# On Bed Form Resistance and Bed Load Transport in Vegetated Channels

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}), and H is flow depth (m). For channels with submerged vegetation having H/h

_{v}= 3, where h

_{v}is vegetation height, bed resistance is reduced to just 10% of the bare bed value [14,15].

## 2. Experimental Setup

#### 2.1. Flume Setup

^{2}/4, where N is the number of stems per unit bed area, and d is the outsider diameter of stem. The vegetation concentrations for these three configurations are 0.033, 0.014, and 0.005, respectively.

_{50}= 0.45 mm and 1.6 mm, were used. The standard deviations of sediment mixtures defined as ${\sigma}_{g}={({d}_{84}/{d}_{16})}^{0.5}$, were 1.47 and 1.35 for sediment sizes of 0.45 mm and 1.6 mm, respectively. Sediment mixtures with the value of ${\sigma}_{g}$ less than 1.6 are considered as weakly non-uniform [24]. The density of sediment, ${\rho}_{s}$, is equal to 2650 kg/m

^{3}. Before each experimental run, sediment was saturated and placed evenly on bed surface at a depth of 10 cm (Figure 2).

#### 2.2. Water and Bed Slopes

#### 2.3. Bed Load Transport Rate

^{2}/s). All the measurements were taken after flow had reached steady state and are summarized in Table 1.

#### 2.4. Bed Surface Elevation

## 3. Data Processing

#### 3.1. Grain Resistance

^{2}), $\rho $ is water density (kg/m

^{3}), H

_{s}is the equilibrium flow depth (m), S is bed slope, $\kappa $ is von Karman constant (=0.41), ${k}_{sg}$ is grain roughness height (=2.5d

_{50}), and $U$ is mean flow velocity (m/s) [=Q/(BH)], where B is channel width (m). According to the above two equations, Equation (1) is used to calculate the equilibrium flow depth, and Equation (2) is used to calculate the grain resistance, ${\tau}_{g}$.

#### 3.2. Sidewall Resistance

_{v}is mean pore velocity though the vegetation [V

_{v}= Q/(BH(1 − ϕ))] [12,22]; ${f}_{w1}$ and ${f}_{w2}$ are the Darcy–Weisbach friction coefficient for the glass and stainless steel sidewalls, respectively. These coefficients can be obtained by using the Colebrook equation:

^{−5}m. R

_{e}is flow Reynolds number defined as ${\mathrm{R}}_{e}=4r\xb7{V}_{v}/\upsilon $, where r is the total hydraulic radius (m) (see Equation (10)), and $\upsilon $ is the kinematic viscosity of water (m

^{2}/s).

#### 3.3. Bed Form Resistance

^{3}), $\forall $ is the volume of water (m

^{3}) [$\forall ={A}_{bed}\xb7H$] in which ${A}_{bed}$ is bed surface area (m

^{2}), ${\tau}_{bf}$ is bed form resistance (N/m

^{2}), A

_{w}

_{1}and A

_{w}

_{2}are the areas of the glass and the stainless steel sidewall surface, respectively (m

^{2}), and F

_{D}is the vegetation drag force (N). For a reach of unit length and width, B, the bed surface area is ${A}_{bed}=B(1-\varphi )$ and the glass and stainless steel sidewall surface areas are A

_{w}

_{1}= A

_{w}

_{2}= H.

_{D}, can be calculated as:

- 1.
- Calculate the modified vegetation-related hydraulic radius, ${r}_{vm}$, using Equation (9).
- 2.
- Calculate Reynolds number for the pseudo-fluid model, ${R}_{e}^{\prime}$, using Equation (8).
- 2.
- Calculate the drag coefficient for the pseudo-fluid model, ${{C}_{D}}^{\prime}$, using Equation (7).
- 4.
- Calculate the vegetation drag coefficient, ${C}_{D}$, using Equation (6).
- 5.
- Calculate the vegetation drag force, ${F}_{D}$, using Equation (5).
- 6.
- Calculate the bed form resistance, ${\tau}_{bf}$, using Equation (4).
- 7.
- Calculate the Darcy–Weisbach bed form friction coefficient using ${f}_{bf}=8{\tau}_{bf}/(\rho {V}_{v}^{2})$.
- 8.
- Recalculate the Darcy–Weisbach bed friction coefficient using ${f}_{b}={f}_{g}+{f}_{bf}$.
- 9.
- Repeat step #4 until the difference between the calculated and the assumed values of ${f}_{b}$ is within a desired tolerance.

_{v}, ${C}_{D}$, and ${\tau}_{bf}$ values for all the experiments are shown in Table 2. All the experimental flows are subcritical with Froude number (F

_{r}) ranging from 0.162 to 0.343. This approach of separating the total bed resistance into grain and bed form resistances was also applied for estimating bed load transport rate in one-dimensional hydrodynamic model [32].

#### 3.4. Bed Form Height

_{avg}, for ϕ = 0.014 and 0.033 are equal to 7.0 and 5.7 mm, respectively. This shows that bed form height was slightly decreased with the increasing of vegetation concentration.

_{avg}value is equal to 12 mm larger than that for other $\varphi $ values. For d

_{50}= 0.45 mm, as shown in Table 2, the ΔZ value is slightly increased because smaller sized sand dunes were observed. When d

_{50}=1.6 mm, as shown in Figure 5b and Table 2, the ΔZ values are increased as flow velocity is increased due to the increasing of sand dunes’ sizes. This implies that sand dunes were developed through the sparse vegetation as flow velocity increases.

_{avg}versus $\varphi $ (Figure 6) indicated that the ΔZ

_{avg}is decreased rapidly as the $\varphi $ value increased from 0.005 to 0.014, and then decreased gradually as the $\varphi $ value increased from 0.014 to 0.033. This trend is consistent with the evolution of bed form from sand dunes at low vegetation concentration to fully developed scour holes around each vegetation stem at high concentration.

## 4. Empirical Relations and Methods

#### 4.1. Bed Form Resistance Relation

_{bf}is bed form drag coefficient for bed form, Δ is the height of bed form, λ is the length of bed form, and U

_{bf}is the mean velocity within the height of bed form for bed surface free of vegetation, which is proportional to the vegetation concentration and mean pore velocity as:

_{g}/τ

_{c}− 1. Bed form is at low energy regime (i.e., ripple, dune) when T < 25, and high energy regime (i.e., dynamic flat bed) when T ≥ 25. For ripples and dunes at T < 25, the ratio of bed form height and length is:

#### 4.2. Bed Load Transport Relation

_{b}is bed load particle velocity in non-vegetated channel, ${\tau}_{g}^{*}$ is non-dimensional value of grain resistance [${\tau}_{g}^{*}={\tau}_{g}/\{({\rho}_{s}-\rho )g{d}_{50}\}$], G

_{s}is the specific gravity of the sediment and equals 2.65, C

_{M}is the added mass coefficient for sediment particles in water, which has a theoretical value of 0.5 for spherical particles [38], ${\omega}_{1}$ and ${\omega}_{2}$ are coefficients that correlate bed load saltation length to non-dimensional bed shear stress originated in Equations (12) and (13) in Shim and Duan [37]. Bed load velocity in vegetated channels is assumed to be proportional to the one in non-vegetated channels as:

_{d}is the dynamic friction angle. For mild sloped channel bed similar to the experimental conditions cited in this paper, η=1.0 is used in Equation (23). In additions the non-dimensional grain shear replaced the non-dimensional bed shear stress in the original equation [34]. However, Equation (23) is applicable to non-vegetated channel, incorporating the effect of vegetation roots in bed load layer, we assume the thickness of bed load in mild sloped vegetated channel is analogous to Equation (23) as:

_{1}are coefficients that differentiate the calculation of bed load thickness in vegetated and non-vegetated channel. Then, Equation (22) can be written as:

_{M}= 0.5 for spherical particles. The dynamic friction angle is assumed to be ${35}^{\circ}$ for medium size sand. Shim and Duna [37] found ${\omega}_{1}=26.3\mathrm{and}{\omega}_{2}=34.6$ from experimental data. In this study, the non-dimensional critical shear stress is 0.034. Unfortunately, there is no measurements of saltation particle length in vegetated channel. Therefore, we assumed these constants are valid for vegetated channels. Then, Equation (26) can be further simplified as:

_{1}and D

_{2}are empirical coefficients that need to be determined by observed data.

## 5. Downhill Simplex Method to Determine the Coefficients

^{2}) was calculated to find the best match between the predictions from Equations (19) and (27) and the experimental data measured (observed). The NSE and R

^{2}values were calculated using the following two equations:

#### 5.1. Optimal Coefficient Set in Bed Form Resistance Relation

_{1}= 205.156 and C

_{2}= 2.484. Unfortunately, the maximum NSE value was only 0.034 and R

^{2}was 0.117. This indicates Equation (19) is not an appropriate function for bed form resistance. In order to improve the NSE value, we modified the constants in Equation (19) and added calibration coefficients while maintaining the original variables. The modified equation can be written as:

_{1}= 0.238, C

_{2}= 0.519, C

_{3}= 1.533, C

_{4}= 0.638, and C

_{5}=−1.034. The corresponding R

^{2}value for this optimum coefficient set was 0.890 as shown in the scatter plot (Figure 8). These values of the NSE and R

^{2}show that the predicted Equation (30) and the observed non-dimensional bed form resistances are correlated well for all the datasets. The positive exponential for vegetation concentration indicates bed form resistance increases with vegetation concentration.

#### 5.2. Optimal Coefficient Set for Bed Load Transport Relation

_{1}and D

_{2}) on 1.0-NSE values throughout the DSM iterations is shown in Figure 9. Apparently, both coefficients converge to constants at the maximum NSE value. The optimum coefficient values for the minimum 1.0-NSE value of 0.019 (or the maximum NSE value of 0.981) are D

_{1}= 1.919 and D

_{2}= −0.168. The corresponding R

^{2}value for the above optimum coefficient set is 0.981 as shown in the scatter plot (Figure 10). The resultant equation of the bed load transport rate with the optimal coefficients is as below:

_{2}= −0.168 in Equation (31) is negative, meaning the bed load transport rate is reducing with vegetation concentration. If the total bed resistance remains as a constant, the increase in vegetation concentration will increase bed form resistance (Equation (30)) so that grain resistance will be reduced. Consequently, the bed load transport rate reduces with vegetation concentration as seen in Equation (31). However, D

_{2}= −0.168, which is close to zero. Therefore, to quantify the influence of the vegetation concentration on the prediction of the bed load transport rate, Equation (31) is reevaluated using DSM optimization without the ϕ parameter. The values of NSE and R

^{2}are 0.653 and 0.985, respectively. Although the maximum values of R

^{2}with and without the incorporation of the vegetation concentration are approximately the same, the maximum NSE value with the incorporation of the vegetation concentration (NSE = 0.981) is greater than the one without the vegetation parameter (NSE = 0.653). This means that the vegetation concentration has a moderate effect on the prediction of bed load transport (Equation (31).

## 6. Discussion

_{D}, which varies with flow Reynold number, and approaches a constant for a single cylinder in fully turbulent flow. In this study, C

_{D}is required in each experiment for quantifying the drag force induced by vegetation stems. In order to determine the resistance on vegetation stems (Equation (5)), the grain resistance and the sidewall resistance were calculated first by Equations (2) and (3), respectively. Second, the total bed shear stress, consisting of grain resistance and bed form resistance, were assumed in Step #4. By knowing the grain and sidewall resistances, the vegetation drag coefficient C

_{D}was calculated in Step #4.1–4.4. Third, the vegetation resistance and bed form resistance were calculated in Step #4.5 and Step #4.6, respectively. Fourth, we converted bed form resistance to the Darcy–Weisbach resistance coefficient (Step #4.7). In Step #4.8-4.9, we recalculated the bed resistance. This bed resistance must be equal to the assumed bed resistance; otherwise, the assumed bed resistance was adjusted until it converged. This procedure calculates the bed form resistance after knowing the grain resistance and vegetation drag resistance. The C

_{D}value is dependent on the vegetation stem Reynolds number (${\mathrm{Re}}_{d}={V}_{v}d/\nu $) (Figure 11), flow Reynolds number (${\mathrm{Re}}_{H}=UH/\nu $) (Figure 12), and vegetation concentration (ϕ) (Figure 13).

_{D}ranges from 0.75 to 1.18 for vegetation concentration from 0.002 to 0.033, vegetation stem Reynolds number (780 < ${\mathrm{Re}}_{d}$ < 6844), and flow Reynolds number (3444 < ${\mathrm{Re}}_{H}$ < 67,675). The stem Reynolds number indicates the turbulence strength near the stems, while the flow Reynolds number measures the turbulence in the main flow. As we have seen, flow is fully turbulent, with the minimum Reynolds number of 3444. Therefore, the C

_{D}coefficient varies within a narrow range from 0.75 to 1.18 based on the trial–error calculation. Figure 11 and Figure 12 show that C

_{D}reduces as the Reynolds number increases, which is consistent with other studies [20,23]. Figure 13 shows that C

_{D}increases with vegetation concentration due to the impact of overlapped vorticity structures.

## 7. Conclusions

^{2}values. The results showed that vegetation concentration has moderate impacts on bed form resistance and bed load transport. As vegetation concentration increases, bed form resistance will increase, while the bed load transport rate will reduce. This explains that a high-density vegetated channel blocks bed load transport to downstream reaches. Nevertheless, these conclusions were drawn from the limited laboratory experimental data, and require additional data of sediment transport in a densely vegetated channel and field to verify their applicability.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notation

a | vegetation frontal area per unit volume (m^{−1}); |

$aH$ | vegetation roughness density (-); |

${A}_{bed}$ | bed surface area (m^{2}); |

B | channel width (m); |

${C}_{D}$ | drag coefficient for a cylindrical emergent stem; |

${C}_{D}^{\prime}$ | drag coefficient for the pseudo-fluid model; |

C_{bf} | bed form drag coefficient (-); |

d | vegetation stem diameter (mm); |

d_{16,}d_{84} | sizes for which 16% and 84% of the sediment are finer than d_{16} and d_{84}, respectively (mm); |

d_{50} | median sediment size (mm); |

F_{D} | vegetation drag force (N); |

Fr | Froude number (-); |

g | gravity acceleration (m/s^{2}); |

G_{s} | specific gravity of the sediment (-); |

H | flow depth (m); |

H_{s} | equilibrium flow depth (m); |

m | total number of data; |

N | number of stems per unit bead area (m^{−2}); |

NSE | Nash–Sutcliffe efficiency coefficient (-); |

n | total number of bed elevation points; |

O | observed value of non-dimensional bed form resistance or non-dimensional bed load transport rate; |

P | predicted value of non-dimensional bed form resistance or non-dimensional bed load transport rate; |

Q | flow rate (m^{3}/s); |

${q}_{b}$ | bed load transport rate (m^{2}/s); |

${q}_{b}^{*}$ | non-dimensional bed load transport; |

R^{2} | coefficient of determination (-); |

${\mathrm{Re}}^{\prime}$ | Reynolds number for the pseudo-fluid model (-); |

${\mathrm{Re}}_{H}$ | flow Reynolds number (-); |

${\mathrm{Re}}_{d}$ | vegetation stem Reynolds number (-); |

r | total hydraulic radius (m); |

${r}_{v}$ and ${r}_{vm}$ | vegetation-related, and modified vegetation-related hydraulic radii respectively (m); |

S | bed slope (-); |

S_{s} | vegetation stem spacing (mm); |

T | bed mobility factor (-); |

$U$ | mean flow velocity (m/s); |

U_{bf} | mean velocity within the height of bed form (m/s); |

u_{b} | bed load particle velocity in non-vegetated channel (m/s); |

u_{b-veg} | bed load particle velocity in vegetated channel (m/s); |

V_{v} | mean pore velocity (m/s); |

X, Y, and Z | position of the bed points (points clouds) in space and distance (depth) (m); |

${Z}_{i}$ | bed elevation at any point i (m); |

$\overline{Z}$ | mean bed elevation of original bed surface (m); |

Δ | height of bed form; |

ΔZ | bed form height (mm); |

ΔZ_{avg} | average of the bed form height for each ϕ value (mm); |

ζ_{b} | bed load layer thickness in non-vegetated channel; |

ζ_{b-veg} | bed load layer thickness in vegetated channel; |

$f$ | total Darcy–Weisbach friction coefficient (-); |

${f}_{b}$ | bed Darcy–Weisbach friction coefficient=${f}_{g}+{f}_{bf}$(-); |

${f}_{g},{f}_{bf}$ | grain and bed form Darcy–Weisbach friction coefficients, respectively (-); |

ϕ | vegetation concentration (-); |

ϕ_{d} | dynamic friction angle; |

$\gamma $ | specific weight of water (N/m^{3}); |

λ | length of bed form; |

$\mu $ | water dynamic viscosity (N$\xb7$s/m^{2}); |

$\nu $ | kinematic viscosity of water (m^{2}/s); |

${k}_{sg},{k}_{sw1},{k}_{sw2}$ | grain, glass, and stainless steel roughness heights (m) |

$\kappa $ | von Karman constant = 0.41 (-); |

$\rho ,{\rho}_{s}$ | water and sediment density, respectively (kg//m^{3}); |

${\sigma}_{g}$ | standard deviation of sediment mixture; |

${\tau}_{b}$ | total bed resistance (N/m^{2}) = ${\tau}_{g}+{\tau}_{bf}$; |

${\tau}_{g},{\tau}_{bf}$ | grain and bed form resistances, respectively (N/m^{2}); |

${\tau}_{w1}$, ${\tau}_{w2}$ | glass and stainless steel sidewall resistances, respectively (N/m^{2}); |

${\tau}_{bf}^{*}$, ${\tau}_{g}^{*}$, ${\tau}_{c}^{*}$ | non-dimensional bed form, grain, and critical resistances, respectively; |

$\forall $ | volume of water (m^{3}); |

β_{1}, β_{2}, α_{1}, α_{2}, α_{3}, α_{4}, C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, η, D_{1}, D_{2}, ω_{1}, ω_{2} | coefficients. |

## References

- Fenzl, R.N. Hydraulic Resistance of Broad Shallow Vegetated Channels. Ph.D. Thesis, University of California, Davis, CA, USA, 1962. [Google Scholar]
- Temple, D.M.; Robinson, K.M.; Ahring, R.M.; Davis, A.G. Stability Design of Grassed-Lined Open Channels; US Department of Agriculture, Agricultural Research Service: Washington, DC, USA, 1987. [Google Scholar]
- Shi, Z.; Pethick, J.S.; Pye, K. Flow structure in and above the various heights of a saltmarsh canopy: A laboratory flume study. J. Coast. Res.
**1995**, 11, 1204–1209. [Google Scholar] - Zhang, M.-L.; Li, C.; Shen, Y.-M. A 3D non-linear k–ε turbulent model for prediction of flow and mass transport in channel with vegetation. Appl. Math. Model.
**2009**, 34, 1021–1031. [Google Scholar] [CrossRef] - Yang, J.Q.; Chung, H.; Nepf, H.M. The onset of sediment transport in vegetated channels predicted by turbulent kinetic energy. Geophys. Res. Lett.
**2016**, 43, 11261–11268. [Google Scholar] [CrossRef][Green Version] - Yang, J.Q.; Nepf, H.M. A Turbulence-Based Bed-Load Transport Model for Bare and Vegetated Channels. Geophys. Res. Lett.
**2018**, 45, 10428–10436. [Google Scholar] [CrossRef] - Yang, J.Q.; Nepf, H.M. Impact of Vegetation on Bed Load Transport Rate and Bedform Characteristics. Water Resour. Res.
**2019**, 55, 6109–6124. [Google Scholar] [CrossRef] - Zhao, T.; Nepf, H.M. Turbulence Dictates Bedload Transport in Vegetated Channels Without Dependence on Stem Diameter and Arrangement. Geophys. Res. Lett.
**2021**, 48, e2021GL095316. [Google Scholar] [CrossRef] - Zanke, U.; Roland, A.; Wurpts, A. Roughness Effects of Subaquaeous Ripples and Dunes. Water
**2022**, 14, 2024. [Google Scholar] [CrossRef] - Engelund, F. Hydraulic Resistance for Flow over Dunes. In Progress Report 44; Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark: Lyngby, Denmark, 1977. [Google Scholar]
- Jordanova, A.A.; James, C.S. Experimental Study of Bed Load Transport through Emergent Vegetation. J. Hydraul. Eng.
**2003**, 129, 474–478. [Google Scholar] [CrossRef] - Kothyari, U.C.; Hashimoto, H.; Hayashi, K. Effect of tall vegetation on sediment transport by channel flows. J. Hydraul. Res.
**2009**, 47, 700–710. [Google Scholar] [CrossRef] - López, F.; García, M. Open-channel flow through simulated vegetation: Suspended sediment transport modeling. Water Resour. Res.
**1998**, 34, 2341–2352. [Google Scholar] [CrossRef] - Al-Asadi, K.; Duan, J.G. Three-Dimensional Hydrodynamic Simulation of Tidal Flow through a Vegetated Marsh Area. J. Hydraul. Eng.
**2015**, 141, 06015014. [Google Scholar] [CrossRef] - Al-Asadi, K.; Duan, J.G. Assessing methods for estimating roughness coefficient in a vegetated marsh area using Delft3D. J. Hydroinformatics
**2017**, 19, 766–783. [Google Scholar] [CrossRef][Green Version] - Wang, X.; Huai, W.; Cao, Z. An improved formula for incipient sediment motion in vegetated open channel flows. Int. J. Sediment Res.
**2021**, 37, 47–53. [Google Scholar] [CrossRef] - Wang, X.; Li, S.; Yang, Z.-H.; Huai, W.-X. Incipient sediment motion in vegetated open-channel flows predicted by critical flow velocity. J. Hydrodyn.
**2022**, 34, 63–68. [Google Scholar] [CrossRef] - Hashimoto, H.; Hirano, M. Gravity Flow of Sediment-Water Mixtures in a Steep Open Channel. Doboku Gakkai Ronbunshu
**1996**, 1996, 33–42. [Google Scholar] [CrossRef][Green Version] - Specht, F.-J. Einfluß von Gerinnebreite und Uferbewuchs auf die Hydraulisch-Edimentologischen Verhältnisse Naturnaher Fließgewässer; TU Braunschweig: Braunschweig, Germany, 2002. [Google Scholar]
- Nepf, H.M. Hydrodynamics of vegetated channels. J. Hydraul. Res.
**2012**, 50, 262–279. [Google Scholar] [CrossRef][Green Version] - Follett, E.M.; Nepf, H.M. Sediment patterns near a model patch of reedy emergent vegetation. Geomorphology
**2012**, 179, 141–151. [Google Scholar] [CrossRef][Green Version] - Cheng, N.-S.; Nguyen, H.T. Hydraulic Radius for Evaluating Resistance Induced by Simulated Emergent Vegetation in Open-Channel Flows. J. Hydraul. Eng.
**2011**, 137, 995–1004. [Google Scholar] [CrossRef] - Cheng, N.-S. Calculation of drag coefficient for arrays of emergent circular cylinders with pseudofluid model. J. Hydraul. Eng.
**2013**, 139, 602–611. [Google Scholar] [CrossRef] - Parker, G. Transport of gravel and sediment mixtures. In Sedimentation Engineering: Processes, Measurements, Modelling and Practice; Garcia, M., Ed.; ASCE: Reston, VA, USA, 2008; pp. 165–251. [Google Scholar]
- Mankoff, K.D.; Russo, T.A. The Kinect: A low-cost, high-resolution, short-range 3D camera. Earth Surf. Process. Landforms
**2012**, 38, 926–936. [Google Scholar] [CrossRef] - Caviedes-Voullième, D.; Juez, C.; Murillo, J.; García-Navarro, P. 2D dry granular free-surface flow over complex topography with obstacles. Part I: Experimental study using a consumer-grade RGB-D sensor. Comput. Geosci.
**2014**, 73, 177–197. [Google Scholar] [CrossRef] - Kahn, S.; Bockholt, U.; Kuijper, A.; Fellner, D.W. Towards precise real-time 3D difference detection for industrial applications. Comput. Ind.
**2013**, 64, 1115–1128. [Google Scholar] [CrossRef] - Azzari, G.; Goulden, M.L.; Rusu, R.B. Rapid Characterization of Vegetation Structure with a Microsoft Kinect Sensor. Sensors
**2013**, 13, 2384–2398. [Google Scholar] [CrossRef][Green Version] - Andersen, M.R.; Jensen, T.; Lisouski, P.; Mortensen, A.K.; Hansen, M.K.; Gregersen, T.; Ahrendt, P.J.A.U. Kinect depth sensor evaluation for computer vision applications. In Electrical and Computer Engineering, Tech. Rep.; ECE-TR-6, Department of Engineering, Aarhus University: Aarhus, Denmark; Available online: http://eng.au.dk/fileadmin/DJF/ENG/PDF-filer/Tekniske_rapporter/Technical_Report_ECE-TR-6-samlet.pdf (accessed on 11 October 2022).
- Le Bouteiller, C.; Venditti, J.G. Sediment transport and shear stress partitioning in a vegetated flow. Water Resour. Res.
**2015**, 51, 2901–2922. [Google Scholar] [CrossRef] - Einstein, H.A.; Banks, R.B. Fluid resistance of composite roughness. Trans. Am. Geophys. Union
**1950**, 31, 603–610. [Google Scholar] [CrossRef] - Bai, Y.; Duan, J.G. Simulating unsteady flow and sediment transport in vegetated channel network. J. Hydrol.
**2014**, 515, 90–102. [Google Scholar] [CrossRef] - Sturm, T. Open Channel Hydraulics; McGraw-Hill: Boston, MA, USA, 2001. [Google Scholar]
- Zanke, U.; Roland, A. Sediment Bed-Load Transport: A Standardized Notation. Geosciences
**2020**, 10, 368. [Google Scholar] [CrossRef] - Zhang, S.; Duan, J.G.; Strelkoff, T.S. Grain-Scale Nonequilibrium Sediment-Transport Model for Unsteady Flow. J. Hydraul. Eng.
**2013**, 139, 22–36. [Google Scholar] [CrossRef] - Shim, J.; Duan, J.G. Experimental study of bed-load transport using particle motion tracking. Int. J. Sediment Res.
**2017**, 32, 73–81. [Google Scholar] [CrossRef] - Shim, J.; Duan, J. Experimental and theoretical study of bed load particle velocity. J. Hydraul. Res.
**2018**, 57, 62–74. [Google Scholar] [CrossRef] - Fernandez-Luque, R.; van Beek, R. Erosion and transport of bed-load sediment. J. Hydraul. Res.
**1976**, 14, 127–144. [Google Scholar] [CrossRef] - Nelder, J.A.; Mead, R. A Simplex Method for Function Minimization. Comput. J.
**1965**, 7, 308–313. [Google Scholar] [CrossRef] - Xu, B.; Heidari, A.A.; Kuang, F.; Zhang, S.; Chen, H.; Cai, Z. Quantum Nelder-Mead Hunger Games Search for optimizing photovoltaic solar cells. Int. J. Energy Res.
**2022**, 46, 12417–12466. [Google Scholar] [CrossRef] - Al-Asadi, K.; Abbas, A.A.; Hamdan, A.N. Optimization of the Hydrological Tank Model by Downhill Simplex Method. Int. J. Civ. Eng.
**2020**, 18, 1433–1450. [Google Scholar] [CrossRef] - Alvarez-Vázquez, L.J.; Júdice, J.J.; Martínez, A.; Rodríguez, C.; Vázquez-Méndez, M.E.; Vilar, M.A. On the optimal design of river fishways. Optim. Eng.
**2011**, 14, 193–211. [Google Scholar] [CrossRef] - Han, L.; Neumann, M. Effect of dimensionality on the Nelder–Mead simplex method. Optim. Methods Softw.
**2006**, 21, 1–16. [Google Scholar] [CrossRef] - ASCE Committee on Sedimentation. Sedimentation Engineering: Processes, Measurements, Modeling, and Practice; ASCE Manuals and Reports on Engineering Practice No. 110; Garcia, M.H., Ed.; American Society of Civil Engineers: Reston, VA, USA, 2008; ISBN 13:978-0-7844-0814-8. [Google Scholar]
- Einstein, H.A. Formulas for the Transportation of Bed Load. Trans. Am. Soc. Civ. Eng.
**1942**, 107, 561–577. [Google Scholar] [CrossRef] - Meyer-Peter, E.; Muller, R. Formulas for bed load transport. In Proceedings of the second Meeting of International Association for Hydraulic Structures Research, Stockholm, Sweden, 7 June 1948; pp. 39–64. [Google Scholar]
- Bagnold, R.A. An Approach to the Sediment Transport Problem from General Physics; U.S. Geological Survey Professional Paper 4221; U.S. Government Printing Office: Washington, DC, USA, 1966. [Google Scholar]
- Parker, G. Surface-based bedload transport relation for gravel rivers. J. Hydraul. Res.
**1990**, 28, 417–436. [Google Scholar] [CrossRef] - Duan, J.G.; Scott, S. Selective bed-load transport in Las Vegas Wash, a gravel-bed stream. J. Hydrol.
**2007**, 342, 320–330. [Google Scholar] [CrossRef] - Wilcock, P.R.; Crowe, J.C. Surface-based Transport Model for Mixed-Size Sediment. J. Hydraul. Eng.
**2003**, 129, 120–128. [Google Scholar] [CrossRef]

Run | d (mm) | d_{50}(mm) | Stem Spacing (S _{s}) (mm) | N | $\mathit{\varphi}$ | S (%) | H (cm) | Q × 10^{3}(m ^{3}/s) | q_{b} × 10^{6}(m ^{2}/s) |
---|---|---|---|---|---|---|---|---|---|

1 | 16 | 0.45 | 78 | 164.366 | 0.033 | 0.805 | 11.4 | 11.34 | 0.95 |

2 | 16 | 0.45 | 78 | 164.366 | 0.033 | 1.14 | 15.3 | 18.44 | 2.77 |

3 | 16 | 0.45 | 78 | 164.366 | 0.033 | 1.705 | 17.3 | 25.18 | 7.68 |

4 | 16 | 1.6 | 78 | 164.366 | 0.033 | 1.4 | 15.1 | 18.87 | 0.51 |

5 | 16 | 1.6 | 78 | 164.366 | 0.033 | 1.55 | 15.9 | 21.95 | 3.02 |

6 | 16 | 1.6 | 78 | 164.366 | 0.033 | 1.78 | 16.8 | 25.18 | 6.17 |

7 | 16 | 0.45 | 120 | 69.4444 | 0.014 | 0.62 | 8.8 | 9.92 | 0.38 |

8 | 16 | 0.45 | 120 | 69.4444 | 0.014 | 0.77 | 11.52 | 16.36 | 2.02 |

9 | 16 | 0.45 | 120 | 69.4444 | 0.014 | 1.05 | 15.3 | 25.66 | 4.99 |

10 | 16 | 1.6 | 120 | 69.4444 | 0.014 | 0.94 | 13.87 | 23.32 | 1.47 |

11 | 16 | 1.6 | 120 | 69.4444 | 0.014 | 1.08 | 15.29 | 29.05 | 2.57 |

12 | 16 | 1.6 | 120 | 69.4444 | 0.014 | 1.3 | 16.36 | 34.67 | 12.65 |

13 | 16 | 0.45 | 200 | 25 | 0.005 | 0.51 | 11.4 | 21.50 | 3.53 |

14 | 16 | 0.45 | 200 | 25 | 0.005 | 0.62 | 12.2 | 25.66 | 4.14 |

15 | 16 | 0.45 | 200 | 25 | 0.005 | 0.74 | 13.8 | 31.57 | 8.19 |

16 | 16 | 1.6 | 200 | 25 | 0.005 | 0.54 | 13.5 | 29.05 | 1.58 |

17 | 16 | 1.6 | 200 | 25 | 0.005 | 0.65 | 14.5 | 34.67 | 3.82 |

18 | 16 | 1.6 | 200 | 25 | 0.005 | 0.85 | 15.9 | 40.61 | 5.74 |

Run | ${\mathit{\tau}}_{\mathit{g}}$ (N/m ^{2})
| ${\mathit{\tau}}_{\mathit{w}1}$ (N/m ^{2})
| ${\mathit{\tau}}_{\mathit{w}2}$ (N/m ^{2})
| V_{v}(cm/s) | C_{D} | ${\mathit{\tau}}_{\mathit{b}\mathit{f}}$ (N/m ^{2})
| ΔZ (mm) | F_{r} |
---|---|---|---|---|---|---|---|---|

1 | 0.334 | 0.079 | 0.081 | 17.15 | 1.18 | 3.264 | 5.5 | 0.162 |

2 | 0.484 | 0.107 | 0.110 | 20.78 | 1.15 | 6.279 | 5.4 | 0.170 |

3 | 0.712 | 0.148 | 0.153 | 25.09 | 1.12 | 11.513 | 5.6 | 0.193 |

4 | 0.884 | 0.114 | 0.118 | 21.54 | 1.16 | 8.708 | 6.1 | 0.177 |

5 | 1.037 | 0.136 | 0.140 | 23.80 | 1.13 | 9.204 | 5.2 | 0.191 |

6 | 1.209 | 0.156 | 0.162 | 25.84 | 1.12 | 11.029 | 6.5 | 0.201 |

7 | 0.360 | 0.096 | 0.099 | 19.06 | 1.12 | 2.944 | 7.7 | 0.205 |

8 | 0.528 | 0.139 | 0.143 | 24.00 | 1.04 | 4.220 | 7.3 | 0.226 |

9 | 0.731 | 0.180 | 0.186 | 28.35 | 1.01 | 7.905 | 7.8 | 0.231 |

10 | 1.089 | 0.183 | 0.190 | 28.42 | 1.00 | 5.324 | 6.3 | 0.244 |

11 | 1.339 | 0.225 | 0.234 | 32.12 | 0.97 | 6.138 | 5.7 | 0.262 |

12 | 1.646 | 0.271 | 0.283 | 35.82 | 0.96 | 7.770 | 7.0 | 0.283 |

13 | 0.691 | 0.224 | 0.233 | 31.60 | 0.95 | 2.759 | 9.2 | 0.299 |

14 | 0.853 | 0.270 | 0.282 | 35.23 | 0.94 | 3.607 | 13.0 | 0.322 |

15 | 1.012 | 0.308 | 0.323 | 38.32 | 0.93 | 5.069 | 15.0 | 0.329 |

16 | 1.245 | 0.277 | 0.289 | 36.05 | 0.92 | 2.527 | 5.4 | 0.313 |

17 | 1.524 | 0.332 | 0.347 | 40.05 | 0.92 | 3.260 | 10.3 | 0.336 |

18 | 1.818 | 0.369 | 0.387 | 42.78 | 0.92 | 5.844 | 21.7 | 0.343 |

Investigator | d (mm) | d_{50} (mm) | $\mathit{\varphi}$ | S (%) | H (cm) | V_{v} (cm/s) | ${\mathit{\tau}}_{\mathit{b}}$ (N/m^{2})
| F_{r} | q_{b} × 10^{6}(m ^{2}/s) |
---|---|---|---|---|---|---|---|---|---|

Kothyari et al. [12] | 2.0 to 5.0 | 0.55 to 5.9 | 0.002 to 0.012 | 1.7 to 20.8 | 2.78 to 6.08 | 33.8 to 94.9 | 1.70 to 59.43 | 0.44 to 1.78 | 0.5 to 8121 |

Jordanova & James [11] | 5.0 | 0.45 | 0.0314 | 1.18 to 1.84 | 2.05 to 11.1 | 15.5 to 18.5 | 0.51 to 1.32 | 0.16 to 0.37 | 1.89 to 6.94 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Duan, J.G.; Al-Asadi, K. On Bed Form Resistance and Bed Load Transport in Vegetated Channels. *Water* **2022**, *14*, 3794.
https://doi.org/10.3390/w14233794

**AMA Style**

Duan JG, Al-Asadi K. On Bed Form Resistance and Bed Load Transport in Vegetated Channels. *Water*. 2022; 14(23):3794.
https://doi.org/10.3390/w14233794

**Chicago/Turabian Style**

Duan, Jennifer G., and Khalid Al-Asadi. 2022. "On Bed Form Resistance and Bed Load Transport in Vegetated Channels" *Water* 14, no. 23: 3794.
https://doi.org/10.3390/w14233794