## Abstract

Rough-strip energy dissipators (R-SEDs) at the bottom of spillway bends have energy dissipation and flow diversion effects on bend flow. In this paper, twelve Plackett-Burman tests were conducted. Energy dissipation rate and coefficient of variation of superelevation were used as evaluation indices. Influencing factors (i.e., relative height, height ratio, spacing, angle and thickness of R-SEDs as well as centerline radius, width, angle and bottom slope of the bend) of R-SEDs' energy dissipation and flow diversion effects were analyzed using Minitab 21.1 in combination with the entropy weight method and the TOPSIS method. Relative height of R-SED and bend angle were significant factors affecting energy dissipation effects; height ratio of R-SED and centerline radius of the bend were significant factors affecting flow diversion effects; the coefficient of variation of superelevation had a larger weight (0.549) than the energy dissipation rate (0.451), indicating that R-SEDs' flow diversion effects were larger than their energy dissipation effects; height ratio of R-SED, centerline radius of the bend and bend angle were significant factors affecting overall energy dissipation and flow diversion effects and selected as key factors for further steepest climbing tests and response surface design.

## HIGHLIGHTS

R-SEDs were proposed for curved spillways, with a simple structure to facilitate construction.

The Plackett–Burman design as the experimental method reduced the number of experiments and effectively facilitated the selection of key influencing factors.

A new index coefficient of variation of superelevation was established for evaluating flow diversion effects of R-SEDs, with simple calculation for application.

## LIST OF SYMBOLS

*h*_{r}Relative height of R-SED

*λ*Height ratio of R-SED

*α*R-SED spacing

*θ*R-SED angle

*δ*R-SED thickness

*R*Centerline radius of the bend

*B*Bend width

*β*Bend angle

*i*Bottom slope of the bend

Average water depth of the inlet cross-section of the bend

*h*_{1}Height of the R-SED at the concave bank

*h*_{2}Height of the R-SED at the convex bank

*M*Marking point of an enlarged view of the R-SED at the concave bank

*N*Marking point of an enlarged view of the R-SED at the convex bank

*L*_{1}Length of inlet straight section

*L*_{2}Length of outlet straight section

*h*Water depth at each measurement point

*Q*Discharge flow rate of the spillway

*C*_{0}Flow rate coefficient of the right triangular thin-walled weir

*H*The head over the right triangular thin-walled weir

*P*_{1}Height of the right triangular thin-walled weir

*B*_{1}Width of the flow diversion canal at the upstream of the right triangular thin-walled weir

*p*_{ij}Proportion of the

*i*th evaluation object under the*j*th index*E*_{j}Entropy value of the

*j*th index*W*_{j}Weight of the

*j*th index- ,
Euclidean distance between each object and the ideal solution

Relative closeness of each object to the ideal solution

*η*Energy dissipation rate of R-SEDs

*E*_{1}Total mechanical energy per unit weight of water in the upstream flow cross-section

*Z*_{1}Minimum elevation of the spillway bottom of the upstream flow cross-section

*H*_{1}Average water depth of the upstream flow cross-section

*v*_{1}Average flow velocity of the upstream flow cross-section

*E*_{2}Total mechanical energy per unit weight of water in the downstream flow cross-section

*Z*_{2}Minimum elevation of the spillway bottom of the downstream flow cross-section

*H*_{2}Average water depth of the downstream flow cross-section

*v*_{2}Average velocity of the downstream flow cross-section

*φ*Kinetic energy correction factor

*g*The acceleration of gravity

- SCV
Coefficient of variation of superelevation

*σ*_{i}Standard deviation of transverse water surface superelevation Δ

*y*for all calculated sections under the*i*th condition*μ*_{i}Average transverse water surface superelevation Δ

*y*of all the calculated sections under the*i*th condition- Δ
*y*_{ij} The difference between the water level at the concave bank of the bend and the horizontal plane where the center of the water surface is located at the

*j*th calculated cross-section under the*i*th condition*k*Superelevation coefficient

*v*_{ij}Average velocity of the

*j*th calculated section under the*i*th condition*B*_{i}Water surface width of the open channel based on the centerline water surface elevation under the

*i*th condition*r*_{i}Centerline radius of the bend under the

*i*th condition- Δ
*y* Transverse water surface superelevation of the bend

*W*_{(η)}Weight of the energy dissipation rate

*W*_{(SCV)}Weight of the coefficient of variation of superelevation

## INTRODUCTION

Influenced by topographic and geological conditions, engineering characteristics, construction conditions and economic indicators (Seo & Shin 2018; Yang *et al.* 2019; Damarnegara *et al.* 2020), some spillways or natural river channels must have corners, thus forming bends. The water flowing through the bend is called bend flow, which is different from straight-section water flow. When water flows through a bend, uneven distribution of water flow (Zhang *et al.* 2015) and cross-sectional flow velocity (Pradhan *et al.* 2018) occurs on both banks of the bend due to centrifugal inertia forces. Thus, secondary flows are developed (Seyedashraf & Akhtari 2015). These unfavorable flow structures lead to sediment movement, riverbed evolution and river channel deformation (Olsen 2003).

The bend flow is one of the topics in the field of hydraulics research. In 1876, Thomson (1876) first proposed the problem of bend circulating flow through experimental research. Currently, various research methods of bend flow have been developed, mainly including model tests, numerical simulations and theoretical studies. The research on bend flow is mainly divided into the study of basic water flow characteristics and the study of engineering measures to improve the flow pattern of bend flow. The study of the basic characteristics of the bend flow is focused on the water depth distribution (Seyedashraf & Akhtari 2015; Zhang *et al.* 2015; Qin *et al.* 2016; Zhou *et al.* 2017; Maatooq & Hameed 2020), flow velocity distribution (Han *et al.* 2011; Seyedashraf & Akhtari 2015; Vaghefi *et al.* 2015; Qin *et al.* 2016; Zhou *et al.* 2017; Moncho-Esteve *et al.* 2018; Pradhan *et al.* 2018; Schreiner *et al.* 2018; Seo & Shin 2018; Hu *et al.* 2019; Kim *et al.* 2020; Yan *et al.* 2020) and secondary flow evolution (Booij 2003; Huai *et al.* 2012; Ramamurthy *et al.* 2013; Seyedashraf & Akhtari 2015; Engel & Rhoads 2016; Gu *et al.* 2016; Zhou *et al.* 2017; Moncho-Esteve *et al.* 2018; Schreiner *et al.* 2018; Seo & Shin 2018; Hu *et al.* 2019; Shaheed *et al.* 2021). Based on the understanding of the basic bend flow characteristics, scholars have improved the bend flow pattern using engineering measures such as permeable spurs (Yang *et al.* 2019), guide walls (Zhang *et al.* 2015), vanes (Ranjan *et al.* 2006; Han *et al.* 2011) and riprap (Martín-Vide *et al.* 2010). However, compared with the research on the basic characteristics of bend flow, the research on related engineering measures is still rare. Thus, in this paper, a simpler rough-strip energy dissipator (R-SED) was added to the bend of curved spillways. The simple shape and convenient construction of R-SEDs can effectively solve the adverse hydraulic phenomena in the curved spillway, which will facilitate the safe and stable operation of the curved spillway and the continuous downstream water supply (such as industrial and irrigation water).

The research on the R-SED was first conducted based on the hydraulic model test (the geometric scale of the model was 1:50) of the curved spillway of Project 635 Reservoir in Xinjiang, China, and the research was mainly based on the laboratory test. The actual operation of the spillway reveals that the water depth and flow velocity of the concave and convex banks of the spillway differed greatly at a discharge flow rate of 800 m^{3}/s. The water flow in the bend was splashing and turbulent. In order to solve the above adverse water flow problem, the R-SED was arranged at the bottom of the bend, exhibiting good energy dissipation and flow diversion effects in the bend (Li 2016). Due to the different scaling ratios (geometric ratios) of different curved spillway test models and the special and different geological conditions of each project, laboratory-scale model dimensions were adopted for the research and analysis in this study.

The R-SED has both energy dissipation and flow diversion effects on the bend flow, and the influencing factors affecting these effects are generally divided into R-SED layout parameters (such as relative height of R-SED, height ratio of R-SED, R-SED spacing, R-SED angle and R-SED thickness) and curved spillway engineering parameters (such as centerline radius of the bend, bend width, bend angle and bottom slope of the bend). In the existing R-SED study, Li (2016) mainly analyzed the results of the R-SED in the hydraulic model test of the curved spillway of Project 635 Reservoir in Xinjiang, China, using the single-factor test method. Li (2016) only considered the influence of three parameters of the R-SED arrangement (i.e., height ratio of R-SED, R-SED angle and R-SED thickness) on the energy dissipation and flow diversion effect without including the influence of the engineering parameters of the curved spillway. Li *et al.* (2020) mainly analyzed the influence law of six factors (height ratio of R-SED, R-SED spacing, R-SED angle, centerline radius of the bend, bend width and discharge flow rate) on the effect of energy dissipation and flow diversion of R-SEDs using the orthogonal test method (six factors and three levels), and the energy dissipation rate and the water flux dispersion coefficient for evaluation energy dissipation and flow diversion effects were adopted, respectively. However, the influence considered influencing factors are still not comprehensive and the calculation formula of the water flux dispersion coefficient is complicated for practical application.

*et al.*2020) and a new evaluation index of the R-SED's flow diversion effect was established, i.e., coefficient of variation of superelevation. The newly established formula for calculating the coefficient of variation of superelevation is simple, easy to understand and convenient for practical engineering applications. The study can be used for the next steepest climbing test and response surface test design in order to select the key factors and provide reference for the R-SED design of similar curved spillways. The research on the parameter optimization of the R-SED in curved spillways is divided into two parts, as shown in Figure 1.

## METHODS

### Test apparatus

*L*

_{1}= 0.7 m) can ensure the smooth flow of inlet water into the bend section. According to Table 2, eight bends of different sizes were designed and prepared for the test. The outlet straight section (

*L*

_{2}= 1.5 m) can smoothly connect with the flow out of the bend. The water circulation system consists of a water pump, a rectangular water storage tank, a water measuring weir, an underground water reservoir and underground water flow diversion pipelines.

Level . | hr (cm)
. | λ (–)
. | α (°)
. | θ (°)
. | δ (cm)
. | R (cm)
. | B (cm)
. | β (°)
. | i (–)
. |
---|---|---|---|---|---|---|---|---|---|

−1 | 0.4 | 1 | (1/6) β | 15 | (1/100) B | 150 | 50 | 30 | 0.02 |

1 | 0.8 | 2 | (1/3) β | 30 | (1/50) B | 300 | 100 | 60 | 0.04 |

Level . | hr (cm)
. | λ (–)
. | α (°)
. | θ (°)
. | δ (cm)
. | R (cm)
. | B (cm)
. | β (°)
. | i (–)
. |
---|---|---|---|---|---|---|---|---|---|

−1 | 0.4 | 1 | (1/6) β | 15 | (1/100) B | 150 | 50 | 30 | 0.02 |

1 | 0.8 | 2 | (1/3) β | 30 | (1/50) B | 300 | 100 | 60 | 0.04 |

*Note:* is the average water depth of the inlet cross-section of the bend.

Test Number . | h (cm)
. _{r} | λ (–)
. | α (°)
. | θ (°)
. | δ (cm)
. | R (cm)
. | B (cm)
. | β (°)
. | i (–)
. |
---|---|---|---|---|---|---|---|---|---|

1 | 0.4 | 2 | (1/6) β | 15 | (1/100)B | 300 | 100 | 60 | 0.02 |

2 | 0.8 | 1 | (1/6) β | 15 | (1/50)B | 300 | 100 | 30 | 0.04 |

3 | 0.4 | 2 | (1/3) β | 15 | (1/50)B | 150 | 50 | 30 | 0.04 |

4 | 0.4 | 1 | (1/6) β | 15 | (1/100)B | 150 | 50 | 30 | 0.02 |

5 | 0.4 | 1 | (1/3) β | 30 | (1/50)B | 150 | 100 | 60 | 0.02 |

6 | 0.4 | 1 | (1/6) β | 30 | (1/50)B | 300 | 50 | 60 | 0.04 |

7 | 0.4 | 2 | (1/3) β | 30 | (1/100)B | 300 | 100 | 30 | 0.04 |

8 | 0.8 | 2 | (1/3) β | 15 | (1/50)B | 300 | 50 | 60 | 0.02 |

9 | 0.8 | 2 | (1/6) β | 30 | (1/100)B | 150 | 50 | 60 | 0.04 |

10 | 0.8 | 1 | (1/3) β | 15 | (1/100)B | 150 | 100 | 60 | 0.04 |

11 | 0.8 | 2 | (1/6) β | 30 | (1/50)B | 150 | 100 | 30 | 0.02 |

12 | 0.8 | 1 | (1/3) β | 30 | (1/100)B | 300 | 50 | 30 | 0.02 |

Test Number . | h (cm)
. _{r} | λ (–)
. | α (°)
. | θ (°)
. | δ (cm)
. | R (cm)
. | B (cm)
. | β (°)
. | i (–)
. |
---|---|---|---|---|---|---|---|---|---|

1 | 0.4 | 2 | (1/6) β | 15 | (1/100)B | 300 | 100 | 60 | 0.02 |

2 | 0.8 | 1 | (1/6) β | 15 | (1/50)B | 300 | 100 | 30 | 0.04 |

3 | 0.4 | 2 | (1/3) β | 15 | (1/50)B | 150 | 50 | 30 | 0.04 |

4 | 0.4 | 1 | (1/6) β | 15 | (1/100)B | 150 | 50 | 30 | 0.02 |

5 | 0.4 | 1 | (1/3) β | 30 | (1/50)B | 150 | 100 | 60 | 0.02 |

6 | 0.4 | 1 | (1/6) β | 30 | (1/50)B | 300 | 50 | 60 | 0.04 |

7 | 0.4 | 2 | (1/3) β | 30 | (1/100)B | 300 | 100 | 30 | 0.04 |

8 | 0.8 | 2 | (1/3) β | 15 | (1/50)B | 300 | 50 | 60 | 0.02 |

9 | 0.8 | 2 | (1/6) β | 30 | (1/100)B | 150 | 50 | 60 | 0.04 |

10 | 0.8 | 1 | (1/3) β | 15 | (1/100)B | 150 | 100 | 60 | 0.04 |

11 | 0.8 | 2 | (1/6) β | 30 | (1/50)B | 150 | 100 | 30 | 0.02 |

12 | 0.8 | 1 | (1/3) β | 30 | (1/100)B | 300 | 50 | 30 | 0.02 |

### Test program

The Plackett–Burman design is a two-level partial factorial experimental design method, which is mainly used for experimental designs with a large number of factors and where the significance of these factors relative to the response variable is not determined (Abdel-Fattah *et al.* 2005; Ghanem *et al.* 2017). The method mainly analyzed two levels of each factor and determined the significance of each factor by comparing the difference between the two levels of each factor with the overall difference. Two levels (i.e., high (1) and low (−1)) were selected for each influencing factor in the experimental design, with the high level two times larger than the low level.

*h*), height ratio (

_{r}*λ*), spacing (

*α*), angle (

*θ*) and thickness (

*δ*)) and spillway bend parameters (i.e., centerline radius (

*R*), width (

*B*), angle (

*β*) and bottom slope (

*i*)). Based on the research needs and findings of our group (Li 2016; Li

*et al.*2020), the values of each factor were selected. The test factors and levels are shown in Table 1 and the symbols are illustrated in Figure 3.

The R-SEDs were installed at the bottom of the spillway bend. Each R-SED extended continuously from the concave bank to the convex bank, close to the bottom of the bend. The roughness of the bend bottom increased after the R-SEDs were added, and the water flow between adjacent R-SEDs collided and swirled, thus strengthening the energy dissipation effect of R-SEDs. Without the R-SED, the water flow in the spillway bend showed a phenomenon of increased flow depth at the concave bank and decreased flow depth at the convex bank; the discharge water flow continuously scoured the sidewall of the bend at the concave bank, which was not conducive to the structural safety and stability. Thus, in order to better improve the uneven distribution of water flow between the concave and convex banks of the bend and improve the flow diversion effect of R-SEDs, the height (*h*_{1}) of the R-SED at the concave bank was designed to be larger than the height (*h*_{2}) at the convex bank, i.e., *h*_{1} > *h*_{2}. Therefore, the R-SED had a trapezoidal longitudinal section.

The relative height of R-SED (*h _{r}*) is the ratio between the height of the R-SED at the concave bank (

*h*

_{1}) to the average water depth at the inlet cross-section of the bend (), i.e.,

*h*

_{r}*=*

*h*

_{1}/. The height ratio of R-SED (

*λ*) is the ratio of the height of the R-SED at the concave bank (

*h*

_{1}) to the height at the convex bank (

*h*

_{2}), i.e.,

*λ*

*=*

*h*

_{1}/

*h*

_{2}. R-SED spacing (

*α*) is the angle corresponding to the arc length between two adjacent R-SED centerlines along the bend centerline direction. R-SED angle (

*θ*) is the angle between the centerline of the R-SED and the direction perpendicular to the bend centerline. R-SED thickness (

*δ*) is the horizontal distance between the upstream and downstream faces of the R-SED, which is defined as the ratio of the horizontal distance to the bend width (

*B*) for nondimensionalization. The centerline radius of the bend (

*R*) is the distance between the bend centerline and the center of the curvature of the bend. Bend width (

*B*) is the horizontal distance between the two banks of the spillway. Bend angle (

*β*) is the angle corresponding to the bend flow path along the bend centerline. Bottom slope of the bend (

*i*) is the slope along the spillway bottom. The schematic diagram of these parameters is shown in Figure 3.

### Measurement arrangement

- (a)
Water depth measurement

The water level measurement probe was used to measure the water depth, with an accuracy of 0.1 mm. A total of 19 water depth measurement cross-sections were arranged along the spillway model, i.e., 0#–18#. Each cross-section was arranged with five measurement points (i.e., A–E). Considering the viscous resistance of the sidewall to the water flow, two near-bank measurement points (A and E) were located at 1 cm from the sidewalls of the concave and convex banks, respectively.

- (b)
Flow velocity measurement

The hourly average flow velocity was measured using a Pitot tube. A total of 10 measurement cross-sections (i.e., 0#, 2#, 4#, 6#, 8#, 10#, 12#, 14#, 16# and 18#) were selected as flow velocity measurement sections and three measurement points (i.e., A, C and E) were selected for each section. The vertical measurement position of each measurement point was located at 2 *h*/3 from the bottom (*h* is the water depth at the measurement point).

- (c)
Discharge flow rate measurement

*Q*is the discharge flow rate (L/s);

*H*is the head over the weir (m);

*C*

_{0}is the flow rate coefficient of the weir, which is related to the size of the opening and can be calculated by:where

*P*

_{1}is the height of the weir (m);

*B*

_{1}is the width of the flow diversion canal at the upstream of the weir (m).

### Construction of evaluation indices

The R-SED has a dual effect of energy dissipation and flow diversion on the bend flow. Evaluation indices need to be constructed to assess the energy dissipation effect and flow diversion effects of the R-SED under twelve test scenarios.

#### Energy dissipation rate

*η*) was introduced as an evaluation index. The change interval of energy dissipation rate is (0, 1) and a larger energy dissipation rate indicates a larger energy dissipation effect. The energy dissipation rate is calculated through:where

*η*

_{(i)}is the energy dissipation rate under the

*i*th condition (%),

*E*

_{1}is the total mechanical energy per unit weight of water in the upstream flow cross-section (m),

*Z*

_{1}is the minimum elevation of the upstream flow cross-section (m),

*H*

_{1}is the average depth of the upstream flow cross-section (m),

*v*

_{1}is the average velocity of the upstream flow cross-section (m/s),

*E*

_{2}is the total mechanical energy per unit weight of water in the downstream flow cross-section (m),

*Z*

_{2}is the minimum elevation of the downstream flow cross-section (m),

*H*

_{2}is the average depth of the downstream flow cross-section (m),

*v*

_{2}is the average velocity of the downstream flow cross-section (m/s);

*α*for the kinetic energy correction factor, taken as 1.0 and

*g*is the acceleration of gravity, taken as 9.81 m/s

^{2}.

Cross-sections 4# and 12# were selected as the upstream and downstream sections of the bend, respectively. The horizontal plane where the bottom elevation of cross-section 12# was located was taken as the reference plane. The energy dissipation rate in the 12 test scenarios was calculated using Equations (3)–(5). The calculation results of the energy dissipation rate are shown in Table 3.

Test Number . | h (cm)
. _{r} | λ (–)
. | α (°)
. | θ (°)
. | δ (cm)
. | R (cm)
. | B (cm)
. | β (°)
. | i (–)
. | η (–)
. | SCV (–)
. |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.4 | 2 | (1/6) β | 15 | (1/100)B | 300 | 100 | 60 | 0.02 | 0.535 | 0.188 |

2 | 0.8 | 1 | (1/6) β | 15 | (1/50)B | 300 | 100 | 30 | 0.04 | 0.557 | 0.473 |

3 | 0.4 | 2 | (1/3) β | 15 | (1/50)B | 150 | 50 | 30 | 0.04 | 0.318 | 0.357 |

4 | 0.4 | 1 | (1/6) β | 15 | (1/100)B | 150 | 50 | 30 | 0.02 | 0.153 | 0.516 |

5 | 0.4 | 1 | (1/3) β | 30 | (1/50)B | 150 | 100 | 60 | 0.02 | 0.461 | 0.413 |

6 | 0.4 | 1 | (1/6) β | 30 | (1/50)B | 300 | 50 | 60 | 0.04 | 0.662 | 0.318 |

7 | 0.4 | 2 | (1/3) β | 30 | (1/100)B | 300 | 100 | 30 | 0.04 | 0.335 | 0.140 |

8 | 0.8 | 2 | (1/3) β | 15 | (1/50)B | 300 | 50 | 60 | 0.02 | 0.656 | 0.211 |

9 | 0.8 | 2 | (1/6) β | 30 | (1/100)B | 150 | 50 | 60 | 0.04 | 0.623 | 0.301 |

10 | 0.8 | 1 | (1/3) β | 15 | (1/100)B | 150 | 100 | 60 | 0.04 | 0.643 | 0.605 |

11 | 0.8 | 2 | (1/6) β | 30 | (1/50)B | 150 | 100 | 30 | 0.02 | 0.419 | 0.380 |

12 | 0.8 | 1 | (1/3) β | 30 | (1/100)B | 300 | 50 | 30 | 0.02 | 0.402 | 0.306 |

Test Number . | h (cm)
. _{r} | λ (–)
. | α (°)
. | θ (°)
. | δ (cm)
. | R (cm)
. | B (cm)
. | β (°)
. | i (–)
. | η (–)
. | SCV (–)
. |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.4 | 2 | (1/6) β | 15 | (1/100)B | 300 | 100 | 60 | 0.02 | 0.535 | 0.188 |

2 | 0.8 | 1 | (1/6) β | 15 | (1/50)B | 300 | 100 | 30 | 0.04 | 0.557 | 0.473 |

3 | 0.4 | 2 | (1/3) β | 15 | (1/50)B | 150 | 50 | 30 | 0.04 | 0.318 | 0.357 |

4 | 0.4 | 1 | (1/6) β | 15 | (1/100)B | 150 | 50 | 30 | 0.02 | 0.153 | 0.516 |

5 | 0.4 | 1 | (1/3) β | 30 | (1/50)B | 150 | 100 | 60 | 0.02 | 0.461 | 0.413 |

6 | 0.4 | 1 | (1/6) β | 30 | (1/50)B | 300 | 50 | 60 | 0.04 | 0.662 | 0.318 |

7 | 0.4 | 2 | (1/3) β | 30 | (1/100)B | 300 | 100 | 30 | 0.04 | 0.335 | 0.140 |

8 | 0.8 | 2 | (1/3) β | 15 | (1/50)B | 300 | 50 | 60 | 0.02 | 0.656 | 0.211 |

9 | 0.8 | 2 | (1/6) β | 30 | (1/100)B | 150 | 50 | 60 | 0.04 | 0.623 | 0.301 |

10 | 0.8 | 1 | (1/3) β | 15 | (1/100)B | 150 | 100 | 60 | 0.04 | 0.643 | 0.605 |

11 | 0.8 | 2 | (1/6) β | 30 | (1/50)B | 150 | 100 | 30 | 0.02 | 0.419 | 0.380 |

12 | 0.8 | 1 | (1/3) β | 30 | (1/100)B | 300 | 50 | 30 | 0.02 | 0.402 | 0.306 |

#### Coefficient of variation of superelevation

*SCV*) was introduced as an evaluation index. The coefficient of variation of superelevation varied between (0, 1) and is negatively correlated with flow diversion effects, i.e., a smaller coefficient of variation of superelevation indicates better flow diversion effects of the R-SED. The coefficient of variation of superelevation is expressed as:where SCV

_{(i)}is the coefficient of variation of superelevation under the

*i*th condition;

*σ*is the standard deviation of the superelevation of transverse water surface (Δ

_{i}*y*) of all calculated sections under the

*i*th condition (m);

*μ*is the mean value of superelevation of transverse water surface of all calculated sections under the

_{i}*i*th condition (m); Δ

*y*is the difference between the water surface at the concave bank of the bend and the horizontal surface of the center point at the

_{ij}*j*th calculated section under the

*i*th condition (m);

*k*is the superelevation coefficient, taken as 0.5 for a simple circular-curved bend of a rectangular open channel;

*v*is the average velocity of the

_{ij}*j*th calculated section under the

*i*th condition (m/s);

*B*is the water surface width of the open channel based on the centerline water surface elevation under the

_{i}*i*th condition (m);

*g*is the acceleration of gravity, taken as 9.81 m/s

^{2};

*r*is the radius of the center of the bend under the

_{i}*i*th condition (m). The calculation of transverse water surface superelevation of the bend (Δ

*y*) is illustrated in Figure 6.

The addition of R-SEDs in the bend can induce the secondary distribution of water flow in the bend and the straight section downstream. To fully measure the flow diversion effect of R-SEDs, cross-section 4# at the bend and cross-section 18# at the straight section downstream were selected as the inlet and outlet cross-sections for the calculation of the coefficient of variation of superelevation. The calculation results are shown in Table 3.

### Multi-objective evaluation based on entropy weight method and TOPSIS method

- (a)
Establishing a comprehensive performance evaluation system

Twelve groups of Plackett–Burman tests were used as feasibility study schemes, and the energy dissipation rate and the coefficient of variation of superelevation were used as target variables to construct the original matrix .

- (b)
Determining the weight of each evaluation index using the entropy weight method

*et al.*2021). The matrix

*Y*was derived from the nondimensionalization of the original matrix and then the entropy weight

*W*of each evaluation index was determined. A larger entropy weight indicates that the evaluation index is more important. The calculation formula is expressed as:where

_{j}*i*denotes the number of evaluation objects (

*i*= 1, 2, …,

*m*);

*j*denotes the number of evaluation indices (

*j*= 1, 2, …,

*n*);

*p*is the proportion of the

_{ij}*i*th evaluation object under the

*j*th index;

*E*is the entropy value of the

_{j}*j*th index.

- (c)
Obtaining the overall ranking of each program using the TOPSIS method

*et al.*2019). Firstly, the original matrix is normalized to obtain a matrix and the matrix

*B*is multiplied by the entropy weight

*W*to obtain the weighting matrix

_{j}*Z*. Secondly, the Euclidean distance and between each object and the ideal solution as well as the relative closeness of each object is calculated. A larger relative closeness indicates that the solution is closer to the ideal solution and that the rating is better. Finally, the solutions are comprehensively ranked according to the relative closeness of each solution to form a decision basis. The calculation equations are as follows:

### Statistical analysis of data

Minitab 21.1 software was used for statistical analysis of the data, including the energy dissipation rate, the coefficient of variation of superelevation and relative closeness.

## RESULTS AND DISCUSSION

### Analysis of influencing factors of energy dissipation effects

#### Analysis of variance for the energy dissipation rate

Analysis of variance (ANOVA) was performed on the nine factors (predictor variables) using the energy dissipation rate as the response variable, as shown in Table 4. The significance level was selected as 0.05 and the factor was significant when *P* < 0.05. Table 4 shows that the probability *P* corresponding to the main effect was less than 0.05, rejecting the original hypothesis and indicating that the total effect of the regression was significant. Particularly, the relative height of R-SED and the bend angle were significant.

Source . | DF . | Adj SS . | Adj MS . | F-value
. | P-value
. | Significance . |
---|---|---|---|---|---|---|

Model | 9 | 0.280806 | 0.031201 | 19.43 | 0.050 | * |

Linear | 9 | 0.280806 | 0.031201 | 19.43 | 0.050 | * |

h _{r} | 1 | 0.058277 | 0.058277 | 36.3 | 0.026 | * |

λ | 1 | 0.000005 | 0.000005 | 0.00 | 0.960 | |

α | 1 | 0.001497 | 0.001497 | 0.93 | 0.436 | |

θ | 1 | 0.000129 | 0.000129 | 0.08 | 0.803 | |

δ | 1 | 0.012136 | 0.012136 | 7.56 | 0.111 | |

R | 1 | 0.023419 | 0.023419 | 14.59 | 0.062 | |

B | 1 | 0.001587 | 0.001587 | 0.99 | 0.425 | |

β | 1 | 0.161996 | 0.161996 | 100.90 | 0.010 | * |

i | 1 | 0.021759 | 0.021759 | 13.55 | 0.067 | |

Error | 2 | 0.003211 | 0.001605 | |||

Total | 11 | 0.284017 | ||||

R-sq = 0.9887 | R-sq (adj) = 0.9378 |

Source . | DF . | Adj SS . | Adj MS . | F-value
. | P-value
. | Significance . |
---|---|---|---|---|---|---|

Model | 9 | 0.280806 | 0.031201 | 19.43 | 0.050 | * |

Linear | 9 | 0.280806 | 0.031201 | 19.43 | 0.050 | * |

h _{r} | 1 | 0.058277 | 0.058277 | 36.3 | 0.026 | * |

λ | 1 | 0.000005 | 0.000005 | 0.00 | 0.960 | |

α | 1 | 0.001497 | 0.001497 | 0.93 | 0.436 | |

θ | 1 | 0.000129 | 0.000129 | 0.08 | 0.803 | |

δ | 1 | 0.012136 | 0.012136 | 7.56 | 0.111 | |

R | 1 | 0.023419 | 0.023419 | 14.59 | 0.062 | |

B | 1 | 0.001587 | 0.001587 | 0.99 | 0.425 | |

β | 1 | 0.161996 | 0.161996 | 100.90 | 0.010 | * |

i | 1 | 0.021759 | 0.021759 | 13.55 | 0.067 | |

Error | 2 | 0.003211 | 0.001605 | |||

Total | 11 | 0.284017 | ||||

R-sq = 0.9887 | R-sq (adj) = 0.9378 |

#### Pareto chart of energy dissipation rate

*t*-values of each effect from the

*t*-test are taken as vertical coordinates; the critical values of the

*t*-values are determined according to the selected significance level; the effects with absolute values exceeding the critical values are selected, indicating that these effects are significant. In this study, the energy dissipation rate was used as the response variable and a significance level was set as 0.05. Thus, the Pareto chart of each factor was obtained, as shown in Figure 7. From Figure 7, the relative height of R-SED and the bend angle were significant factors, which was consistent with the results in Table 4. In terms of their significance, the factors influencing the energy dissipation effect of R-SEDs were in the following order: bend angle > relative height of R-SED > centerline radius of the bend > bottom slope of the bend > R-SED thickness > bend width > R-SED spacing > R-SED angle > height ratio of R-SED.

#### Main effect plot of energy dissipation rate

*B*; centerline radius of the bend, 300 cm; bend width, 100 cm; bend angle, 60°; bottom slope of the bend, 0.04. The energy dissipation rate was larger at a smaller R-SED spacing, i.e.,

*α*= (1/6)

*β*.

#### Analysis of the interaction between the factors

### Analysis of influencing factors of flow diversion effects

#### Analysis of variance of the coefficient of variation of superelevation

The coefficient of variation of superelevation was used as the response variable. The significance level was selected as 0.05. The ANOVA was performed on the nine factors, as shown in Table 5. Table 5 shows that the *P-*value corresponding to the main effect of the coefficient of variation of superelevation was less than 0.05, indicating that the original hypothesis was rejected and that the total effect of the regression was considered significant. The height ratio of R-SED and the centerline radius of the bend were significant.

Source . | DF . | Adj SS . | Adj MS . | F-Value
. | P-Value
. | Significance . |
---|---|---|---|---|---|---|

Model | 9 | 0.205751 | 0.022861 | 20.02 | 0.048 | * |

Linear | 9 | 0.205751 | 0.022861 | 20.02 | 0.048 | * |

h _{r} | 1 | 0.009858 | 0.009858 | 8.63 | 0.099 | |

λ | 1 | 0.092728 | 0.092728 | 81.22 | 0.012 | * |

α | 1 | 0.001762 | 0.001762 | 1.54 | 0.34 | |

θ | 1 | 0.020137 | 0.020137 | 17.64 | 0.052 | |

δ | 1 | 0.0008 | 0.0008 | 0.7 | 0.491 | |

R | 1 | 0.07317 | 0.07317 | 64.09 | 0.015 | * |

B | 1 | 0.003022 | 0.003022 | 2.65 | 0.245 | |

β | 1 | 0.001546 | 0.001546 | 1.35 | 0.365 | |

i | 1 | 0.002729 | 0.002729 | 2.39 | 0.262 | |

Error | 2 | 0.002283 | 0.001142 | |||

Total | 11 | 0.208034 | ||||

R-sq = 0.9890 | R-sq (adj) = 0.9396 |

Source . | DF . | Adj SS . | Adj MS . | F-Value
. | P-Value
. | Significance . |
---|---|---|---|---|---|---|

Model | 9 | 0.205751 | 0.022861 | 20.02 | 0.048 | * |

Linear | 9 | 0.205751 | 0.022861 | 20.02 | 0.048 | * |

h _{r} | 1 | 0.009858 | 0.009858 | 8.63 | 0.099 | |

λ | 1 | 0.092728 | 0.092728 | 81.22 | 0.012 | * |

α | 1 | 0.001762 | 0.001762 | 1.54 | 0.34 | |

θ | 1 | 0.020137 | 0.020137 | 17.64 | 0.052 | |

δ | 1 | 0.0008 | 0.0008 | 0.7 | 0.491 | |

R | 1 | 0.07317 | 0.07317 | 64.09 | 0.015 | * |

B | 1 | 0.003022 | 0.003022 | 2.65 | 0.245 | |

β | 1 | 0.001546 | 0.001546 | 1.35 | 0.365 | |

i | 1 | 0.002729 | 0.002729 | 2.39 | 0.262 | |

Error | 2 | 0.002283 | 0.001142 | |||

Total | 11 | 0.208034 | ||||

R-sq = 0.9890 | R-sq (adj) = 0.9396 |

#### Pareto chart of coefficients of variation of superelevation

#### Main effect plot of coefficients of variation of superelevation

*β*; R-SED angle, 30°; centerline radius of the bend, 300 cm; bend angle, 60°), the coefficient of variation of superelevation was smaller, while flow diversion effects of R-SEDs were larger. However, the relative height of R-SED, R-SED thickness, bend width and bottom slope of the bend were positively correlated with the coefficient of variation of superelevation. The slopes of the regression line of these four factors were close, indicating that they had a similar effect on the coefficient of variation of superelevation. In order to minimize the coefficient of variation of superelevation, these four factors were taken as small as possible: relative height of R-SED, 0.4 cm; R-SED thickness, (1/100)

*B*; bend width, 50 cm; bottom slope of the bend, 0.02.

#### Analysis of the interaction between influencing factors

### Analysis of influencing factors on the overall energy dissipation and flow diversion effects

#### Determination of the weights of each evaluation index using the entropy weight method

The weights of each evaluation index were calculated using Equations (10)–(13) and the calculated parameters are shown in Table 6. The weight of the energy dissipation rate (*W*_{(η)}) and the coefficient of variation of superelevation (*W*_{(SCV)}) were 0.451 and 0.549, respectively, i.e., *W*_{(SCV)}> *W*_{(η)}. This indicates that in these 12 Plackett–Burman tests, the coefficient of variation of superelevation carried more information than the energy dissipation rate and that the flow diversion effect of R-SEDs on the bend flow was larger than the energy dissipation effect.

Test number . | η
. | SCV . | η*
. | SCV*
. | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0.535 | 0.188 | 0.752 | 0.896 | 0.097 | 0.137 | 0.943 | 0.930 | 0.451 | 0.549 |

2 | 0.557 | 0.473 | 0.795 | 0.283 | 0.103 | 0.043 | ||||

3 | 0.318 | 0.357 | 0.324 | 0.532 | 0.042 | 0.081 | ||||

4 | 0.153 | 0.516 | 0.000 | 0.191 | 0.000 | 0.029 | ||||

5 | 0.461 | 0.413 | 0.606 | 0.412 | 0.078 | 0.063 | ||||

6 | 0.662 | 0.318 | 1.000 | 0.616 | 0.129 | 0.094 | ||||

7 | 0.335 | 0.140 | 0.358 | 1.000 | 0.046 | 0.153 | ||||

8 | 0.656 | 0.211 | 0.989 | 0.847 | 0.128 | 0.129 | ||||

9 | 0.623 | 0.301 | 0.924 | 0.653 | 0.120 | 0.100 | ||||

10 | 0.643 | 0.605 | 0.964 | 0.000 | 0.125 | 0.000 | ||||

11 | 0.419 | 0.380 | 0.524 | 0.483 | 0.068 | 0.074 | ||||

12 | 0.402 | 0.306 | 0.490 | 0.643 | 0.068 | 0.098 |

Test number . | η
. | SCV . | η*
. | SCV*
. | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0.535 | 0.188 | 0.752 | 0.896 | 0.097 | 0.137 | 0.943 | 0.930 | 0.451 | 0.549 |

2 | 0.557 | 0.473 | 0.795 | 0.283 | 0.103 | 0.043 | ||||

3 | 0.318 | 0.357 | 0.324 | 0.532 | 0.042 | 0.081 | ||||

4 | 0.153 | 0.516 | 0.000 | 0.191 | 0.000 | 0.029 | ||||

5 | 0.461 | 0.413 | 0.606 | 0.412 | 0.078 | 0.063 | ||||

6 | 0.662 | 0.318 | 1.000 | 0.616 | 0.129 | 0.094 | ||||

7 | 0.335 | 0.140 | 0.358 | 1.000 | 0.046 | 0.153 | ||||

8 | 0.656 | 0.211 | 0.989 | 0.847 | 0.128 | 0.129 | ||||

9 | 0.623 | 0.301 | 0.924 | 0.653 | 0.120 | 0.100 | ||||

10 | 0.643 | 0.605 | 0.964 | 0.000 | 0.125 | 0.000 | ||||

11 | 0.419 | 0.380 | 0.524 | 0.483 | 0.068 | 0.074 | ||||

12 | 0.402 | 0.306 | 0.490 | 0.643 | 0.068 | 0.098 |

#### Determination of the overall ranking of each program using the TOPSIS method

The relative closeness was a comprehensive evaluation index of the energy dissipation and flow diversion effects of R-SEDs. Based on the calculation results in Table 6, the relative closeness of each scenario to the ideal solution was calculated using Equations (14)–(18). Then, the overall ranking of the energy dissipation and flow diversion effect of R-SEDs on the bend flow under each test scenario was obtained, as shown in Table 7. Table 7 shows that, among the 12 test scenarios, Scenario 8 had the largest relative closeness and the highest overall rating.

Test number . | η
. | SCV . | 1-SCV . | . | . | . | . | . | . | . | Rank . |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.535 | 0.188 | 0.812 | 0.306 | 0.354 | 0.138 | 0.194 | 0.035 | 0.140 | 0.802 | 2 |

2 | 0.557 | 0.473 | 0.527 | 0.319 | 0.229 | 0.144 | 0.126 | 0.084 | 0.109 | 0.564 | 6 |

3 | 0.318 | 0.357 | 0.643 | 0.182 | 0.280 | 0.082 | 0.154 | 0.103 | 0.073 | 0.415 | 11 |

4 | 0.153 | 0.516 | 0.484 | 0.088 | 0.211 | 0.040 | 0.116 | 0.159 | 0.021 | 0.118 | 12 |

5 | 0.461 | 0.413 | 0.587 | 0.264 | 0.256 | 0.119 | 0.140 | 0.083 | 0.092 | 0.524 | 9 |

6 | 0.662 | 0.318 | 0.682 | 0.379 | 0.297 | 0.171 | 0.163 | 0.043 | 0.148 | 0.776 | 4 |

7 | 0.335 | 0.140 | 0.860 | 0.192 | 0.375 | 0.087 | 0.206 | 0.084 | 0.121 | 0.589 | 5 |

8 | 0.656 | 0.211 | 0.789 | 0.375 | 0.344 | 0.169 | 0.189 | 0.017 | 0.160 | 0.904 | 1 |

9 | 0.623 | 0.301 | 0.699 | 0.356 | 0.305 | 0.161 | 0.167 | 0.040 | 0.141 | 0.780 | 3 |

10 | 0.643 | 0.605 | 0.395 | 0.368 | 0.172 | 0.166 | 0.094 | 0.111 | 0.127 | 0.532 | 8 |

11 | 0.419 | 0.380 | 0.620 | 0.240 | 0.270 | 0.108 | 0.148 | 0.085 | 0.087 | 0.507 | 10 |

12 | 0.402 | 0.306 | 0.694 | 0.230 | 0.302 | 0.104 | 0.166 | 0.078 | 0.096 | 0.553 | 7 |

Test number . | η
. | SCV . | 1-SCV . | . | . | . | . | . | . | . | Rank . |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.535 | 0.188 | 0.812 | 0.306 | 0.354 | 0.138 | 0.194 | 0.035 | 0.140 | 0.802 | 2 |

2 | 0.557 | 0.473 | 0.527 | 0.319 | 0.229 | 0.144 | 0.126 | 0.084 | 0.109 | 0.564 | 6 |

3 | 0.318 | 0.357 | 0.643 | 0.182 | 0.280 | 0.082 | 0.154 | 0.103 | 0.073 | 0.415 | 11 |

4 | 0.153 | 0.516 | 0.484 | 0.088 | 0.211 | 0.040 | 0.116 | 0.159 | 0.021 | 0.118 | 12 |

5 | 0.461 | 0.413 | 0.587 | 0.264 | 0.256 | 0.119 | 0.140 | 0.083 | 0.092 | 0.524 | 9 |

6 | 0.662 | 0.318 | 0.682 | 0.379 | 0.297 | 0.171 | 0.163 | 0.043 | 0.148 | 0.776 | 4 |

7 | 0.335 | 0.140 | 0.860 | 0.192 | 0.375 | 0.087 | 0.206 | 0.084 | 0.121 | 0.589 | 5 |

8 | 0.656 | 0.211 | 0.789 | 0.375 | 0.344 | 0.169 | 0.189 | 0.017 | 0.160 | 0.904 | 1 |

9 | 0.623 | 0.301 | 0.699 | 0.356 | 0.305 | 0.161 | 0.167 | 0.040 | 0.141 | 0.780 | 3 |

10 | 0.643 | 0.605 | 0.395 | 0.368 | 0.172 | 0.166 | 0.094 | 0.111 | 0.127 | 0.532 | 8 |

11 | 0.419 | 0.380 | 0.620 | 0.240 | 0.270 | 0.108 | 0.148 | 0.085 | 0.087 | 0.507 | 10 |

12 | 0.402 | 0.306 | 0.694 | 0.230 | 0.302 | 0.104 | 0.166 | 0.078 | 0.096 | 0.553 | 7 |

#### Selection of main influencing factors

The relative closeness was used as the response variable and a significance level of 0.05 was selected. An ANOVA was conducted to examine the importance of these nine influencing factors on the relative closeness, as shown in Table 8. Table 8 shows that the model was significant (*P* < 0.05). Particularly, the height ratio of R-SED, the centerline radius of the bend and the bend angle were significant for the relative closeness.

Source . | DF . | Adj SS . | Adj MS . | F-Value
. | P-Value
. | Significance . |
---|---|---|---|---|---|---|

Model | 9 | 0.479869 | 0.053319 | 22.79 | 0.043 | * |

Linear | 9 | 0.479869 | 0.053319 | 22.79 | 0.043 | * |

h _{r} | 1 | 0.031554 | 0.031554 | 13.49 | 0.067 | |

λ | 1 | 0.072006 | 0.072006 | 30.77 | 0.031 | * |

α | 1 | 0.000079 | 0.000079 | 0.03 | 0.871 | |

θ | 1 | 0.012849 | 0.012849 | 5.49 | 0.144 | |

δ | 1 | 0.008324 | 0.008324 | 3.56 | 0.2 | |

R | 1 | 0.143629 | 0.143629 | 61.38 | 0.016 | * |

B | 1 | 0.000064 | 0.000064 | 0.03 | 0.884 | |

β | 1 | 0.206204 | 0.206204 | 88.13 | 0.011 | * |

i | 1 | 0.00516 | 0.00516 | 2.21 | 0.276 | |

Error | 2 | 0.00468 | 0.00234 | |||

Total | 11 | 0.484549 | ||||

R-sq = 0.9903 | R-sq (adj) = 0.9469 |

Source . | DF . | Adj SS . | Adj MS . | F-Value
. | P-Value
. | Significance . |
---|---|---|---|---|---|---|

Model | 9 | 0.479869 | 0.053319 | 22.79 | 0.043 | * |

Linear | 9 | 0.479869 | 0.053319 | 22.79 | 0.043 | * |

h _{r} | 1 | 0.031554 | 0.031554 | 13.49 | 0.067 | |

λ | 1 | 0.072006 | 0.072006 | 30.77 | 0.031 | * |

α | 1 | 0.000079 | 0.000079 | 0.03 | 0.871 | |

θ | 1 | 0.012849 | 0.012849 | 5.49 | 0.144 | |

δ | 1 | 0.008324 | 0.008324 | 3.56 | 0.2 | |

R | 1 | 0.143629 | 0.143629 | 61.38 | 0.016 | * |

B | 1 | 0.000064 | 0.000064 | 0.03 | 0.884 | |

β | 1 | 0.206204 | 0.206204 | 88.13 | 0.011 | * |

i | 1 | 0.00516 | 0.00516 | 2.21 | 0.276 | |

Error | 2 | 0.00468 | 0.00234 | |||

Total | 11 | 0.484549 | ||||

R-sq = 0.9903 | R-sq (adj) = 0.9469 |

## CONCLUSIONS

In this paper, based on 12 groups of Plackett–Burman tests, the R-SEDs set at the bottom of the curved spillway were studied using the entropy weight method and the TOPSIS method. Statistical analysis of the test results was performed using Minitab 21.1 software. The conclusions are drawn as follows:

- (1)
The relative height of R-SED and the bend angle had significant effects on the energy dissipation effect of the R-SEDs. The height ratio and angle of R-SED had no significant effect on the energy dissipation effect of the R-SEDs. R-SED spacing was a negatively correlated factor, i.e., a smaller spacing of the R-SED can induce better the energy dissipation effects of the R-SEDs. The relative height of R-SED, R-SED thickness, the centerline radius of the bend, bend width, bend angle and bottom slope of the bend were positively correlated with the energy dissipation rate. The factors affecting the energy dissipation effect of the R-SED were in the following order: bend angle > relative height of R-SED > centerline radius of the bend > bottom slope of the bend > R-SED thickness > bend width > R-SED spacing > R-SED angle > height ratio of R-SED.

- (2)
The height ratio of R-SED and the centerline radius of the bend had significant effects on the flow diversion effect of R-SEDs. The height ratio of R-SED, R-SED spacing, R-SED angle, the centerline radius of the bend and the bend angle had a negative correlation with the coefficient of variation of superelevation. The relative height of R-SED, R-SED thickness, the bend width and the bottom slope of the bend showed a positive correlation with the coefficient of variation of superelevation. The factors influencing the flow diversion effect of R-SEDs were in the following order: height ratio of R-SED > centerline radius of the bend > R-SED angle > relative height of R-SED > bend width > bottom slope of the bend > R-SED spacing > bend angle > R-SED thickness.

- (3)
The weights of the energy dissipation rate and the coefficient of variation of superelevation were 0.451 and 0.549, respectively, indicating that the flow diversion effect of the R-SEDs was larger than their energy dissipation effect. The height ratio of R-SED, the centerline radius of the bend and the bend angle had significant effects on the overall energy dissipation and flow diversion effects of the R-SEDs. These three factors will be selected as the key factors for further steepest climbing tests and response surface test design. The factors influencing the overall energy dissipation and flow diversion effect of R-SEDs were in the following order: bend angle > centerline radius of the bend > height ratio of R-SED > relative height of R-SED > R-SED angle > R-SED thickness > bottom slope of the bend > R-SED spacing > bend width.

- (4)
The R-SED performance achieved the highest overall rating with the following parameter selection:

*h*= 0.8 cm,_{r}*λ*= 2,*α*= (1/3)*β*,*θ*= 15°,*δ*= (1/50)*B*,*R*= 300 cm,*B*= 50 cm,*β*= 60° and*i*= 0.02. This can be used as a reference for the R-SED design of similar curved spillway projects.

## ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51769037) and the University Research Program Innovation Team Project of Xinjiang Uygur Autonomous Region (Grant No. XJEDU2017T004).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*The Study on Energy Dissipation and River Diversion Effect on Water Flow from Roughness Section on Carved Spillway*