Drag Coefficient of Emergent Vegetation in a Shallow Nonuniform Flow Over a Mobile Sand Bed

Widely distributed in natural rivers and coasts, vegetation interacts with fluid flows and sediments in a variable and complicated manner. Such interactions make it difficult to predict associated drag forces during sediment transport. This paper investigates the drag coefficient for an emergent vegetated patch area under nonuniform flow and mobile bed conditions, based on an analytical model solving the momentum equation following our previous work (Zhang et al., 2020, https://doi.org/10.1029/2020WR027613). Emergent vegetation was modeled with rigid cylinders arranged in staggered arrays of different vegetation coverage ∅. Laboratory flume tests were conducted to measure variations in both the water and bed surfaces along a vegetated patch on a sand bed. Based on the experimental and theoretical analyses, a dimensionless drag model integrating both terms of flow properties and bed effects is proposed to predict the drag coefficient Cd over a mobile bed. The calculated values of Cd exhibit two different trends, that is, nonmonotonically or monotonically increasing along the streamwise direction, due to the combined effect of water surface gradient and bed slope. The morphodynamic response of the mobile bed to nonuniform flow manifests as an evolution in the bed slope within the vegetated patch. Ongoing scouring directs the flow's energy toward overcoming the rising Cd and bed slope, leading to a relatively stable stage with a low sediment transport rate. This study advances the existing understanding of the drag coefficient's role over a mobile bed within nonuniform flows. It also enhances the applicability of vegetation drag models in riverine restoration.


Introduction
Aquatic vegetation is prevalent in various aquatic environments, including waterways, rivers, lakes, and tidal zones.It plays a crucial role in shaping the morphological evolution of streams and influencing their sediment transport capacity (Huai et al., 2021;Ielpi et al., 2022;Nepf, 2012a;Swartz et al., 2022).Vegetation exerts a substantial influence on river morphological evolution by introducing additional drag to flow, resulting in reduced flow velocity and energy, consequently affecting the sediment transport capacity of the flowing water (Calvani et al., 2023;Le Bouteiller & Venditti, 2014;Li et al., 2022).Additionally, vegetation-induced wake (Kondziolka & Nepf, 2014) and vegetation patterns (Li et al., 2022;Schwarz et al., 2018) also play a significant role in controlling stream morphodynamics, contributing to the alteration of landscape patterns within aquatic systems.Consequently, comprehending the impact of the vegetation drag on stream morphodynamics holds paramount importance for river restoration projects (Broome et al., 1988;Orth et al., 2020;Temmink et al., 2022;Xu et al., 2022).
The morphodynamic evolution of river systems is intricately linked to the sediment load (Popovic et al., 2021;Wu et al., 2023).Vegetation drag predominantly influences the movement of bed-load sediment (Cavalcante et al., 2021;Follett & Nepf, 2012;Wu et al., 2021a;Zong & Nepf, 2011).The presence of vegetation often results in a reduction in local flow velocity, facilitating sediment retention.However, recent studies have highlighted that vegetation-generated turbulence can also enhance the onset and transport of sediment (Tinoco & Coco, 2016;Yang & Nepf, 2018, 2019).Existing models for bed-load transport in vegetated channels are primarily based on two approaches: the bed shear stress theory (Etminan et al., 2018;Le Bouteiller & Venditti, 2015;Lu et al., 2021), and the turbulent kinetic energy (TKE) theory (Shan et al., 2020;Yang et al., 2016;Yang & Nepf, 2018, 2019).These approaches have estimated the bed-load transport rate by determining the shear stress in the presence of vegetation or by calculating the vegetation-generated TKE (Jordanova & James, 2003;Lu et al., 2021;Shan et al., 2020).Therefore, an accurate description of vegetation drag is the key to applying the two categories of models to predict sediment transport rate in a vegetated environment, given its central role in obtaining the near bed TKE and the vortex shear stress caused by vegetation stems (Li et al., 2022;Lu et al., 2021).The central scientific challenge is to devise a precise method to quantify the vegetation drag, represented by the drag coefficient C d , particularly under conditions of sediment transport, and to understand how this drag coefficient responds to varied vegetation properties.
Regarding uniform flow (from a spatially averaged perspective), numerous studies have been conducted to investigate the vegetation drag coefficient C d under fixed bed conditions (Aberle & Järvelä, 2013;Cheng, 2013;Kothyari et al., 2009;Nepf, 1999;Tanino & Nepf, 2008).The majority of studies have observed a general trend where the vegetation array's C d tends to decrease with higher Reynolds numbers.Multiple models for predicting canopy drag coefficients have been proposed in the literature (Cheng & Nguyen, 2011;Etminan et al., 2017;Nepf, 1999;Tanino & Nepf, 2008).In conditions involving a mobile bed, the Reynolds number and streamwise bed elevation indirectly lead to changes in the vegetation drag coefficient.This drag coefficient is notably sensitive in predicting sediment transport, in contrast to previous studies mostly employing the bulk drag coefficient (Lu et al., 2021;Wu et al., 2021a;Yang & Nepf, 2018, 2019).For instance, Yang and Nepf (2018) initially estimated the drag coefficient as approximately 1 in their sediment transport model.There were studies (Lu et al., 2021;Wu et al., 2021a) that utilized a fixed-bed empirical equation where Re v represents the vegetation array Reynolds number (Cheng & Nguyen, 2011), which resulted in a constant drag coefficient value under similar flow velocity and vegetation coverage conditions.Using a different approach, Yang and Nepf (2019) employed an empirical equation developed for fixed beds: ] ( Etminan et al., 2017), incorporating parameters such as d (the diameter of a rigid cylinder), U p (pore velocity serving as an indicator of the velocity to which the canopy elements are exposed), and ζ (the ratio of average velocity between adjacent cylinders to the pore velocity).Alternatively, Shan et al. (2020) introduced a new iterative method to derive a C d that adapts to mobile beds, considering variations in flow velocity and flow adjustment length.The aforementioned methods for estimating the drag coefficient of vegetation in sediment transport either treat it as a constant value or employ empirical equations derived from fixed beds to calculate C d .These approaches neglect the longitudinal behavior of the drag coefficient within the context of sediment transport.Hence, it is crucial to account for its variation along the streamwise direction.
In addition to the impact of vegetation under uniform flow conditions, turbulent flow through a vegetation canopy generates various coherent vortical structures, consequently giving rise to a nonlinear distribution of Reynolds stress (Afzalimehr & Dey, 2009;Lima & Izumi, 2014;Wang et al., 2019), a phenomenon which disrupts the expected uniform flow.Furthermore, a natural vegetation array commonly induces a nonuniform water surface along the streamwise direction by introducing additional drag (Crompton et al., 2020;García-Serrana et al., 2017;Li & Shen, 1973;Wang et al., 2023;Zhang et al., 2020).In nonuniform flow conditions (a gradually varied flow), the vegetation drag experiences the influence of flow nonuniformity, resulting in a nonmonotonic variation of the drag coefficient (Wang et al., 2015;Zhang et al., 2020).Moreover, subsequent research reveals that the bending deformation of flexible vegetation interacts with this flow nonuniformity, leading to a reconfiguration of drag along the vegetated patch.This reconfiguration exhibits both blockage and sheltering effects (Zhang et al., 2020).Beyond the coupling effect between vegetation flexibility and flow nonuniformity, the gradient of the water surface directly affects the flow nonuniformity over the fixed bed.For example, transient waves following a dam break (Melis et al., 2019) and extreme rainfall events (Wang et al., 2018) influence the water surface gradient under nonuniform flow conditions, consequently affecting the nonmonotonic streamwise distribution of the drag coefficient.
The preceding studies have demonstrated that flow nonuniformity is influenced by conditions at the interface of vegetated nonuniform flows, such as vegetation bending or flow surface disturbances, consequently leading to modifications in the vegetation drag profile along the streamwise direction for the identical vegetation elements.It is important to note that all these experiments were conducted under fixed bed conditions.However, under actual conditions with a mobile bed, the riverbed surface experiences substantial morphological alterations (Wang & Cirpka, 2021;Wu et al., 2020;Yang & Nepf, 2019), accompanied by variations in bed shear stress (Etminan et al., 2018;Jiang et al., 2022;Tseng & Tinoco, 2021;Yang et al., 2015).
Understanding of the physical mechanism underlying the interaction between flow nonuniformity and vegetation drag remains limited, especially concerning the influence of the bed surface on nonuniformity and vegetation drag.Therefore, there is a pressing need to precisely quantify this effect in the context of sediment transport within nonuniform flow conditions.This study seeks to bridge this knowledge gap by predicting the drag coefficient over a mobile sand bed through analytical and experimental research.
In this paper, we initially formulate a dimensionless drag coefficient equation for a mobile sand bed under nonuniform flow conditions utilizing the momentum equation.Then, we examine variations in drag coefficients along the streamwise direction by conducting a series of hydrodynamic experiments, focusing on the low sediment transport stage.Finally, we scrutinize the reasons behind the drag coefficient variation considering each term in the model and evaluate the temporal evolution of the drag coefficient.All the analyses presented in this study are limited to emergent vegetation arrays in nonuniform open-channel flows under mobile sand bed conditions.

Governing Equation for Flow Over a Mobile Bed
In the case of nonuniform flow within a dense emergent canopy under a steady flow rate Q, we can employ the assumption of locally uniform flow conditions.This assumption is supported by the relatively independent streamwise velocity profiles (as shown in Supporting Information S1), which achieve a local force equilibrium between the flow driving and drag terms over a given length-scale dx in the streamwise direction (Thompson et al., 2011;Wang et al., 2015;Zhang et al., 2020).In contrast to broad uniform flow conditions, this approach for handling gradually varied flow aligns with the mathematical concept of solving definite integrals.This methodology is equivalent to solving the averaged momentum equation for the entire emergent canopy (Wang et al., 2023).Additionally, we account for the influence of bed slope and friction in the context of a mobile bed.The equilibrium relationship of forces within a small streamwise length-scale dx is represented as where γ = ρg is the specific weight of the water, ρ is the water density, g is the acceleration of gravity, B is the channel width, H is the water depth as a function of the streamwise coordinate x in the channel.∅ is defined as the coverage of vegetation, where ∅ = πmd 2 /4 and m is the number of cylinders per unit bed area.S f is the energy gradient per unit of stream volume, and τ bed is the bed shear stress.Here, F d is the drag force exerted by the vegetation per unit bed area (Tanino & Nepf, 2008).F d is defined as Water Resources Research where C d is the spatially averaged drag coefficient of the vegetation, and a = md represents the spatially averaged frontal area per unit vegetation volume.U = Q/[BH(1 ∅)] is the reference velocity taken as the pore velocity which represents the spatially averaged flow velocity.Nonuniform flow is commonly approximated using the Saint-Venant equation (SVE) for steady conditions, making several key assumptions: (a) the flow is onedimensional along the streamwise direction; (b) the pressure distribution approximates hydrostatic conditions, thereby neglecting vertical acceleration; and (c) the bed slope gradient due to scouring is not excessively steep (de Saint-Venant, 1871; Melis et al., 2019).Under these conditions, the steady-state form of the SVE is employed to solve for the energy slope S f , represented as where S 0 signifies the bed slope.This framework is implemented in a one-dimensional (laterally averaged) model.
Determining bed shear stress on vegetated beds poses challenges due to the significant impact of vegetation on the near-bed velocity profile and turbulence intensities (Etminan et al., 2018).In this investigation, for a small streamwise length-scale dx, the bed shear stress τ bed within the vegetated patch can be estimated using the linear stress model, as proposed by Yang et al. (2015).The formula for estimating bed shear stress τ bed within the vegetated patch is as follows: Here, ν is the kinematic viscosity of water, and H v represents the thickness of the viscous layer.Yang et al. (2015) introduced the approximation ) for estimating τ bed , where C f represents the bed skin friction coefficient.In the high stem Reynolds number of this study Re d = Ud/ν ≥ 4/C f , which implies that the thickness of the viscous layer is smaller than d/2.Therefore, 4, and τ bed becomes equal to ρC f U 2 , which formally renders the impact of vegetation on τ bed as negligible in the viscous layer (Yang & Nepf, 2018).In the above equations, the bed skin friction coefficient C f can be estimated semiempirically using water depth H and the sediment size d s (Julien, 2010;Shan et al., 2020) (5) Consequently, (6)

Calculation of C d on a Mobile Bed
Substituting Equations 2, 3, and 6 and the continuity equation ( dUH dx = 0) into Equation 1for the conservation of momentum yields For the present study, the morphodynamic evolution of the bed (as characterized by the bed elevation H bed ) is significant and the water surface H sur is variable in the vegetated patch area.Therefore, under nonuniform flow and mobile bed conditions, the water depth H(x) = H sur H bed and the bed slope S 0 = dH bed /dx.Substituting the expressions of H(x) and S 0 into Equation 7 results in the drag coefficient for a cylinder array  (Wang et al., 2015;Zhang et al., 2020). 1 gH ( dH bed dx ) is the elevation component representing the elevation head loss per unit of streamwise distance, and 1 gH C f represents the bed friction component which is the skin stress on the sediment grains on the bed surface.The influence of each term in Equation 8 on C d is discussed in Section 5.1.In this experimental investigation, we primarily observe alterations in the channel bed morphology resulting from sediment erosion, particularly at the trailing edge of the vegetated patch.The overall increase in bed slope precedes changes in local bedforms in affecting bed elevation.
Consequently, the influence of bedform drag is encompassed within the bed slope term 1 gH ( dH bed dx ) , as discussed in Section 5.2.

Experiment Setup
The experiments were conducted in a 0.3 m-wide and 12.4 m-long flume at the Hydraulics Laboratory of Beijing Forestry University, Beijing, China.Laboratory flume tests were carried out in the flume with model emergent vegetation of different ∅ and Q as documented in Table 1, and a bed that was set flat (initial bed slope S 0 = 0).The discharge was controlled by a centrifugal pump and measured by an electromagnetic flowmeter.The values of Q were selected to maintain emergent vegetation throughout this study, which was measured with a flow meter with 0.1 L/s accuracy.This selection resulted in a nonuniform flow condition where downstream water depth was controlled by adjusting tail-gates and varying the canopy density, as illustrated in Figure 1.In the present study,  the tail-gate was adjusted to be consistent for experimental run conditions and remained unchanged throughout the experiments.
The rigid emergent vegetation was modeled by acrylic circular cylinders with a diameter of d = 6.0 mm and fixed in a 5 mm-thick perforated plastic board, which is similar to the arrangement of vegetation on floodplains described in previous studies (Lightbody & Nepf, 2006;Liu et al., 2021;Wu et al., 2021a).The minimum distance l c between nearby holes is larger than d/2 to ensure that the effect of the viscous layer is negligible.The densest coverage, ∅ = 0.30, was selected for illustration as shown in Figure 2a with l c > d/2 = 3 mm.The rigid elements were arranged in a staggered configuration with four different coverages: ∅ = 0.30, 0.15, 0.10, and 0.06.These coverages were selected based on the observed range of ∅ (0.05-0.85) in marshes and sea buckthorn flexible dams (Nepf, 2012b;Yang et al., 2014).Details are provided in Table 1.
The bed sediment used in this experiment was a natural sand with a grain density of ρ s = 2.65 g/cm 3 (the bulk density was 1.65 g/cm 3 ), a median diameter of D 50 = 0.854 mm, and a standard deviation ≈D 90 /D 10 = 1.82,where D n (mm) is the diameter of the sediment exceeding n percentage of the material's mass.The particle size distribution of the sediment (Figure 2b) was measured using a Mastersizer 3000 laser diffraction instrument.The value of the median diameter D 50 is adopted to represent the sediment size d s throughout the following analysis (Wu et al., 2021a).
At the beginning of each experiment, a 3.0 cm-thick layer of sand was placed on top of the plastic boards on the flume bed and flattened.This sand layer covered a 1.5 m-long area in the middle section of the flume.To establish an initially uniform bed slope, a plastic board with fixed cylinders was first placed on the flume bottom.Then, two triangular ramps (each 3.5 cm high) were attached upstream and downstream of the plastic board, as depicted in Figure 1.The sediment layer was subsequently deposited between these ramps, as illustrated in Figure 2c.
To ensure the sand layer was flattened in the densest vegetated patch (∅ = 0.30), the required mass of sediment was first calculated using the porosity (1 ∅), the volume of the vegetated patch, and the wet density of the sand.
Then, this amount of the sediment was evenly spread over the top of the boards by a set of square sieves (Figure 2c).The soil-sieves were 0.3 m wide with 1.2 mm (≈D 90 grain size) pre-drilled holes, and a sheet was used to control the uniform fall of the sediment.Finally, the sediment bed was saturated with seepage water prior to the start of the test (Wang et al., 2014).Following the earlier procedures for leveling the sand bed, careful control was exercised to ensure that the initial sand layer remained within a range of ±2 mm elevation, particularly within the emergent canopy.This uniform elevation was established before the discharge phase and was monitored by a depth gauge with a resolution of 1 mm.The chosen sediment layer thickness allowed for the study of overall bed morphology and vegetation drag, representing conditions associated with weak sediment transport rates, as depicted in the stable stage shown in Figure S2 in Supporting Information S1.

Sediment Collection: Stable Stage of Sediment Transport
The mass of sediment transported by water flow per unit time per unit channel width was defined as the instantaneous bed-load transport rate, denoted as Q s (Wu et al., 2021a;Yang & Nepf, 2018).Sediment was collected every 1 min with an interval time of 30 s using a 15-cm wide sampler settled on the middle of the flume at the downstream end of the vegetated patch.The sediment collected at every minute was put into an aluminum box (the box weight is w 1 ).All samples were dried to constant weight in a 105℃ oven, then weighed to get w 2 .The dry mass of the sediment was calculated as (w 2 w 1 ).The Q s was calculated as the mass of the collected sediment divided by the time for the collection and the width of the sampler, that is, Q s = (w 2 w 1 )/(0.15× 60) g•(m•s) 1 .We conducted repeated bed-load measurements until the bed-load transport rates exhibited fluctuations of less than 10%, then recorded the elapsed time as a duration time (Yang & Nepf, 2018).
The recording of a duration period serves as an indicator of the attainment of a stable stage in bed-load transport.
The measurements of sediment transport rate (Q s ) exhibited an initial decrease, followed by a relatively stable phase over time for all experimental runs (refer to Figure S2 in Supporting Information S1).Here, the stable stage is defined as a period characterized by low sediment transport rates, with less than 10% fluctuations, which has previously been established as a relatively stable stage (Church et al., 1998;Hassan & Church, 2000).The initial stage is defined as a moment when sediment transport begins at the trailing edge, coinciding with the point when the flow transitions into a nonuniform phase.During this initial stage, the bed surface takes on a morphology similar to a flatbed.

Measurements of Water Surface and Bed Surface
After the bed-load transport rate reached the stable stage, the values of H sur (x) and H bed (x) for each experimental run were determined using images captured by a side view camera equipped with a 4800 × 3200 resolution lens.These images portrayed profiles of the water and the bed surfaces, respectively.The camera was positioned outside the glass window of the flume, with the lens oriented parallel to both the water surface and bed surface, as illustrated in Figure 3.To mitigate the excessive scouring of the sand layer near the flume's sidewall, we strategically introduced stems adjacent to the immediate sidewall.This arrangement helped reduce the scouring of the riverbed at the sidewall, ensuring that the sediment transport across the entire cross-section remained uniform (Wu et al., 2021a).Throughout each experimental run, the camera recorded the variations of the water and bed surfaces (Figure 3).Meanwhile, we employed a depth gauge to measure the bed elevation at the leading and trailing edges of the vegetated patch, which served as a calibration point for the image data.The color images captured by the camera were transformed into grayscale images to clearly and accurately identify the current sand bed-fluid interface.In the present study, the symbol x denotes the streamwise distance, where x = 0 marks the starting position of water flow entering the vegetated patch, and x = L represents the endpoint of the vegetated patch.The measured values of H sur (0), H bed (0), H sur (L), and H bed (L) are summarized in Table 1 when the bedload transport rate attains the stable stage.
We employed the same methodology to record the water surface and bed elevation for all experimental runs at the initial stage for comparison.Herein, the bed surface globally maintained a flat profile with H bed (x) = 3.0 cm.The measured water surface values of H 0 and H L are listed in Table 2, where H 0 and H L represent the water level of leading and trailing edges at the initial stage, respectively.The water surface H sur (x) and the bed surface H bed (x) for ∅ = 0.15 and ∅ = 0.06 were captured by camera, as illustrated in Figure 3 for four discharges.Additional images from other runs are available in Figure S3 in Supporting Information S1.For analyzing images, the vegetated patch along the streamwise direction was divided into 200 grids, denoted as m g = 200 for each run.Thus, the length of a measurement cell along the streamwise direction is calculated as dx = L/m g .The water depth at the x = 0 position served as the initial water surface H sur (0), and the bed elevation at this position was referred to as H bed (0).The values of H sur (0) and H bed (0) served as the boundary conditions, which were calibrated using a depth gauge to ensure a deviation of less than 0.1%.The downstream H sur (x) and H bed (x) were measured with a dx-spatial resolution grid for each image.Consequently, this method allows for measuring the bed and water surfaces across the entire vegetated patch, achieving a high spatial resolution approaching 0.3 cm/grid.Moreover, it is worth noting that this method has been applied to measure water surface profiles in the nonuniform flow over a fixed bed (Wang et al., 2015(Wang et al., , 2018;;Zhang et al., 2020).

Models of Water Surface and Bed Surface
The actual water surface H sur (x) and bed surface H bed (x) are variable in nonuniform flow conditions with a mobile bed, as depicted in Figure 3.We employ the measured H sur (x) and H bed (x), obtained from the flume experiments conducted with varying vegetation coverage, to estimate ( dH sur dx ) and ( dH bed dx ) .For the water surface in nonuniform flow, H sur (x) varies along the streamwise direction due to factors such as flow resistance, variations in bottom slope, and other boundary conditions (Chanson, 1999).In our observations, we note that H sur (x) consistently decreases along the streamwise direction in cases of gradually varied flows of the M2 type (Subramanya, 2009;Wang et al., 2015Wang et al., , 2023;;Zhang et al., 2020).According to the present experiment and previous study (Zhang et al., 2020), a quadratic polynomial is employed to characterize the water surface where x = 0 represents the beginning position at the leading edge of the vegetated patch.A regression function of the water surface was applied to determine a 1 , b 1 and c 1, similar to the approach used by Zhang et al. (2020).
Notably, we observed that H bed (x) is generally decreases with x in the present experiment, which is consistent with the morphodynamic evolution of a bed in conditions of high plant coverage (Le Bouteiller & Venditti, 2014).
Deriving a reasonable estimation of ( dH bed dx ) for each grid point from the H bed (x) image is more intricate compared to quantifying the gradient of the water surface.This complexity arises from the fact that approximations of the differential slope are sensitive to local bedform variations in the measured bed surface, assuming constant scouring within the cross-section.Considering the overall decline in H bed (x), ensuring a smooth measurement of H bed (x) is essential to align local derivative approximation align with the overall bed surface shape.This approach helps to mitigate uncertainties associated with local measurements.Theoretically, bed elevation H bed (x) in nonuniform shallow flow can be approximated as a continuous and smooth function.The function must be differentiable and reliably extracted from bed measurements without overfitting to local bedforms.The function must also satisfy two primary constraints: (a) describing the overall decrease in bed elevation within the vegetated patch, that is, dH bed (x)/dx < 0; and (b) ensuring that from the trailing edge to the leading edge of the vegetation array, the bed erosion rate in the sand layer becomes smaller, and the bed change becomes flatter than at the trailing edge of the vegetation array, that is, d 2 H bed (x)/dx 2 < 0. Numerous functions meet these two constraints, including polynomial or logarithmic functions (discussed in Section 5.1).For practicality and to enable easy comparison with the water surface gradient, we choose a quadratic polynomial function to describe the bed, that is where the condition a 2 < 0 represents the bed erosion, and c 2 is a constant.The parameters a 2 , b 2 , and c 2 were obtained by fitting the measured H bed (x) data with the least squares method for all experimental runs.The boundary conditions for bed elevation include H bed (0), representing the bed elevation at the starting point and the bed elevation at the endpoint (H bed (L) < 3 cm).The numerical values of a 2 , b 2 , and c 2 are influenced by factors such as the flow rate Q and vegetation coverage ∅.

Water Surface and Bed Surface Characteristics
The measured nonuniform water surface and bed surface were modeled by Equations 9 and 11, and the outcome of the regression analysis is summarized in Table 1. Figure 4 illustrates the agreement between the measured elevations of the water surface and the bed surface, alongside the fitted values of H sur (x) and H bed (x) for nine tests involving Q and ∅ at the stable stage.Additional agreement data from different runs can be found in Figure S4 within Supporting Information S1.Several observations can be made in the subfigures where ∅ is held constant.First, the upstream water level rises with Q. Subsequently, the downstream water level experiences a slight increase when maintaining a constant tail-gate opening size and distance with the downstream ramp.Simultaneously, there is a more pronounced degree of bed surface erosion and a lower bed elevation along the streamwise direction.Conversely, when maintaining a constant Q, an improved fit to Equation 9 for the water surface is achieved as ∅ increases.This higher ∅ value corresponds to a flow exhibiting more nonuniformity.The bed elevation at the trailing edge is notably lower during the stable stage.In this context, it remains uncertain whether the relation between bed elevation and streamwise distance is governed by a quadratic function, and whether the relation varies across erosion characteristics.Here, the suitability of a quadratic function for describing the relation between H bed (x) and x is assessed through experimentation.The fitting results demonstrate that Equations 9 and 11 are suitable for modeling the observed nonuniform boundaries in this experiment characterized by high water surface gradient and trailing erosion.Indeed, this regression method has limited applications to water surface in experimental tests with a low water surface gradient (e.g., R1.3).For the bed surface, this method may not effectively capture local bedforms resulting from erosion or deposition (e.g., R4.1 and R4.2).The effect of neglecting local bedforms is discussed in Section 5.2.Hence, the quadratic function model remains a viable approach for capturing the overarching trends of the water and bed surfaces along the streamwise direction.This approach is especially relevant in situations where bed erosion occurs at the trailing edge of the canopy, and the hydraulic gradient increases due to the declining water surface.

Drag Coefficient C d of the Stable Stage
Based on the fitted water surface and bed surface in Section 4.1, C d was computed from Equation 8. Figure 5 presents the streamwise distribution of the normalized drag coefficient ) and x + = x/L is the normalized length, representing a longitudinally averaged drag coefficient (Wang et al., 2018(Wang et al., , 2023;;Zhang et al., 2020).The curves in Figure 5a exhibit a parabolic form characterized by an initial increase followed by a decrease.Figure 5b shows a similar change trend, with an initial increase that approaches a maximum and then transitions into a decrease.This trend is demonstrated in all the tests of Q = 0.8 L/s as well as runs R2.1 and R2.4 under the condition of Q = 1.2 L/s.Sediment motion was observed in these tests, although there was only minimal erosion of the sand bed.Moreover, the bed slope S 0 = dH bed /dx was much smaller than the water surface gradient S w = dH sur /dx in Equation 8. Thus, the bed slope exerted little influence on C d , and dH sur /dx dominated the change of S f , which is consistent with the results obtained in previous studies for the fixed bed in nonuniform flow (Wang et al., 2015;Zhang et al., 2020).
Figures 5c and 5d mostly present a globally increasing trend, although C d /〈C d 〉 decreases slightly at the leading edge of the vegetated patch (Figure 5c), which is different from the previously observed parabolic variation.Specifically, for tests with ∅ = 0.15 and an increasing discharge that approaches 1.2 L/s, or even 1.5 L/s, there is a trend for C d /〈C d 〉 to first decrease and then increase (Figure 5c).In Figure 5d, there is a monotonically increasing curve for all tests with Q = 1.8 L/s, as well as for runs R3.3 and R3.4 with Q = 1.5 L/s.When maintaining a constant discharge Q, the rate of increase for C d /〈C d 〉 decreases as ∅ increases, corresponding to flatter curves (Figure 5d).At the leading edge of the vegetated patch, an increase in ∅ leads to an increase in C d /〈C d 〉, demonstrating a blockage effect as described by Etminan et al. (2017).Conversely, at the trailing edge of the patch, a sheltering effect is observed, with C d /〈C d 〉 decreasing as ∅ increases.This is most likely due to the increase in TKE with increasing vegetation coverage ∅ (Nepf, 2012b).This increase introduces additional kinetic energy into the boundary layer for the adjacent downstream vegetation which delays the separation of the boundary layer (Etminan et al., 2017;Zhang et al., 2020) and stabilizes the bed-load transport (Yang & Nepf, 2019).The nonuniformity of flow reinforces this effect at the vegetated patch under the condition of high discharge.As a result, erosion at the trailing edge becomes more intense, which contributes to the formation of the longitudinal bed morphology.In such scenarios (Figures 5c and 5d), the net water depth H(x) experiences a gradual decline, with a slight increase observed at the trailing edge (refer to Figure S5 in Supporting Information S1).This phenomenon can be attributed to the reduction in the bed surface elevation, and it is essential to note that S f is influenced by the combined impact of both dH sur /dx and dH bed /dx.When the bed slope S 0 approaches or exceeds the water surface gradient S w in Equation 8, the drag coefficient monotonically increases along the streamwise direction.

Comparison With C d init of the Initial Stage
To verify the variation of vegetation drag for the mobile sand bed, we studied the drag coefficient at the initial stage C d init when the sand motion was initiated.This drag coefficient was calculated according to Equation 8 by setting dH bed /dx = 0.The best-fit parameters for the water surface are summarized in Table 2. Likewise, C d init / 〈C d init 〉 represents the normalized drag coefficient at the initial stage.Figure 6 presents the nonmonotonic change of C d init / 〈C d init 〉 against the normalized distance x + for all tests at the initial stage.A universal nonmonotonic trend of first increasing and then decreasing is observable in Figures 6a through 6c.The drag coefficients for R1.1, R3.2, and R4.4 show a weak nonmonotonic change in Figure 6d, which presents a similar variation shape of C d /〈C d 〉 in Figure 5b.All the subfigures in Figure 6 exhibit a consistent trend of initially increasing and then decreasing.This suggests that the drag coefficient at the initial stage shares similar characteristics with that for nonuniform flow under fixed bed conditions.

Water Resources Research
10.1029/2023WR036535 ZHANG ET AL.

Factors Influencing C d
The contributions of all the terms in Equation 8to C d can be expressed as where the vegetation effect term E veg reflects the combined effects of vegetation size and its coverage density on the drag coefficient, which is expressed as The pressure term P* is ZHANG ET AL.
the advection term A* is the bed slope term S 0 * is and the bed friction term C f * is The coverage ∅ and the averaged frontal area a are constant for each run, that is, E veg is a constant.Hence, the variation of C d is determined by Figure 7 shows the interplay among terms P*, A*, S 0 * , and C f * along the streamwise direction for runs R2.1, R1.2, R2.3, and R4.3.These cases represent four typical shapes of curves at the stable stage analyzed in Figures 5a-5d, respectively.The shape of [P * A * + S 0 the advection term A* as the bed-load transport continues in the streamwise direction.Eventually, the change of bed surface elevation affects the net water depth value and reverses the trend for the change of the pressure term P*.As the run approaches its conclusion, P* dominates the variation of , finally resulting in a different trend of C d changing from a nonmonotonic relation to a monotonic increase.
After elucidating the impact of every factor on C d , it is evident that Equations 13b, 13c, and 13d necessitate the estimation of longitudinal gradients, which are critical for computing dH sur dx and dH bed dx from images over a mobile bed.The adoption of quadratic functions (Equations 9 and 11) provides a reasonable description of the variations in H sur and H bed .We have opted for a logarithmic function that complies with the constraints (dH sur (x)/dx < 0 and d 2 H sur (x)/dx 2 < 0, dH bed (x)/dx < 0 and d 2 H bed (x)/dx 2 < 0) to investigate the sensitivity of the fitting functions concerning the drag coefficient.The imaged H sur (x) and H bed (x) were fitted as follows: where dH sur (x) dx = d 1 / (x e 1 ) , dH bed (x) dx = d 2 / (x e 2 ) , the parameters d 1 , e 1 , f 1 for H sur (x) and d 2 , e 2 , f 2 for H bed (x) were determined through nonlinear regression for each run, and their values are detailed in Table S2 in Supporting Information S1.The logarithmic model of boundary surfaces effectively captures the overall trend of the water and bed surfaces.Comparisons between the measured and fitted water surface and bed elevation are illustrated in Figure S6 in Supporting Information S1.Equation 14a provides a good fit with a correlation coefficient of R 2 ∈(0.85, 0.99) for the nonuniform water surface.However, for the bed surface, there is evidence of overfitting in the curves for R2.3, R3.1, R4.1, and R1.1, particularly at the trailing edge of the canopy (refer to Figure S6).For The drag coefficient model, Equation 12, coupled with the logarithmic functions (Equations 14a and 14b), is utilized to calculate the streamwise variation of the drag coefficient.Nonmonotonic variations in drag coefficient, similar to those depicted in Figures 5a and 5b, have previously been illustrated by Wang et al. (2015).We have selected runs R2.2 and R4.3 from Figures 5c and 5d to compare the drag coefficients for the logarithmic and quadratic functions (Figure 8).Both models for the drag coefficient exhibit the same increasing trend (with mean relative errors of 3.4% and 4.9%, respectively), and the rate of increase aligns with the behavior of each term in Figure 7. Indeed, the characteristics of bed elevation and water surface in nonuniform flow over a mobile bed, where gradients intensify along the streamwise direction, necessitate functions that satisfy the specified constraints in Equation 12to calculate the drag coefficient C d .The quadratic function selected represents one of the equations with a solid fitting performance.

The Influence of Bedforms
We examine the influence of bedforms, which results in a division of the total bed shear stress into two components: skin friction and form drag induced by the bedforms.Following Wiberg and Nelson (1992) and Le Bouteiller and Venditti (2015), the total bed shear stress is written as where C bf is the drag coefficient for the bedforms, ∆ and λ are the height and length of bedforms.U bf is the mean velocity for the vertical height k s < z < ∆, where k s = 2.5D 50 represents the grain roughness for the sand bed.U bf is given based on a logarithmic velocity (Le Bouteiller & Venditti, 2015), that is where κ = 0.408 is the von Karman constant.Substituting Equations 2, 3, 15a, and 15b into Equation 1, the vegetation drag coefficient C d including the form drag due to the bedforms can be obtained as where the bedform term C bf * is In order to compute C d from Equation 15c, values of C bf , ∆, and λ for the bedforms need to be determined.The bedform drag coefficient C bf of 0.17 was adopted from previous studies of asymmetric ripples (Le Bouteiller & Venditti, 2015;Wiberg & Nelson, 1992).For this study, the bedform height ∆ and length λ were employed to calculate C bf * based on the dynamic equilibrium location for sediment transport and the initial bed surface, as shown in Figure 9.This approach considers all deposition and erosion for the bedform.The bedform height is an absolute height |∆|.The bedforms are determined based on a relative zero crossing (Figure 9).In practice, this approach reasonably overestimates the drag for bedforms.
The nonuniform flow through the vegetated patch produces a strong driving pressure (P* A*).The pressure and advection terms enhance the coupled dynamics of flow and sediment motions, which drives the formation of pool and riffle on the mobile sand bed (de Almeida & Rodríguez, 2011).The riffle and pool are found along the vegetated patch at the stable stage, as shown in Figure 9.The persistence of this pool-riffle pattern is selfsustaining with respect to both flowing water and changes in bed terrain.This phenomenon can be attributed to the interplay of spatial patterns in flow velocity and bed-load transport, as discussed by Hassan et al. (2022), as well as the downstream control exerted by riffles in non-vegetated sections.As water enters the deeper pool (located at the trailing edge of the patch), its velocity decreases, subsequently reducing the initiation of sediment transport.This may result in a dynamic equilibrium in the movement of sediment from the riffle to the pool during the stable stage.
Here test R4.).However, it is noteworthy that bed erosion predominantly results in a general decrease in bed elevation along the streamwise direction rather than the formation of sand ripples.It is essential to highlight that in this study, the impact of the bedforms primarily manifests as an increased slope along the bed.These changes in bedform characteristics are accounted for within the bed slope term S 0 .
Additionally, the influence of the measured bed elevation data on the fitting parameters of the quadratic function is thoroughly discussed in Text S7 of the Supporting Information S1.The bed surface model effectively captures the overall changes in bed elevation and further predicts the longitudinal behavior of the measured drag coefficient.

Temporal Variation and the Approach to the Stable Drag Coefficient
The flow passing through the mobile sand bed with vegetation can be fundamentally regarded as a vegetated flow problem with temporally changing boundary conditions (Huai et al., 2021;Yang et al., 2021).Here, the most intense scouring test under the condition of Q = 1.8 L/s is selected for exploring the variation of the drag coefficient with time t. Figure 11 shows the change of the bed-load transport rate Q s over time under the condition of Q = 1.8 L/s and ∅ = 0.1, that is, test R4.3.Q s quantifies the bed erosion occurring as water flows through the vegetated patch, and this erosion rate gradually decreases from 20.2 to 3.6 g • (m • s) 1 .At 25 min and 30 s, Q s reached a relatively stable value with subsequent fluctuations smaller than 10%.The bed-load transport rate in this stage is meager, which indicates that the sediment was continuously and stably scoured within the finite vegetation canopy (Wu et al., 2021a(Wu et al., , 2021b)), similar to results for a natural river bank (Huai et al., 2021).to the sediment transport is insignificant (p > 0.05) for C d at the leading edge of the vegetated patch.However, bed-load transport in the nonuniform flow changed the trend of C d .This is likely due to the fact that the vegetationgenerated TKE is more intense at the trailing edge of the vegetated patch (refer to Figure S8 in Supporting Information S1), which enhances sediment transport and dynamic alterations in flow nonuniformity over time.Consequently, the drag exerted by the upstream vegetation on the water volume may diminish in response to balance the downstream TKE and establish a stable, low sediment transport rate within the vegetated patch (Huai et al., 2020;Yang & Nepf, 2018).The findings herein offer quantitative insights into the longitudinal vegetation drag experienced on a mobile sand bed.Such data can be applied to assess bed-load transport resulting from turbulence generated by vegetation in shallow nonuniform flows.

Model Application and Limitation
Overall, an increase in the vegetation drag and bed slope reduces sediment transport rates.This serves as a primary mechanism for promoting stability in nonuniform vegetated flows, paralleling the sediment transport dynamics observed in gravel-bed stream channels (Church et al., 1998).Within the vegetated patch, bed elevation development exhibits a self-organized process driven by the interplay of vegetation drag (Le Bouteiller & Venditti, 2014) and flow nonuniformity (Wang et al., 2023).Future research can leverage this drag coefficient model to characterize the drag of various  Water vegetation types and predict the bed-load transport rates in nonuniform flows.Additionally, the experimental conditions in this study are in alignment with the underlying assumptions.Despite the flume width being fixed at 0.3 m, the experimental runs cover a broad spectrum of aspect ratios (width/flow depth), ranging from 3.75 to 18.75.This range is consistent with the magnitude reported by Yang et al. (2015), which spans from 9.23 to 17.14.The experimental boundary conditions, including the vegetated patch length and tail-gate, remained constant throughout this study.However, a comprehensive investigation into the impact of these boundary conditions on the drag coefficient over a mobile bed requires further exploration.
The drag model developed in this study represents a step forward in predicting the drag coefficient within emergent vegetation over a mobile bed.Indeed, it exhibits certain limitations and assumptions.First, it is designed to operate within the vegetation canopy represented by cylinder arrays, which have a vertically uniform distribution of biomass.The model does not account for the potential influence of variations in vegetation morphology on shear stress and flow nonuniformity.Second, the thickness of the viscous layer H v exceeds the sediment size s , with a calculated range of H v from 1.4 to 1.9 mm for the densest run R4.1.This justifies the applicability of the linear-stress model to rough sand beds in our study.Under high Reynolds number conditions of Re d ≥ 4/C f , Equation 4 simplifies to a constant formula for the shear stress over a bare bed.This simplification reflects the reduction of the viscous sublayer due to vegetation-generated turbulence (Zhao & Nepf, 2021).Notably, while the quantification of C f was not the primary focus of this study, the C f applied in Equation 6 is derived from a semiempirical approach for sand beds in the presence of vegetation (Liu et al., 2021;Shan et al., 2020;Yang & Nepf, 2019).This approach takes into account variations in velocity U and water depth H to capture streamwise changes of C f in nonuniform flow conditions.Lastly, this study presents a one-dimensional drag model that assumes a uniform scouring transverse cross-section.This model is designed for application in shallow nonuniform flows with very low sediment transport rates, where the bed elevation generally exhibits an overall declining trend.Under such conditions, variations in local bedforms are incorporated within the slope term S 0 * .
However, if local bedforms act as controls on the sediment particle travel distances (Singh et al., 2023;Wu et al., 2021c), further enhancements to the model are warranted to account for the influence of these bedforms on the drag coefficient.

Conclusion
The emergent vegetation array creates a nonuniform water surface by introducing additional drag to the flow.Meanwhile, it exerts a significant influence on sediment retention and transport.This work investigated the drag coefficient of rigid vegetation in shallow nonuniform flow over a mobile sand bed.The drag coefficient C d was calculated based on laboratory flume experiments utilizing a momentum-based approach.This approach was applied under conditions characterized by a low sediment transport rate.The principal findings of this research highlight the following: 1.A general formula for the drag coefficient C d of a rigid vegetation array in nonuniform flow was proposed, which, for the first time, integrates the terms of the flow properties and bed elevation effects.Overall, the longitudinal variation of C d exhibits either a monotonically increasing or a nonmonotonic trend as x increases.This behavior is influenced by both the bed slope S 0 and the water surface gradient S w .2. These two trends of C d are determined by the pressure term P*, which is sensitive to variations in the bed slope term S 0 * .The increase of S 0 * affects flow nonuniformity and eventually reverses the change trend of P*, resulting in two different trends for C d .The driving term enhances the coupled dynamics of flow and sediment transport, resulting in the self-maintenance of the mobile bed topography.3.For flow through a vegetation array on a sand bed, the drag coefficient C d shifts from a nonmonotonic change at the initial stage to a monotonically increasing one.This change occurs as the bed slope term becomes larger than the advection term due to the bed-load transport.The energy of the nonuniform flow is expended in overcoming both the bed slope and the increasing C d , which ultimately leads to a stable, low sediment transport rate.
Future research should encompass testing the drag model across a broader spectrum of hydraulic gradients, sediment particle sizes, and diverse types of vegetation.Such comprehensive investigations will contribute to a more robust understanding of the relationship between C d and its associated factors (E veg , P*, A*, S 0 * , and C f * ).
Additionally, there is a need for further exploration into bed shear stress and sediment transport flux along nonuniform flows in the presence of vegetation.
Water Resources Research 10.1029/2023WR036535 ZHANG ET AL.

Figure 1 .
Figure 1.The schematic of the rigid vegetated patch in the test section of the flume for nonuniform flow on the mobile sand bed for R4.3 (not to scale).

Figure 2 .
Figure 2. (a) The schematic of the vegetation elements arranged in a staggered configuration within a rectangular area for ∅ = 0.30.(b) The grain size distribution of sediments used in the experiments.(c) The experimental flume with the vegetation model and the sand-laying device (a set of square sieves).

Figure 3 .
Figure 3. Illustrations for the vegetation zone in the test section of the flume, with images of the water surface and bed elevation taken by a sideview camera for ∅ = 0.15 and ∅ = 0.06 in the experiments.Subfigures on the left and the right columns represent the stable stage and the initial stage of each run, respectively.

Figure 4 .
Figure 4.The modeled and measured water surface H sur (x) and bed surface H bed (x) along the vegetated patch in the streamwise direction for nine tests, varying Q and ∅.The black open triangles indicate the measured water level, the black open circles represent the measured data of the bed surface, the red lines and blue lines are the fitted quadratic functions for the water surface and the bed surface of each run, respectively.

Figure 5 .
Figure 5.The streamwise variations of the normalized drag coefficient C d /〈C d 〉 with x + for the rigid vegetation over a mobile bed at the stable stage using data from all experimental tests.Subfigures (a) and (b) represent the weak bed-load transport conditions showing a nonmonotonic change trend.Subfigures (c) and (d) are for the moderate transport conditions and mostly show an overall increasing trend.

Figure 6 .
Figure 6.The streamwise variations of the normalized drag coefficient C d init / 〈C d init 〉 with x + for the rigid vegetation over a mobile bed at the initial stage.Subfigures (a), (b), and (c) show a strong nonmonotonic trend.Subfigure (d) shows a weak nonmonotonic trend.
, Equation 14b fits well with the experimental data.Overall, the logarithmic function serves a similar role as the quadratic function in describing the water surface and bed elevation of the nonuniform flow.

Figure 7 .
Figure 7. Analysis of the shape of C d along the normalized streamwise coordinate x + for Runs R2.1, R1.2, R2.3, and R4.3.Solid lines denote the cases with different Q and ∅.

Figure 8 .
Figure 8.A comparison of the drag coefficient using the logarithmic and quadratic functions for R2.2 and R4.3.

Figure 9 .
Figure 9.The spatial variation of the bedform profile along the streamwise distance x.The measured and modeled water surface and the bed topography profile are illustrated with different colors.The blue area indicates the water volume, and the yellow region represents the sand bed layer.The peak for the bedform greater than the initial bed thickness (z = 3 cm) is used to estimate the bedform height ∆ and length λ based on the equilibrium location for the sediment transport.

Figure 10 .
Figure 10.The variation of the bedform effect C bf * along the streamwise distance for the R4.3 case.(a) The variations of bed The variations of the mobile sand bed surface and the water surface are recorded by the camera used in the experiment.The video file for the R4.3 case is attached in Supporting Information S1 Data (https://doi.org/10.6084/m9.figshare.20710330.v3).To explore the variation of the drag coefficient with time under mobile sand bed conditions, we extracted the pictures at t = 16 min and 30 s, and at 22 min and 30 s.The values of C d,t1 and C d,t2 represent the drag coefficients of vegetation at these two moments, respectively.These values are calculated according to Equation 8 and shown in Figure 12.C d,t1 and C d,t2 present a similar monotonically increasing trend, which is consistent with the change of C d stable (the drag coefficient at the stable stage).The findings emphasize the importance of the bed slope term S 0 * as long as the sand bed experiences erosion.This, in turn, impacts the water depth H and leads to a reversal in the trend of C d , particularly noticeable at the trailing edge of the vegetated patch.Figure 12 also shows the variation of C d init with x + at the initial stage for case R4.3.We compared drag coefficients between the temporal variation conditions (mobile bed) and those during the initial stage (resembling a fixed bed), with the same roughness characteristics (D 50 = 0.854 mm).We observed that the values of C d,t1 , C d, t2 , C d stable , and C d init are similar at the leading edge of the vegetated patch, while the difference between the first three values and C d init mainly appears at the patch's trailing edge.This finding shows that the change in S 0 * due

Figure 11 .
Figure 11.The measured temporal bed-load transport rate Q s over time for the R4.3 case under the discharge of Q = 1.8 L/s.

Figure 12 .
Figure 12.Variations of drag coefficients with x + .C d,t1 , C d,t2 , C d stable , and C d init represent drag coefficients of vegetation at t = 16 min and 30 s, 22 min and 30 s, the stable stage, and the initial stage, respectively.C d init shows a nonmonotonic trend, while the other drag coefficients all show a monotonically increasing trend. )

Table 1
Parameters of Water Surface and Bed Surface Profiles at the Stable Stage of Each Run in the ExperimentQ

Table 2
Parameters of Each Run at the Initial Stage in the Experiment ZHANG ET AL.