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Study on Energy Dissipation Characteristics of Skimming Flow on Stepped Spillways

臺階式溢洪道滑行水流消能特性之研究PENGHUI MA

馬　朋　輝College of Water Resources and

Architectural Engineering, Northwest A & F University, Shaanxi, China;

Key Laboratory of Agricultural Soil and Water Engineering in Arid and

Semiarid Areas of Ministry of Education, Northwest A & F University, China,

Ph.D. Student

YAJIN HU

胡　亞　瑾College of Water Resources and

Architectural Engineering, Northwest A & F University, Shaanxi, China;

Key Laboratory of Agricultural Soil and Water Engineering in Arid and

Semiarid Areas of Ministry of Education, Northwest A & F University, China,

Ph.D. Student

HANSHENG LIU*

劉　韓　生College of Water Resources and

Architectural Engineering, Northwest A & F University, China,

Professor

ABSTRACT

Stepped spillway has attracted the attention of the hydraulic engineering researchers and has been widely studied and applied for its remarkable energy dissipation effect, greatly reducing the size of stilling basin at the end of the spillway, saving engineering quantity and investment. In order to reflect the increased energy dissipation effect of steps on stepped spillways, this paper compared stepped spillways with smooth spillways of the same shape, and introduced the concepts of step energy dissipation rate and step energy dissipation head ratio. Based on the hydraulic model test of Simutasi Hydropower Station, the variation characteristics of the total energy dissipation rate, smooth energy dissipation rate, step energy dissipation rate and step energy dissipation head ratio along the way and their relationship with the relative critical water depth were systematically analyzed. It was revealed that the decrease of the total energy dissipation rate of the stepped spillway with the increase of the discharge per unit width (or relative critical water depth) is caused by the decrease of the smooth energy dissipation rate while the step energy dissipation rate remains unchanged. The research results have important practical significance for understanding, analyzing and applying stepped spillways.

Keywords: Stepped spillway, Experimental study, Total energy dissipation rate, Smooth energy dissipation rate, Step energy dissipation rate, Step energy dissipation head ratio.

摘　　　　　要

臺階式溢洪道由於消能效果顯著、能大大減小溢洪道末端消力池的尺寸、節省工程量和投資而引起水利工程界的重視，並得到較為廣泛的研究和應用。為了反映臺階式溢洪道

* Corresponding author: professor/ College of Water Resources and Architectural Engineering, Northwest A & F University/ Weihui Road, No.23, Yangling 712100, Shaanxi, PR China / hanshengliuxn@163.com

臺灣水利　第 69 卷　第 2 期民國 110 年 6 月出版

Taiwan Water ConservancyVol. 69, No. 2, June 2021DOI: 10.6937/TWC.202106/PP_69(2).0008

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1. INTRODUCTION

In order to intercept water flow, raise water level or adjust water storage capacity, water retaining structures such as sluice or dam need to be built in water conservancy projects. The water retaining structure makes the water level difference between upstream and downstream, which makes the flow through sluice or dam form huge energy. In order to eliminate the energy, the traditional engineering uses smooth spillway or overflow dam for trajectory energy dissipation or built stilling basin at downstream for hydraulic jump energy dissipation. In order to prevent cavitation damage, it is necessary to make strict regulations on the local unevenness of the smooth spillway or overflow dam, especially the local unevenness in the vertical flow direction (Tian et al., 2003).

Over the past 30 years, with the appearance and application of roller-compacted concrete (RCC) dam construction technology, the configuration design of traditional spillway and overflow dam has been greatly influenced. Stepped spillway and overflow dam are flood discharge and energy dissipation facilities developed with the development of RCC dam construction technology. They are developed on the basis of smooth spillway and have received strong attention from hydraulic researchers around the world. Compared with traditional energy dissipaters, there are several reasons for the popularity of stepped spillways: (1) the energy dissipation rate is significant; (2) the depth and dimensions required for the stilling basin are

reduced compared with traditional measures (Peyras et al., 1992); and (3) they are compatible with RCC dams (Chanson, 1993).

Different from the traditional smooth spillways, stepped spillways involve a series of cuts into the spillway face from the crest to the toe. The flow on stepped spillways can be divided into three flow patterns according to the slope of spillway, step height and discharge per unit width, i.e. nappe flow, skimming flow and transition flow. In hydraulic engineering, the design of stepped spillway is usually based on the presence of skimming flow. The nappe flow often occurs when the discharge per unit width is small, and the transition flow should be avoided as much as possible due to its unstable flow pattern.

Researchers have done a lot of experimental study on stepped spillways, and the current research focuses on the energy dissipation characteristics of the skimming flow over stepped spillways (Chen et al., 2003; Chinnarasri and Wongwises, 2006; Felder and Chanson, 2011; Guenther et al., 2013; Matos, 2009; Ohtsu et al., 2004; Shahheydari et al., 2015; Wu et al., 2012; Yu et al., 2019; Zhang et al., 2005). However, no consistent conclusion has been reached due to the complexity of flow on stepped spillways. And all the studies are on the total energy dissipation rate of stepped spillway, which cannot reflect the energy dissipation increased by the raised steps.

In fact, the smooth spillway also has certain energy dissipation effect due to the inherent roughness. For stepped spillway, only the energy

中臺階所增加的消能作用，本文將臺階式溢洪道和同體型的光滑溢洪道進行對比，引入了臺階消能率和臺階消能水頭比。通過斯木塔斯水電站水工模型試驗，系統分析了臺階式溢洪道總消能率、光滑消能率、臺階消能率及臺階消能水頭比沿程變化規律及其與相對臨界水深的關係，並且揭示了臺階式溢洪道總消能率隨單寬流量(或相對臨界水深)的增大而降低是由於臺階消能率不變而光滑消能率降低所導致的。研究結果對於認識、分析和應用臺階式溢洪道具有重要的現實意義。

關鍵詞： 臺階式溢洪道，試驗研究，總消能率，光滑消能率，臺階消能率，臺階消能水頭比。

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dissipation rate beyond the smooth spillway can reflect the energy dissipation effect of increasing steps. If the excess is small, it indicates that the energy dissipation effect of steps is not significant, and whether it is necessary to adopt the stepped spillway remains debatable. If the excess is large, it shows that the step energy dissipation effect is prominent, which greatly improves the necessity of adopting the stepped spillway.

The total energy dissipation rate of the stepped spillway cannot reflect the change of energy dissipation rate of the spillway after the step is added, that is, cannot reflect the energy dissipation contribution of the step. Therefore, it is not enough to study only the total energy dissipation rate of stepped spillway, but also the energy dissipation rate beyond the smooth spillway, which is very important for understanding, analyzing and applying stepped spillways.

Therefore, in this paper, the variation characteristics of energy dissipation rate of stepped spillway and the corresponding smooth spillway along the way and the influence of relative critical water depth on them were analyzed. Then, by comparing the energy dissipation rate of the stepped spillway with that of the smooth spillway, the step energy dissipation rate and step energy dissipation head ratio were defined, and the variation characteristics along the way and the influence of the relative critical water depth on them were analyzed. Furthermore, the role of steps in energy dissipation of stepped spillway was analyzed and demonstrated, which provides a new idea for further study on energy dissipation characteristics of stepped spillway.

2. METHODOLOGY

2.1. Experiment Design, Measurement and Flow Pattern

Experiment Design

A series of hydraulic model tests were carried out on the stepped spillway of Simutasi Hydropower Station on Akeyazi River in Xinjiang, China. The dam height of Simutasi Hydropower Station was 103 m, the chute of spillway adopts rectangular section, with width of 9 m, drop of 75 m, slope ratio of 1:1.25 (θ = 38.7o). All the model tests were designed according to the gravity similarity criterion, and the model scales were 1:30. Model tests under different working conditions were carried out for three step heights of 0.5 m, 1.0 m and 2.0 m. See Table 1 for specific test data.

The minimum Reynolds numbers of the tests were all more than 500, indicating that the flow patterns were turbulent. In the test, the water depth of the stepped spillway was measured by the ruler several times and the average value was taken, and the discharge was measured by the rectangular sharp-edged weir.

Experimental Set-up and Measurement

The test model is designed according to the gravity similarity criterion, and the scale of the model is 1: 30. The test system is composed of upstream water tank, inlet section, stepped chute section, flat bottom stilling basin, return channel and rectangular sharp-edged weir. See Figure 1

Table 1. Model test data

Test itemSlope θ/( o )

Step height d/(m)

Discharge per unit width q/(m2/s)

Relative critical water depth

hk /d

Total length of step L/(m)

Minimum Reynolds number

Remin

Test 1 38.7 0.5 21.96~46.58 7.3~12.1 120 6.43 × 106

Test 2 38.7 1.0 21.20~62.18 3.6~7.3 120 5.78 × 106

Test 3 38.7 2.0 21.96~46.58 1.8~3.0 120 6.23 × 106

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for model test layout. In order to observe the flow pattern conveniently, the inlet section, the stepped chute section and the stilling basin section are made of plexiglass, with a total height of 2.5 m and a stepped spillway width of 0.3 m.

In the test, the water depth of the cross section is measured at the top of each step with a ruler with millimeter scale, and the measuring direction is perpendicular to the datum line formed by the convex angle of the connecting step. The flow discharge in the test is measured by the rectangular sharp-edged weir in the downstream return channel, and the flow discharge is calculated as follows (Liu et al., 2011):

(1)

(2)

where Q is the flow discharge (m3/s); m0 is the discharge coefficient considering the influence of the head of the traveling velocity head; B is the width of the rectangular sharp-edged weir (m); g is the gravity acceleration (m/s2); H is the weir head of the rectangular sharp-edged weir (m); P is the weir height of the rectangular sharp-edged weir (m).

Flow Pattern

Stepped spillways are associated with three typical flow patterns, nappe flow (Chanson, 1995; Peyras et al., 1992), transition flow (Chanson, 1996; Torabi et al., 2013), and skimming flow regimes (Chanson, 1994a; Chamani and Rajaratnam, 1999a), based on the relative critical water depth hk /d (hk

is the critical water depth at the spillway inlet, d is the step height) and the slope of the spillway. When the relative critical water depth is small, the water naturally falls to each step. There is always a cavity between the drop tongue and the step water pad, and free water surface is formed in the step cavity, which is called nappe flow. When the relative critical water depth is large, the inside of each step is filled with water without any cavity, and a horizontal vortex is formed between the corner of each step and the main flow. Near the main flow, the rotation direction of the vortex is consistent with that of the main flow, which is called skimming flow. Transition flow is a kind of flow pattern formed between nappe flow and skimming flow, which have cavities in some steps, but vortex flow in other steps. According to the observation and empirical formula, the flow patterns in the tests were all skimming flow.

With regard to the boundary between the nappe

Fig. 1. Layout of the experimental model.

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flow and the skimming flow, Chanson (1994a), Chamani and Rajaratnam (1999b), Ohtsu et al. (2004) and Tian (2005) have studied the flow pattern of stepped spillway, and have given some empirical formulas to divide the nappe flow and the skimming flow. Yasuda's formula is between Chanson's and Tian's formula, and the empirical formulas for the lower limit of skimming flow and the upper limit of nappe flow are respectively as follows:

(3)

(4)

where

(5)

where hk is the critical water depth (m); d is the step height (m); θ is the slope of stepped spillway ( o ); q is the discharge of per unit width (m2/s).

2.2. Definition and Calculation of Energy Dissipation Rates

The stepped spillway evolved from the smooth spillway by setting a series of steps. Figure 2 shows the schematic diagram of stepped spillway and its corresponding smooth spillway. In Figure 2, the section 1"-1" of stepped spillway and the section 1-1 of the corresponding smooth spillway are mutually corresponding sections. The water depth h1" of the

stepped spillway 1"-1" section and the water depth h1 of the smooth spillway 1-1 section are mutually corresponding water depths. The velocity v1" of the stepped spillway 1"-1" section and the velocity v1 of the smooth spillway 1-1 section are mutually corresponding velocities. The water head E1" of the stepped spillway 1"-1" section and the water head E1 of the smooth spillway 1-1 section are mutually corresponding water heads.

Total Energy Dissipation Rate

The total energy dissipation rate ηtotal of a certain section of stepped spillway is the ratio of the accumulated energy dissipation head of stepped spillway (the total head of weir crest minus the residual head of a certain section of stepped spillway) to the total head of spillway weir crest, which can be written as:

(6)

where

(7)

(8)

where E0 is the total head of spillway weir crest (m); E1" is the residual head of a certain section of stepped spillway (m); Hw is the height of weir crest (m); h0 is the water depth at the weir crest (m); α

Fig. 2. Schematic diagram of stepped spillway and corresponding smooth spillway.

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is the energy coefficient; v0 is the average velocity of weir crest flow (m/s); g is the acceleration of gravity (m/s2); H1" is the height of a certain section of stepped spillway from the datum (m); h1" is the water depth of a certain section of stepped spillway (m); θ is the spillway slope ( o ); v1" is the average velocity of a certain section of stepped spillway (m/s).

In many open channel applications where the channel is of regular cross section with fairly straight alignment, the energy coefficient (α) is assumed to equal unity because the effect of nonuniform velocity distribution on the computed velocity head and momentum is small (Chow, 1959). To determine the effect that the nonuniform velocity distribution has on the computed velocity head and momentum, Equation 9 was used to determine the energy coefficient:

(9)

where v is the velocity for an incremental area (ΔA) in the velocity profile (m/s); V is the mean velocity (m/s); A is the area (m2). Assuming two-dimensional flow that is uniform across the width, the area (A) can be replaced by the flow depth.

The characteristic of energy coefficient along the path is that the energy coefficient increases upstream of the free-surface inception point and reaches the maximum value at the inception point; then downstream of the free-surface inception point, the energy coefficient decreases rapidly and tends to a certain constant value.

According to the study of the skimming, nonaerated flow on stepped spillway over roller compacted concrete dam with slope of 53o, Meireles et al. (2012) considered that the energy coefficient is nearly independent of relative critical water depth, hk /d, for rather steep slopes (45o < θ < 53o), and they proposed Equation (10) for the energy coefficient upstream of the free-surface inception point, as follows:

(10)

for x/Li < 1.0.Hunt et al. (2014) made a detailed study on the

energy coefficient of stepped spillway with three slopes of 14o, 18.4o and 26.6o, and they came to a conclusion contradicts with Meireles et al.’s (2012) findings by indicating that the energy coefficient is a function of the step height to critical water depth ratio d/hk, the stepped spillway slope θ, and the normalized length x/Li. They proposed Equation (11) for the energy coefficient upstream of the free-surface inception point, as follows:

(11)

for 0.1 ≤ x/Li ≤ 1.0, 0.10 ≤ d/hk

≤ 1.1, and 10o ≤ θ ≤ 30o.Hunt et al. (2014) also proposed Equation (12)

for the energy coefficient downstream of the free-surface inception point, as follows:

(12)

for x/Li > 1.0, 0.10 ≤ d/hk ≤ 1.1, and 10o ≤ θ ≤ 30o.

Li is the stream-wise coordinate at the inception point (m), originating at the upstream end of the spillway, and Equation (13) is used for its calculation according to Chanson (1994b).

(13)

where ks is the surface roughness: ks = d cos θ (m); F* is the Froude number defined in terms of roughness height: F*

= q/ g(sin θ)ks3 .

Based on the literature review, it is found that the current systematic research on the energy coefficient along the path is carried out for flat slope and steep slope, respectively. Through the comparison of Hunt et al.'s (2014) research, it is found that there is little difference between the energy coefficients along path between the slope of 18.4o and 26.6o. Because the cross section distribution of velocity along the path is not

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measured in the model tests, for the purposes of this study, Equations (11) and (12) are selected for the basis of determining the residual head of a certain section of stepped spillway, which has been greatly improved compared with taking the energy coefficient of stepped spillway with different flow rates and different step heights as a fixed value.

Smooth Energy Dissipation Rate

The smooth energy dissipation rate ηsmooth of a certain section of corresponding smooth spillway is the ratio of the accumulated energy dissipation head of corresponding smooth spillway (the total head of weir crest minus the residual head of a certain section of corresponding smooth spillway) to the total head of spillway weir crest, which can be written as:

(14)

where

(15)

where E1 is the residual water head of the same section on the corresponding smooth spillway (m); H1 is the height of the same section on the corresponding smooth spillway from the datum, H1

= H1" (m); h1 is the water depth of the same section on the corresponding smooth spillway (m); α is the energy coefficient and is taken as 1.05 according to the Design Specification for Spillway (SL 253-2018); v1 is the average velocity of the same section on the corresponding smooth spillway (m/s).

Step Energy Dissipation Rate

The step energy dissipation rate ηstep of the stepped spillway is the difference between the total energy dissipation rate and the smooth energy dissipation rate. That is, the cumulative energy dissipation head of a certain section of the stepped spillway minus the cumulative energy dissipation head of the corresponding section of the smooth spillway, accounting for the ratio of the total head of

the spillway weir crest, and can be written as:

(16)

Step Energy Dissipation Head Ratio

The step energy dissipation head ratio ∆Hstep

/∆Htotal for a certain section of stepped spillway is the ratio of the accumulated energy dissipation head increment relative to smooth spillway due to existence of steps to the accumulated total energy dissipation head, and can be written as:

(17)

The water depth along the stepped spillway was measured by ruler for many times to take the average value. The average velocity was obtained by dividing the discharge per unit width by the average water depth. The residual head, energy dissipation head and total energy dissipation rate of stepped spillway can be obtained by further calculation. Similarly, the residual head, energy dissipation head and smooth energy dissipation rate of a certain section of smooth spillway can be calculated according to the water depth and discharge per unit width. The water depth of the smooth spillway was calculated by using the subsection summation method (SL 253-2018).

(18)

(19)

(20)

(21)

where ∆l1-2 is the segment length (m); h1 and h2 are the water depths at the beginning and end of the segment respectively, i.e. the water depths at sections 1-1 and 2-2 of the smooth spillway in

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Figure 2 (m); v1 and v2 are the average velocity at the beginning and end of the segment respectively, i.e. the average velocities at sections 1-1 and 2-2 of the smooth spillway in Figure 2 (m/s); α is the energy coefficient and is taken as 1.05; i = sinθ; J is the average friction gradient in the segment; n is the roughness coefficient of spillway and its value is taken as 0.014; v is the average velocity in the segment (m/s); R is the average hydraulic radius in the segment (m); R1 and R2 are the hydraulic radius at the beginning and end of the segment respectively, i.e. the hydraulic radius at sections 1-1 and 2-2 of the smooth spillway in Figure 2 (m).

3. RESULTS AND DISCUSSION

3.1. Characteristics of total energy dissipation rate

The water depths of stepped spillways with three different step heights under different working

conditions were measured and the average velocities were calculated. Then the total energy dissipation rate along the stepped spillway was calculated, and its longitudinal variation characteristics and its relationship with the relative critical water depth were analyzed.

The longitudinal variation of the total energy dissipation rate of the stepped spillway (ηtotal) was shown in Figure 3. The solid legends in the figure represent the total energy dissipation rate of stepped spillway. Figure 3(a) analyzed the effect of relative critical water depth on the total energy dissipation rate based on the data of Test 1 (d = 0.5 m, q =

21.96 m2/s, 31.91 m2/s and 46.58 m2/s). Figure 3(b) analyzed the effect of relative critical water depth on the total energy dissipation rate based on the data of Test 2 (d = 1.0 m, q = 21.20 m2/s, 35.72 m2/s, 46.58 m2/s and 62.18 m2/s). Figure 3(c) analyzed the effect of relative critical water depth on the total energy dissipation rate based on the data of Test 3 (d = 2.0 m,

Fig. 3. Variation characteristics of total and smooth energy dissipation rates.

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q = 21.96 m2/s, 31.91 m2/s and 46.58 m2/s). Fig. 3(d) analyzed the influence of the relative critical water depth on the total energy dissipation rate based on some data from Tests 1 to 3 (q = 46.58 m2/s, d = 0.5 m, 1.0 m and 2.0 m).

From Figure 3, it can be concluded that the total energy dissipation rate gradually increases along the stepped spillway, and the increasing gradient also gradually increases, which indicating that the energy dissipation effect of the stepped spillway gradually increases along the way.

From Figure 3(a)-(c), it can be concluded that the total energy dissipation rate decreases with the increase of relative critical water depth when the step height is fixed. Specially, when the relative critical water depth increases from 7.3 to 12.1, the total energy dissipation rates of stepped spillway with a step height of 0.5 m decrease by 11.46%~ 33.47% (the range of x/L is 0.17~1.0). When the relative critical water depth increases from 3.6 to 7.3, the total energy dissipation rates of stepped spillway with a step height of 1.0 m decrease by 24.35%~ 42.85% (the range of x/L is 0.13~1.0). When the relative critical water depth increases from 1.8 to 3.0, the total energy dissipation rates of stepped spillway with a step height of 2.0 m decrease by 8.91%~31.31 % (the range of x/L is 0.12~1.0).

From Figure 3(d), it can be concluded that the total energy dissipation rate decreases with the increase of relative critical water depth when the discharge per unit width is fixed. Specially, when the relative critical water depth increases from 3.0 to 12.1, the total energy dissipation rates of stepped spillway with a discharge per unit width of 46.58 m2/s decrease by 6.24%~13.15% (the range of x/L is 0.12~1.0).

To sum up, the total energy dissipation rate gradually increases along the stepped spillway, and the increasing gradient also gradually increases. The total energy dissipation rate decreases with the increase of relative critical water depth.

3.2. Characteristics of smooth energy dissipation rate

The steps of the stepped spillway were removed to form a smooth spillway. The water depth and average velocity under different working conditions were calculated by the above-mentioned subsection summation method. Then the smooth energy dissipation rate was calculated, and its longitudinal variation characteristics and its relationship with the critical water depth were analyzed.

The longitudinal variation of the smooth energy dissipation rate (ηsmooth) was shown in Figure 3. The hollow legends in the figure represent the smooth energy dissipation rate.

From Figure 3, it can be concluded that the smooth energy dissipation rate increases along the stepped spillway, and the increasing gradient also gradually increases, which indicating that the energy dissipation effect of the smooth spillway gradually increases along the way, and the internal shear effect and the friction effect of the side wall increase along the way.

From Figure 3(a)-(c), it can be concluded that the smooth energy dissipation rate decreases with the increase of critical water depth. When the critical water depth increases from 3.7 to 6.0, the smooth energy dissipation rates of the smooth spillway corresponding to the stepped spillway with a step height of 0.5 m decrease by 39.00%~45.25% (the range of x/L is 0.17~1.0). When the critical water depth increases from 3.6 to 7.3, the smooth energy dissipation rates of the smooth spillway corresponding to the stepped spillway with a step height of 1.0 m decrease by 29.15%~54.57% (the range of x/L is 0.13~1.0). When the critical water depth increases from 3.7 to 6.0, the smooth energy dissipation rates of the smooth spillway corresponding to the stepped spillway with a step height of 2.0 m decrease by 33.33%~44.63% (the range of x/L is 0.12~1.0). The reason why the smooth energy dissipation rate decreases with the increase of critical water depth is that the increase

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percentage of weir crest head is larger than that of energy dissipation head.

To sum up, the smooth energy dissipation rate increases along the way, and the increasing gradient also gradually increases. The smooth energy dissipation rate decreases with the increase of critical water depth.

3.3. Characteristics of step energy dissipation rate

From the application point of view, the total energy dissipation rate of stepped spillway is convenient for engineering application, but from the perspective of energy dissipation characteristics, a single total energy dissipation rate index cannot precisely reflect the energy dissipation characteristics of stepped spillway and cannot reflect the added value of energy dissipation rate of stepped spillway compared with smooth spillway. Therefore, the stepped energy dissipation rate was introduced in this paper. The larger the value is, the more

necessary the stepped spillway is to be applied, and it can also reflect the increased energy dissipation effect of the step.

The longitudinal variation of the step energy dissipation rate (ηstep) was shown in Figure 4. As can be seen from Figure 4, the step energy dissipation rate increases gradually along the way and increase linearly. The average value of linear correlation coefficient is 0.9909, the minimum value is 0.9804, and the maximum value is 0.9987. The linear correlation is strong, which shows that the energy loss of the steps at different positions is the same. This may be due to the fact that the spillway is set with uniform and equal height steps along the way. Under the skimming flow pattern, the energy loss increased by each step is the same. The step energy dissipation rate of a section of the stepped spillway is actually the sum of the energy dissipation rates of the previous steps. Therefore, the increasing gradient of the total energy dissipation rate along the stepped

Fig. 4. Variation characteristics of step energy dissipation rate.

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spillway is caused by the increasing gradient of the smooth energy dissipation rate due to the increase of the internal shear action and the friction action of the side wall.

From Figure 4(a)-(c), it can be concluded that the step energy dissipation rate at the same position basically does not change with the change of the relative critical water depth. Specifically, when the relative critical water depth increases from 7.3 to 12.1, the maximum standard deviation of the step energy dissipation rate of the same section is 1.99 % when the step height is 0.5 m. When the relative critical water depth increases from 3.6 to 7.3, the maximum standard deviation of the step energy dissipation rate of the same section is 1.85 % when the step height is 1.0 m. When the relative critical water depth increases from 1.8 to 3.0, the maximum standard deviation of the step energy dissipation rate of the same section is 2.63 % when the step height is 2.0 m. Therefore, it can be considered that the step energy dissipation rate is not affected by the relative critical water depth when the step height is constant. When the relative critical water depth increases, the total head of the spillway weir crest increases. As the step energy dissipation rate is constant, so the energy dissipation head of the stepped spillway beyond the smooth spillway actually increases, which cannot be reflected by the total energy dissipation rate.

From Figure 4(d), it can be concluded that the step energy dissipation rate decreases with the increase of the relative critical water depth when the discharge per unit width is constant, i.e. the higher the step height is, the greater the step energy dissipation rate is. When the relative critical water depth increases from 3.0 to 12.1, the step energy dissipation rates of stepped spillway with a discharge per unit width of 46.58 m2/s decrease by 8.07%~15.83% (the range of x/L is 0.12~1.0). When the discharge per unit width is constant, the total head of the stepped spillway weir crest is constant, so the energy dissipation head of the stepped

spillway beyond the smooth spillway increases with the increase of the step height.

It is generally believed that the total energy dissipation rate of the stepped spillway decreases with the increase of the discharge per unit width. The main reasons are as follows: the stepped spillway forms vortex in the stepped area under the skimming flow pattern, so the water can be divided into upper mainstream water and vortex water in the lower stepped area. The upper mainstream water flows on the pseudo-bottom formed at the top of the step, and there is still a velocity gradient like the smooth spillway. The shearing action between the water flow layers and between the water flow and the side wall consumes the energy of the water flow. The smooth energy dissipation rate decreases with the increase of the discharge per unit width (or critical water depth), while the step energy dissipation rate is not affected by the discharge per unit width (or relative critical water depth when d is certain), resulting in the total energy dissipation rate decreases with the increase of the discharge per unit width (or relative critical water depth).

To sum up, the step energy dissipation rate increases gradually along the stepped spillway and increase linearly. It can be considered that the energy loss of the steps at different positions is the same. The step energy dissipation rate at the same section is not affected by the relative critical water depth when the step height is constant. The step energy dissipation rate decreases with the increase of the relative critical water depth when the discharge per unit width is constant.

3.4. Characteristics of step energy dissipation head ratio

The step energy dissipation rate reflects the proportion of the increased energy dissipation head of the step to the total head of the spillway crest. To further illustrate the contribution of the step to the total energy dissipation, the step energy dissipation head ratio was introduced.

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The longitudinal variation of the step energy dissipation head ratio (∆Hstep/∆Htotal) was shown in Figure 5. From Figure 5, it can be concluded that the step energy dissipation head ratio decreases along the way. The reason is that the steps at different positions on the stepped spillway play the same energy dissipation role, and the cumulative increase value of the energy dissipation head increases linearly along the way. However, the energy dissipation effect of the downstream part of the smooth spillway increases due to the increasing gradient of the smooth energy dissipation rate increases along the way.

From Figure 5(a)-(c), it can be concluded that the step energy dissipation head ratio increases with the increase of the relative critical water depth when the step height is constant, which is caused by the constant step energy dissipation and the decreased smooth energy dissipation. Specially, when the relative critical water depth increases

from 7.3 to 12.1, the step energy dissipation head ratios of stepped spillway with a step height of 0.5 m increase by 1.21%~21.58% (the range of x/L is 0.17~1.0). When the relative critical water depth increases from 3.6 to 7.3, the step energy dissipation head ratios of stepped spillway with a step height of 1.0 m increase by 2.53%~22.06% (the range of x/L is 0.13~1.0). When the relative critical water depth increases from 1.8 to 3.0, the step energy dissipation head ratios of stepped spillway with a step height of 2.0 m increase by 1.79%~20.17% (the range of x/L is 0.12~1.0).

From Figure 5(d), it can be concluded that the step energy dissipation head ratio decreases with the increase of the relative critical water depth when the discharge per unit width is constant, which is caused by the constant smooth energy dissipation and the decreased step energy dissipation. When the relative critical water depth increases from 3.0 to 12.1, the step energy dissipation head ratios of stepped

Fig. 5. Variation characteristics of step energy dissipation head ratio.

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spillway with a discharge per unit width of 46.58 m2/s decrease by 1.17%~3.40% (the range of x/L is 0.12~1.0).

To sum up, the step energy dissipation head ratio decreases along the way. The step energy dissipation head ratio increases with the increase of the relative critical water depth when the step height is constant. The step energy dissipation head ratio decreases with the increase of the relative critical water depth when the discharge per unit width is constant.

4. CONCLUSIONS

In order to reflect the contribution of steps to energy dissipation, the stepped spillway and the smooth spillway with the same shape were compared and analyzed. Based on the analysis of the total energy dissipation rate and smooth energy dissipation rate, the step energy dissipation rate and the step energy dissipation head ratio were introduced, and the longitudinal variation characteristics and its relationship with the relative critical water depth were analyzed.

The total energy dissipation rate, smooth energy dissipation rate and step energy dissipation rate increase gradually along the way, while the step energy dissipation head ratio decreases gradually along the way. The increasing gradient of total energy dissipation rate and smooth energy dissipation rate increases along the way, while the increasing gradient of step energy dissipation rate does not change along the way, i.e. the energy dissipation effect of steps at different positions is the same. The increasing gradient of the total energy dissipation rate along the stepped spillway is caused by the increasing gradient of the smooth energy dissipation rate due to the increase of the internal shear action and the friction action of the side wall.

The total energy dissipation rate and smooth energy dissipation rate decrease with the increase of the relative critical water depth. The step energy

dissipation rate is not affected by the relative critical water depth when the step height is constant. The step energy dissipation rate decreases with the increase of the relative critical water depth when the discharge per unit width is constant. The step energy dissipation head ratio increases with the increase of the relative critical water depth when the step height is constant. The step energy dissipation head ratio decreases with the increase of the relative critical water depth when the discharge per unit width is constant. The decrease of the total energy dissipation rate of stepped spillway under large discharge per unit width is caused by the decrease of the smooth energy dissipation rate and the constant of the step energy dissipation rate.

In view of the highly linear correlation of the step energy dissipation rate along the way, the stepped spillway can be compared with the smooth spillway of the same shape, and the relative hydraulic parameters of the stepped spillway (i.e. compared with smooth spillway, the change of relevant hydraulic parameters due to the presence of steps) can be introduced to systematically discuss the characteristics of the relative hydraulic parameters, which has important practical significance for understanding, analyzing and applying the stepped spillway.

ACKNOWLEDGEMENTS

This research was supported by the Special Fund for Agro-scientific Research in the Public In t e re s t (Gran t numbers : 201503105 and 201503125). The authors gratefully acknowledge the anonymous reviewers and editors for their careful reviews and suggestions, which significantly contributed to the manuscript improvement.

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Received: 109/03/31

Revised: 109/04/21

Accepted: 109/06/08