# Flow Resistance Equation in Sand-bed Rivers and Its Practical Application in the Yellow River

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## Abstract

**:**

## 1. Introduction

_{50}[17]. Notably, Wu and Wang [18] asserted that the Froude number F, grain bed shear stress ${\tau}^{\prime}$ and critical bed shear stress ${\tau}_{c}$ should be considered in the expression of A. They developed an empirical flow resistance equation suitable for the flow regime from stationary flat beds to moving plane beds by a third-order polynomial curve fitting based on flume and field data.

_{s}to viscous sublayer thickness $\delta $. Accordingly, V/${u}_{\ast}$ is related to the hydraulic radius R and the Nikuradse roughness height k

_{s}. Then, Chezy’s coefficient C can be represented by a function of R, $\chi $, and k

_{s}. Consequently, the bed roughness can be obtained by solving k

_{s}. Li and Liu [21] expressed k

_{s}in the logarithmic velocity distribution equation as the product of D

_{50}and a proportionality coefficient $\alpha $, and derived the graphical relation between $\mathrm{log}10(1/\alpha )$ and ${\mathrm{log}10(V/V}_{c})$ (V

_{c}is the incipient velocity of the bed-material load, an indicator of bed material movement) based on field data including the Yellow River, the Yangtze River and the Ganjiang River. They used the relation between $\alpha $ and the relative velocity V/V

_{c}to reflect the influence of the flow intensity and movement of the bed material on flow resistance. van Rijn [22] investigated bed-form height and length based on flume and field data and developed an equation for ${k}_{s}$ by adding grain roughness height ${k}_{s,\mathrm{grain}}$ and form roughness height ${k}_{s,\mathrm{form}}$. ${k}_{s,\mathrm{grain}}$ is calculated by representative diameters and some alternative equations were also developed (e.g., [23,24]). ${k}_{s,\mathrm{form}}$ is calculated by geometric parameters of the bed-form, but they are hard to observe in natural rivers. Niazkar et al. [4] considered that ${k}_{s}$ is closely related to the settling velocity, F, Shields parameter ${\tau}_{\ast}$ and particle size, and they obtained the expression of ${k}_{s}$ based on a large amount of flume and field data.

^{1/6}J

^{1/2}and a negative correlation with F. Actually, the data of n used to fit the equation proposed by Ma et al. [28] are calculated by Manning’s roughness equation. Under the condition where R

^{1/6}J

^{1/2}generally have a little change in sand-bed rivers, the equation proposed by Ma et al. [28] naturally has a good performance in the LYR, and limited information about flow resistance thus can be determined from the equation.

## 2. Materials and Methods

#### 2.1. Data Collection

^{−4}to 7$\times $10

^{−4}were considered [30]. (3) The data samples with a Nikuradse roughness height (the impact of vegetation and river training structures were not considered, and the hydraulic radius was replaced by the water depth during the calculation of the Nikuradse roughness height) greater than the corresponding water depth were removed. (4) The Q (flow discharge) ~V, Q~H and Q~J relations of the data samples of each gauge station are plotted individually, and points that obviously deviate from the point group were deleted. (5) The data samples from the GC, LK and LJ gauge stations with Q values lower than 500 m

^{3}/s and n higher than 0.04 were removed. Accordingly, 930 data samples were finally collected.

^{−4}−10

^{−3}. The secondary filtering process was undertaken according to the following principles: data samples with water depths deeper than 0.1 m [22] and width-depth ratios greater than 6 [31] were reserved; data samples with a specific gravity equal to 2.65 were selected; data samples from rivers and canals whose mean velocity has a qualitatively negative correlation with n were selected. Finally, 280 data samples from different rivers and canals (the ACOP Canal (ACP), the CHOP Canals (CHO), the Mississippi River (MIS), the Rio Grande Conveyance Channel (RGC), the Rio Grande (RGR) and the Rio Grande near Bernalillo (RIO)) were selected. The total samples of the above six datasets were 676, and 396 samples were removed. The ratio of the removed samples to the total samples is 58.58%.

^{3}) indicates that the LYR data mainly represent the data of sediment-laden flows under high-concentration scenarios. The arithmetic mean sediment concentration of the Brownlie data (1.09 kg/m

^{3}) indicates that the Brownlie data mainly represent the data of sediment-laden flows under low-concentration scenarios. The LYR data were randomly divided into two subsets at a ratio of 1:2. A total of 317 samples (calibration data) were chosen to calibrate the proposed flow resistance equation, and the remaining 613 LYR samples and Brownlie data (validation data) were used for equation validation. The range of the flow and sediment conditions of the calibration and validation data are shown in Table 1.

#### 2.2. Establishment of the Flow Resistance Equation

_{s}. They set $\chi $ to be 1 (since the research conditions are mainly in hydraulically rough conditions) and obtained the expression of C through a combination of the expression of V/${u}_{\ast}$ and Chezy’s equation:

_{c}, and the relationship reflects the influence of different bed forms on flow resistance.

_{c}as Li and Liu [21] did to reflect the flow intensity and movement of the bed material in the modification of $\alpha $. Specifically, $\alpha $ is expressed as a function of a comprehensive influence coefficient Z composed of $\kappa $, ${\delta}_{\mathrm{m}}/{D}_{50}$ and V/V

_{c}:

_{50}for nonuniform bed-material load; and ${n}_{\mathrm{d}}$ is the Manning’s roughness coefficient corresponding to the grain roughness. When calculating ${n}_{\mathrm{d}}$ in Equation (4), ${n}_{\mathrm{d}}$ = D

_{50}

^{1/6}/A is adopted, and A is calculated by the following equation [17]:

_{0}(set to 1 m) is a coefficient used to keep the equation compatible with the theory of dimensional homogeneity. ${Re}_{\mathrm{vd}}$ in Equation (4) is the Reynolds number computed by the incipient velocity of the bed-material load, ${Re}_{\mathrm{vd}}{=V}_{\mathrm{c}}{D/\nu}_{\mathrm{m}}$. ${\nu}_{\mathrm{m}}$ is the kinematic viscosity coefficient of sediment-laden flows [36]:

_{v}is the sediment concentration by volume; d

_{50}is the median size of the suspended load, which is set to 0.025 mm in this study when lack of d

_{50}values; and $\nu $ is the kinematic viscosity coefficient of the water. The values of $\nu $ under different temperatures can be obtained by interpolation using the data listed in Table A.2 of Chang [37]. $\kappa $ is expressed as [36]

_{1}and b

_{2}are two undetermined coefficients. b

_{1}and b

_{2}can be calibrated via a curve fitting of $\mathrm{log}10(1/\alpha )$ and Z based on field data. Notably, sediment concentration is indispensable in the calculation of $\kappa $, ${\delta}_{\mathrm{m}}$, and ${V}_{\mathrm{c}}$. Thus, the influence of sediment concentration on flow resistance is reflected indirectly by Equation (2), making it possible to increase the estimation accuracy of Manning’s roughness coefficient in sediment-laden flows under high-concentration scenarios.

#### 2.3. The Selection of b_{1} and b_{2}

_{1}and b

_{2}. b

_{1}and b

_{2}are determined via the following steps: (1) calculate $\kappa $, ${\delta}_{\mathrm{m}}$ and ${V}_{\mathrm{c}}$ using Equations (7), (8) and (4), respectively; (2) manually set the values of b

_{1}and b

_{2}, and calculate Z using Equation (9); (3) calculate $\mathrm{log}10$(1/$\alpha $) by Equation (1); (4) conduct a third-order polynomial curve fitting between $\mathrm{log}10$(1/α) and Z (Z > 0) and obtain the empirical equation of the relationship between $\mathrm{log}10$(1/α) and Z; and (5) choose the b

_{1}and b

_{2}values that give the best fitting result. Notably, we mainly focus on the flow resistance law under a movable bed; thus, we only select the data samples of calibration data with the calculated V

_{c}smaller than V. In the actual calculation process, a grid search strategy is used to select b

_{1}and b

_{2}, with values ranging from −2 and 2 at an increment of 0.01, and 401 × 401 relationships between log10(1/α) and Z are obtained. From all of the experiments, b

_{1}= 0.48 and b

_{2}= 0.13 are found to give the best result; thus, Equation (9) can be expressed as follows:

## 3. Results and Discussion

#### 3.1. Validation of the Proposed Flow Resistance Equation

_{50}, sediment concentration and water temperature, the calculated Manning’s roughness coefficient is computed by combining the proposed Equation (11), Equation (1) and ${n=H}^{1/6}/C$. Equations developed by Zhao and Zhang [25] (referred to as the Zhao-Zhang equation hereafter), Wu and Wang [18] (referred to as the Wu-Wang equation hereafter) and Ma et al. [28] (referred to as the Ma-Xia equation hereafter) are selected for comparison with the proposed equation. The performance of each equation is quantified by the correlation coefficient r between the actual and calculated n and the mean absolute percentage error of n (MAPE

_{n}):

_{n}and r, the proposed Equation (11) performs the best (MAPE

_{n}: 14.88%; r: 0.88). Moreover, for the proposed Equation (11), Z ranges from 0.0137 to 0.5592 during the validation process, which means that nearly the whole application range (0.0101–0.5749) of the proposed Equation (11) is validated. Notably, the validation data samples that do not meet with the applicable range of the validated equations are removed. The percentage of the removed data samples for the proposed Equation (11), the Zhao-Zhang equation, the Wu-Wang equation and the Ma-Xia equation are 0.34%, 0.56%, 0.34% and 0%, respectively, which indicates that the validated equations have little difference in applicability to the validation data. Figure 3 shows that when the actual n is smaller than 0.02, the scatters in Figure 3a are distributed on both sides of the forty-five-degree line (hereafter referred to as the perfect agreement line). Most of the data points in Figure 3b,d are distributed under the perfect agreement line, indicating that the n calculated by the Zhao-Zhang equation and Ma-Xia equation are lower than the actual value of n for n values less than 0.02. The majority of the data points in Figure 3c are distributed above the perfect agreement line, indicating that the n calculated by the Wu-Wang equation is generally higher than the actual value of n for n values less than 0.02. When the actual n increases, although the deviation of the proposed Equation (11) increases (Figure 3a), the data points are still relatively closely distributed on both sides of the perfect agreement line. The data points of the other three equations have obvious negative biases (Wu-Wang and Ma-Xia equations) or relatively large deviations (Zhao-Zhang equation).

_{n}: 13.07%; r: 0.78). The MAPE

_{n}of the proposed Equation (11) is 14.35%, which is 1.1 times that of the Wu-Wang equation. The percentages of the removed data samples of the proposed Equation (11), the Zhao-Zhang equation, the Wu-Wang equation and the Ma-Xia equation are 0.71%, 1.79%, 0% and 0%, respectively.

_{n}: 15.12%; r: 0.85). The Ma-Xia equation achieves a similar performance with a slightly higher MAPE

_{n}. The Zhao-Zhang equation obtains a better r than that of the proposed Equation (11), but the MAPE

_{n}is much higher. The performance of the Wu-Wang equation on the LYR is limited, with the MAPE

_{n}nearly two times those of the proposed Equation (11) and the Ma-Xia equation. The percentage of the removed data samples of the proposed Equation (11), the Zhao-Zhang equation, the Wu-Wang equation and the Ma-Xia equation are 0.16%, 0%, 0.49% and 0%, respectively.

_{n}. For the LYR validation data, the proposed Equation (11) achieves the highest accuracy, and the performance of the Wu-Wang equation is limited. Considering that the LYR data mainly represent the data of sediment-laden flows under high-concentration scenarios and that the Brownlie data mainly represent the data of sediment-laden flows under low-concentration scenarios, the proposed Equation (11) is the only validated flow resistance equation that achieves a satisfactory accuracy for sediment-laden flows under both high- and low-concentration scenarios in sand-bed rivers.

#### 3.2. The Optimum Main Channel Width for Sediment Transportation in the LYR

_{m}. The computational conditions need to be confirmed before calculating the B

_{m}~J and B

_{m}~n relations. In the HG reach, the average n, slope, D

_{50}, and d

_{50}are 0.011, 2 × 10

^{−4}, 0.125 mm and 0.021 mm, respectively. The water temperature conditions are set to 26 ℃, 28 ℃, 30 ℃ and 32 ℃ according to the actual situation of the main flood season in the LYR. ${\omega}_{0}$ at 26 ℃, 28 ℃, 30 ℃ and 32 ℃ is 0.195 cm s

^{−1}, 0.202 cm s

^{−1}, 0.208 cm s

^{−1}and 0.215 cm s

^{−1}, respectively. We select the current channel-forming discharge (4000 m

^{3}/s) as the flow discharge. The suspended load carrying capacity is calculated through the combination of the water continuity equation, Equation (14) and the comprehensive stability index of the riverbed Z

_{w}in the HG reach, which is applied to river pattern recognition. The expression of Z

_{w}was proposed by Zhang et al. [41]:

_{w}is set to 4 in this study. Combining the water continuity equation, Manning’s roughness equation, Equation (14) and Equation (15), the suspended load carrying capacities at 26 °C, 28 °C, 30 °C and 32 °C are 27.80 kg/m

^{3}, 26.08 kg/m

^{3}, 24.79 kg/m

^{3}and 23.44 kg/m

^{3}, respectively. Treating the suspended load carrying capacity as the sediment concentration, the computational conditions are shown in Table 5.

_{m}~J and B

_{m}~n relations are listed as follows:

- Give the main channel width B
_{m}. - Assume a water depth H and calculate the cross-sectional area.
- Calculate the mean velocity V using Equation (14).
- Calculate the flow discharge Q
_{1}with V and the cross-sectional area. - If abs(Q − Q
_{1}) $\ge $ 0.01, repeat steps 2–4. - If abs(Q − Q
_{1}) < 0.01: (1) assume a slope J and calculate Z from Equation (10); (2) calculate log10(1/α) from Equation (11); (3) derive J_{1}from Equation (1) and Chezy’s equation; (4) if abs(J − J_{1}) $\ge $ 1$\times $10^{−4}, repeat substeps (1)–(3); (5) if abs(J − J_{1}) < 1$\times $10^{−4}, calculate n with ${n=H}^{1/6}/C$.

_{m}~J and B

_{m}~n relations are obtained (Figure 5). As shown in Figure 5a, the slope monotonically increases with the main channel width. However, as shown in Figure 5b, n decreases first and then increases; thus, a critical point (minimum point) of n can be found in each line in Figure 5b. Sha and Jiang [42] studied the eco-erosion problem of very shallow flow on a slope surface and revealed that the evolution of the flow path follows the maximum entropy principle of thermodynamics, indicating that water flow tends to reach a morphology of a plane and section with minimum resistance. Li et al. [43] investigated the development patterns of sand bars in alluvial rivers and believed that the plane morphology of sand bars always tends to reach the minimum resistance to ensure morphological stability under the given flow, sediment and boundary conditions. In this study, following the concept of minimum resistance, it is speculated that the ratio of the energy consumed by resistance to the total energy is the smallest and that the sediment transport efficiency reaches a maximum when n reaches the critical value under the same flow discharge, sediment concentration, particle size and water temperature conditions. We regard the main channel width under the critical n as the stable main channel width and define it as the optimum main channel width for sediment transportation B

_{m_optimum}. As shown in Table 6, B

_{m_optimum}is 770 m, 800 m, 830 m and 850 m when the water temperature is 26 °C, 28 °C, 30 °C and 32 °C, respectively. Different channel width results under the same flow discharge (4000 m

^{3}/s) were obtained by previous researchers using various methods. Jiang et al. [44] plotted the relation between the flow discharge and stable main channel width, and found that the stable main channel width was about 600 m in the LYR. Ma and Liu [45] obtained the design channel widths at HYK (1080 m) and GC (806 m) by hydraulic computations. Xu [46] found that the stable main channel width ranges from 500 m to 1000 m in the LYR through the combination of flow resistance equation, sediment transport equation and regime equation. Ma [47] discovered that the stable main channel widths range from 580 m to 987 m when the sediment concentrations were 30~120 kg/m

^{3}. Ma [11] asserted that the optimum main channel width is 360~460 m in the LYR. We find that the stable main channel widths calculated by this study basically match the existing results except for the result obtained by Ma [11], which is smaller than other results. This finding is consistent with the current river training width in the LYR (1000 m from HYK to GC [48]). The determination of the stable main channel width is crucial in the determination of the river training width and the strategy for the LYR. Due to the importance of this issue, the determination of the width needs to be carefully argued by different methods, as done by [11,44,45,46,47]. This study provides an alternative approach for theoretically estimating the optimum main channel width (stable main channel width) for the river training strategy for the LYR.

## 4. Conclusions

_{c}) (an indicator of the flow intensity and bed material movement), $\kappa $ (an indicator of the energy consumption condition in sediment-laden flows) and ${\delta}_{\mathrm{m}}/{D}_{50}$ (an indicator of the relative friction condition in the near-wall region). The specific relation between $\alpha $ and Z is established by conducting a third-order polynomial curve fitting, as shown by Equation (11). The proposed equation is validated and compared with the Zhao-Zhang equation, Wu-Wang equation, and Ma-Xia equation based on a reliable river dataset containing LYR data and Brownlie data. The proposed equation achieves the highest accuracy in terms of the mean absolute percentage error and correlation coefficient between the calculated and actual Manning’s roughness coefficient. The proposed equation is the only equation among the validated flow resistance equations that is suitable for sediment-laden flows under both high- and low-concentration scenarios in sand-bed rivers. Moreover, the graphical relation between the main channel width B

_{m}and n in the typical wandering reach (the HG reach) in the LYR was obtained through the combination of Equation (11) and the suspended load carrying capacity equation (Equation (14)). We suggest that when n reaches a minimum, the ratio of the energy consumed by resistance to the total energy reaches a minimum and that the sediment transport efficiency reaches a maximum. The main channel width under the minimum n is treated as the optimum main channel width for sediment transportation B

_{m_optimum}. B

_{m_optimum}in the HG reach ranges from 770 m to 850 m with water temperatures ranging from 26 ℃ to 32 ℃. The optimum main channel width (stable main channel width) calculated by this study basically matches the existing results and is consistent with the current river training width of the LYR. Considering the determination of the stable main channel width is crucial in the determination of the river training width and the strategy for the LYR, the proposed equation has great potential as a theoretical tool that can be used to support the determination of the river training strategy for the LYR. Moreover, since the river training width needs to be adjusted if the runoff and sediment conditions in the LYR change, this paper also provides a helpful tool for researchers and practitioners to calculate the stable main channel width under future flow discharge, sediment concentration, particle sizes and water temperatures conditions, supporting the determination of the river training width.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Scatter plot between $\mathrm{log}10(1/\alpha )$ and $Z={\kappa}^{0.48}{{(\delta}_{\mathrm{m}}/{D}_{50})}^{0.13}{\mathrm{log}10(V/V}_{\mathrm{c}})$, and the fitting result.

**Figure 3.**Predicted n plotted against actual n for the validation data: (

**a**) The proposed Equation (11); (

**b**) Zhao-Zhang equation; (

**c**) Wu-Wang equation; (

**d**) Ma-Xia equation.

**Figure 4.**Predicted n plotted against actual n with the LYR validation data: (

**a**) The proposed Equation (11); (

**b**) Zhao-Zhang equation; (

**c**) Wu-Wang equation; (

**d**) Ma-Xia equation.

**Figure 5.**The B

_{m}~J and B

_{m}~n curves of the LYR: (

**a**) B

_{m}~J curves of the LYR; (

**b**) B

_{m}~n curves of the LYR (the circles represent critical points).

Category | Name | Number | V/(m/s) | H/(m) | J/10^{−4} | D_{50}/(mm) | S/(kg/m^{3}) |
---|---|---|---|---|---|---|---|

calibration | LYR | 317 | 0.41~3.33 | 0.5~6.27 | 1~5 | 0.029~0.27 | 0.537~299 |

validation | LYR | 613 | 0.37~3.2 | 0.52~6.3 | 1~6.6 | 0.03~0.371 | 0.325~311 |

validation | ACP | 100 | 0.35~1.29 | 0.91~4.27 | 1.01~1.66 | 0.084~0.313 | 0.013~1.008 |

validation | CHO | 30 | 0.8~1.6 | 1.31~3.41 | 1.15~2.54 | 0.09~0.311 | 0.148~1.318 |

validation | MIS | 19 | 0.69~2.42 | 4.82~17.28 | 1.03~1.34 | 0.213~0.684 | 0.026~0.329 |

validation | RGC | 8 | 0.81~1.52 | 0.92~1.51 | 5.3~8 | 0.18~0.28 | 0.674~2.7 |

validation | RGR | 85 | 0.33~2.38 | 0.16~1.46 | 6.9~10 | 0.174~0.655 | 0.011~7.273 |

validation | RIO | 38 | 0.62~2.38 | 0.33~1.46 | 7.4~8.9 | 0.207~0.368 | 0.464~4.557 |

Condition | The Proposed Equation (11) | Zhao-Zhang Equation | Wu-Wang Equation | Ma-Xia Equation |
---|---|---|---|---|

MAPE_{n}/% | 14.88 | 23.46 | 26.02 | 20.39 |

r | 0.88 | 0.82 | 0.84 | 0.80 |

Condition | The Proposed Equation (11) | Zhao-Zhang Equation | Wu-Wang Equation | Ma-Xia Equation |
---|---|---|---|---|

MAPE_{n}/% | 14.35 | 29.63 | 13.07 | 27.98 |

r | 0.69 | 0.63 | 0.78 | 0.73 |

Condition | The Proposed Equation (11) | Zhao-Zhang Equation | Wu-Wang Equation | Ma-Xia Equation |
---|---|---|---|---|

MAPE_{n}/% | 15.12 | 20.69 | 31.97 | 16.92 |

r | 0.85 | 0.87 | 0.75 | 0.85 |

Number | Q/(m^{3}/s) | S/(kg/m^{3}) | D_{50}/(mm) | d_{50}/(mm) | T/(℃) | ${\mathsf{\omega}}_{0}/(\mathbf{cm}/\mathbf{s})$ |
---|---|---|---|---|---|---|

1 | 4,000 | 27.80 | 0.125 | 0.021 | 26 | 0.195 |

2 | 4,000 | 26.08 | 0.125 | 0.021 | 28 | 0.202 |

3 | 4,000 | 24.79 | 0.125 | 0.021 | 30 | 0.208 |

4 | 4,000 | 23.44 | 0.125 | 0.021 | 32 | 0.215 |

Number | Q/(m^{3}/s) | S/(kg/m^{3}) | T/(℃) | B_{m_optimum}/(m) | n | J/10^{−4} | V/(m/s) | H/(m) |
---|---|---|---|---|---|---|---|---|

1 | 4000 | 27.80 | 26 | 770 | 0.0107 | 1.70 | 2.18 | 2.38 |

2 | 4000 | 26.08 | 28 | 800 | 0.0107 | 1.74 | 2.16 | 2.31 |

3 | 4000 | 24.79 | 30 | 830 | 0.0107 | 1.78 | 2.14 | 2.25 |

4 | 4000 | 23.44 | 32 | 850 | 0.0107 | 1.81 | 2.13 | 2.21 |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cai, R.; Zhang, H.; Zhang, Y.; Zhang, L.; Huang, H. Flow Resistance Equation in Sand-bed Rivers and Its Practical Application in the Yellow River. *Water* **2020**, *12*, 727.
https://doi.org/10.3390/w12030727

**AMA Style**

Cai R, Zhang H, Zhang Y, Zhang L, Huang H. Flow Resistance Equation in Sand-bed Rivers and Its Practical Application in the Yellow River. *Water*. 2020; 12(3):727.
https://doi.org/10.3390/w12030727

**Chicago/Turabian Style**

Cai, Rongrong, Hongwu Zhang, Yu Zhang, Luohao Zhang, and Hai Huang. 2020. "Flow Resistance Equation in Sand-bed Rivers and Its Practical Application in the Yellow River" *Water* 12, no. 3: 727.
https://doi.org/10.3390/w12030727