*2.3. Calculation Method for the Lateral Distribution of Water and Sediment Factors* 2.3.1. Transverse Distribution of the Velocity

Through analyzing a large volume of measured data from the Yellow River, we observed that the transverse distribution of velocity was mainly related to water depth. Therefore, the formula for the velocity distribution along the transverse direction was as follows:

$$\frac{V\_{\text{i}}}{V} = \mathcal{C}\_{1} \left(\frac{h\_{\text{i}}}{h}\right)^{\frac{2}{3}} \tag{7}$$

where *V* and *V*<sup>i</sup> represent the average velocity of the section and the velocity at any point, respectively; *h* represents the average section and water depth at any point, respectively; *C*<sup>1</sup> is the section shape coefficient of ~1, which can be obtained by mass conservation.

$$\mathbf{C}\_{1} = \frac{\mathbf{Q}}{\int\_{a}^{b} \frac{V}{h^{2/3}} h\_{\mathrm{i}}^{5/3} \mathrm{d}y} \tag{8}$$

where *Q* is the average flow of the section, *y* is the transverse coordinate, and *a* and *b* are the starting point distances between the two ends of the river width of the section (*b* > *a*).

Hundreds of measured data sets from the lower Yellow River were used to verify Equation (7). The average velocity range of the section was 0.10–3.56 m/s, of which the data range covered the pre-flood and flood seasons. The results are shown in Figure 3. Equation (7) could be used to accurately calculate the transverse distribution law of velocity before flood season and during the flood season. This formula has comprehensive considerations and convenient applications.

considerations and convenient applications.

1

=

*b a <sup>Q</sup> <sup>C</sup> <sup>V</sup> h y <sup>h</sup>*

where *Q* is the average flow of the section, *y* is the transverse coordinate, and *a* and *b* are the starting point distances between the two ends of the river width of the section (*b* > *a*). Hundreds of measured data sets from the lower Yellow River were used to verify Equation (7). The average velocity range of the section was 0.10–3.56 m/s, of which the data range covered the pre-flood and flood seasons. The results are shown in Figure 3. Equation (7) could be used to accurately calculate the transverse distribution law of velocity before flood season and during the flood season. This formula has comprehensive

5/3 2/3 <sup>i</sup> d

(8)

**Figure 3.** Verification of transverse velocity distribution by flow data before, and during flood season: (**a**) Tiexie, 24 April 1966 (before flood season); (**b**) Peiyu, 23 July 1966 (flood season); (**c**) Guanzhuangyu, 1 April 1966 (before flood season); (**d**) Huayuankou, 23 May 1966 (flood season); (**e**) Tiexie, 2 July 1966 (flood season); (**f**) Guanzhuangyu, 24 July 1966 (flood season). **Figure 3.** Verification of transverse velocity distribution by flow data before, and during flood season: (**a**) Tiexie, 24 April 1966 (before flood season); (**b**) Peiyu, 23 July 1966 (flood season); (**c**) Guanzhuangyu, 1 April 1966 (before flood season); (**d**) Huayuankou, 23 May 1966 (flood season); (**e**) Tiexie, 2 July 1966 (flood season); (**f**) Guanzhuangyu, 24 July 1966 (flood season).

#### 2.3.2. Transverse Distribution of Sediment Concentration 2.3.2. Transverse Distribution of Sediment Concentration

As the non-uniformity of water depth and resistance distribution along the river width of the wandering channel is prominent, the distribution of sediment transport capacity along the river width is quite different, resulting in obvious differences in the distribution of sediment concentration along the river width in the process of sediment transport. Through analyzing a large volume of measured data of the Yellow River, Enhui et al. [25] observed that the transverse distribution law of sediment concentration is not only related to hydraulic factors and sediment concentration but also closely related to the composition of suspended sediment. The finer the suspended sediment composition, the more uniform the transverse distribution of the sediment concentration. Therefore, in addition to introducing the sediment concentration factor, the suspension index *ω*/*ku*\* was also introduced to reflect the thickness of the suspended sediment composition, and the As the non-uniformity of water depth and resistance distribution along the river width of the wandering channel is prominent, the distribution of sediment transport capacity along the river width is quite different, resulting in obvious differences in the distribution of sediment concentration along the river width in the process of sediment transport. Through analyzing a large volume of measured data of the Yellow River, Enhui et al. [25] observed that the transverse distribution law of sediment concentration is not only related to hydraulic factors and sediment concentration but also closely related to the composition of suspended sediment. The finer the suspended sediment composition, the more uniform the transverse distribution of the sediment concentration. Therefore, in addition to introducing the sediment concentration factor, the suspension index *ω*/*ku*\* was also introduced to reflect the thickness of the suspended sediment composition, and the transverse distribution formula of sediment concentration was established as follows [25]:

$$\frac{S\_1}{S} = \mathcal{C}\_2 \left(\frac{h\_1}{h}\right)^{(0.1 - 1.6\frac{\omega\_k}{k\omega\_\*} + 1.3\text{Sv})} \left(\frac{V\_1}{V}\right)^{(0.2 + 2.6\frac{\omega\_k}{k\omega\_\*} + \text{Sv})} \tag{9}$$

 ω

s s V V \* \* (0.1 1.6 1.3 ) (0.2 2.6 ) ii i <sup>2</sup> () () *S S Sh V ku ku <sup>C</sup> Sh V* −+ ++ = (9) where *S* and *S*<sup>i</sup> represent the average sediment concentration of the section and at any point, respectively; *S*<sup>V</sup> is the volume sediment concentration, and *S*<sup>V</sup> = *S*/2650; *h*, *h*<sup>i</sup> , *V*<sup>i</sup> , and *V* have the same meaning as previously described. *C*<sup>2</sup> is the section shape coefficient of ~1, which could be obtained from sediment conservation.

ω

$$\mathbf{C}\_{2} = \frac{Q}{\int\_{a}^{b} q\_{\mathrm{i}} \left(\frac{\hbar\_{\mathrm{i}}}{\hbar}\right)^{(0.1-1.6\frac{\omega\_{\mathrm{k}}}{k\mu\_{\mathrm{s}}}+1.3\mathrm{S}\_{\mathrm{V}})} \left(\frac{V}{\mathcal{V}}\right)^{(0.2+2.6\frac{\omega\_{\mathrm{k}}}{k\mu\_{\mathrm{s}}}+\mathrm{S}\_{\mathrm{V}})} \mathrm{d}y} \mathrm{d}y} \tag{10}$$

where *q*<sup>i</sup> is the unit width flow at any point of the section, *ω*<sup>s</sup> is the settling velocity of particles, calculated by Equations (11)–(14) [26]; *k* is the Carmen constant, which was calculated using Equation (15); *u*\* is the average frictional velocity of the section, and *u*<sup>∗</sup> = p *gh J*, *J* = 2‱; *Q* has the same meaning as before.

$$
\omega\_{\rm s} = \omega\_0 (1 - \frac{S\_{\rm V}}{2.25\sqrt{d\_{50}}})^{3.5} (1 - 1.25S\_{\rm V}) \tag{11}
$$

s s V V \* \*

 ω

(10)

*q y h V*

V 3.5 s0 V 50 (1 ) (1 1.25 ) 2.25 *<sup>S</sup> <sup>S</sup>*

*d*

−+ + +

$$
\omega\_0 = 2.6(\frac{\text{d}\_{\text{cp}}}{\text{d}\_{50}})^{0.3} \omega\_{\text{p}} e^{-635d\_{\text{cp}}^{0.7}} \tag{12}
$$

=− − (11)

$$
\omega\_{\rm p} = \frac{1}{18} \frac{\gamma\_{\rm s} - \gamma}{\gamma} g \frac{d\_{50}}{\nu}^2 \tag{13}
$$

$$\frac{d\_{50}}{d\_{\rm cp}} = 0.75\tag{14}$$

$$k = 0.4 - 1.68\sqrt{S\_v}(0.365 - S\_v) \tag{15}$$

More than 150 measured data sets from the lower reaches of the Yellow River were used to verify Equation (9). The average flow sediment concentration of the section ranged from 3 kg/m<sup>3</sup> to 480 kg/m<sup>3</sup> , including both floodplain and non-floodplain flood data (Figure 4). The verification results showed that Equation (9) could not only accurately calculate the transverse distribution law of sediment concentration in non-floodplain floods but also calculate the transverse distribution law of sediment concentration in the floodplain flood main channel, beach, and mixing area. The formula is suitable for both general and high sediment-laden flows, and the factors considered in the formula are comprehensive and easy to apply. used to verify Equation (9). The average flow sediment concentration of the section ranged from 3 kg/m3 to 480 kg/m3, including both floodplain and non-floodplain flood data (Figure 4). The verification results showed that Equation (9) could not only accurately calculate the transverse distribution law of sediment concentration in non-floodplain floods but also calculate the transverse distribution law of sediment concentration in the floodplain flood main channel, beach, and mixing area. The formula is suitable for both general and high sediment-laden flows, and the factors considered in the formula are comprehensive and easy to apply.

*Water* **2022**, *14*, x FOR PEER REVIEW 7 of 18

which could be obtained from sediment conservation.

=

\* *u ghJ* = ,*J* = 2‱; *Q* has the same meaning as before.

ω ω

*a*

*<sup>Q</sup> <sup>C</sup>*

where *S* and *S*i represent the average sediment concentration of the section and at any point, respectively; *S*V is the volume sediment concentration, and *S*V = *S*/2650; *h*, *h*i, *V*i, and *V* have the same meaning as previously described. *C*2 is the section shape coefficient of ~1,

> <sup>2</sup> (0.1 1.6 1.3 ) (0.2 2.6 ) i i <sup>i</sup>() () d *S S <sup>b</sup> ku ku*

where *q*i is the unit width flow at any point of the section, *ω*s is the settling velocity of particles, calculated by Equations (11)–(14) [26]; *k* is the Carmen constant, which was calculated using Equation (15); *u*\* is the average frictional velocity of the section, and

*h V*

ω

**Figure 4.** Verification of formula 9 by flood data with different sediment concentration: (**a**) low sediment concentration flood; (**b**) medium sediment concentration flood; and (**c**) high sediment concentration flood. **Figure 4.** Verification of formula 9 by flood data with different sediment concentration: (**a**) low sediment concentration flood; (**b**) medium sediment concentration flood; and (**c**) high sediment concentration flood.

#### 2.3.3. Lateral Distribution of the Suspended Sediment Composition 2.3.3. Lateral Distribution of the Suspended Sediment Composition

In natural rivers, the distribution of velocity and sediment concentration along the river width is uneven, which also leads to an uneven distribution of suspended sediment In natural rivers, the distribution of velocity and sediment concentration along the river width is uneven, which also leads to an uneven distribution of suspended sediment composition along the river width, and the suspended sediment composition in the mainstream area is generally coarse.

Based on the analysis of the measured data in the lower reaches of the Yellow River, the following formula for the distribution of the average particle size of suspended sediment along the river width was obtained:

$$\frac{d\_{\rm cpi}}{d\_{\rm cp}} = \mathcal{C}\_{\rm 3} (\frac{\mathcal{S}\_{\rm i}}{\mathcal{S}})^{0.6} \left(\frac{V\_{\rm i}}{V}\right)^{0.1} \tag{16}$$

where *d*cp is the average suspended sediment particle size of the section, *d*cpi is the average suspended sediment particle size at any point of the section, and *C*<sup>3</sup> is the section shape coefficient, which could be obtained from the sediment conservation.

$$\mathbf{C}\_{3} = \frac{\mathbf{Q}\mathbf{S}}{\int\_{a}^{b} \left[q\_{\mathrm{i}}\mathbf{S}\_{\mathrm{i}} \left(\frac{\mathbf{S}\_{\mathrm{i}}}{\mathbf{S}}\right)^{0.6} \left(\frac{V\_{\mathrm{i}}}{V}\right)^{0.1}\right] \mathrm{d}y} \tag{17}$$

Based on the suspended sediment gradation data measured by Tiexie–Xinzhai in the lower Yellow River in 1966 and corrected according to the suspended sediment gradation correction method, the average particle size was calculated and verified using Equation (16). The average *d*cp range of the section was 0.01–0.06 mm (Figure 5). The results showed that the calculated results were satisfactory regardless of the size and concentration of suspended sand. correction method, the average particle size was calculated and verified using Equation (16). The average *d*cp range of the section was 0.01–0.06 mm (Figure 5). The results showed that the calculated results were satisfactory regardless of the size and concentration of suspended sand.

composition along the river width, and the suspended sediment composition in the main-

Based on the analysis of the measured data in the lower reaches of the Yellow River, the following formula for the distribution of the average particle size of suspended sedi-

> cpi i i 0.6 0.1 3

*<sup>d</sup> S V <sup>C</sup>*

where *d*cp is the average suspended sediment particle size of the section, *d*cpi is the average suspended sediment particle size at any point of the section, and *C*3 is the section shape

*QS <sup>C</sup> S V qS y S V*

Based on the suspended sediment gradation data measured by Tiexie–Xinzhai in the lower Yellow River in 1966 and corrected according to the suspended sediment gradation

( )()

i i 0.6 0.1 i i [ ( ) ( ) ]d

*d SV* <sup>=</sup> (16)

(17)

cp

*b a*

coefficient, which could be obtained from the sediment conservation.

3

=

*Water* **2022**, *14*, x FOR PEER REVIEW 8 of 18

stream area is generally coarse.

ment along the river width was obtained:

**Figure 5.** Verification of Formula (16) with different suspended sand particle size data: (**a**) *d*cp = 0.0196 mm; (**b**) *d*cp = 0.026 mm; (**c**) *d*cp = 0.0357 mm; (**d**) *d*cp = 0.045 mm; and (**e**) *d*cp = 0.0509 mm. **Figure 5.** Verification of Formula (16) with different suspended sand particle size data: (**a**) *d*cp = 0.0196 mm; (**b**) *d*cp = 0.026 mm; (**c**) *d*cp = 0.0357 mm; (**d**) *d*cp = 0.045 mm; and (**e**) *d*cp = 0.0509 mm.

#### 2.3.4. Sediment-Carrying Capacity of Water Flow 2.3.4. Sediment-Carrying Capacity of Water Flow

Following sediment concentration, the physical properties and turbulent structure change, which impacts the flow energy loss, velocity, and sediment concentration distribution. Therefore, to obtain a sediment-carrying capacity formula suitable for both general flow and high sediment concentration flow, it is necessary to consider the influence of sediment on flow from the energy consumption graph of secondary flow. Hongwu et al. obtained a sediment-carrying capacity formula including all suspended sediments (for alluvial rivers, R ≈ h): Following sediment concentration, the physical properties and turbulent structure change, which impacts the flow energy loss, velocity, and sediment concentration distribution. Therefore, to obtain a sediment-carrying capacity formula suitable for both general flow and high sediment concentration flow, it is necessary to consider the influence of sediment on flow from the energy consumption graph of secondary flow. Hongwu et al. obtained a sediment-carrying capacity formula including all suspended sediments (for alluvial rivers, R ≈ h):

$$S\_\* = 2.5[\frac{(0.0022 + S\_V)V^3}{k\frac{\gamma\_{\rm s} - \gamma\_{\rm m}}{\gamma\_{\rm m}}gh\omega\_{\rm s}}\ln(\frac{h}{6d\_{50}})] \tag{18}$$

m γ Equation (18) adopts kg, m, and s as the unit system.

The verification process of the formula of sediment concentration distribution along the transverse direction, the formula for suspended sediment composition distribution along the transverse direction, and the general formula for flow sediment carrying capacity is detailed in a previous report [25].

The lower Yellow River channel is constantly adjusted with changes in incoming water and sediment and strives to adapt the water and sediment transport in the riverbed to the incoming water and sediment. Adjustment of the riverbed is reflected in the adjustment of the transverse shape of the river section. The cross-section of the lower Yellow River is a compound cross-section. Different water and sediment conditions determine the adjustment form and variation range of the cross-section. At the same time, the adjustment law of the cross-section of the beach and main channel in the lower Yellow River is different. This study mainly focused on the riverbed evolution under flat discharge, and the adjustment of its cross-section is primarily reflected in the adjustment of the main channel. Therefore, the change in the main channel was analyzed here.

#### **3. Results** 3.1.1. Overall Change in the River Cross-Sectional Shape before and after Construction

**3. Results** 

#### *3.1. Asymmetry of Wandering Channel without Engineering Constraints* The year 1960 was selected as the representative year for the free development river

*3.1. Asymmetry of Wandering Channel without Engineering Constraints* 

channel. Therefore, the change in the main channel was analyzed here.

*Water* **2022**, *14*, x FOR PEER REVIEW 9 of 18

Equation (18) adopts kg, m, and s as the unit system.

ity is detailed in a previous report [25].

3.1.1. Overall Change in the River Cross-Sectional Shape before and after Construction bend without engineering constraints, and the year 2019 was selected as the representa-

The verification process of the formula of sediment concentration distribution along the transverse direction, the formula for suspended sediment composition distribution along the transverse direction, and the general formula for flow sediment carrying capac-

The lower Yellow River channel is constantly adjusted with changes in incoming water and sediment and strives to adapt the water and sediment transport in the riverbed to the incoming water and sediment. Adjustment of the riverbed is reflected in the adjustment of the transverse shape of the river section. The cross-section of the lower Yellow River is a compound cross-section. Different water and sediment conditions determine the adjustment form and variation range of the cross-section. At the same time, the adjustment law of the cross-section of the beach and main channel in the lower Yellow River is different. This study mainly focused on the riverbed evolution under flat discharge, and the adjustment of its cross-section is primarily reflected in the adjustment of the main

The year 1960 was selected as the representative year for the free development river bend without engineering constraints, and the year 2019 was selected as the representative year under the action of the limited control boundary. According to the asymmetry index formula proposed above, the asymmetry of the cross-sectional morphology of the freedevelopment river bend without engineering constraints in 1960 and the river channel under the action of the limited control boundary in 2019 in the wandering section of the lower Yellow River was calculated (Figure 6). In 1960, most of the asymmetry indexes of the cross-section between Tiexie and Gaocun in the wandering section of the lower Yellow River were in the range of 1.01–1.50 and the average value of the asymmetry indexes of the entire wandering section was 1.19. Following construction of the project, most of the cross-sectional shape asymmetry indexes of the river reach in 2019 were in the range of 1.10–1.70 and the average value of the asymmetry indexes of the entire wandering section during this period was 1.33, indicating that there was a certain asymmetry in the crosssection shape of the river before and after the construction project and the asymmetry of the cross-section shape of the river after the construction project was greater than that without the project, and that the river after the construction project was constrained by the project boundary and the asymmetry of the cross-sectional shape was more prominent. tive year under the action of the limited control boundary. According to the asymmetry index formula proposed above, the asymmetry of the cross-sectional morphology of the free-development river bend without engineering constraints in 1960 and the river channel under the action of the limited control boundary in 2019 in the wandering section of the lower Yellow River was calculated (Figure 6). In 1960, most of the asymmetry indexes of the cross-section between Tiexie and Gaocun in the wandering section of the lower Yellow River were in the range of 1.01–1.50 and the average value of the asymmetry indexes of the entire wandering section was 1.19. Following construction of the project, most of the cross-sectional shape asymmetry indexes of the river reach in 2019 were in the range of 1.10–1.70 and the average value of the asymmetry indexes of the entire wandering section during this period was 1.33, indicating that there was a certain asymmetry in the cross-section shape of the river before and after the construction project and the asymmetry of the cross-section shape of the river after the construction project was greater than that without the project, and that the river after the construction project was constrained by the project boundary and the asymmetry of the cross-sectional shape was more prominent.

**Figure 6. Figure 6.**  Asymmetry of channel cross-section in wandering section of the lower Yellow River. Asymmetry of channel cross-section in wandering section of the lower Yellow River.

3.1.2. Calculation of the Asymmetry Index of Water and Sediment Factors in the Free Developing River Bend

In the 1950s and the 1960s, the lower reaches of the Yellow River belonged to a typical wandering channel. The channel was wide and shallow, scattered, densely covered with sandbars, the mainstream fluctuated, and the river regime changed sharply. It was difficult to determine the obvious and compliant curved top and transition sections. The planned river regulation in the wandering section of the lower Yellow River began in the 1970s and 1980s. Therefore, the lower Yellow River before 1980 can be regarded as a quasifree development bend in the period without engineering constraints. Therefore, 1979 was selected as the representative year for the free development of the bend before the construction of the wandering section of the lower Yellow River. To study the asymmetry of the distribution of water and sediment factors in a typical section of a free developing river, we selected the typical section based on two principles: (a) the river regime is relatively stable and (b) at the reach where the mainstream swing is obvious, a relatively obvious bend is observed as the bend top section, and the mainstream in the transition section is relatively stable.

Based on these principles, the section between the Heishi River channel in the wandering section of the lower Yellow River was selected as the representative section (Figure 7). According to the river regime map before the flood in 1979 and the measured large section

data of the section, the Liuyuankou and Jiahetan river channels were selected as the representative sections of the transition section, and Heishi, Gucheng, and Mazhai were selected as the representative sections of the bend top section without considering the variation of gradient along the river width, and the gradient value was 2‱and the roughness values were 0.01, 0.012, and 0.015, respectively. According to Manning's formula, the velocity was calculated, the discharge was inversely deduced, and the calculated flat discharge was 5000 m3/s. The flat water level, flat area, and other characteristic parameters of each section were calculated, and the transverse distribution of each water and sediment factor was calculated using Equations (6), (8), (15), and (17). Finally, the scouring and silting balance (*S* = 23 kg/m <sup>3</sup> ) of each section of the Heishi River section was determined using Equations ((1)–(5)) (scouring (*S* = 5 kg/m<sup>3</sup> ), siltation (*S* = 50 kg/m<sup>3</sup> )). The asymmetry indexes of the cross-sectional shape and water sediment factors under the three states and the calculation results of the asymmetry indexes under the three states were not different (Table 1). *Water* **2022**, *14*, x FOR PEER REVIEW 11 of 18

**Figure 7.** Wandering reach of the lower Yellow River (Heishi–Hedao): ① Sanguanmiao control project ② Weitan control project ③ Liuyuankou extension project ④ Dagong project ⑤ WANGan project ⑥ Gucheng project ⑦ Guantai project ⑧ Jiahetan project ⑨ Zhouying project ⑩ Laojuntang project ⑪ Yulin project ⑫ Wuzhuang vulnerable project ⑬ Huangzhai vulnerable project ⑭ Huozhai vulnerable project ⑮ Baocheng vulnerable project ⑯ Hedao project ⑰ Sanhecun project. A. Heishi section, B. Liuyuankou section, C. Gucheng section, D. Jiahetan section, E. Mazhai section, and F. Hedao section. **Figure 7.** Wandering reach of the lower Yellow River (Heishi–Hedao): <sup>1</sup> Sanguanmiao control project <sup>2</sup> Weitan control project <sup>3</sup> Liuyuankou extension project <sup>4</sup> Dagong project <sup>5</sup> WANGan project <sup>6</sup> Gucheng project <sup>7</sup> Guantai project <sup>8</sup> Jiahetan project <sup>9</sup> Zhouying project <sup>10</sup> Laojuntang project <sup>11</sup> Yulin project <sup>12</sup> Wuzhuang vulnerable project <sup>13</sup> Huangzhai vulnerable project <sup>14</sup> Huozhai vulnerable project <sup>15</sup> Baocheng vulnerable project <sup>16</sup> Hedao project <sup>17</sup> Sanhecun project. A. Heishi section, B. Liuyuankou section, C. Gucheng section, D. Jiahetan section, E. Mazhai section, and F. Hedao section.

**Table 1.** Asymmetry index of the Heishi–Hedao river reach. **Asymmetry Index Heishi (Bend Top Section) Liuyuankou (Transition Section) Gucheng (Bend Top Section) Jiahetan (Transition Section) Mazhai (Bend Top Section) Hedao (Transition Section)**  *AS*A 1.37 1.13 1.45 1.14 1.53 1.21 *AS*V 1.22 1.08 1.29 1.10 1.43 1.14 *AS*S 1.07 1.03 1.10 1.02 1.15 1.05 *AS*dcp 1.06 1.02 1.08 1.01 1.09 1.05 *AS*S\* 1.30 1.11 1.38 1.13 1.43 1.19 Table 1 shows that: (a) the asymmetry index of each section under the three scouring and silting states was >1, indicating that the river cross-section and the distribution of water and sediment factors during the free development period had certain asymmetry. (b) There was no obvious difference in the calculation results of asymmetry indexes of each section under the three scouring and silting states, indicating that the overall change in the asymmetry of the channel cross-sectional shape and water and sediment distribution under different scouring and silting states during the period of free development of the river bend was small. (c) Each asymmetry index of the section at the top of the river bend was significantly greater than that of the transition section, indicating that during the period of free development of the river bend, the shape and water and sediment distribution of the

Table 1 shows that: (a) the asymmetry index of each section under the three scouring

(b) There was no obvious difference in the calculation results of asymmetry indexes of each section under the three scouring and silting states, indicating that the overall change in the asymmetry of the channel cross-sectional shape and water and sediment distribution under different scouring and silting states during the period of free development of the river bend was small. (c) Each asymmetry index of the section at the top of the river bend was significantly greater than that of the transition section, indicating that during the period of free development of the river bend, the shape and water and sediment distribution of the section at the top of the river bend were more asymmetric than that of the

transition section.


section at the top of the river bend were more asymmetric than that of the transition section. **Table 1.** Asymmetry index of the Heishi–Hedao river reach.

Table 2 shows the verification of erosion and deposition balance, erosion, and deposition of each section of the Heishi River reach. The results showed that when the flow was 5000 m3/s, the sediment transport of each representative section under different scouring and silting states had the following laws: (a) when the sediment concentration was 23 kg/m<sup>3</sup> , at left and right, the distribution of sediment concentration in the whole reach was relatively uniform, and the river channel was basically in the balance state of erosion and deposition, which is manifested in the micro-scouring state of the section at the top of the bend and the micro silting state of the section at the transition section. (b) When the sediment concentration was 5 kg/m<sup>3</sup> , the whole reach was in the entire cross-section scouring state at the left and right. (c) When the sediment concentration was 50 kg/m<sup>3</sup> , the whole reach was in the full section siltation state. According to the verification results, the calculation results of river flow sediment-carrying capacity under various scouring and silting states reflect the distribution of water and sediment factors in each section.


**Table 2.** Verification of scouring and silting state of each section of Heishi–Hedao river reach (5000 m3/s).

*3.2. Asymmetry of Wandering Channel under Finite Control Boundary*

3.2.1. Asymmetry of Channel Cross-Sectional Shape

The typical reach of the wandering section of the lower Yellow River was selected to study the asymmetry of the river cross-section under the action of finite engineering. The section selection principles were as follows: (a) the river regime of the reach is relatively stable; (b) the curved top section works are closely combined with the water flow, the main sliding works are better, and the river regime in the transition section is relatively stable. Based on these principles, the section between Fanzhuang and Yuanfang in the wandering section of the lower Yellow River was selected as the representative section (Figure 8). According to the river regime map before the flood season in 2018 and the measured large section data, the cross-sectional asymmetry indexes of the bend top section and transition section with good engineering slip in this section were statistically analyzed (Table 3). According to the river regime map, the sections of Sizhuang and Chenqiao are at the top of the bend, and the project is relatively smooth. The three sections of Fanzhuang, Gucheng, and Yuanfang are the transition section between the two curved tops. As shown in Table 3, compared with the transition section, the cross-section at the bend top section

had a stronger asymmetry. This shows that the shape of the river cross-section was more asymmetric under the action of a limited control boundary. *Water* **2022**, *14*, x FOR PEER REVIEW 13 of 18

> **Figure 8.** Wandering reach of the lower Yellow River (Fanzhuang–Yuanfang): ⑱ Liuyuankou extension project ⑲ Dagong project ⑳ Wangan project ㉑ Gucheng project ㉒ Fujunshi project G. Fanzhuang section, H. Sizhuang section, I. Gucheng section, J. Chenqiao section, and K. Yuanfang **Figure 8.** Wandering reach of the lower Yellow River (Fanzhuang–Yuanfang): <sup>18</sup> Liuyuankou extension project <sup>19</sup> Dagong project <sup>20</sup> Wangan project <sup>21</sup> Gucheng project <sup>22</sup> Fujunshi project G. Fanzhuang section, H. Sizhuang section, I. Gucheng section, J. Chenqiao section, and K. Yuanfang section.

> section. 3.2.2. Asymmetric Distribution of Water and Sediment Factors in the River Channel **Table 3.** Asymmetry index of wandering river bend top and transition section (scouring and silting balance, *S* = 32 kg/m<sup>3</sup> ).


the reach and the distribution of water and sediment factors had a certain asymmetry. (2) The calculation results of cross-sectional erosion, deposition balance, and erosion state 3.2.2. Asymmetric Distribution of Water and Sediment Factors in the River Channel

were basically the same, which shows that the asymmetric distribution of river factors changed slightly under the two erosion and deposition states. Compared with these two states, in the state of cross-section deposition, the cross-sectional asymmetry index at the transition increased and the cross-section asymmetry index at the bend top decreased, indicating that in the state of cross-section deposition, the cross-section riverbed deformation at the bend top was small and the cross-section riverbed deformation at the transition was large. (3) Each asymmetry index of the section at the bend top of the river section was larger than that at the transition section, indicating that after the construction project, the distribution of water and sediment factors at the bend top of the river section was more asymmetric than that at the transition section. These results resembled the asymmetric results of the downstream swing channel without engineering constraints. The difference was that after construction of the project, compared with the cross-section scouring and silting balance and scouring state, the cross-To further analyze the influence of the asymmetric change in the cross-sectional shape on the asymmetry of river water and sediment-transport capacity, we selected the Fanzhuang–Yuanfang section of the lower Yellow River as the representative section. The gradient value was 2‱and the flat discharge was 5000 m3/s. The asymmetry indexes of water and sediment factors under various scouring and silting conditions at each section of the Fanzhuang–Yuanfang reach were calculated, as shown in Tables 3–5. The results showed that: (1) the asymmetry indexes of water and sediment factors in the Fanzhuang–Yuanfang reach were >1, indicating that after the construction project, the cross section of the reach and the distribution of water and sediment factors had a certain asymmetry. (2) The calculation results of cross-sectional erosion, deposition balance, and erosion state were basically the same, which shows that the asymmetric distribution of river factors changed slightly under the two erosion and deposition states. Compared with these two states, in the state of cross-section deposition, the cross-sectional asymmetry index at the transition

section riverbed deformation at the bend top was smaller and the cross-section riverbed

increased and the cross-section asymmetry index at the bend top decreased, indicating that in the state of cross-section deposition, the cross-section riverbed deformation at the bend top was small and the cross-section riverbed deformation at the transition was large. (3) Each asymmetry index of the section at the bend top of the river section was larger than that at the transition section, indicating that after the construction project, the distribution of water and sediment factors at the bend top of the river section was more asymmetric than that at the transition section.


**Table 4.** Asymmetry index of wandering river bend top and transition section (scouring state, *S* = 5 kg/m<sup>3</sup> ).

**Table 5.** Asymmetry index of wandering river bend top and transition section (silting state, *S* = 90 kg/m<sup>3</sup> ).


These results resembled the asymmetric results of the downstream swing channel without engineering constraints. The difference was that after construction of the project, compared with the cross-section scouring and silting balance and scouring state, the crosssection riverbed deformation at the bend top was smaller and the cross-section riverbed deformation at the transition was larger in the cross-section silting state.

Table 6 shows the verification of the erosion and deposition balance, erosion, and deposition of each section. The difference between the set sediment concentration and the calculated sediment-carrying capacity of the section under various states was small, indicating that the above calculation reflected the distribution state of water and sediment factors of each section.



Based on the above calculation results, when the flow was 5000 m3/s, under the condition of different sediment concentrations, the sediment transport of each section obeyed the following laws: (a) when the sediment concentration was 32 kg/m<sup>3</sup> at the left and right, the sediment concentration distribution of the entire river section was relatively uniform and the river channel was basically in a balanced state of erosion and deposition. In particular, the section at the top of the bend was in a state of micro-erosion, and the

section at the transition section was in the state of micro-deposition. (b) When the sediment concentration was 5 kg/m<sup>3</sup> , the entire reach was in a full cross-section scouring state on the left and right. (c) When the sediment concentration was 90 kg/m<sup>3</sup> , the entire reach was in the full section siltation state. According to the calculation results of sediment concentration and sediment-carrying capacity under various scouring and silting states at a flow of 5000 m3/s before and after the construction project, the sediment concentration and sediment-carrying capacity under various scouring and silting states improved after the construction of the project, which indicated that the construction of a river regulation project could improve the sediment transport capacity of the river after enhancing the control effect on river regime stability.

#### **4. Discussion**
