $3\times {10}^{-6}$ m²/s and the overall mean $q_m$ is $1.7\times {10}^{-6}{\mathrm{m}}^2/\mathrm{s}$ in the baseline case and $1.6\times {10}^{-6}{\mathrm{m}}^2/\mathrm{s}$ in the vane array case, which has the same order of magnitude as the measured domain-averaged $q_{s,x}$ ($q_{\mathrm{a}\mathrm{v}\mathrm{g}(s,x)}=1.14\times {10}^{-6}{\mathrm{m}}^2/\mathrm{s}$). The two measured values validate the Eulerian-average method applied to laser-scanning bedform tracking data.

To further investigate the local spanwise variations of bedload transport rates along the channel, cross-stream profiles at five $X$ locations ($X=1\mathrm{m}$, $3\mathrm{m}$, $5\mathrm{m}$, $7\mathrm{m}$, and $9\mathrm{m}$) were plotted in Fig. 8, showing the comparison between the baseline case (black dashed line) and the vane array case (red solid line). The result shows a clear reduction of $q_s$ within the vane array region ($Y<1.5\mathrm{m}$) and a slight increase on the unobstructed side ($Y>1.5\mathrm{m}$) at $X=3\mathrm{m}$ for the case with submerged vanes. However, the bedload transport system tends to adjust itself back to the baseline flow condition downstream of the vanes, as two cases show nearly the same profiles at $X>5\mathrm{m}$. A first quantitative estimate of the vane array effect on the sediment transport can be based on the spanwise distribution of the bedform mass flux (see the red line in Fig. 8). Downstream of the center of the array ($X=3\mathrm{m}$), in the unobstructed flow region, at $y_1\sim 2\mathrm{m}$, the mass flux $q_{s,x}$ peaks at $1.34\times {10}^{-6}{\mathrm{m}}^2/\mathrm{s}$, while the corresponding mass flux within the array, at $0.5\mathrm{m}<y_2<1\mathrm{m}$, is in the range of $q_{s,x}=0.58$ to $0.82\times {10}^{-6}{\mathrm{m}}^2/\mathrm{s}$. The percentage difference quantifiable as $[q_{s,x}(y_1)-q_{s,x}(y_2)]/q_{avg(s,x)}$ ranges from 32% to 66%. Conversely, note that the spanwise distribution of $q_{s,x}$ for the baseline in the same location (black dashed line) is homogeneous, as expected. This maximum observable change in the spanwise distribution of the streamwise mass flux reflects all the physical mechanisms occurring at the array scale, including (1) bedform and bedload lateral steering by single vanes (Fig. 5), along with (2) the distortion of the mean flow because of the asymmetric distribution of the vanes in the channel inducing an accelerated flow in the unobstructed side, as opposed to a velocity deficit zone within the array. This inferred flow distortion has been observed along with the formation of forced bars induced by a relatively wide porous wall perpendicular to the flow Redolfi et al. (2021) and by a spanwise array of hydrokinetic turbines Musa et al. (2019). In both cases, the obstructions (i.e., the wall and the turbines) were installed on half-side of the channel, creating an accelerated flow on the unobstructed side and a velocity deficit, i.e., a wake, downstream of the obstruction. The flow distortion affected bedload sediment transport, generating faster bedforms–higher transport on the unobstructed side (where erosion was observed) and slower bedforms–lower transport in the wake (where deposition was observed), overall resulting in a streamwise mass flux imbalance. Therefore, we expect that strongly asymmetric roughness distributions play a significant role in the spanwise modulation of the mass flux, even when single roughness elements are not designed to induce a lateral component of the bed shear stress, sustaining lateral transport or bedform steering, as the vane tested here.

Following the control volume approach discussed in the section “Lateral Bedload Transport Quantification,” we set the upstream boundary of the two CVs right upstream of the submerged vane array at $X=-0.5\mathrm{m}$ and varied the CVs’ length by moving the downstream boundary from $X=0\mathrm{m}$ to $X=10\mathrm{m}$ with a $0.5\mathrm{m}$ increment as shown in Fig. 9(a). This is set to obtain a different, robust discretization of $q_{s,x}$ in control volume subsets. The corresponding x-dependent averages of the lateral bedload transport rate, $q_{s,y}$, according to Eq. (9), are hereinafter, defined as the evolutionary means within the control volumes.

As indicated by Eq. (9), the final form of $q_{s,y}$ depends on the difference of volumetric sediment outflux from ${\mathrm{C}\mathrm{V}}_1$ and ${\mathrm{C}\mathrm{V}}_2$ ($q_{s,x1}-q_{s,x2}$). Fig. 9(b) shows the varying $q_{s,x1}$ (solid black line) and $q_{s,x2}$ (dashed grey line) with the changing size of ${\mathrm{C}\mathrm{V}}_1$ and ${\mathrm{C}\mathrm{V}}_2$ in the vane array case, in fact showing the longitudinal profile of the averaged streamwise sediment flux of the two-channel regions. As discussed, when comparing spanwise heterogeneity in the streamwise mass flux distribution, within the region of $0<X<5\mathrm{m}$, the submerged vanes hinder the streamwise bedload transport, resulting in a lower $q_{s,x2}$ that infers an upward lateral bedload transport. Then, $q_{s,x2}$ starts to increase near the downstream edge of the array (indicated by the right boundary of the shaded area) and surpasses $q_{s,x1}$ further downstream ($X>5\mathrm{m}$) as the system tends to readjust itself back to the uniform condition with no in-stream obstacles.

The difference between $q_{s,x1}$ and $q_{s,x2}$ results in an evolutionary lateral bedload transport as presented in Fig. 10(a). The grey bars show nearly zero lateral transport in the baseline case. However, the black bars display a net positive $q_{s,y}$ from $X=0\mathrm{m}$ to $X=5\mathrm{m}$, with the maximum value $\approx 2\times {10}^{-7}{\mathrm{m}}^2/\mathrm{s}$ at $X=1\mathrm{m}$, followed by a gradual decrease, eventually reaching a net value $q_{s,y}\approx 0$ when $X>5\mathrm{m}$. The positive values along the channel highlight the extent of the desired sediment redirection by deploying a yawed submerged vane array. Compared to the baseline case where $q_{s,y}$ is nearly zero, the value of $q_{s,y}$ within the vane array ranges from $1\times {10}^{-7}$ to $2\times {10}^{-7}{\mathrm{m}}^2/\mathrm{s}$, which is between 9% to 18% of the averaged streamwise transport rate, ($q_{\mathrm{a}\mathrm{v}\mathrm{g}(s,x)}=1.14\times {10}^{-6}{\mathrm{m}}^2/\mathrm{s}$). Note that the $q_{s,y}$ represents the average lateral transport per unit width. To identify the region within the array, most affected by lateral transport under the current setup, we can look at the cumulative, or total, lateral transport $Q_{s,y}$, integrated over the $[0-X]$ range. Following Eq. (10), the total lateral bedload transport flux in the varying size of the control volumes $Q_{s,y}$ is shown in Fig. 10(b). The location of the total lateral transport flux peak is at about $X=2.5\mathrm{m}$, corresponding to 2.3 times the row distance $s_x$, or 0.77 the total length of the array $L_a$ ($L_a=3s_x$). The peak location is consistent with the observed maximum difference between $q_{s,x1}$ and $q_{s,x2}$, as shown by the dotted red line in Fig. 9(b). This occurs near the third row of the vane array, suggesting that perhaps the last set of vanes is too sheltered by upstream wakes to perform optimally. Although the bedform characteristics seem to recover downstream of the array, the vanes’ net effect on the lateral transport, driven primarily by contributions from the upstream vanes of the array, remains until $X=5\mathrm{m}$, which corresponds to the switch point of the evolutionary $q_{s,y}$ shown in Fig. 10(a).

We acknowledge that the correct scaling parameters for the peak location of $Q_{s,y}$ and the overall streamwise extent of the vane $(s_x,L_a)$ are not unambiguously defined, and more experiments will be needed. We also note a weak dampened oscillation of the lateral transport $X>5$, which could be a signature of channel bathymetric distortion and thus deserve further analysis.

To obtain a more local description of lateral transport, we now applied mass conservation of sediments in progressive channel bed areas with a pair of moving control volumes [Fig. 11(a)]. The control volumes are designed under a fixed length, $L=1.5\mathrm{m}$, and a 50% ($0.75\mathrm{m}$) overlapped area between each moving window. In this case, the volumetric sediment influx is no longer the domain-averaged streamwise bedload transport rate as we applied previously [Eq. (3)]. Instead, the sediment influxes at ${\mathrm{C}\mathrm{V}}_1$ ($q_{\mathrm{i}\mathrm{n}(s,x1)}$) and ${\mathrm{C}\mathrm{V}}_2$ ($q_{\mathrm{i}\mathrm{n}(s,x2)}$) might be different (at least within the array region), and they can vary depending on the locations of the two control volumes. Therefore, Eqs. (4) and (5) will need to be revised for the moving control volume approach:

Following the same procedure for obtaining Eq. (6), the mean lateral bedload transport rate in the moving control volumes will be updated as [Eqs. (11) and (12)]:

According to mass conservation, the lateral averaged streamwise bedload transport rate should be constant across the whole channel. That is to say, no matter where the moving control volumes move to, the average streamwise bedload transport rates entering or exiting the control volumes should be the same, which has proven to be $q_{\mathrm{a}\mathrm{v}\mathrm{g}(s,x)}$. Therefore, another mass balance factor, $C_{M(\mathrm{i}\mathrm{n})}$, would be required for the measured sediment influx to correct the potential errors because of the blank gaps between the three laser scanning regions, like introducing $C_M$ to correct the measured sediment outflux:

Finally, Eq. (13) can be rewritten as

Fig. 11(b) presents the local (as opposed to the evolutionary) $q_{s,y}$ along the channel. The gray bars once again illustrate a relatively low lateral transport in the baseline scenario, whereas the black bars depict elevated positive local values of $q_{s,y}$ at the beginning of the upstream vane array region from $X=0\mathrm{m}$ to $X=1\mathrm{m}$. This shift is attributed to transitioning from a uniform downstream-migrating bedform field to an upward-redirected field induced by the yawed submerged vanes. As the vanes’ wakes merge, the central portion of the array experiences a likely reduced flow velocity (Lee et al. 2022a), and the distortion of the bedforms contributing to lateral transport is diminished. At approximately midlength of the array ($X\sim 2.5\mathrm{m}$), the local effect is negligible, and the lateral flux starts to switch, bringing sediment back toward the array side. The switch point ($X\sim 2–2.5\mathrm{m}$) marks the extension of the local vanes-induced lateral mobility, and it corresponds to the maximum total lateral transport flux $Q_{s,y}$ [Fig. 10(b)]. Therefore, it is inferred that this would be the recommended location on the unobstructed channel side where an ideal sediment-capturing system or the uptake for a sediment bypass tunnel can be placed.

Note that this negative $q_{s,y}$ at the exit of the vane array revealed by the lateral flux estimation methods is not reflected by a clear yaw in the bedforms movement in the same direction (see Video S1). We speculate that, in this specific location, bedforms migration might not be the dominant transport mechanism for the lateral flux of sediments, which might be locally mobilized via sliding because of a local increase and distortion of the mean flow and the shear stress. This is supported by the lower order of magnitude of $q_{s,y}$ compared to $q_{s,x}$. This argument reinforces the CV approach. It allows us to capture the lateral flux, which may result from different, weaker transport mechanisms, as opposed to the streamwise flux, which is dominated by bedform dynamics and is, in fact, measured by tracking their kinematics and morphology. This variation in lateral transport directionality is likely an array-scale effect that redistributes sediments in the cross-flow direction to reestablish the baseline bedform transport downstream. The transition is fairly gradual, as we observe here. Then, after the lateral flux reverts toward the array side within the last row, small dampening oscillations are observed along the channel. This behavior can be observed in both the integrated and local fluxes in Figs. 10(b) and 11(b), respectively. This oscillating sediment flux resembles the steady, spatially decaying bed distortion effects observed by Musa et al. (2018b) and Redolfi et al. (2021). Based on the latter work and predictive modeling, we speculate that under specific hydraulic conditions—defined by the channel $B/H$ ratio and defined as resonant in Redolfi et al. (2021)—the spatial oscillations of lateral bedform transport may extend farther downstream. However, it is worth mentioning that our experiments were performed without the envisioned sediment collector system. Effectively removing sediments precisely where the lateral transport initiates and peaks may potentially prevent the observed directional shift in the first place or, at the least, dampen the ensuing oscillations of the lateral flux.