# Estimation of Boundary Shear Stress Distribution in a Trapezoidal Cross-Section Channel with Composite Roughness

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A New Partition Model of a Cross-Section

_{w}, and P

_{b}denote the flow depth, the cross-section area, the side-wall, and bed wetted perimeter regions, respectively. Subscripts “0”, “w”, and “b” denote the standardized section, side-wall region, and bed region, respectively. θ is the angle between the bottom line and the side-wall. k denotes the ratio of hydraulic radius for the side-wall regions (R

_{w}) and bed region (R

_{b}). For a standardized cross-section, the hydraulic radius for the side-wall regions (R

_{w0}) and bed region (R

_{b0}) are Hk

_{0}/2 and H/2, respectively, i.e.:

_{b0}/P

_{w}and k

_{0}is:

_{0}− cosθ ≤ 0, then the standardized cross-section is assumed triangular. The concept of standardized cross-section provides two cross-section types: P

_{b}/P

_{w}≥ P

_{b0}/P

_{w}(Type 1) and P

_{b}/P

_{w}< P

_{b0}/P

_{w}(Type 2). The former type is considered as a wide cross-section compared to a corresponding standardized cross-section with water depth H, while the latter one is considered to be a narrow cross-section. The dividing lines are often different from those of the standardized cross-section. For Type 1, R

_{w1}= Hk

_{1}/2, and R

_{b1}= (A − Hk

_{1}P

_{w}/2)/P

_{b}. For Type 2, H

_{2}= 0.5P

_{b}sinθ/(k

_{2}− cosθ), R

_{b2}= H

_{2}/2, and R

_{w2}= (A − H

_{2}P

_{b}/2)/P

_{w}. Subscripts “1” and “2” denote Type 1 and Type 2, respectively.

_{1}or k

_{2}are calculated, the distribution of the boundary shear stress can be determined. The ratio of the side-wall or bed-region boundary shear force to the total boundary shear force (SF

_{w}or SF

_{b}), in which SF

_{w}+ SF

_{b}= 1, can also be calculated. For Type 1:

_{0}, which is very complex since it is related to secondary flow and turbulent stress in the neighborhood of the corner between the wall and bed [24,25]. Different parameters, such as roughness height k

_{s}(Nikuradse’s equivalent roughness), the Darcy–Weisbach resistance coefficient f, and friction velocity u

_{∗}, have been employed. The Keulegan’s method provided that k

_{0}= 1, which is not suitable for composite roughness cases [18]. The Einstein’s method assumed that the average flow velocities in wall and bed regions should be equal [5,32,33]. If the dividing method that is given in Figure 1 and the Darcy–Weisbach formula are used to calculate the sub-region velocity, based on Einstein’s method k

_{0}should be:

_{w0}and f

_{b0}are resistance coefficients of the wall-region and bed-region in a standardized cross-section. The calculation method for the resistance coefficient is detailed in Appendix A.

_{0}by [28]. Based on the experimental data, an empirical formula k

_{0}= (f

_{w0}/f

_{b0})

^{0.55}was obtained. It should be mentioned that H/2 instead of k

_{0}H/2 as hydraulic radius has been used to calculate f

_{w0}by [28] for convenience. Based on YLM’s method, k

_{0}= u

_{∗}

_{b}/u

_{∗}

_{w}is for smooth boundary cases and k

_{0}= k

_{sw}/k

_{sb}is for rough boundary cases [22,23]. However, YLM’s method is not applicable to the composite roughness cases in which the turbulent flow patterns are different in the bed-region and wall-region (i.e., one is hydraulically smooth and the other is hydraulically rough) since different parameters (e.g., u

_{∗}and k

_{s}) have been used for different flow patterns.

_{b0}, average velocity of bed-region V

_{b0}, and the friction velocity u

_{∗}

_{b0}of the bed-region are equal to those values of the wall-region in a standardized cross-section. In other words, for a standardized cross-section with a smooth boundary, all the methods that are mentioned above provide k

_{0}= 1.

## 3. “Equal Local-Region Velocity” Assumption and Empirical Treatments

#### 3.1. “Equal Local-Region Velocity” Assumption

_{i}. For steady and uniform flow, the kinetic energy equation can be written as:

_{w}” and “MN

_{b}” are normal to Zonal(y) and Zonal(z), respectively. Line “EF” is the zero-shear stress dividing line. The shear stress at points of M and N

_{w}are zero, since the two points are located at the dividing line and the water surface, respectively. The shear stress at point of N

_{b}should also be zero, considering the symmetry of the cross-section.

_{w}” is zero and there is no turbulent source at this line, the shear stress should be zero in this direction. Thus, the second term on the right side of Equation (7) can be neglected as y and z denote the directions of Zonal(y) and line “MN

_{w}”, respectively, in region “EDF”. Likewise, the first term on the right side of (7) can be neglected as y and z denote the directions of line “MN

_{b}” and Zonal(z), respectively, in region “EGF”. Hence, Equation (7) can be simplified as Equation (8a) for region “EDF” or Equation (8b) for region “EGF”.

_{max}of the zonal zone occurs at the “zero-shear stress” dividing line and the average velocity V

_{Zonal}can be expressed as:

_{max}= u

_{mz}= u

_{my}, the average velocities of the corresponding zonal zones are equal based on Equation (9), i.e., V

_{Zonal(y)}= V

_{Zonal(z)}. Though some zonal(y)s are connected to water surface, not the dividing line (for the cases with θ < 90°, i.e., trapezoidal cross-section), we propose an “equal local-region velocity” assumption that the average velocities of two local regions at two sides of a dividing line are equal for both uniform roughness cases and composite roughness cases.

#### 3.2. Empirical Treatments

_{bL}are the hydraulic radius and resistance coefficient of local-region “EGF”, respectively. S′ denotes the local hydraulic slope.

_{bL}= V

_{w}) and Darcy–Weisbach formula, we obtain:

_{w1}is the resistance coefficient of wall-region in a Type 1 cross-section.

_{w2}and f

_{b2}are the resistance coefficients of the wall-region and bed-region in a Type 2 cross-section, respectively.

_{0}= (k

_{0}P

_{w}+ P

_{b0})(H/2), in which (k

_{0}P

_{w}+ P

_{b0}) can be regarded as an equivalent wetted perimeter and H/2 can be regarded as an equivalent hydraulic radius. Based on this “equivalent” concept, the equivalent hydraulic radius for Type 1 should be A/(k

_{0}P

_{w}+ P

_{b}). Then we define:

_{wm}(112 experimental values of Type 1 from smooth boundary cases that are mentioned in Section 4.1, S90, S45, and S68 denote a smooth rectangular cross-section, trapezoidal cross-section with θ = 45°, and trapezoidal cross-section with θ = 68°, respectively), the value of S′/S can be calculated using Equations (11a) and (3a). Figure 2a shows the comparison between S′/S and η. The RMSE (root mean square error) for all three cases, S90, S45, and S68 are 0.21, 0.29, 0.1, and 0.1, respectively. Some data of two parameters differ considerably, especially for the cases of S90 with a width-depth ratio B/H > 8.5. However, for these cases, the values of SF

_{w}are relatively small (SF

_{wm}< 0.15) and a small deviation (|SF

_{wc}− SF

_{wm}|< 0.03) would cause large variations for S′/S. Conversely, the large difference between η and S′/S would not induce large deviation for SF

_{w}. Eliminating these data, Figure 2b shows the strong correlation (R

^{2}= 0.65) between η and S′/S. Therefore, the value of S′/S can be roughly replaced by η.

## 4. New Methods and Comparisons

_{0}based on Equation (4); (ii) calculate k

_{1}based on Equation (11a) (replacing S′/S by η) as P

_{b}/P

_{w}≥ k

_{0}− cosθ or calculate k

_{2}based on Equation (11b) as P

_{b}/P

_{w}< k

_{0}− cosθ (see details in Appendix A); (iii) calculate SF

_{w}based on Equation (3); and (iv) estimate the boundary shear stress in the wall-bed interaction zone as Equation (13), neglecting the effect of secondary currents:

#### 4.1. Smooth Boundary Cases

_{ε}defined in Equation (14a) denotes the percentage of the calculated SF

_{wc}with an absolute error that is less than ε; E

_{n}defined in Equation (14b) represents the systematic deviation of the calculated SF

_{wc}from the experimental value SF

_{wm}.

_{ε}” denotes total number of data and number of data with absolute error |SF

_{wc}− SF

_{wm}| ≤ ε; subscripts “c” and “m” denote the calculated values and measured values, respectively.

_{0.03}= 92.9%, P

_{0.06}= 100%, and E

_{n}= 4.7%, while those of Einstein’s method are P

_{0.03}= 35.7%, P

_{0.06}= 94.6%, and E

_{n}= 16.4%. It indicates that the “equal local-region velocity” assumption actually improved the reproduction performance compared to Einstein’s “equal sub-region velocity” assumption. For all smooth boundary cases, the reproduction performance of the proposed method is P

_{0.03}= 91.4%, P

_{0.06}= 97.9%, and E

_{n}= 3.5%, while that of YLM’s method is P

_{0.03}= 71.6%, P

_{0.06}= 94.3%, and E

_{n}= −3.6%. Overall, the proposed method gives best performance among these three methods.

_{w}; however, all of them include several empirical lumped correction factors. The empirical formulations of these correction factors that were obtained by the regression of experimental data are complex. Besides, similar to YLM’s method, the methods of [26,27] can only apply to smooth boundary cases. The method that was provided by [28], similar to Einstein’s method, can only calculate the average boundary shear stress of a wall-region or bed-region.

#### 4.2. Composite Roughness Cases

_{sw}/k

_{sb}. To calculate the resistance coefficient f, the reference roughness has been assumed: (a) Nikuradse’s equivalent roughness k

_{s}= 0.0015 mm is used for the smooth boundary of C90 cases; and (b) Nikuradse’s equivalent roughness of the plywood boundary is assumed to be k

_{s}= 0.05 mm. Figure 5 indicates that the proposed method also provides acceptable performance of SF

_{w}for composite roughness cases.

_{0.03}= 84.2% and P

_{0.06}= 100%, which is close to that based on [28] with P

_{0.03}= 89.5% and P

_{0.06}= 100%. For Type 1 cross-sections of C45 cases, the proposed method improves the calculation accuracy compared to Einstein’s method. The calculation accuracy based on the proposed method is P

_{0.03}= P

_{0.06}= 100%, compared to P

_{0.03}= 14.3% and P

_{0.06}= 92.9% based on Einstein’s method. Experimental data of C90 and C68 cases have not been employed for comparison since the reference roughness has been assumed.

## 5. Discussion

#### 5.1. The Effect of Secondary Currents and Empirical Treatments

_{Zonal}increases with increasing length of the zonal zone. The gradient of velocity V

_{Zonal}between adjacent zonals would induce secondary currents that transfer energy from high velocity zones (with large zonal length) to low velocity zones (with small zonal length) and then adjust the distribution of velocity and boundary shear stress. Due to the effect of secondary currents, the zero-shear stress dividing lines may not be straight lines as assumed in the new partition model.

_{zonal}); (ii) the calculated local shear stress is relatively smaller compared to the experiment results for a low velocity region (i.e., region with small zonal length L

_{zonal}); and (iii) the mean values of the calculated and experimental results are basically equal. These results show that the secondary currents adjust the distribution of velocity and energy in the separated region and transfer the downstream momentum from a high velocity region to a low velocity region.

#### 5.2. The Improvement of the Proposed Method

_{wc}based on Einstein’s method are relatively larger than SF

_{wm}with a large aspect ratio. It may be caused by the fact that the equal sub-region velocity assumption neglects the non-uniform distribution of velocity in the bed-region for a Type 1 cross-section. Compared to Einstein’s method, the proposed method made some improvements. The relative error (SF

_{wc}/SF

_{wm}− 1) of several data of S90 with a large width-depth ratio (accordingly small value of SF

_{w}) based on the proposed method is larger than 0.2, however, the corresponding absolute error (SF

_{wc}− SF

_{wm}) is actually small, less than 0.02.

_{1}and k

_{2}, while Einstein’s method did not; (ii) the proposed method assumes V

_{w}= V

_{bL}for a Type 1 cross-section, while Einstein’s method neglects the non-uniform distribution of velocity in bed-region and assumes V

_{w}= V

_{b}.

_{1}and k

_{2}, and the distribution of boundary shear stress can be obtained based on Equation (13) for both methods as a dividing line has been specifically given [22]. YLM’s method introduced a “minimum relative distance of energy transportation” assumption to simplify the term for energy transfer, while the proposed method employed a new partition model. However, both of the methods neglect the effect of secondary currents.

_{wc}from SF

_{wm}is strongly related to the ratio between the average velocity of the bed-region and wall-region V

_{b}/V

_{w}: For a standardized cross-section with V

_{b}= V

_{w}, SF

_{wc}≈ SF

_{wm}; for a Type 1 cross-section with V

_{b}> V

_{w}, SF

_{wc}< SF

_{wm}; and for a Type 2 cross-section with V

_{b}< V

_{w}, SF

_{wc}> SF

_{wm}(Figure 7). The most possible explanation is that YLM’s method neglects an important velocity condition that the velocities at two points that are divided by the “zero-shear stress” dividing line should be equal (i.e., u

_{my}= u

_{mz}in Figure 1). Compared to YLM’s method, the proposed method does not show the systematic deviation although the relative error (SF

_{wc}/SF

_{wm}–1) of some cases with a small value of SF

_{wc}(less than 0.1) is larger than 0.2. As mentioned in Section 4, the proposed method gives better performance than YLM’s method for smooth cases.

_{∗}and k

_{s}to determine k

_{1}and k

_{2}for smooth and rough boundary cases, respectively. Since YLM’s method employed different parameters (u

_{∗}and k

_{s}) to determine k

_{1}and k

_{2}for smooth and rough boundary cases, it does not apply to the situation that the turbulent flow patterns are different in the bed-region and wall-region. The proposed method, which employed the equal local-region velocity assumption to calculate k

_{1}and k

_{2}, does not have this limitation.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

_{w}), and corresponding geometry parameters of the cross-sections of S90, S45, S68, C90, C45, and C68 that are mentioned in Section 4 are available from the corresponding author upon reasonable request. All the data that are mentioned are laboratory data.

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Calculation of Resistance Coefficient f

_{s}; R

_{e}

_{∗}denotes the roughness Reynolds number R

_{e}

_{∗}= u

_{∗}k

_{s}/ν, in which ν represents the kinetic viscosity; for smooth boundary cases with R

_{e}

_{∗}< 1, B

_{s}= 2.5lnR

_{e}

_{∗}+ 5.5.

#### Appendix A.2. Calculation of k_{1} and k_{2}

_{1}H/2 and H/2, respectively. Based on Equation (A1) and Darcy–Weisbach formula, we can obtain:

_{1}can be calculated as:

_{0}= 1 for smooth boundary cases.

_{2}can be obtained by:

_{1}and k

_{2}can be obtained based on tentative calculation of Equations (11a) and (11b).

## References

- Rajaratnam, N.; Muralidhar, D. Boundary shear stress distribution in rectangular open channels. Houille Blanche
**1969**, 1, 603–610. [Google Scholar] [CrossRef] - Yuen, K.W.H. A Study of Boundary Shear Stress, Flow Resistance and Momentum Transfer in Open Channels with Simple and Compound Trapezoidal Cross Sections. Ph.D. Thesis, University of Birmingham, Birmingham, UK, 1989. [Google Scholar]
- Khodashenas, S.R.; Paquier, A. A geometrical method for computing the distribution of boundary shear stress across irregular straight open channels. J. Hydraul. Res.
**1999**, 37, 381–388. [Google Scholar] [CrossRef] - Knight, D.W.; Demetriou, J.D.; Hamed, M.E. Boundary shear in smooth rectangular channels. J. Hydraul. Eng.
**1984**, 110, 405–422. [Google Scholar] [CrossRef] - Cacqueray, N.D.; Hargreaves, D.M.; Morvan, H.P. A computational study of shear stress in smooth rectangular channels. J. Hydraul. Res.
**2009**, 47, 50–57. [Google Scholar] [CrossRef] - Yang, S.Q.; Lim, S.Y. Mechanism of energy transportation and turbulent flow in a 3D channel. J. Hydraul. Eng.
**1997**, 123, 684–692. [Google Scholar] [CrossRef] - Ansari, K.; Morvan, H.P.; Hargreaves, D.M. Numerical investigation into secondary currents and wall shear in trapezoidal channels. J. Hydraul. Eng.
**2011**, 137, 432–440. [Google Scholar] [CrossRef] - Khodashenas, S.R.; El kadi Abderrezzak, K.; Paquier, A. Boundary shear stress in open channel flow: A comparison among six methods. J. Hydraul. Res.
**2008**, 46, 598–609. [Google Scholar] [CrossRef] - Flintham, T.P.; Carling, P.A. Prediction of mean bed and wall boundary shear in uniform and compositely rough channels. In International Conference on River Regime; Hydraulics Research Limited: Wallingford, UK, 1988; pp. 267–287. [Google Scholar]
- Khozani, Z.S.; Bonakdari, H.; Zaji, A.H. Using two soft computing methods to predict wall and bed shear stress in smooth rectangular channels. Appl. Water Sci.
**2017**, 7, 3973–3983. [Google Scholar] [CrossRef] - Vazquez, P.M.; Sharifi, S. Modelling boundary shear stress distribution in open channels using a face recognition technique. J. Hydroinform.
**2017**, 19, 157–172. [Google Scholar] [CrossRef] - Wormleaton, P.R.; Merrett, D.J. An improved method of calculation for steady uniform flow in prismatic main channel/flood plain sections. J. Hydraul. Res.
**1990**, 28, 157–174. [Google Scholar] [CrossRef] - Prinos, P.; Townsend, R.D. Comparison of methods for predicting discharge in compound open channels. Adv. Water Resour.
**1984**, 7, 180–187. [Google Scholar] [CrossRef] - Moreta, P.J.; Martin-Vide, J.P. Apparent friction coefficient in straight compound channels. J. Hydraul. Res.
**2010**, 48, 169–177. [Google Scholar] [CrossRef] - Chen, Z.; Chen, Q.; Jiang, L. Determination of apparent shear stress and its application in compound channels. Procedia Eng.
**2016**, 154, 459–466. [Google Scholar] [CrossRef] - Leighly, J.B. Toward a theory of the morphologic significance of turbulence in the flow of water in streams. Prog. Phys. Geogr.
**1932**, 6, 1–22. [Google Scholar] - Nezu, I.; Nakagawa, H.E.D. Turbulence in Open-Channel Fows; Balkema: Rotterdam, The Netherlands, 1993. [Google Scholar]
- Keulegan, G.H. Laws of turbulent flow in open channels. J. Res. Natl. Bur. Stand.
**1938**, 21, 708–741. [Google Scholar] [CrossRef] - Einstein, H.A. Formulas for the transportation of bed load. Trans. ASCE
**1942**, 107, 561–597. [Google Scholar] [CrossRef] - Chien, N.; Wang, Z. Mechanics of Sediment Transport; Science Press: Beijing, China, 1986. (In Chinese) [Google Scholar]
- Yang, S.Q.; Lim, S.Y. Boundary shear stresses distributions in smooth rectangular open channel flows. Proc. Inst. Civ. Eng.-Water Marit. Energy
**1998**, 130, 163–173. [Google Scholar] - Yang, S.Q.; Lim, S.Y. Boundary shear stress distributions in trapezoidal channels. J. Hydraul. Res.
**2005**, 43, 98–102. [Google Scholar] [CrossRef] - Yang, S.Q.; Yu, J.X.; Wang, Y.Z. Estimation of diffusion coefficients, lateral shear stress, and velocity in open channels with complex geometry. Water Resour. Res.
**2004**, 40, W05202. [Google Scholar] [CrossRef] - Shiono, K.; Knight, D.W. Turbulent open-channel flows with variable depth across the channel. J. Fluid Mech.
**1991**, 222, 617–646. [Google Scholar] [CrossRef] - Guo, J.; Julien, P.Y. Shear stress in smooth rectangular open-channel flows. J. Hydraul. Eng.
**2005**, 131, 30–37. [Google Scholar] [CrossRef] - Kabiri-Samani, A.; Farshi, F.; Chamani, M.R. Boundary Shear Stress in Smooth Trapezoidal Open Channel Flows. J. Hydraul. Eng.
**2013**, 139, 205–212. [Google Scholar] [CrossRef] - Javid, S.; Mohammadi, M. Boundary shear stress in a trapezoidal channel. Int. J. Eng.
**2012**, 25, 323–332. [Google Scholar] [CrossRef] - Luo, Y.; Zhu, S.; Cao, B.; Jiang, C.J. A “Standard Cross-section” Method for the calculation of Riverbed and Bank shear stress. Appl. Math. Mech.
**2021**, 42, 915–923. (In Chinese) [Google Scholar] - Han, Y.; Yang, S.Q.; Sivakumar, M.; Qiu, L.C.; Chen, J. Flow Partitioning in Rectangular Open Channel Flow. Math. Probl. Eng.
**2018**, 2018, 11. [Google Scholar] [CrossRef] - Yang, S.Q.; Han, Y.; Lin, P.; Jiang, C.; Walker, R. Experimental study on the validity of flow region division. J. Hydro-Environ. Res.
**2014**, 8, 421–427. [Google Scholar] [CrossRef] - Han, Y.; Yang, S.Q.; Dharmasiri, N.; Sivakumar, M. Experimental study of smooth channel flow division based on velocity distribution. J. Hydraul. Eng.
**2015**, 141, 06014025. [Google Scholar] [CrossRef] - Johnson, J.W. The importance of considering sidewall friction in bed-load investigations. Civil Eng.
**1942**, 12, 329–332. [Google Scholar] - Vanoni, V.A.; Brooks, N.H. Laboratory Studies of the Roughness and Suspended Load of Alluvial Streams; Sedimentation Laboratory Report No. E68; California Institute of Technology: Pasadena, CA, USA, 1957. [Google Scholar]
- Nezu, I.; Nakagawa, H.E.D.; Tominaga, H. Secondary currents in a straight channel flow and the relation to its aspect ratio. Turbul. Shear. Flows
**1985**, 4, 246–260. [Google Scholar] - State Key Laboratory of Hydraulics and Mountain River Engineering (SKLHMRE). Hydraulics; Higher Education Press: Beijing, China, 2015; p. 131. (In Chinese) [Google Scholar]
- Cruff, R.W. Cross-Channel Transfer of Linear Momentum in Smooth Rectangular Channels; Water-Supply Paper 1592-B, United States Geological Survey: Reston, VA, USA, 1965. [Google Scholar]
- Ghosh, S.N.; Roy, N. Boundary shear distribution in open channel flow. J. Hydraul. Div.
**1970**, 96, 967–994. [Google Scholar] [CrossRef] - Kartha, V.C.; Leutheusser, H.J. Distribution of tractive force in open channels. J. Hydraul. Div.
**1970**, 96, 1469–1483. [Google Scholar] [CrossRef] - Knight, D.W.; Macdonald, J.A. Hydraulic resistance of artificial strip roughness. J. Hydraul. Div.
**1979**, 105, 675–690. [Google Scholar] [CrossRef] - Knight, D.W.; Macdonald, J.A. Open channel flow with varying bed roughness. J. Hydraul. Div.
**1979**, 105, 1167–1183. [Google Scholar] [CrossRef] - Myers, W.R.C. Flow resistance in wide rectangular channels. J. Hydraul. Div.
**1982**, 108, 471–482. [Google Scholar] [CrossRef] - Noutsopoulos, G.C.; Hadjipanos, P. Discussion: Boundary Shear in Smooth and Rough Channels. J. Hydraul. Div.
**1982**, 108, 809–812. [Google Scholar] [CrossRef] - Seckin, G.; Seckin, N.; Yurtal, R. Boundary shear stress analysis in smooth rectangular channels. Can. J. Civ. Eng.
**2006**, 33, 336–342. [Google Scholar] [CrossRef] - Knight, D.W. Boundary shear in smooth and rough channels. J. Hydraul. Div.
**1981**, 107, 839–851. [Google Scholar] [CrossRef] - Alhamid, A.I. Boundary Shear Stress and Velocity Distribution in Differentially Roughened Trapezoidal Open Channels. Ph.D. Thesis, University of Birmingham, Birmingham, UK, 1991. [Google Scholar]
- Blasius, H. Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten; Forschungs-Arbeit des Ingenieur-Wesens 131; Springer: Berlin/Heidelberg, Germany, 1913. (In German) [Google Scholar]
- Einstein, H.A. The Bed-Load Function for Sediment Transportation in Open Channel Flows; Technical Bulletin No. 1026, U.S. Department of Agriculture, Soil Conservation Service: Washington, DC, USA, 1950. [Google Scholar]
- Da Silva, A.M.F.; Bolisetti, T. A method for the formulation of Reynolds number functions. Can. J. Civil Eng.
**2000**, 27, 829–833. [Google Scholar] [CrossRef]

**Figure 1.**Partition model of a cross-section: (

**a**) standardized section, (

**b**) Type 1 section, and (

**c**) Type 2 section.

**Figure 2.**Relationship between S’/S calculated based on SF

_{wm}and η for smooth boundary cases: (

**a**) comparisons between them, and (

**b**) linear regression neglecting some data.

**Figure 5.**Comparisons of the measured data and the calculated results for composite roughness cases.

**Figure 6.**Comparisons of the calculated (denoted by fine lines) and measured boundary shear stress distribution: (

**a**) Smooth, θ = 90° and P

_{b}/P

_{w}= 1.56; (

**b**) Smooth, θ = 90° and P

_{b}/P

_{w}= 1.09; (

**c**) Smooth, θ = 90° and P

_{b}/P

_{w}= 0.6; (

**d**) Rough (k

_{s}= 1.04 mm), θ = 90° and P

_{b}/P

_{w}= 0.98; (

**e**) Rough (k

_{s}= 1.85 mm), θ = 90° and P

_{b}/P

_{w}= 0.92; (

**f**) Rough (k

_{s}= 3.17 mm), θ = 90° and P

_{b}/P

_{w}= 0.9, (

**g**) Rough (k

_{s}= 3.17 mm), θ = 63.4° and P

_{b}/P

_{w}= 1.09; (

**h**) Rough (k

_{s}= 3.17 mm), θ = 45° and P

_{b}/P

_{w}= 1.41 (

**i**) Rough (k

_{s}= 3.17 mm), θ = 33.7° and P

_{b}/P

_{w}= 0.89.

**Figure 7.**Limitation of different methods: (

**a**) YLM’s method, (

**b**) Einstein’s method, and (

**c**) the proposed method.

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**

Luo, Y.; Zhu, S.; Yang, F.; Gao, W.; Yan, C.; Yan, R.
Estimation of Boundary Shear Stress Distribution in a Trapezoidal Cross-Section Channel with Composite Roughness. *Water* **2022**, *14*, 2530.
https://doi.org/10.3390/w14162530

**AMA Style**

Luo Y, Zhu S, Yang F, Gao W, Yan C, Yan R.
Estimation of Boundary Shear Stress Distribution in a Trapezoidal Cross-Section Channel with Composite Roughness. *Water*. 2022; 14(16):2530.
https://doi.org/10.3390/w14162530

**Chicago/Turabian Style**

Luo, You, Senlin Zhu, Fan Yang, Wenxiang Gao, Caiming Yan, and Rencong Yan.
2022. "Estimation of Boundary Shear Stress Distribution in a Trapezoidal Cross-Section Channel with Composite Roughness" *Water* 14, no. 16: 2530.
https://doi.org/10.3390/w14162530