# Analysis of Multifactor Influence Model on Energy Dissipation Rate of Rough-Strips Energy Dissipator in the Spillway Bend

Authors: Honghong Zhang; Zhenwei Mu; Fan Fan; Fanqi Li
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The rough-strips energy dissipator (R-SED) is applied to the bottom of the spillway bend and can play the role of energy dissipation and flow stabilization. In this study, based on 18 sets of orthogonal tests and the principle of dimensional analysis, a multifactor influence model of R-SED’s energy dissipation rate was proposed. A dimensionless factork was introduced, which can reflect the comprehensive characteristics of the geometric dimensions of R-SEDs. The multifactor influence model of the energy dissipation rate considered nine factors, including bend radius of curvature Rc, bend width B, flow velocity of the bend inlet v, R-SED’s average height hL, R-SED’s arrangement angle θ, R-SED’s arrangement spacing ∆L, fluid density ρ, dynamic viscosity coefficient μ, and gravitational acceleration g. The residual sum of squares of the model (RSS) was 6.6% and the correlation coefficient R was 83.2% (>80%), indicating the universality and feasibility of the model. The independent variables of the multifactor model of the energy dissipation rate were ranked according to the Pearson value in descending order: (ΔL/Rc) > (θ) > (B/Rc) > (hL/Rc) > (1/Fr2). This indicates that R-SEDs’ layout parameters showed larger effects on the multifactor model of the energy dissipation rate, compared with the engineering layout parameters of the spillway. The maximum relative error between the predicted value of the multifactor model and the measured value of the validation group was 6.28%, indicating good agreement. In the orthogonal tests, scenario 5 had the highest energy dissipation rate (44.83%) with k = 0.023; scenario 16 had the largest k value (0.043), with an energy dissipation rate of 40.78%. The multifactor influence model of R-SEDs’ energy dissipation rate proposed in this paper was a semi-theoretical and semi-empirical calculation formula, which can provide reference and support for similar practical engineering designs.


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## Body

## 1. Introduction

There are five bends in the spillway of the “635” Reservoir in Xinjiang, China, and both of the stilling basins are on the bends. When the spillway discharges 800 m3/s, the depth and velocity of the concave and convex banks differ greatly. The straight sections between bends, as a result of the small bend radius, the steep slope, and the large circulation intensity, cannot meet the length of the adjustment section needed for the longitudinal velocity distribution to normal after the water flows out of the bends. A combination of the rough-strips energy dissipator (R-SED) and the diversion piers is used to solve the problem of flow chaos in the bends (Figure 1). The diversion piers divert flow by intervening in the water body and adjusting the direction and momentum distribution to ensure that the water flow in the diffusion sections of the chutes evenly diffuses. The R-SED, at the same time, makes a significant energy dissipation effect, a relatively small engineering volume, and generally not large head loss, which effectively solves the energy dissipation and diversion problems of the curved spillway of steep slope jet. In previous studies at home and abroad, however, there are few relevant analyses on the use of the R-SED technology for curved spillways, accordingly, it is necessary to develop research on it.Figure 1
Joint energy dissipator of the rough-strips energy dissipator (R-SED) and the diversion piers in the curved section of the spillway of “635” reservoir in Xinjiang, China.The problem of bend flow is complex and multiple, which has always been a challenging problem in the field of hydraulics research [1]. Due to the unique geometric characteristics and boundary conditions of the bend, when the water flow enters the bend from a straight section, the water surface in the concave bank of the bend is raised, and that of the convex bank is lowered, resulting in the water surface transverse gradient ratio and superelevation, even the empty-bottom phenomenon of the convex bank in supercritical flow, which is not conducive to the normal operation of the actual project. For the first time in 1876, Thomson [2] proposed the problem of bend circulation through an experimental study. Since then, many scholars from all around the world have conducted research on the bend flow.In terms of prototype observation, some scholars make actual measurements on the observation data of natural bends [3–5], and taking into account the geographical location restrictions and measurement accuracy, most scholars now use indoor test method. For physical model tests, some scholars studied macroscopic features, such as the structure, turbulence characteristics, hydraulic characteristics of bend flow by conducting different types of model tests and combining with theoretical analysis [6–14]; and in terms of numerical simulation, some scholars have established two-dimensional or three-dimensional bend flow mathematical models by using a variety of algorithms, such as the finite volume method, and compared with the model test to verify, supplement the model test shortcomings, and analyze the characteristics of bend flow from a microscopic point of view [5, 8–11, 13–26]. Previous studies have mostly focused on hydraulic characteristics of flow in bends, and some scholars have put forward engineering auxiliary measures to improve flow conditions in bends, such as vanes [27, 28], guide walls [29], spur dikes [30, 31], riprap [32], etc. The R-SED studied in this paper, with a simple shape and economic cost, is also a kind of energy dissipation measure applied to spillway bend. The flow regime of the energy dissipator with or without the R-SED is compared in Figure 2.Figure 2
Comparison of flow regime under R-SED conditions and non-R-SED conditions: (a) non-R-SED condition (curved section), (b) R-SED condition (curved section), (c) non-R-SED condition (downstream exit section) and (d) R-SED condition (downstream exit section).
(a)(b)(c)(d)For the research of the R-SED, based on the spillway regulation project of “635” in Xinjiang, the research team carried out research and analysis on hydraulic characteristics of the R-SED in an earlier stage by means of theoretical analysis, physical model test, and numerical simulation, and some research results [33, 34] have been obtained. However, this has not yet reached the ultimate goal of the subject research, and there are few studies on the calculation method of the R-SED.In this paper, a multifactor influence model of the R-SED energy dissipation rate is proposed. Based on 18 sets of orthogonal tests and dimensional analysis principle, the relationship between the nine main influencing factors, such as spillway engineering parameters and the R-SED layout parameters, and the energy dissipation rate is comprehensively analyzed, and an R-SED multifactor influence model is established. By solving the model, a calculation method for the R-SED is initially proposed, and the model accuracy is verified through tests. This will promote the current research status of the subject, and more importantly, realize the formulation of the R-SED, thus providing a theoretical reference for practical engineering.


## 2. Materials and Methods

### 2.1. Test Equipment

The experiment was conducted in the Key Laboratory of Hydraulic Engineering and Hydraulics, College of Water Conservancy and Civil Engineering, Xinjiang Agricultural University, China. The test equipment was designed according to Chinese standards, i.e., Specification for Normal Hydraulic Model Test (SL 155–2012) and Test Regulation for Special Hydraulic Model (SL 156∼165–95) [35, 36]. The physical model test device contains the main body of the spillway model and the self-circulating water supply system. The overall schematic diagram of the test device is illustrated in Figure 1. The main part of the spillway model is made of 3 mm plexiglass plate, and the whole process is made of a rectangular section, which is composed of three parts: the straight section at the entrance, the 60° curved section and the straight section at the exit. The slope ratio along the way is 0.025, in which the total length L1 of the entrance straight section is 60 cm, the 60° bend section is a thin-walled bend, and the total length L2 of the exit straight section is 140 cm.Test measurement instruments were selected and used according to the Chinese standard “Calibration Method of Common Instruments for Hydraulic and River Model Test” (SL 233–2016) [37]. The measurement content of the test device includes three parts: flow discharge, water depth, and flow velocity. A right triangle thin-walled weir is used to measure the flow discharge; a water level probe is used to measure the water depth with an accuracy of 0.1 mm, and a total of 51 water depth measurement sections are set along the flow direction, and each section is arranged with measurement points in every 5 cm from the left bank to the right bank, with a total of 11 measuring points; a pitot tube is used to measure the flow velocity, and a total of 13 flow velocity measurement sections are set along the model, 6 measurement points are evenly selected for each section, and the measurement positions are 1/3 away from the bed surface. Figure 3 shows the test measurement cross-section and measurement point settings.Figure 3
Test model structure and measuring points layout.The R-SED used in this paper is arranged at a 60° bend, extending from the concave bank to the convex bank, and close to the bottom of the spillway. The spacing, angle, height, and other parameters determine the layout of the R-SED under different working conditions. The detailed layout parameters are presented in Figure4.Figure 4
R-SED layout and section shape.


### 2.2. Working Condition Design

There are many parameters affecting the layout pattern of the R-SED and the setting of the spillway project. The orthogonal test is a multifactor and multilevel design method, which can effectively reduce the number of test groups without affecting the test results. Based on the previous test experience and results of the subject, three factors, i.e., the heighth1 − h2, the spacing ∆L and the angle θ of the R-SED, are selected as the R-SED arrangement parameters, and three factors, i.e., bend radius of curvature Rc, bend width B and discharge flow Q, are selected as the engineering parameters of the spillway. Each factor (6 factors in total) sets three levels according to actual engineering conditions. The specific factors and levels of the orthogonal test are given in Table 1. The geometric dimensions of the bend are illustrated in Figure 5. The design of the test conditions adopts the orthogonal table L18 (37), a total of 18 conditions are set, and the specific test conditions are given in Table 2.Table 1
Factors and levels of orthogonal test.
Levelh1 − h2/(cm)∆L/(cm)θ/(rad)Rc/(cm)B/(cm)Q/(L/s)11.6–0.8180.314140502022.0–1.0240.3841706022.532.4–1.2300.4542008025Figure 5
Geometric dimensions of the bend.Table 2
Design of orthogonal test.
Test no.h1 − h2/(cm)∆L/(cm)θ/(rad)Rc/(cm)B/(cm)Q/(L/s)11.6–0.8180.314140502021.6–0.8240.3841706022.531.6–0.8300.454200802542.0–1.0180.314170602552.0–1.0240.384200802062.0–1.0300.4541405022.572.4–1.2180.3841408022.582.4–1.2240.454170502592.4–1.2300.3142006020101.6–0.8180.4542006022.5111.6–0.8240.3141408025121.6–0.8300.3841705020132.0–1.0180.3842005025142.0–1.0240.4541406020152.0–1.0300.3141708022.5162.4–1.2180.4541708020172.4–1.2240.3142005022.5182.4–1.2300.3841406025


### 2.3. Principle of Dimensional Analysis

Considering the universality of the formula, this paper usesπ theorem in dimensional analysis to solve. Content of the π theorem can be expressed as:If a physical process containsn physical quantities such as x1, x2, …, xn, etc., the physical process can generally be expressed as the following functional relationship, namely,(1)fx1,x2,...,xn=0,where m physical quantities can be selected as basic physical quantities, then the physical process must be reduced to a relational formula composed of (n − m) dimensionless quantities:(2)Fπ1,π2,...,πn−m=0.


## 2.1. Test Equipment

The experiment was conducted in the Key Laboratory of Hydraulic Engineering and Hydraulics, College of Water Conservancy and Civil Engineering, Xinjiang Agricultural University, China. The test equipment was designed according to Chinese standards, i.e., Specification for Normal Hydraulic Model Test (SL 155–2012) and Test Regulation for Special Hydraulic Model (SL 156∼165–95) [35, 36]. The physical model test device contains the main body of the spillway model and the self-circulating water supply system. The overall schematic diagram of the test device is illustrated in Figure 1. The main part of the spillway model is made of 3 mm plexiglass plate, and the whole process is made of a rectangular section, which is composed of three parts: the straight section at the entrance, the 60° curved section and the straight section at the exit. The slope ratio along the way is 0.025, in which the total length L1 of the entrance straight section is 60 cm, the 60° bend section is a thin-walled bend, and the total length L2 of the exit straight section is 140 cm.Test measurement instruments were selected and used according to the Chinese standard “Calibration Method of Common Instruments for Hydraulic and River Model Test” (SL 233–2016) [37]. The measurement content of the test device includes three parts: flow discharge, water depth, and flow velocity. A right triangle thin-walled weir is used to measure the flow discharge; a water level probe is used to measure the water depth with an accuracy of 0.1 mm, and a total of 51 water depth measurement sections are set along the flow direction, and each section is arranged with measurement points in every 5 cm from the left bank to the right bank, with a total of 11 measuring points; a pitot tube is used to measure the flow velocity, and a total of 13 flow velocity measurement sections are set along the model, 6 measurement points are evenly selected for each section, and the measurement positions are 1/3 away from the bed surface. Figure 3 shows the test measurement cross-section and measurement point settings.Figure 3
Test model structure and measuring points layout.The R-SED used in this paper is arranged at a 60° bend, extending from the concave bank to the convex bank, and close to the bottom of the spillway. The spacing, angle, height, and other parameters determine the layout of the R-SED under different working conditions. The detailed layout parameters are presented in Figure4.Figure 4
R-SED layout and section shape.


## 2.2. Working Condition Design

There are many parameters affecting the layout pattern of the R-SED and the setting of the spillway project. The orthogonal test is a multifactor and multilevel design method, which can effectively reduce the number of test groups without affecting the test results. Based on the previous test experience and results of the subject, three factors, i.e., the heighth1 − h2, the spacing ∆L and the angle θ of the R-SED, are selected as the R-SED arrangement parameters, and three factors, i.e., bend radius of curvature Rc, bend width B and discharge flow Q, are selected as the engineering parameters of the spillway. Each factor (6 factors in total) sets three levels according to actual engineering conditions. The specific factors and levels of the orthogonal test are given in Table 1. The geometric dimensions of the bend are illustrated in Figure 5. The design of the test conditions adopts the orthogonal table L18 (37), a total of 18 conditions are set, and the specific test conditions are given in Table 2.Table 1
Factors and levels of orthogonal test.
Levelh1 − h2/(cm)∆L/(cm)θ/(rad)Rc/(cm)B/(cm)Q/(L/s)11.6–0.8180.314140502022.0–1.0240.3841706022.532.4–1.2300.4542008025Figure 5
Geometric dimensions of the bend.Table 2
Design of orthogonal test.
Test no.h1 − h2/(cm)∆L/(cm)θ/(rad)Rc/(cm)B/(cm)Q/(L/s)11.6–0.8180.314140502021.6–0.8240.3841706022.531.6–0.8300.454200802542.0–1.0180.314170602552.0–1.0240.384200802062.0–1.0300.4541405022.572.4–1.2180.3841408022.582.4–1.2240.454170502592.4–1.2300.3142006020101.6–0.8180.4542006022.5111.6–0.8240.3141408025121.6–0.8300.3841705020132.0–1.0180.3842005025142.0–1.0240.4541406020152.0–1.0300.3141708022.5162.4–1.2180.4541708020172.4–1.2240.3142005022.5182.4–1.2300.3841406025


## 2.3. Principle of Dimensional Analysis

Considering the universality of the formula, this paper usesπ theorem in dimensional analysis to solve. Content of the π theorem can be expressed as:If a physical process containsn physical quantities such as x1, x2, …, xn, etc., the physical process can generally be expressed as the following functional relationship, namely,(1)fx1,x2,...,xn=0,where m physical quantities can be selected as basic physical quantities, then the physical process must be reduced to a relational formula composed of (n − m) dimensionless quantities:(2)Fπ1,π2,...,πn−m=0.


## 3. Results and Discussion

### 3.1. Model Building

Based on the basic principle ofπ theorem above, a multifactor influence model of the R-SED energy dissipation rate was established:Step 1.
Determine the parameters that influence the physical process.
This is the most decisive step. The physical phenomenon of water movement in bend with the R-SED should be considered comprehensively and objectively. The internal friction force and gravity effect of the fluid should be taken into consideration when finding out the main influencing factors of energy dissipation rate of the R-SED.
The energy dissipation rateη is a key index to measure the energy dissipation effect. Based on a large number of tests and preliminary research results, there are 9 parameters which mainly influence the energy dissipation rate, i.e., bend radius of curvature Rc, bend width B, flow velocity of the bend inlet v, R-SED’s average height hL, R-SED’s arrangement angle θ, R-SED’s arrangement spacing ∆L, fluid density ρ, dynamic viscosity coefficient μ and gravitational acceleration g. Write each influencing factor into the following functional relationship:(3)Fη,Rc,B,v,hL,θ,ΔL,ρ,μ,g=0.Step 2.
Determine basic physical quantities and basic dimensions.
Generally, choose three basic physical quantities, namely,m = 3, and their dimensions must be independent. The dimensions of the above influencing factors are listed, as given in Table 3.
In order to ensure the independence of the dimensions, three physical quantities of geometric quantity, kinematic quantity and dynamic quantity are selected respectively.Rc, v, and ρ are chosen as basic physical quantities in this paper, and [L, T, M] are selected as basic dimensions. Since the number of physical quantities n = 10, there are n-m = 10 − 3 = 7 equations composed of dimensionless quantities, namely,(4)F1π1,π2,π3,π4,π5,π6,π7=F1ηRcx1vy1ρz1,BRcx2vy2ρz2,hLRcx3vy3ρz3,θRcx4vy4ρz4,ΔLRcx5vy5ρz5,μRcx6vy6ρz6,gRcx7vy7ρz7=0.Table 3
Dimension of each parameter of dimensional analysis model.
RcBvhLθ∆LρμgLLLT−1L—LML−3ML−1T−1LT−2Step 3.
Determine the exponentxi, yi, and zi of the dimensionless number π.
Sinceπ is a dimensionless number, according to the principle of dimensional harmony, the dimensions of the numerator and denominator of each factor in the formula should be harmonious.
Forπ1, η itself is a dimensionless number, then(5)π1=η.
Forπ2, there should be B=Rcx2vy2ρz2, choose [L, T, M] as the basic dimensions, then(6)π2=BRc.
Similarly, forπ3, there should be hL=Rcx3vy3ρz3, choose [L, T, M] as the basic dimensions, then(7)π3=hLRc.
Calculateπ4, π5, π6, and π7 in this way, and get Π4=θ, Π5=∆L/Rc, Π6=μ/(vρRc)(8)π7=gRc−1v2Step 4.
Write the dimensionless equation.
Substituting the obtained dimensionless numberπi into the dimensionless (4), and get(9)F1π1,π2,π3,π4,π5,π6,π7=F1η,BRc,hLRc,θ,ΔLRc,μRcvρ,gRc−1v2=0,where μ/Rcvρ=1/Re, i.e., the Reynolds number; g/Rc−1v2=1/Fr2, i.e., the reciprocal of the square of the Froude number. Then, the above formula can be written as(10)η=fBRc,hLRc,θ,ΔLRc,1Re,1Fr2.
The above equation refers to the dimensionless model affected by multifactor of energy dissipation rate of the R-SED.Step 5.
Draw the model to be solved.
The physical problem can be expressed by the power function of the variables related to it, and (10) can be reduced to the power function form:(11)η=kBRcahLRcbθcΔLRcd1Ree1Fr2f.
(11) is a curve model. To further simplify the model, the logarithm of (11) is taken as:(12)lnη=lnk+alnBRc+blnhLRc+clnθ+dlnΔLRc+eln1Re+fln1Fr2.
To convert to a linear model, replace the variables in (15). Let:y=lnη, x0=lnk, x1=ln(B/Rc), x2=ln(hL/Rc)(13)x3=lnθx4=lnΔLRcx5=ln1Rex6=ln1Fr2.
(12) can be simplified as:(14)y=x0+ax1+bx2+cx3+dx4+ex5+fx6.
The above equation is the model to be solved for multiple regression analysis influenced by multifactor of the energy dissipation rate.


### 3.2. Model Solution

Multiple regression analysis can explain the change of another variable through the change of several variables, determine the quantitative relationship between the variables, and get the corresponding mathematical model. The entrance and exit sections of the bend are selected as the water passing section, and the plane of the bottom elevation of the exit section is taken as the calculation datum plane to calculate the energy dissipation rateη under various working conditions. Combined with the results of the orthogonal test, multiple regression analysis of (14) was carried out to solve the model. The orthogonal test results are given in Table 4.Table 4
Orthogonal test results.
Test no.B/(m)hL/(m)θ/(rad)∆L/(m)1/Re (×10−5)1/Fr2/(−)η/(%)k/(−)10.500.0120.3140.183.410.5720.830.02020.600.0120.3840.243.400.6126.370.01930.800.0120.4540.303.770.6238.940.01740.600.0150.3140.183.140.5728.210.02550.800.0150.3840.244.600.6544.830.02360.500.0150.4540.303.090.5920.910.02270.800.0180.3840.184.250.4930.480.03780.500.0180.4540.242.800.6327.980.03390.600.0180.3140.303.760.6144.300.018100.600.0120.4540.183.410.5933.640.029110.800.0120.3140.243.830.5426.170.015120.500.0120.3840.303.300.6828.010.015130.500.0150.3840.182.780.6433.950.031140.600.0150.4540.243.790.5826.210.027150.800.0150.3140.304.150.6235.380.015160.800.0180.4540.184.600.6640.780.043170.500.0180.3140.243.030.6435.390.023180.600.0180.3840.303.170.5422.200.022The coefficients of (14) are solved by SPSS software multiple regression analysis module, and the regression results are presented in (17). It should be pointed out that the regression analysis shows that the power corresponded to Reynolds number parameter (1/Re) has the trend toward infinity, which has little influence on the dependent variable, so the Reynolds number Re parameter is ignored.(15)y=−0.562+0.315x1+8.558x2−0.170x3−0.169x4+1.027x6.After obtaining the fitting result based on the measured data, the characteristic values of the result parameters need to be analyzed to determine the feasibility of the prediction model. According to the theory of least square method, the smaller theRSS (residual sum of squares) is, the better the fitting effect is. In the regression model of (15), RSS = 0.066. In order to obtain a better fit, analyze the size of the correlation coefficient R. The closer R is to 1, the better the fitting effect is. Here, the correlation coefficient of the regression model is R = 0.832 (>0.80), which is relatively high, and has certain feasibility, which can provide reference for the follow-up study.Converting (17) to a power function form:(16)η=0.570BRc0.315hLRc8.558θ−0.170ΔLRc−0.1691Fr21.027.Basing on the results of multiple regression model, the correlation between the five dimensions andη was analyzed. The larger the absolute value of the Pearson value of the independent variable is, the greater its correlation with the dependent variable is. The fitting results show that the Pearson value corresponded to (B/Rc), (hL/Rc), (θ), (ΔL/Rc), and (1/Fr2) are 0.161, 0.118, 0.451, 0.526, and 0.005, respectively. It can be seen that (θ) and (ΔL/Rc) have a greater impact on η in the tests of this paper, while (1/Fr2) have the least influence. It shows that the layout of the R-SED has a great influence on the energy dissipation rate, and that the influence of the layout parameters of the spillway is relatively small, which verifies the effectiveness of the R-SED arrangement at the bend of the spillway. Due to the centrifugal force of the bend, the water level difference of the concave and convex bank before the installation of the R-SED is large, the flow regime is turbulent, and the concentrated flow causes severe erosion on the concave bank side wall of the spillway, affecting the stability of the building; after setting the R-SED, the water flow hits it after entering the bend so that the energy is dissipated, the kinetic energy and the velocity distribution are evenly distributed, the water level difference of the both banks is reduced, and the flow regime is improved.


### 3.3. Model Validation

The “635” spillway regulation project in Xinjiang uses the R-SED as energy dissipation measure. The project has been in operation for many years and the response has been satisfactory. The subject research has been based on this project. Considering that it is difficult to obtain engineering measured data, this paper takes the physical model test related to the “635” spillway regulation project in Xinjiang as an example to verify (16).The verification group model is also divided into three parts: the straight section at the entrance, the 60° curved section and the straight section at the exit. The straight sections at the entrance and exit adopt the same rectangular cross-section model as the orthogonal test, and the slope ratio along the way is 0.025. The R-SED adopts a new trapezoidal section. A total of five sets of working conditions are set in the model test, with the same flow discharge of 22.5 L/s. The design of working conditions adopts the principle of single variable method, and the specific combination of working condition design parameters are given in Table5.Table 5
Design of retesting working conditions of the verification groups.
Test no.h1 − h2/(cm)∆L/(cm)θ/(rad)Rc/(cm)B/(cm)Q/(L/s)11.6–0.8140855022.521.6–0.8180855022.531.6–0.8220855022.542.0–1.0180855022.552.4–1.2180855022.5At the same time, the outlet and inlet sections of the bend are selected as the water passing sections, and the plane of floor elevation at the bend outlet is taken as the calculation datum plane. The energy dissipation rates of the five verification groups are calculated, and the measured values of the energy dissipation rate of the multifactor influence model are obtained. Each value of the verification groups is substituted into (16) to calculate and obtain the calculated value of the energy dissipation rate of the multifactor influence model. Analyze and compare the measured values of the model energy dissipation rate with the calculated ones, as shown in Figure 6. The maximum relative error is calculated to be 6.28%. Considering the external factors, such as air temperature and water temperature during the test and factors that cannot be controlled artificially, it is believed that the retest results agree well with the predicted results. Therefore, the multifactor influence model of energy dissipation rate can provide formula reference for practical engineering. Considering the specificity and uniqueness of each project, as well as the natural factors that the laboratory cannot simulate, some influencing factors are not mentioned in this article, such as the slope ratio, and will be further studied in the future.Figure 6
Comparison of the calculated values of the prediction model verification groups and the actual values of the retest.


### 3.4. Establishment of Dimensionless Factor

The layout of the R-SED affects the flow characteristics of the curved spillway. Considering the layout parameters, such as the average heighthL, the spacing ∆L, and the angle θ of the R-SED, the dimensionless factor k is introduced to reflect the comprehensive characteristics of the geometric dimensions of the R-SED:(17)k=hLΔLsinθ,where hL is the average height of the concave and convex side of the R-SED, cm; θ is the arrangement angle of the R-SED, °; ∆L is the arrangement spacing of the R-SED, cm. The k values of 18 sets of orthogonal experiments were calculated, as shown in Table 4. The results demonstrate that when the energy dissipation rate was maximum, k = 0.023. The precise correlation between the k value and the energy dissipation rate η will be described in detail in future studies.


## 3.1. Model Building

Based on the basic principle ofπ theorem above, a multifactor influence model of the R-SED energy dissipation rate was established:Step 1.
Determine the parameters that influence the physical process.
This is the most decisive step. The physical phenomenon of water movement in bend with the R-SED should be considered comprehensively and objectively. The internal friction force and gravity effect of the fluid should be taken into consideration when finding out the main influencing factors of energy dissipation rate of the R-SED.
The energy dissipation rateη is a key index to measure the energy dissipation effect. Based on a large number of tests and preliminary research results, there are 9 parameters which mainly influence the energy dissipation rate, i.e., bend radius of curvature Rc, bend width B, flow velocity of the bend inlet v, R-SED’s average height hL, R-SED’s arrangement angle θ, R-SED’s arrangement spacing ∆L, fluid density ρ, dynamic viscosity coefficient μ and gravitational acceleration g. Write each influencing factor into the following functional relationship:(3)Fη,Rc,B,v,hL,θ,ΔL,ρ,μ,g=0.Step 2.
Determine basic physical quantities and basic dimensions.
Generally, choose three basic physical quantities, namely,m = 3, and their dimensions must be independent. The dimensions of the above influencing factors are listed, as given in Table 3.
In order to ensure the independence of the dimensions, three physical quantities of geometric quantity, kinematic quantity and dynamic quantity are selected respectively.Rc, v, and ρ are chosen as basic physical quantities in this paper, and [L, T, M] are selected as basic dimensions. Since the number of physical quantities n = 10, there are n-m = 10 − 3 = 7 equations composed of dimensionless quantities, namely,(4)F1π1,π2,π3,π4,π5,π6,π7=F1ηRcx1vy1ρz1,BRcx2vy2ρz2,hLRcx3vy3ρz3,θRcx4vy4ρz4,ΔLRcx5vy5ρz5,μRcx6vy6ρz6,gRcx7vy7ρz7=0.Table 3
Dimension of each parameter of dimensional analysis model.
RcBvhLθ∆LρμgLLLT−1L—LML−3ML−1T−1LT−2Step 3.
Determine the exponentxi, yi, and zi of the dimensionless number π.
Sinceπ is a dimensionless number, according to the principle of dimensional harmony, the dimensions of the numerator and denominator of each factor in the formula should be harmonious.
Forπ1, η itself is a dimensionless number, then(5)π1=η.
Forπ2, there should be B=Rcx2vy2ρz2, choose [L, T, M] as the basic dimensions, then(6)π2=BRc.
Similarly, forπ3, there should be hL=Rcx3vy3ρz3, choose [L, T, M] as the basic dimensions, then(7)π3=hLRc.
Calculateπ4, π5, π6, and π7 in this way, and get Π4=θ, Π5=∆L/Rc, Π6=μ/(vρRc)(8)π7=gRc−1v2Step 4.
Write the dimensionless equation.
Substituting the obtained dimensionless numberπi into the dimensionless (4), and get(9)F1π1,π2,π3,π4,π5,π6,π7=F1η,BRc,hLRc,θ,ΔLRc,μRcvρ,gRc−1v2=0,where μ/Rcvρ=1/Re, i.e., the Reynolds number; g/Rc−1v2=1/Fr2, i.e., the reciprocal of the square of the Froude number. Then, the above formula can be written as(10)η=fBRc,hLRc,θ,ΔLRc,1Re,1Fr2.
The above equation refers to the dimensionless model affected by multifactor of energy dissipation rate of the R-SED.Step 5.
Draw the model to be solved.
The physical problem can be expressed by the power function of the variables related to it, and (10) can be reduced to the power function form:(11)η=kBRcahLRcbθcΔLRcd1Ree1Fr2f.
(11) is a curve model. To further simplify the model, the logarithm of (11) is taken as:(12)lnη=lnk+alnBRc+blnhLRc+clnθ+dlnΔLRc+eln1Re+fln1Fr2.
To convert to a linear model, replace the variables in (15). Let:y=lnη, x0=lnk, x1=ln(B/Rc), x2=ln(hL/Rc)(13)x3=lnθx4=lnΔLRcx5=ln1Rex6=ln1Fr2.
(12) can be simplified as:(14)y=x0+ax1+bx2+cx3+dx4+ex5+fx6.
The above equation is the model to be solved for multiple regression analysis influenced by multifactor of the energy dissipation rate.


## 3.2. Model Solution

Multiple regression analysis can explain the change of another variable through the change of several variables, determine the quantitative relationship between the variables, and get the corresponding mathematical model. The entrance and exit sections of the bend are selected as the water passing section, and the plane of the bottom elevation of the exit section is taken as the calculation datum plane to calculate the energy dissipation rateη under various working conditions. Combined with the results of the orthogonal test, multiple regression analysis of (14) was carried out to solve the model. The orthogonal test results are given in Table 4.Table 4
Orthogonal test results.
Test no.B/(m)hL/(m)θ/(rad)∆L/(m)1/Re (×10−5)1/Fr2/(−)η/(%)k/(−)10.500.0120.3140.183.410.5720.830.02020.600.0120.3840.243.400.6126.370.01930.800.0120.4540.303.770.6238.940.01740.600.0150.3140.183.140.5728.210.02550.800.0150.3840.244.600.6544.830.02360.500.0150.4540.303.090.5920.910.02270.800.0180.3840.184.250.4930.480.03780.500.0180.4540.242.800.6327.980.03390.600.0180.3140.303.760.6144.300.018100.600.0120.4540.183.410.5933.640.029110.800.0120.3140.243.830.5426.170.015120.500.0120.3840.303.300.6828.010.015130.500.0150.3840.182.780.6433.950.031140.600.0150.4540.243.790.5826.210.027150.800.0150.3140.304.150.6235.380.015160.800.0180.4540.184.600.6640.780.043170.500.0180.3140.243.030.6435.390.023180.600.0180.3840.303.170.5422.200.022The coefficients of (14) are solved by SPSS software multiple regression analysis module, and the regression results are presented in (17). It should be pointed out that the regression analysis shows that the power corresponded to Reynolds number parameter (1/Re) has the trend toward infinity, which has little influence on the dependent variable, so the Reynolds number Re parameter is ignored.(15)y=−0.562+0.315x1+8.558x2−0.170x3−0.169x4+1.027x6.After obtaining the fitting result based on the measured data, the characteristic values of the result parameters need to be analyzed to determine the feasibility of the prediction model. According to the theory of least square method, the smaller theRSS (residual sum of squares) is, the better the fitting effect is. In the regression model of (15), RSS = 0.066. In order to obtain a better fit, analyze the size of the correlation coefficient R. The closer R is to 1, the better the fitting effect is. Here, the correlation coefficient of the regression model is R = 0.832 (>0.80), which is relatively high, and has certain feasibility, which can provide reference for the follow-up study.Converting (17) to a power function form:(16)η=0.570BRc0.315hLRc8.558θ−0.170ΔLRc−0.1691Fr21.027.Basing on the results of multiple regression model, the correlation between the five dimensions andη was analyzed. The larger the absolute value of the Pearson value of the independent variable is, the greater its correlation with the dependent variable is. The fitting results show that the Pearson value corresponded to (B/Rc), (hL/Rc), (θ), (ΔL/Rc), and (1/Fr2) are 0.161, 0.118, 0.451, 0.526, and 0.005, respectively. It can be seen that (θ) and (ΔL/Rc) have a greater impact on η in the tests of this paper, while (1/Fr2) have the least influence. It shows that the layout of the R-SED has a great influence on the energy dissipation rate, and that the influence of the layout parameters of the spillway is relatively small, which verifies the effectiveness of the R-SED arrangement at the bend of the spillway. Due to the centrifugal force of the bend, the water level difference of the concave and convex bank before the installation of the R-SED is large, the flow regime is turbulent, and the concentrated flow causes severe erosion on the concave bank side wall of the spillway, affecting the stability of the building; after setting the R-SED, the water flow hits it after entering the bend so that the energy is dissipated, the kinetic energy and the velocity distribution are evenly distributed, the water level difference of the both banks is reduced, and the flow regime is improved.


## 3.3. Model Validation

The “635” spillway regulation project in Xinjiang uses the R-SED as energy dissipation measure. The project has been in operation for many years and the response has been satisfactory. The subject research has been based on this project. Considering that it is difficult to obtain engineering measured data, this paper takes the physical model test related to the “635” spillway regulation project in Xinjiang as an example to verify (16).The verification group model is also divided into three parts: the straight section at the entrance, the 60° curved section and the straight section at the exit. The straight sections at the entrance and exit adopt the same rectangular cross-section model as the orthogonal test, and the slope ratio along the way is 0.025. The R-SED adopts a new trapezoidal section. A total of five sets of working conditions are set in the model test, with the same flow discharge of 22.5 L/s. The design of working conditions adopts the principle of single variable method, and the specific combination of working condition design parameters are given in Table5.Table 5
Design of retesting working conditions of the verification groups.
Test no.h1 − h2/(cm)∆L/(cm)θ/(rad)Rc/(cm)B/(cm)Q/(L/s)11.6–0.8140855022.521.6–0.8180855022.531.6–0.8220855022.542.0–1.0180855022.552.4–1.2180855022.5At the same time, the outlet and inlet sections of the bend are selected as the water passing sections, and the plane of floor elevation at the bend outlet is taken as the calculation datum plane. The energy dissipation rates of the five verification groups are calculated, and the measured values of the energy dissipation rate of the multifactor influence model are obtained. Each value of the verification groups is substituted into (16) to calculate and obtain the calculated value of the energy dissipation rate of the multifactor influence model. Analyze and compare the measured values of the model energy dissipation rate with the calculated ones, as shown in Figure 6. The maximum relative error is calculated to be 6.28%. Considering the external factors, such as air temperature and water temperature during the test and factors that cannot be controlled artificially, it is believed that the retest results agree well with the predicted results. Therefore, the multifactor influence model of energy dissipation rate can provide formula reference for practical engineering. Considering the specificity and uniqueness of each project, as well as the natural factors that the laboratory cannot simulate, some influencing factors are not mentioned in this article, such as the slope ratio, and will be further studied in the future.Figure 6
Comparison of the calculated values of the prediction model verification groups and the actual values of the retest.


## 3.4. Establishment of Dimensionless Factor

The layout of the R-SED affects the flow characteristics of the curved spillway. Considering the layout parameters, such as the average heighthL, the spacing ∆L, and the angle θ of the R-SED, the dimensionless factor k is introduced to reflect the comprehensive characteristics of the geometric dimensions of the R-SED:(17)k=hLΔLsinθ,where hL is the average height of the concave and convex side of the R-SED, cm; θ is the arrangement angle of the R-SED, °; ∆L is the arrangement spacing of the R-SED, cm. The k values of 18 sets of orthogonal experiments were calculated, as shown in Table 4. The results demonstrate that when the energy dissipation rate was maximum, k = 0.023. The precise correlation between the k value and the energy dissipation rate η will be described in detail in future studies.


## 4. Conclusions

In this paper, based on 18 sets of orthogonal experiments, through the dimensional analysis principle, a multifactor influence model of the energy dissipation rate is established, and SPSS software is used to solve the model. The main conclusions are as follows:(i)
A total of 18 sets of orthogonal tests with 6 factors and 3 levels were conducted. This not only effectively reduced the number of tests, but also comprehensively considered the effects of multiple factors and levels on the R-SED’s energy dissipation rate. Thus, this can provide data support for the derivation of multifactor influence models of the energy dissipation rate.(ii)
The multifactor influence model of the energy dissipation rate takes into account nine influencing factors, i.e., bend radius of curvatureRc, bend width B, the flow velocity of the bend inlet v, R-SED’s average height hL, R-SED’s arrangement angle θ, R-SED’s arrangement spacing ∆L, the fluid density ρ, dynamic viscosity coefficient μ, and gravitational acceleration g, with a certain universality, which can be widely used in engineering equipped with the R-SED. The model regression analysis shows that the value corresponding to the Reynolds number covariate (1/Re) was close to infinity in power, i.e., the energy dissipation rate was not sensitive to the variation of Reynolds number Re.(iii)
The residual sum of squares of the energy dissipation rate multifactor model was 6.6% and the correlation coefficientR was 83.2% (>80%), indicating that the proposed model was feasible and valid. The multiple regression analysis shows that the independent variables of the multifactor model of the energy dissipation rate were ranked according to the Pearson value in descending order: (ΔL/Rc) > (θ) > (B/Rc) > (hL/Rc) > (1/Fr2). This indicates that the multifactor model of the energy dissipation rate was more influenced by R-SEDs’ layout than by the engineering layout parameters of the spillway.(iv)
Based on the design conditions of the verification groups, the multifactor influence model is verified and analyzed. The calculated values of the prediction model are highly consistent with the actual values, with the maximum relative error of 6.28%. The result is accurate and reliable, which can be used to calculate the layout of the R-SED in practical projects.(v)
The dimensionless factork considered the average height, spacing and angle of the R-SED, and was an effective factor to reflect the comprehensive characteristics of the geometric dimensions of R-SEDs. Among the 18 sets of orthogonal tests, scenario 5 had the highest energy dissipation rate (44.83%), with k = 0.023; scenario 16 had the highest k value (0.043), with an energy dissipation rate of 40.78%. However, the correlation between the k value and the energy dissipation rate η needs a lot of experiments to be clear, and will be explained in detail in future studies.The R-SED proposed in this paper can effectively solve the adverse hydraulic phenomena caused by the bend flow, which can provide an application example for similar projects in the future. The multifactor influence model of the energy dissipation rate is proposed to realize the formulation of the R-SED, which provides a theoretical reference for the calculation and construction in actual engineering.



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