Study on the Coefficient of Apparent Shear Stress along Lines Dividing a Compound Cross-Section

Abstract

A compound channel’s discharge capacity and boundary shear force can be predicted as a sum of the discharge capacity of different sub-regions once the apparent shear stress of the dividing line is reasonably quantified. The apparent shear stress was usually expressed as a coefficient multiplied by the difference between two squared velocities of two adjacent regions. This study investigated the range of the coefficient values and their influencing factors. Firstly, the optimal values of the coefficient were obtained based on experimental data. Then, comparisons between the optimal values and several parameters used in quantifying the apparent shear stress were conducted. The results show that the coefficient is mainly related to a morphological parameter of the floodplain and the ratio of resistance coefficients between the floodplain and the main channel. An empirical formula to calculate the coefficient was developed and introduced to calculate the flow discharge and boundary shear stress. Experimental data, including 142 sets of test data of symmetric-floodplain cases and 104 sets of one-floodplain cases, have been used to examine the prediction accuracy of discharges and boundary shear stress. For all these tests, the ranges of water depth of the main channel and the total width of the compound cross-section are about 0.05~0.30 m and 0.3~10 m, respectively; the Q range and the range of Froude numbers of the main channel flow are about 0.0033~1.11 m³/s and 0.3~2.3, respectively. Comparison with other methods and experimental data from both rigid and erodible compound channels indicated that the proposed method not only provided acceptable accuracy for the computation of discharge capacity and boundary shear stress of compound channels in labs but also gave insights for calculating discharge capacity in natural compound channels.

1. Introduction

Flow interaction between different regions in a compound cross-section is an essential factor in discharge capacity in a compound channel [1,2]. In calculating the discharge capacity of a compound channel, a cross-section is often divided into several sub-regions, and the apparent shear stress of a dividing line is often used to quantify the flow interaction in calculating flow discharge in a compound channel [3,4].

Different dividing methods were used in the previous studies to increase the prediction accuracy of discharge capacity. Generally, these methods can be divided into two categories: (1) employing the concept of zero-shear-stress dividing lines to divide the sub-regions of the cross-section or adjust the area of the sub-regions, such as the “inclined interface” dividing method [5] or area-adjust method [6]; (2) using vertical and horizontal dividing lines and calculating the apparent shear stress of these dividing lines. Some studies indicate that the dividing line of zero shear stress may exist [7,8]. However, the dividing line of zero shear stress varies with different cross-sectional shapes and water levels. Thus, the methods of the first category to predict the discharge capacity have significant limitations. For the methods of the second category, a compound cross-section can often be divided into three or four sub-regions. For example, Huthoff et al. [9], Chen et al. [10], Khatua et al. [11], and Khuntia et al. [12] divided a cross-section into three parts: two floodplains; and one main channel. However, the cross-section has been divided into four sub-regions in the Yang et al. [13] method, with two sub-regions divided by a horizontal dividing line in the main channel. The comparison between Yang’s and Huthoff’s methods, which employed a constant coefficient, indicated that the method with four sub-regions might have produced better calculation accuracy [14].

In previous studies, most empirical expressions of the apparent shear stress include the coefficients and the velocity or square velocity difference between the adjacent sub-regions [10]. The empirical expressions with velocity differences are usually obtained by empirical regression, such as Prinos and Townsend [15]. The empirical expressions with square velocity differences have a more theoretical basis. For example, considering the relationship between the eddy motion and turbulent fluctuations, Huthoff et al. [9] assumed that lateral velocity fluctuations and streamwise velocity were proportional to velocity differences between the floodplain and central channel and the average velocity of the floodplain and main channel, respectively. Thus, the apparent shear stress is proportional to the square velocity difference. Both studies by Yang et al. [13] and Huthoff et al. [9] showed that employing a specific value or small value range for the coefficient was suitable for some cases. In contrast, the study by Luo et al. [14] indicates that variable coefficients may increase the prediction accuracy of discharge capacity.

In the present study, the experimental data from different flume tests were used to calculate the optimal values of the coefficient. Then, these optimal values’ value range and influence factors were analyzed. Finally, an empirical expression for a vertical dividing line has been proposed, and a comparison of the performance of the prediction of discharge and boundary shear force of different regions between different methods has been discussed.

2. Methods

The study of Luo et al. [14] (“Luo’s method” in the following text) indicates that the dividing method with four sub-regions [13] (“Yang’s method” in the subsequent text) produces better accuracy than the dividing method with three sub-regions [9] (“Huthoff’s method” in the following text). Thus, the four sub-regions dividing method is used to calculate the optimal values. In Figure 1, B_t, B, P, H, and h represent the width of the cross-section, width of the main channel, wetted perimeter, flow depth of the main channel, and height of the floodplain, respectively; subscripts “1”, “2”, “3”, and “4” denote sub-regions 1, 2, 3 and 4 respectively; subscript “mc” represents the main channel, including sub-regions 2 and 4.

Based on Luo’s method, the calculating methods of the apparent shear stress for a horizontal dividing line and two vertical dividing lines are different. To calculate the apparent shear stress of a horizontal dividing line, an equilibrium state has been employed that the apparent shear stresses of the two vertical dividing lines were zero compared to the actual state that the apparent shear stresses of two vertical dividing lines were not zero. The water level, shape, size, and boundary roughness of the compound cross-section are the same for both states. The differences between the two states are the velocities of sub-region 2 and sub-region 4 and, thus, the apparent shear stress of the horizontal dividing line. For the equilibrium state, the apparent shear stress of the horizontal dividing line is equal to the streamwise direction component of flow gravity of sub-region 2. Using this equilibrium state as a reference and assuming that the coefficient is the same for both states, the apparent shear stress of a horizontal dividing line τ_a24 can be calculated as follows:

$${\tau }_{\mathrm{a}24}B=\gamma A_{2}S(u_2^2-u_4^2)/(u_2^{\mathrm{e}2}-u_4^{\mathrm{e}2})$$

(1)

In which τ_a denotes the mean apparent shear stress; γ, A, and S represent the unit weight of flow, cross-section area, and hydraulic slope, respectively; u and u^e represent sub-region average velocities in actual and equilibrium states, respectively. The calculation method for velocities $u_2^e$ and $u_4^e$ can refer to Luo et al. [14].

In reference to Yang’s method, the expression of apparent shear stress of a vertical dividing line is

$${\tau }_{\mathrm{a}2\mathrm{i}}=\frac{1}{2}\alpha \rho \left({u_2}^2-{u_{\mathrm{i}}}^2\right) (\mathrm{i}=1\mathrm{or}3)$$

(2)

In which α and ρ denote the momentum transfer coefficient and flow density, respectively; subscript “2i” denotes dividing lines between sub-regions i and 2.

The expressions for the force equilibrium of four sub-regions are

$$\gamma A_{1}S-P_1{{\tau }_{\mathrm{b}}}_1+{\tau }_{\mathrm{a}12}(H-h)=0$$

(3)

$$\gamma A_{2}S-{\tau }_{\mathrm{a}12}(H-h)-{\tau }_{\mathrm{a}23}(H-h)-B{\tau }_{\mathrm{a}24}=0$$

(4)

$$\gamma A_{3}S-P_3{\tau }_{\mathrm{b}3}+{\tau }_{\mathrm{a}23}(H-h)=0$$

(5)

$$\gamma A_{4}S-P_4{\tau }_{\mathrm{b}4}+B{\tau }_{\mathrm{a}24}=0$$

(6)

In which τ_b represents the boundary sheer stress; subscripts “12”, “23” and “24” denote dividing lines between sub-regions 1 and 2, sub-regions 2 and 3, and sub-regions 2 and 4, respectively. P₄ is different from P_mc. For symmetric-floodplain cases, P₄ = P_mc; for one-floodplain cases, P₄ = P_mc-(H-h).

For a symmetry compound cross-section, A₁ = A₃, τ_a12 = τ_a23; Formula (5) is unnecessary for a one-floodplain cross-section. There are four unknown velocities, u₁, u₂, u₃, and u₄, and an unknown coefficient α in Equations (3)–(6).

According to the Darcy–Weisbach equation and the concept of frictional flow velocity, boundary shear stress can be expressed as

$${{\tau }_b}_{\mathrm{i}}=\rho u_{\mathrm{i}*}^2=\frac{\rho }{8}{f}_{\mathrm{i}}{u}_{\mathrm{i}}^2 (\mathrm{i}=1,2,3\mathrm{or}4)$$

(7)

In which ρ, u_* represent water density and friction velocity; f is the Darcy–Weisbach resistance coefficient.

To simplify the calculation processes (i.e., using the same expression as Equation (7)), α can be expressed as

$$\alpha =\frac{1}{4}\mu f_{\mathrm{mc}}$$

(8)

In which μ denotes a revised version of momentum transfer coefficient α; f is the Darcy–Weisbach resistance coefficient.

Substituting Equation (8) into Equation (2) yields the following expression:

$${\tau }_{\mathrm{a}2\mathrm{i}}=\frac{\rho }{8}\mu f_{\mathrm{mc}}\left({u_2}^2-{u_{\mathrm{i}}}^2\right)(\mathrm{i}=1\mathrm{or}3)$$

(9)

To simplify the calculation, the downstream component of the gravity of the water body in the sub-regions can be represented by the following formula:

$$\gamma A_{\mathrm{i}}S=\frac{\rho }{8}{P}_{\mathrm{i}}{f}_{\mathrm{i}}{u}_{\mathrm{i}0}^2 (\mathrm{i}=1,2,3\mathrm{or}4)$$

(10)

In which u_i0 can be calculated based on experimental parameters, including ρ, γ, A, P, f, and S.

Substituting Equations (1), (7), (9), and (10) into Equations (3)–(6) yields

$${u_2}^2=\frac{\frac{A_2}{A_4}+\frac{2K_1{A}_1}{A_4(K_1+K_2)}+\frac{A_2}{A_4}\frac{1}{K_3}}{2\frac{K_1}{{u_{40}}^2}\frac{K_2}{K_1+K_2}+\frac{A_2}{A_4}\frac{1}{{u_{40}}^2{K}_3}}$$

(11)

$${u_1}^2=\frac{\frac{A_1}{A_4}{u_{40}}^2+K_1{u_2}^2}{K_1+K_2}$$

(12)

$${u_4}^2=\frac{E_1+\frac{A_2}{A_4}{u_2}^2}{K_3}$$

(13)

In the above equation, $E_1={u_2}^{\mathrm{e}2}-{u_4}^{\mathrm{e}2}$, $K_1=(\mu f_{\mathrm{mc}}{l}_{12})/(f_4{P}_4)$, $K_2=(f_1{P}_1/f_4{P}_4)$, and $K_3=E_1/{u_{40}}^2+A_2/A_4$. f_mc is not equal to f₄ because the hydraulic radius is different for the main channel and sub-region 4.

As sub-region velocities are calculated based on Equations (11)–(13), the apparent shear stress can be calculated based on Equation (2). Then, the boundary shear stress can be calculated based on Equations (3), (5), and (6).

Equations (14) and (15) are two expressions to calculate the discharge error e_Q and the apparent shear stress error e_SF. We define the optimal value of μ₀ when e_Q or e_SF reaches the minimum value.

$$e_{\mathrm{Q}}=\sqrt{{(Q_{\mathrm{mcC}}-Q_{\mathrm{mcM}})}^2+{(Q_{\mathrm{fpC}}-Q_{\mathrm{fpM}})}^2+{(Q_{\mathrm{tC}}-Q_{\mathrm{tM}})}^2}$$

(14)

$$e_{\mathrm{SF}}=\sqrt{{({\tau }_{\mathrm{a}12\mathrm{C}}-{\tau }_{\mathrm{a}12\mathrm{M}})}^2}$$

(15)

In which Q represents flow discharge; subscripts “t”, “M”, and “C” denote total flow, measured data, and calculated value, respectively; subscript “fp” denotes floodplain, including sub-regions 1 and 3.

3. Results

3.1. Existence of μ₀

Experimental data for sub-region discharges are relatively abundant. Part of them is used to determine μ₀, including data from Knight and Demetriou [3] (K&D), Yuen [16] (Yuen, including two series, Yuen1 and Yuen2), Myers and Brennan [17] (FCF, including seven series: S01~S03; S06~S08; and S10), and Atabay [18] (Atabay, including three series: ROS; ORH; and ROA). There are 103 sets of data used to determine the optimal value μ₀ based on minimum discharge error e_Q. The Q range is about 0.0067~1.11 m³/s; the range of the Froude number is about 0.51~2.30.

Compared to experimental data on sub-region discharge, experimental data on boundary shear force are relatively scarce. These data include K&D [15], Yuen1 (part data from Yuen) [15], Prinos and Townsend [15] (P&T), and ROS-S (part data from Atabay), and all of them have been used to calculate μ₀. There are 47 sets of data used to determine the optimal value μ₀ based on minimum apparent shear stress error e_SF. The Q range is about 0.0067~0.08 m³/s; the range of the Froude number is about 0.30~1.63.

Table 1. Summary of the experimental data to determine μ₀.

Sources	Cross-Section Shape	Q (m3/s)	μ0	α	Amount of Data
Sub-region discharges (Q)
K&D	S	0.0067~0.029	0.42~5.85	0.0018~0.0298	14
Yuen	S	0.013~0.055	3.39~10.55	0.0111~0.0368	12
FCF	S/O	0.24~1.11	3.7~28.66	0.0112~0.1053	43
Atabay	S/O	0.018~0.08	1.25~20.01	0.0047~0.0636	34
Boundary shear force (SF)
K&D	S	0.0067~0.029	0.58~5.52	0.0025~0.0265	14
Yuen (Yuen 1)	S	0.013~0.035	1.37~1.81	0.0046~0.0069	4
P&T	S	0.012~0.026	1.72~5.01	0.0086~0.2746	16
Atabay (ROS-S)	S	0.018~0.08	2.91~5.54	0.0115~0.0204	13

shows the data used to determine μ₀, in which “S” and “O” denote symmetry and one floodplain, respectively. For most cases, the error has minimum values as μ increases from 0.1 to 50 (α increases from 0.001 to 0.3), as shown in Figure 2, which proves the existence of μ₀. For all the tests mentioned in Table 1, μ₀ ranges from 0.42 to 28.66, and α ranges from 0.0018 to 0.2746.

3.2. Influencing Factors and Empirical Formulas

The coefficients used in previous formulas consist of different geometrical and hydraulic parameters [10]. The geometrical parameters can be divided into three categories: (1) ratio of width or depth between floodplain and main channel, such as B_t/B, (B_t − B)/B, H/h, (H − h)/h or (H − h)/H; (2) shape of main channel or compound cross-section, such as B/H or B_t/H; and (3) shape of the sub-region of a floodplain, such as (H − h)/P₁ or (H − h)/P₃. The latter two categories consider the variation in width and water level compared to the first category. The hydraulic parameters are relative roughness (n_fp/n_mc) or relative resistant coefficient (f_fp/f_mc).

The relationships between μ₀ and the following eight parameters: (B_t − B)/B; (H − h)/H; B/H, B_t/H; (H − h)/P; f_fp, f_mc; and f_fp/f_mc have been examined (Figure 3). The results show that (1) the value range of μ₀ is about 0.1~30, and the corresponding value range of the optimal α is about 0.001~0.3 (Table 1), (2) the most relevant parameter is (H − h)/P₁; the second is B_t/H; both consider the variation in width and water level, and (3) the relationship between μ₀ and f_fp/f_mc should not be neglected.

The parameters B_t/H and μ₀ are more relevant compared to the parameter B/H, indicating that μ₀ is closely relevant to ratio P₁/H, where P₁ ≈ 0.5(B_t − B) for symmetry-floodplains cases, or P₁ ≈ B_t − B for one-floodplain cases. Therefore, an empirical formula to calculate μ₀ can be developed by employing (H − h)/P_fp and f_fp/f_mc:

$${\mu }_0=0.610{\left(f_{\mathrm{mc}}/f_{\mathrm{fp}}\right)}^{-0.544}{\left(\left(H-h\right)/P_{\mathrm{fp}}\right)}^{-0.586}$$

(16)

Figure 3i shows that the empirical formula Equation (16) provides acceptable prediction accuracy of μ₀ calculated based on e_Q and e_SF, and R² is about 0.67. The comparison also indicates that the optimal μ₀ from both minimum e_Q and e_SF are consistent; i.e., the method using force equilibrium of four sub-regions to calculate both discharges and boundary shear forces is reasonable, and reasonable prediction of both discharges and boundary shear force can be obtained if the μ₀ has been given the appropriate value.

3.3. Comparisons between Different Methods

Based on Equations (11)–(13) and employing the empirical formula Equation (16), both sub-region discharges and boundary shear force of a compound cross-section can be calculated. Except for the experimental data shown in Table 1, other experimental data such as Bousmar [19] (Bousmar), Macintosh [20] (Macintosh, including two series RCC and TCC), Mohanty and Khatua [21] (M&K), Khatua et al. [11] (Khatua), Al-Khatib et al. [22] (Khatib), and Patra et al. [23] (Patra) have been added into the discussion (

Table 2. Summary of the experimental data and calculated coefficient.

Sources	H (m)	Bt (m)	Q (m3/s)	μ0	α	Amount of Data
Experimental data for Symmetric-floodplain
K&D	0.1~0.15	0.3~0.61	0.0067~0.029	1.05~3.22	0.0045~0.016	14
Yuen	0.1~0.15	0.45	0.013~0.055	1.19~2.54	0.0035~0.0094	12
FCF (S01~S03, S07, S08, S10)	0.17~0.30	3.3~10	0.24~1.11	2.08~19.81	0.0063~0.073	37
P&T	0.13~0.15	1.07~1.17	0.012~0.026	2.48~5.37	0.012~0.027	16
Atabay (ROS ORH)	0.07~0.17	1.21	0.018~0.08	2.63~13.06	0.01~0.053	34
M&K	0.07~0.12	3.95	0.015~0.11	5.61~20.52	0.024~0.105	6
Khatua	0.14~0.22	0.44	0.0087~0.039	1.09~3.03	0.0061~0.018	10
Patra	0.11~0.14	1.89	0.048~0.096	3.36~5.51	0.014~0.025	13
Experimental data for One-floodplain
FCF(S6)	0.18~0.30	6.3	0.26~0.93	3.46~11.52	0.011~0.043	6
Atabay (ROA)	0.07~0.11	1.213	0.018~0.055	2.28~4.97	0.0083~0.025	22
Bousmar	0.05~0.08	0.8	0.0078~0.016	3.42~13.08	0.02~0.084	4
Macintosh	0.07~0.11	1.51~1.62	0.019~0.06	2.37~14.24	0.0086~0.062	60
Khatib	0.05~0.11	0.3	0.0033~0.014	1.15~2.10	0.014~0.03	12

).

For symmetric-floodplain cases, there are 142 sets of data: the range of H is about 0.07~0.30 m; the range of B_t is about 0.3~10 m; the Q range is about 0.0067~1.11 m³/s; the range of Froude number is about 0.30~2.30; μ₀ ranges from 1.05 to 20.52; α ranges from 0.0035 to 0.105. For the one-floodplain case, there are 104 sets of data: the range of H is about 0.05 to 0.30 m; the range of B_t is about 0.3~6.3 m; the range of Q is about 0.0033~0.93 m³/s; the range of Froude number is about 0.48~1.68; μ₀ ranges from 1.15 to 14.24; α ranges from 0.0083 to 0.084. For the tests mentioned in the manuscript, B_t/B = 1.704~14.417, and H/h = 1.088~5.5.

Figure 4 shows that the method proposed in this study provides acceptable prediction accuracy of total and main channel discharges for experimental data with total discharge in the range from 0.0033 to 1.11 m³/s (Table 2).

Three parameters were defined to represent the prediction accuracy of discharges and boundary shear forces. NRMSE, defined in Equation (17), represents the relative average error compared to the difference between the measured discharge’s maximum and minimum values. The parameter P_ε% defined in Equation (18) denotes the percentage of the calculated discharges with the absolute value of the relative error less or equal to ε%. The parameter P_ξ defined in Equation (19) denotes the percentage of the calculated SF_mc (the ratio of the main channel shear force to the total shear force) with an absolute error less or equal to ξ.

$$NRMSE=\frac{\sqrt{\frac{1}{N}{\displaystyle \sum _{\mathrm{i}=1}^N{(Q_{\mathrm{C}}-Q_{\mathrm{M}})}^2}}}{Q_{\max \mathrm{M}}-Q_{\min \mathrm{M}}}$$

(17)

$$P_{\epsilon \%}=P\left(\left|\frac{Q_{\mathrm{C}}-Q_{\mathrm{M}}}{Q_{\mathrm{M}}}\right|\times 100\%\le \epsilon \%\right)$$

(18)

$$P_{\xi }=P(\left|S{F_{\mathrm{mc}}}_{\mathrm{C}}-S{F_{\mathrm{mc}}}_{\mathrm{M}}\right|\le \xi )$$

(19)

Table 3. Comparison of prediction accuracy of Q, Q_mc, and SF_mc based on different methods.

	Method	Khatua	Huthoff	Yang α = 0.04	Yang α = 0.02	Luo	Proposed Method
Parameter
P3%(Q)		56.50	41.87	41.46	46.34	53.66	54.88
P5%(Q)		70.73	64.23	60.98	68.29	77.23	77.24
P10%(Q)		89.84	87.80	96.74	93.90	98.78	97.15
NRMSE(Q) × 100		1.6	1.4	0.93	1	0.51	0.59
P3%(Qmc)		37.5	48.81	18.45	33.93	48.48	51.79
P5%(Qmc)		56.55	70.83	30.36	57.74	82.14	81.55
P10%(Qmc)		85.71	86.90	75.00	88.69	98.80	96.43
NRMSE(Qmc) × 100		3.3	2.4	2.4	2	0.99	0.96
P0.01(SFmc)		23.13	23.88	11.94	21.64	31.34	23.88
P0.03(SFmc)		56.72	43.28	38.06	55.22	66.42	61.94
P0.05(SFmc)		73.88	76.12	73.13	82.09	86.57	82.84

shows the prediction accuracy of discharges and boundary shear forces based on different methods. Figure 5 compares the prediction accuracy of Q and Q_mc in different μ ranges based on different methods. The results indicate that employing a varied value for μ, like Luo’s and proposed methods, introduces better prediction accuracy for discharge and boundary shear force than using a specific value. The results further indicate that using a varied value for μ is essential for the computation of both discharge and boundary shear force.

4. Discussion

4.1. Comparison between Different Empirical Formulas

μ₀ is closely relevant to both (H − h)/P_fp (Figure 3) and H/P_fp (R² is about 0.36, equal to that of (H − h)/P_fp). Many experimental results showed that the apparent shear stress varied greatly as the water level changed in the same cross-section (i.e., h/P_fp varied insignificantly) [15,24]. Based on the discussion by Luo et al. [14], the apparent shear stress was closely related to (H − h)/P_fp. Therefore, (H − h)/P_fp was used in the proposed method instead of H/P_fp. As H/P_fp is used instead of (H − h)/P_fp, another empirical expression can also be obtained:

$${\mu }_0=0.819{\left(f_{\mathrm{mc}}/f_{\mathrm{fp}}\right)}^{-0.480}{\left(H/P_{\mathrm{fp}}\right)}^{-0.890}$$

(20)

Equation (20) also provides an acceptable prediction accuracy of μ₀ calculated based on e_Q and e_SF, and R² is about 0.67.

Table 4. Prediction accuracy of Q and Q_mc with different empirical formulas.

	Formulas	Equation (16)	Equation (20)	Equation (21)	Equation (22)
Parameters
P3%(Q)		54.88	52.03	54.07	55.28
P5%(Q)		77.24	76.83	78.86	78.45
P10%(Q)		97.15	95.53	97.15	96.34
NRMSE(Q) × 100		0.59	0.44	0.44	0.46
P3%(Qmc)		51.79	42.86	42.26	47.02
P5%(Qmc)		81.55	69.64	77.38	77.38
P10%(Qmc)		96.43	94.64	98.21	95.25
NRMSE(Qmc) × 100		0.96	1.14	1.06	1.08

shows the calculating accuracy of discharges when different empirical formulas were employed. Calculation accuracy changes slightly when Equation (20) is used instead of Equation (16).

Equation (8) introduces μ as a revised momentum transfer coefficient α to simplify the calculating processes. Referring to the method of developing an empirical formula for the optimal value μ₀, the empirical formula for the optimal value α₀ can also be developed:

$${\alpha }_0=0.00398{\left(f_{\mathrm{mc}}/f_{\mathrm{fp}}\right)}^{-0.452}{((H-h)/P_{\mathrm{fp}})}^{-0.496}$$

(21)

Table 4 indicates that using empirical formulas of α₀ or μ₀ does not change the prediction accuracy of Q and Q_mc.

4.2. Overfitting Problem and the Origin of the Error

Dividing a compound cross-section into three or four sub-regions is helpful in including fewer geometrical parameters in calculating the apparent shear stress. The proposed method and Luo’s method employed only one geometrical parameter, (H − h)/P_fp, and Huthoff’s and Yang’s methods employed no geometrical parameters. However, most previous methods of calculating the apparent shear stress employed two or more geometrical parameters [10]. For example, Prinos and Townsend [15] used two geometrical parameters, (H − h)/H and (B_t − B)/B; Knight and Hamed [24] employed two geometrical parameters, B_t/B and (H − h)/H.

As more parameters are employed to develop the empirical formula of μ₀, overfitting problems may occur. Considering that both P_fp/H and (H − h)/P_fp are closely relevant to μ₀, three parameters, including (H − h)/P_fp, H/h, and f_fp/f_mc, were employed to develop an empirical formula of μ₀:

$${\mu }_0=0.182{\left(f_{\mathrm{mc}}/f_{\mathrm{fp}}\right)}^{-0.454}{((H-h)/P_{\mathrm{fp}})}^{-0.843}{\left(H/h\right)}^{1.434}$$

(22)

Equation (22) provides better prediction accuracy of μ₀, and R² is about 0.72. However, the calculation accuracy of main channel discharges (Q_mc) decreases slightly compared to the proposed method (see Table 4).

The proposed method improves the prediction accuracy by employing a varied coefficient to calculate the apparent shear stresses. However, the flow interaction between the floodplain and the main channel, as well as the flow interaction between sub-region 2 and sub-region 4, involve three-dimensional (3D) secondary flow [24]. The secondary flow transports water and flow momentum across the vertical and horizontal dividing lines. The local flow structure is the main reason for the apparent shear stress. The proposed method considers only the average velocity of adjacent sub-regions and neglects the 3D characteristics of flow interaction. It is the error origin for all the methods dividing the compound cross-section into several sub-regions.

4.3. Erodible Channels

The proposed method is based on experimental data with a rigid cross-section; however, it provides some insights into the prediction of adjustment of discharge capacity for natural compound channels. Natural channels vary with time due to erosion, deposition processes, or human activities [25]. The erosion and deposition processes change the size of sub-regions, and destroying the armor layer in the main channel or vegetation encroachment on the floodplain adjusts the resistance coefficient f_fp/f_mc ratio. The proposed method comprehensively considers the impact of these two changes on the flow capacity. Fu et al. [26] studied the discharge capacity in a compound channel under erosion processes in which the main channel’s size and roughness varied with time. There are two series with different initial bank full depths: Series A and Series B. Figure 6 compares Q_M and Q_C based on the proposed method. It shows that the proposed method can reflect the changes in the flow capacity of a compound channel during the riverbed’s natural erosion and armoring processes.

5. Conclusions

The apparent shear stress of the dividing lines can be calculated by multiplying a revised version of the momentum transfer coefficient with the square velocity difference between the adjacent sub-regions. An optimal coefficient value in calculating apparent shear stress was obtained when the discharge error or error of the apparent shear stress reached a minimum value.

This study shows that the μ₀ range for all experiments mentioned above is about 0~30, and the most relevant geometrical parameter is (H − h)/P_fp. The relative resistant coefficient (f_fp/f_mc) is also essential. An empirical formula to calculate μ₀ has been built using these two parameters. The comparison between the calculated and measured μ₀ indicates that the reasonable prediction of discharges and boundary shear force can be obtained if the μ₀ has been given the appropriate value.

Comparison with other methods and experimental data indicates that the proposed method produces better prediction accuracy for discharge and boundary shear force than those using a specific value. For erodible compound cross-sections, the proposed method also provides an acceptable prediction of discharges. However, the limitation is that the data used for examination are not enough. The proposed method, which comprehensively considers the impact of variation size and roughness of the cross-section on the flow capacity, has the potential to be used in predicting discharge in natural rivers.

We should mention that the ranges of water depth of the main channel and the total width of the compound cross-section are about 0.05~0.30 m and 0.3~10 m, respectively; the Q range and the range of Froude numbers of the main channel flow are about 0.0033~1.11 m³/s and 0.3~2.3, respectively, for all the tests used for the analysis. Once more experimental datasets are collected (e.g., with larger Q values), further validation of the proposed method will be conducted.