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

Abstract

The estimation of boundary shear stress distribution in a channel is a challenge in some hydraulic or environment investigations. A new partition model of a trapezoidal cross-section in a prismatic channel with composite roughness has been introduced based on a concept of a standardized cross-section using the “zero-shear stress” division lines. Based on this new model, an “equal local-region velocity” assumption, which can be regarded as an improvement of Einstein’s (1942) “equal velocity” assumption, has been proposed that is based on a discussion on the mechanism of energy transfer and velocity distribution at two sides of a dividing line. This assumption along with some empirical treatments have been employed to establish a new method to estimate the boundary shear stress of a side-wall or bed. Comparisons show that the proposed method can be applied to composite roughness cases and provides better prediction performance compared to other methods.

1. Introduction

The knowledge of boundary shear forces is very important when investigating hydraulic resistance, sediment transport, and channel migration. The distribution of boundary shear stress is influenced by three main factors including the shapes of flow cross-section, distribution of boundary roughness, and different hydraulic parameters, such as Froude and Reynold numbers [1,2].

The methods for the computation of boundary shear forces can generally be divided into four categories, geometrical methods [3], empirical methods [4], sophisticated turbulent models [5], and analytical methods [6]. Sophisticated turbulent models require a large number of empirical constants, however, these methods do not guarantee more accurate results and are a computational high cost for engineering design purposes [5,7]. The geometrical methods, such as the Merged Perpendicular Method and the Normal Depth Method, give no physical explanations and up till now no evidence shows that these methods can be applied to non-uniform boundary roughness distribution [3,8].

The empirical methods are mainly based on regression analysis of data that are derived from laboratory experiments in single and compound channels. Ref. [4] derived empirical equations that give the percentage of the total shear force that is carried by the sub-boundaries for smooth rectangular channels. Refs. [2,9] extended Knight’s method [4] for the rough trapezoidal channels. Ref. [10] employed the genetic programming and algorithm artificial neural network to find the best models for the prediction of boundary shear stress; while [11] used a face recognition technique. Another choice to calculate sub-region shear stress is to employ empirical formulations for the apparent shear stress of the dividing line. Most of the empirical formulas employ the velocity differences between adjacent sub-regions as a core parameter to represent the apparent shear stress [12,13,14,15]. However, the choices of geometric and hydraulic parameters, for example, the ratio of channel width and the ratio of manning coefficients, are subjective and arbitrary.

These analytical methods are often related to the partitioning of a flow. The dividing method that was proposed by [16] was based on its isovels and orthogonal trajectories, which are not easily available using simple analytical functions [17]. The method of [18], which uses a bisector of the angle to divide the flow region into two side-wall regions and a bed region, neglects the effect of non-uniform roughness and the bed width-flow depth ratio. Though the method of [19] does not provide a specific dividing method, it provides an idea that the partition may be related to the surplus energy transport [6,20]. Following this idea, Ref. [6] developed an analytical approach to determine the division lines of zero Reynolds shear stress, based on a “minimum relative distance of energy transportation” assumption (YLM’s method). This approach has been employed to investigate the sidewall and bed shear stress for trapezoidal open-channel flows [6,21,22]. However, there are not enough data so far to support the hypothesis of using roughness height as the characteristic length in applying YLM’s method to the differential roughened boundary cases [6,23]. The secondary flow and turbulent shear force are two main parts that cause the apparent shear stress along the dividing line [24]. Ref. [25] used two lump empirical parameters to calculate the two shear stresses that were mentioned above for the smooth rectangular channels. Refs. [26,27] used the same methodology to study the boundary shear stress distribution in smooth trapezoidal channels. Ref. [28] established an empirical formulation for the apparent shear stress of the dividing line employing the difference of the sub-region boundary shear stress based on a “momentum transfer-equilibrium deviation” assumption. Except for Einstein’s method [19] and Luo’s method [28] that was mentioned above, most of the analytical methods provide good results only for smooth channels or hydraulically smooth turbulent flows.

Since some experimental findings have proven the existence of the zero-shear stress dividing lines [29], the mechanism of energy transfer and velocity distribution can be discussed in two zones at two sides of the dividing lines. Based on this discussion, an “equal local-region velocity” assumption, which can be regarded as an improvement of Einstein’s equal velocity assumption [19], has been proposed. Using this assumption, a new method to estimate the boundary shear stress has been proposed and examined. This method can be used for prismatic and trapezoidal shape channels with non-uniform boundary roughness.

The outline of the paper is: (i) in Section 2, a new partition model of a trapezoidal cross-section has been employed; (ii) in Section 3, an “equal local-region velocity” assumption along with corresponding empirical treatments have been proposed based on a discussion on the mechanism of energy transfer and velocity distribution at two sides of a dividing line; (iii) in Section 4, comparisons for smooth boundary cases and composite roughness cases are provided; and (iv) in Section 5 and Section 6, the discussions and conclusions are given.

2. A New Partition Model of a Cross-Section

Some experimental studies have confirmed the existence of the “zero-shear stress” division lines and the dividing line is close to a straight line with one point at the corner of wall-bed junction and the other point at the vertical center line or water surface for uniform boundary roughness cases [29,30,31]. We assume this concept of division lines also applies to the composite roughness cases. Thus, a standardized section can be defined that the zero-shear force lines are two straight lines from the inner corners and are linked together at the water surface [28]. Figure 1a shows the standardized cross-section, in which H, A, P_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 Hk₀/2 and H/2, respectively, i.e.:

$$k_0=R_{\mathrm{w}0}/R_{\mathrm{b}0}$$

(1)

The relationship between P_b0/P_w and k₀ is:

$$P_{\mathrm{b}0}/P_{\mathrm{w}}=k_0-\cos \theta$$

(2)

As k₀ − 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 = Hk₁/2, and R_b1 = (A − Hk₁P_w/2)/P_b. For Type 2, H₂ = 0.5P_bsinθ/(k₂ − cosθ), R_b2 = H₂/2, and R_w2 = (A − H₂P_b/2)/P_w. Subscripts “1” and “2” denote Type 1 and Type 2, respectively.

As long as k₁ or k₂ 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:

$$S{F}_{\mathrm{w}}=\frac{R_{\mathrm{w}1}{P}_{\mathrm{w}}}{A}=\frac{k_1}{\cos \theta +2P_{\mathrm{b}}/P_{\mathrm{w}}}$$

(3a)

For Type 2:

$$S{F}_{\mathrm{w}}=1-\frac{R_{\mathrm{b}2}{P}_{\mathrm{b}}}{A}=1-\frac{{\left(P_{\mathrm{b}}/P_{\mathrm{w}}\right)}^2}{\left(k_2-\cos \theta \right)\left(\cos \theta +2P_{\mathrm{b}}/P_{\mathrm{w}}\right)}$$

(3b)

Different methods have been developed to determine k₀, 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 k₀ = 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 k₀ should be:

$$k_0=f_{\mathrm{w}0}/f_{\mathrm{b}0}$$

(4)

In which f_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.

The reason to use the resistance coefficient f instead of manning’s n or Chezy coefficient C is that f can be calculated based on the ratio of roughness height to hydraulic radius, which were usually provided by many related flume tests. The ratio of the resistance coefficient between the sub-regions has also been used to determine k₀ by [28]. Based on the experimental data, an empirical formula k₀ = (f_w0/f_b0)^0.55 was obtained. It should be mentioned that H/2 instead of k₀H/2 as hydraulic radius has been used to calculate f_w0 by [28] for convenience. Based on YLM’s method, k₀ = u_∗b/u_∗w is for smooth boundary cases and k₀ = 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.

For the smooth boundary cases, the hydraulic radius R_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 k₀ = 1.

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

3.1. “Equal Local-Region Velocity” Assumption

The kinetic energy equation can be obtained by multiplying both sides of Reynolds equations by velocity components u_i. For steady and uniform flow, the kinetic energy equation can be written as:

$$0=\rho u_i\left({\phi }_i-\frac{1}{\rho }\frac{\partial p}{\partial x_i}\right)+\frac{\partial }{{\partial }_j}\left[u_i\left({\tau }_{ij}-\frac{\rho u_i{u}_j}{2}\right)\right]-{\tau }_{ij}\frac{\partial u_i}{\partial x_j}$$

(5)

where i and j = x, y, z; the subscript “x” denotes the streamwise x-direction, the subscript “y” and “z” represent two directions that are vertical to each other in the plane of a cross-section; ρ, p, u, φ, and τ represent water density, mean pressure, mean velocity (time average), body forces, and shear stress, respectively.

The first term on the right side of Equation (5) refers to the external energy sources, the second term is involved with the transfer of energy, and the third term refers to the energy dissipation. Based on the magnitude analysis, the second term can be simplified as (see details in [6]):

$$E=\frac{\partial \left[u({\tau }_{xy}/\rho -uv)\right]}{\partial y}+\frac{\partial \left[u({\tau }_{xz}/\rho -uw)\right]}{\partial z}$$

(6)

where “u”, “v”, and “w” are the mean velocity components relative to the x, y, and z directions, respectively.

Equation (6) indicates that both of shear stress and secondary currents cause energy transfer and thus velocity re-distribution. Since the sub-regions are divided by zero-shear stress lines based on the partition model, the secondary currents should be limited in separate regions, which is consistent with the experimental data for uniform roughness cases that were obtained from University of Wollongong [29,34] (see [29]). Therefore, the average velocity of a local region would not change as the effect of secondary flow is neglected. Neglecting the effect of secondary current, Equation (6) can be simplified as:

$$E=\frac{\partial (u{\tau }_{xy}/\rho )}{\partial y}+\frac{\partial (u{\tau }_{xz}/\rho )}{\partial z}$$

(7)

In Figure 1, Zonal(z) and Zonal(y) are perpendicular to bed and wall, respectively, and meet at point M of the dividing line. Lines “MN_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.

Since shear stress at two ends of line “MN_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”.

$$E\approx \frac{\partial (u{\tau }_{xy}/\rho )}{\partial y}$$

(8a)

$$E\approx \frac{\partial (u{\tau }_{xz}/\rho )}{\partial z}$$

(8b)

where the y-direction is normal to the side-wall, and the z-direction is normal to the bed.

Ref. [6] obtained Equation (8) from Equation (6) based on the “minimum relative distance of energy transportation” assumption, while in the present study Equation (8) is obtained using the shear stress condition based on the new partition model. Equation (8) indicates that the velocity distribution in Zonal(z) or Zonal(y) should be a 2-dimensional (2D) vertical distribution. Referring to the exponential law of vertical velocity distribution in an open channel with large width-depth ratio [35], the maximum velocity u_max of the zonal zone occurs at the “zero-shear stress” dividing line and the average velocity V_Zonal can be expressed as:

$$V_{\mathrm{Z}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}=\frac{u_{\max }}{m+1}$$

(9)

in which m denotes the exponent of the exponential law of velocity distribution. The value of m, which is related to Reynold’s number, is about 1/7.

As Zonal(z) and Zonal(y) meet at point M of the dividing line, the point velocities at two sides of the dividing line are equal, i.e., u_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

Different empirical treatments have been made to Type 1 and Type 2 cross-sections considering the effect of secondary currents on zero-shear stress dividing lines, which are discussed in Section 5.

For Type 1, the average velocities of local regions “EGF” and “EDF” are assumed to be equal. Based on the Darcy–Weisbach formula, the local-region velocity can be expressed as:

$$V_{\mathrm{b}\mathrm{L}}=\sqrt{\frac{8}{f_{\mathrm{b}\mathrm{L}}}\frac{H}{2}{S}^{\prime }}$$

(10)

in which H/2 and f_bL are the hydraulic radius and resistance coefficient of local-region “EGF”, respectively. S′ denotes the local hydraulic slope.

S′ is employed to reflect the effect of width-depth ratio in a Type 1 cross-section on the zero-shear stress dividing line. Since the flow in the center part of a bed-region (i.e., the region covered by “EGE′G′”) provides downstream shear stress to two local regions, the value of S′ should be larger than the average hydraulic slope S of the whole cross-section. Based on the “equal local-region velocity” assumption (i.e., V_bL = V_w) and Darcy–Weisbach formula, we obtain:

$$k_1=\frac{f_{\mathrm{w}1}{S}^{\prime }}{f_{\mathrm{b}\mathrm{L}}S}$$

(11a)

in which f_w1 is the resistance coefficient of wall-region in a Type 1 cross-section.

For Type 2, the interaction between two half wall-regions and bed-region is strong, hence the average velocities of a half wall-region and a half bed-region are assumed to be equal:

$$f_{\mathrm{w}2}/f_{\mathrm{b}2}=R_{\mathrm{w}2}/R_{\mathrm{b}2}$$

(11b)

in which f_w2 and f_b2 are the resistance coefficients of the wall-region and bed-region in a Type 2 cross-section, respectively.

For a standardized cross-section, A₀ = (k₀P_w + P_b0)(H/2), in which (k₀P_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/(k₀P_w + P_b). Then we define:

$$\eta =\frac{A}{(k_0{P}_{\mathrm{w}}+P_{\mathrm{b}})(H/2)}$$

(12)

in which η denotes the ratio of equivalent hydraulic radius for Type 1 to that of a corresponding standardized cross-section.

Based on the experimental data SF_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 (R² = 0.65) between η and S′/S. Therefore, the value of S′/S can be roughly replaced by η.

4. New Methods and Comparisons

As η has been used to represent the value of S′/S, a new method based on the “equal local-region velocity” assumption has been established to determine the dividing lines and calculate boundary shear stress: (i) determine the value of k₀ based on Equation (4); (ii) calculate k₁ based on Equation (11a) (replacing S′/S by η) as P_b/P_w ≥ k₀ − cosθ or calculate k₂ based on Equation (11b) as P_b/P_w < k₀ − 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:

$$\tau =\rho g{L}_{Zonal}S$$

(13)

in which L denotes the length of the zonal zone that is perpendicular to the boundary; g is gravitational acceleration.

The performance of different methods has been compared using two different cases: Smooth boundary cases and composite roughness cases.

4.1. Smooth Boundary Cases

The experimental data of smooth boundary cases have been used for comparisons between different methods. These experimental data (112 data for Type 1 and 28 for Type 2) includes: (i) smooth rectangular cases (S90) with 0.3 ≤ B/H ≤ 50: 69 experimental values (50 values of Type 1 and 19 values of Type 2) from the research work of [4,36,37,38,39,40,41,42,43]; (ii) smooth trapezoidal cases with θ = 45° (S45) with 0.3 ≤ B/H ≤ 15: 56 experimental values (50 values of Type 1 and 6 values of Type 2) from the research work of [2,37]; and (iii) smooth trapezoidal θ = 68° (S68) with 0.7 ≤ B/H ≤ 6.5: 15 experimental values (12 values of Type 1 and 3 values of Type 2) from the research work of [9].

There are two parameters that have been defined to represent the systematic deviation and accuracy of reproduction of measurements: P_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.

$$P_{\epsilon }=\frac{N_{\epsilon }}{N}\times 100\%$$

(14a)

$$E_{\mathrm{n}}=\frac{1}{N}{\displaystyle \sum _{i=1}^N\frac{S{F}_{\mathrm{wc}}-S{F}_{\mathrm{wm}}}{S{F}_{\mathrm{wm}}}\times 100}\%$$

(14b)

where “N” and “N_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.

Among the analytical methods, Einstein’s method and YLM’s method are most simple ones. Figure 3 shows that the proposed method performs better compared to Einstein’s method and YLM’s method. For Type 2, both the proposed method and Einstein’s method are calculated based on Equation (11b). For Type 1, the performance indicators of the proposed method are P_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.

Several other analytical methods have also been used to calculate the boundary shear stress for both trapezoidal and rectangular cross-sections, for example, the methods of [26,27,28] (Figure 4). They also provide good reproduction of SF_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

The composite roughness cases include two different categories, the first category is that both the bed and side-wall are hydraulically rough, but the roughness for the two are different; the second category is that the side-wall or bed is hydraulically smooth, while the other is hydraulically rough. The experimental data of composite roughness cases include: (i) composite rectangular cases (C90) with 1.5 ≤ B/H ≤ 15: 41 experimental values (38 values of Type 1 and 3 values of Type 2) from the research work of [39,40,44]; (ii) composite trapezoidal cases with θ = 45° (C45) with 0.7 ≤ B/H ≤ 9: 38 experimental values (14 values of Type 1 and 24 values of Type 2) from the research work of [45]; and (iii) composite trapezoidal θ = 68° (C68) with 0.85 ≤ B/H ≤ 10: 41 experimental values (36 values of Type 1 and 5 values of Type 2) from the research work of [9]. For the C90 and C68 cases, the values of roughness have been provided in the form of side-wall to bed roughness k_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.

Few methods can be used to predict boundary shear stress for composite roughness cases. Among them are the Einstein’s method, methods of [9,28]. The empirical formulation that was established by [9] does not apply to C90 cases [28]. For C45 cases, the calculation accuracy based on the proposed method is P_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

The effect of secondary currents is to adjust the distribution of boundary shear stress. As the velocity distribution in Zonal(z) or Zonal(y) is 2D distribution, the average velocity V_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.

Ref. [37] measured the boundary shear stress distribution in rectangular/trapezoidal channels. Figure 6 shows the comparisons between the measured and calculated shear stress: (i) the calculated local shear stress is relatively larger compared to experimental results for a high velocity region (i.e., region with large zonal length L_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.

For Type 1 and Type 2, different empirical treatments have been employed. For Type 1, the effect of secondary currents has been neglected and the influence of a center part of the bed-region is considered. For Type 2, the average velocities of a half wall-region and a half bed-region are assumed to be equal considering that the effect of the secondary currents in the wall-region on both the wall-region and bed-region cannot be neglected. These empirical treatments have been proven to be effective for smooth and composite roughness cases (see Section 4).

5.2. The Improvement of the Proposed Method

Figure 7 shows that the values of SF_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.

The results indicate that the “equal local-region velocity” assumption can be regarded as an improvement of Einstein’s equal velocity assumption [19]. There are two differences between Einstein’s assumption and the proposed method: (i) the proposed method provides two specific dividing lines that are determined by k₁ and k₂, 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.

Both YLM’s method and the proposed method define two “zero-shear stress” dividing lines by k₁ and k₂, 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.

For YLM’s method, the deviation of the calculated SF_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.

Besides, YLM’s method used characteristic lengths that were represented by u_∗ and k_s to determine k₁ and k₂ for smooth and rough boundary cases, respectively. Since YLM’s method employed different parameters (u_∗ and k_s) to determine k₁ and k₂ 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 k₁ and k₂, does not have this limitation.

6. Conclusions

Based on the experimental evidence of zero-shear stress dividing line, a new partition model of a cross-section has been introduced. The kinetic energy equation has been employed to analyze the flow characteristics in the bed-wall interaction zone and an “equal local-region velocity” assumption has been proposed. Based on this assumption, the zero-shear stress dividing line can be determined and then the distribution of boundary shear stress can be calculated.

The proposed method has established a new partition model of a cross-section with the assumption of the existence of the zero-shear stress dividing lines. There are two types of cross-section that are introduced referring to a standardized cross-section. In this model, the interaction or energy transfer between the sub-regions can be neglected since these sub-regions are cut by zero-shear stress dividing lines.

The proposed method can be used to estimate the average shear stress of a bed or wall region. The comparisons show that the proposed method provides better reproduction performance for all smooth boundary cases compared to other methods and can also be applied to composite roughness cases. However, more composite roughness cases are needed for further justification of this method. The proposed method can also provide a preliminary distribution of shear stress neglecting the effect of the secondary currents.