A hydraulic-stepped spillway was designed using an aeration basin to provide aerated flow to the stepped spillway utilizing a hydraulic jump. However, the flow through the entrance of the stepped spillway might separate from the first step top and impact the downstream steps at a large unit discharge, causing a so-called jet flow. A new experimental study was conducted to better understand the jet flow in the hydraulic-jump-stepped spillway with comparisons with conventional stepped spillways. The results showed that the critical condition required for the formation of the jet flow was close to the geometric parameters of the upstream aeration basin. Among these parameters, the height of the reverse step had a more significant effect on the local flow pattern, thus lowering the risk of jet flow. The relationships between the critical condition and the geometry of the aeration basin suggested that the Froude number at the entrance of the stepped spillway was the key parameter forming the jet flow. Compared with conventional stepped spillways, the hydraulic-jump-stepped spillway could effectively extend the practical application for large unit discharges by providing a better understanding of jet flow conditions.

  • A jet flow for large unit discharges in a hydraulic-jump-stepped spillway.

  • Investigation of the critical condition of the jet flow as compared with conventional stepped spillways.

  • The critical condition affected by the geometry of the aeration basin.

  • An effective element to smooth the surface profile.

  • A linear relation between the critical condition and the Froude number at the brink of the first step.

Graphical Abstract

Graphical Abstract
Graphical Abstract

Stepped spillways have become increasingly popular flood-release structures that can dissipate the energy of high-velocity flow (Boes & Hager 2003a, 2003b; Stojnic et al. 2021). With the increase in the unit discharge, the flow conditions in stepped spillways are classified as three typical flow patterns, i.e., nappe flow, transition flow, and skimming flow (Chamani & Rajaratnam 1999; Renna & Fratino 2010; Biethman et al. 2021). In nappe flow, the water bounces from one step to another, with the steps acting as a succession of free-falling nappes. In skimming flow, the water flows down the spillway as a coherent stream, skimming over the step edges and being cushioned by the recirculating vortices trapped between the mainstream and the steps. The disappearance of the cavity beneath the free-falling nappes indicates the onset of the skimming flow. In addition to the three typical flow patterns, there are some special flow patterns (e.g., water fins, spray flow, shooting flow, and water droplets) observed locally due to the step geometry of stepped spillways (Han et al. 2006; Pfister et al. 2006b; Zare & Doering 2012; Hunt & Kadavy 2017). The resultant cavity (i.e., cavity flow, a separation bubble or roller) may induce improper flow behavior, resulting in cavitation damage, structure erosion, flow instability, and the reduction of overflow discharge (Yu et al. 2008; Wang & Chen 2009; Amromin 2018; Wehrmeister & Ota 2021). In addition, stepped spillways are now used in newly built dams with a combination of upstream apparatuses, such as flaring gate piers, a labyrinth weir, a piano key weir, a deflector, a step aerator, and a ski jump aeration basin (Pfister et al. 2006a; Silvestri et al. 2013; Qian et al. 2016; Dong et al. 2019; Tullis et al. 2021; Terrier et al. 2022). For these new types of stepped spillways, the resultant jet flow and cavity will occur locally for some circumstances compared with conventional stepped spillways with ogee-crested weirs and flat steps.

Given that the inception point of the air entrainment moves downstream with increasing unit discharge, a backwater reach for large unit discharges is prone to cavitation damage, and the maximum unit discharge of a conventional stepped spillway is restricted to 50–60 m2/s at a prototype-scale (Chanson 2015). We have proposed a hydraulic-jump-stepped spillway (HJSS) that increased the unit discharge by 100 m2/s in prototype (Wu et al. 2018; Zhou et al. 2020, 2021). Figure 1 shows a definition sketch of the HJSS, which consisted of a weir, an aeration basin, and a stepped spillway. The aeration basin used in this study supplied the aerated flow to the stepped spillway utilizing a hydraulic jump. However, with the increasing unit discharge, the flow through the entrance of the stepped spillway might separate from the first step top and impact the downstream steps with a large unit discharge, causing a jet flow.

Figure 1

Definition sketch of the hydraulic-jump-stepped spillway (HJSS).

Figure 1

Definition sketch of the hydraulic-jump-stepped spillway (HJSS).

Close modal

Therefore, the main objective of the study was to experimentally investigate the jet flow in a hydraulic-jump-stepped spillway (HJSS) as compared with a conventional stepped spillway. Our intent was to analyze the critical condition for the onset of the jet flow. In this analysis, the hydraulic parameters and geometric parameters of the aeration basin were used to propose an empirical equation based on the experimental data.

Experimental setup

All of the experiments were conducted at the High Speed Flow Laboratory in Hohai University. The experimental setup consisted of a pump, an approach conduit, a large feeding basin, a test model, and a flow return system. The test model included an approach flow channel, a weir, an aeration basin, a stepped spillway, and a tailwater channel, and their widths B were all 0.275 m (Figure 2). The physical model, made of Perspex, was designed at a scale of 1:40 according to an existing stepped spillway project based on the Froude similarity criterion.

Figure 2

Experimental setup.

Figure 2

Experimental setup.

Close modal

The approach flow channel was 1.0 m long. The weir was a standard Waterways Experiment Station (WES) weir with the profile of y = 1.1255x1.85. The height from the crest of the weir to the bottom of the aeration basin P was 0.63 m; the slope of the overflow surface was 1V:0.75H, and the toe of the weir was connected to the bottom of the aeration basin by an ogee curve with a radius of 0.275 m. The aeration basin upstream of the stepped spillway was added to provide the aerated flow for the stepped spillway, with la and ha being the length and the height of the aeration basin, respectively. The stepped spillway was made up of 24 steps. The single step length a was 0.11 m, the single step height b was 0.09 m, and the stepped spillway slope θ was 39.3°. To stabilize the flow in the aeration basin, a reverse step was added with the same length and width as the downstream single step, with hs being the height of the reverse step.

Experimental methodology

The discharges were measured by a V-notch weir downstream of the test model with a relative error below 2%. The flow depth hb upstream of the first step brink was measured by a point gauge, resulting in the Froude number Frb at the brink Frb = Q/(Bhb3/2g1/2), with g being the acceleration of gravity. The visual observation of flow patterns and the hydraulic data measurements were carried out for a wide range of unit discharges, 0.036 ≤ q ≤ 0.403 m2/s (i.e., the relative critical flow depths of 0.56 ≤ hc/b ≤ 2.83, corresponding to a Reynolds number Re > 2 × 104 and a Weber number We > 100. Therefore, the scale effects associated with the flow behaviors should be negligible, according to Boes & Hager (2003a). In this study, we focused on the flow through the entrance of the stepped spillway, with particular emphasis on the effect of the aeration basin. The detailed geometric parameters of these experimental cases are listed in Table 1. The critical condition for the onset of the jet flow (hc/b, Frb), along with the relevant Fr and We, is also listed in Table 1.

Table 1

Experimental cases and geometric parameters of the aeration basin and the critical condition for the onset of the jet flow

Casesla (10−2 m)ha (10−2 m)hs (10−2 m)(hc/b)cri(Frb)criRe (104)We (102)
M111 87.5 18.0 1.24 0.41 11.60 9.32 
M112 87.5 18.0 4.5 1.46 0.51 14.76 14.80 
M113 87.5 18.0 9.0 1.84 0.77 20.95 30.96 
M114 87.5 18.0 13.5 2.34 0.82 30.12 52.65 
M121 87.5 27.0 0.96 0.20 7.91 3.43 
M122 87.5 27.0 4.5 1.14 0.26 10.18 5.75 
M123 87.5 27.0 9.0 1.30 0.36 12.38 9.21 
M124 87.5 27.0 13.5 1.50 0.49 15.46 15.28 
M211 105.0 18.0 1.34 0.38 13.00 10.32 
M212 105.0 18.0 4.5 1.70 0.49 18.67 19.81 
M213 105.0 18.0 9.0 2.11 0.67 25.69 37.07 
M214 105.0 18.0 13.5 2.41 0.65 31.44 47.80 
M221 105.0 27.0 1.02 0.43 8.71 6.52 
M222 105.0 27.0 4.5 1.20 0.42 11.05 8.81 
M223 105.0 27.0 9.0 1.39 0.34 13.75 10.32 
M224 105.0 27.0 13.5 1.67 0.42 18.13 17.01 
M311 122.5 18.0 1.69 0.59 18.40 21.92 
M312 122.5 18.0 4.5 1.96 0.68 23.08 32.55 
M313 122.5 18.0 9.0 2.43 0.74 31.82 52.63 
M314 122.5 18.0 13.5 2.83 0.87 40.05 80.08 
M321 122.5 27.0 1.20 0.41 11.05 8.67 
M322 122.5 27.0 4.5 1.36 0.41 13.31 11.07 
M323 122.5 27.0 9.0 1.63 0.69 17.46 22.60 
M324 122.5 27.0 13.5 2.07 0.57 25.02 32.06 
Casesla (10−2 m)ha (10−2 m)hs (10−2 m)(hc/b)cri(Frb)criRe (104)We (102)
M111 87.5 18.0 1.24 0.41 11.60 9.32 
M112 87.5 18.0 4.5 1.46 0.51 14.76 14.80 
M113 87.5 18.0 9.0 1.84 0.77 20.95 30.96 
M114 87.5 18.0 13.5 2.34 0.82 30.12 52.65 
M121 87.5 27.0 0.96 0.20 7.91 3.43 
M122 87.5 27.0 4.5 1.14 0.26 10.18 5.75 
M123 87.5 27.0 9.0 1.30 0.36 12.38 9.21 
M124 87.5 27.0 13.5 1.50 0.49 15.46 15.28 
M211 105.0 18.0 1.34 0.38 13.00 10.32 
M212 105.0 18.0 4.5 1.70 0.49 18.67 19.81 
M213 105.0 18.0 9.0 2.11 0.67 25.69 37.07 
M214 105.0 18.0 13.5 2.41 0.65 31.44 47.80 
M221 105.0 27.0 1.02 0.43 8.71 6.52 
M222 105.0 27.0 4.5 1.20 0.42 11.05 8.81 
M223 105.0 27.0 9.0 1.39 0.34 13.75 10.32 
M224 105.0 27.0 13.5 1.67 0.42 18.13 17.01 
M311 122.5 18.0 1.69 0.59 18.40 21.92 
M312 122.5 18.0 4.5 1.96 0.68 23.08 32.55 
M313 122.5 18.0 9.0 2.43 0.74 31.82 52.63 
M314 122.5 18.0 13.5 2.83 0.87 40.05 80.08 
M321 122.5 27.0 1.20 0.41 11.05 8.67 
M322 122.5 27.0 4.5 1.36 0.41 13.31 11.07 
M323 122.5 27.0 9.0 1.63 0.69 17.46 22.60 
M324 122.5 27.0 13.5 2.07 0.57 25.02 32.06 

Flow patterns

For the conventional stepped spillway, the hydraulic condition for the formation of each flow pattern was expressed by the ratio of the critical flow depth (hc) to the step height b. The onset of skimming flow can be predicted by Boes & Hager (2003b):
(1)

Substituting θ = 39.3° (the same as the stepped spillway slope of the HJSS) into Equation (1), the critical hc/b for the onset of skimming flow in the conventional stepped spillway was equal to 0.80. A skimming flow was always observed in the conventional stepped spillway when hc/b > 0.8 according to the variation of the flow patterns.

Figure 3 shows the flow patterns over the stepped spillway of the HJSS at different dimensionless unit discharges for case M211. At hc/b = 0.76 and 1.17, a nappe flow and skimming flow were observed along the stepped spillway, as shown in Figure 3(a) and 3(b), respectively. From that moment, the flow pattern over the stepped spillway was similar to the conventional stepped spillway. At hc/b = 1.37, the aeration basin upstream acted as a deflector. The flow separated from the top of the first step and entered the downstream steps in the form of a jet, as seen in Figure 3(c). With the further increase in the unit discharge, the cavity expanded, as shown in Figure 3(d). The separated jet directly bypassed the several steps and impacted the foregoing steps. Moreover, the jet flow resulted in significant air entrainment and impacted the downstream steps. Some instabilities, water level rises, and erosion on the steps were observed in the jet impact region, although the flow quickly regained stability and skimmed over the steps along the pseudo-bottom (i.e., skimming flow). Furthermore, compared with the conventional stepped spillway, a local jet flow was also observed at large unit discharges in other cases of the HJSS. From a practical point of view, this local jet flow should be avoided.

Figure 3

Flow patterns over the stepped spillway of the HJSS at different hc/b for case M211: (a) hc/b = 0.76; (b) hc/b = 1.17; (c) hc/b = 1.37; (d) hc/b = 1.60.

Figure 3

Flow patterns over the stepped spillway of the HJSS at different hc/b for case M211: (a) hc/b = 0.76; (b) hc/b = 1.17; (c) hc/b = 1.37; (d) hc/b = 1.60.

Close modal

Reverse step effect

For the HJSS, the reverse step added in the aeration basin affected the flow patterns over the stepped spillway. Figure 4 shows the flow patterns with a reverse step at different hc/b for case M213, which merely added a reverse step as compared with case M211 in Figure 3.

Figure 4

Flow through the first step with a reverse step in the aeration basin at different hc/b for case M213: (a) hc/b = 0.76; (b) hc/b = 1.17; (c) hc/b = 1.37; (d) hc/b = 1.60.

Figure 4

Flow through the first step with a reverse step in the aeration basin at different hc/b for case M213: (a) hc/b = 0.76; (b) hc/b = 1.17; (c) hc/b = 1.37; (d) hc/b = 1.60.

Close modal

It was observed that, for any hc/b, the local flow pattern on the stepped spillway with a reverse step in the aeration basin always adhered to the top of the first step, and the jet flow appeared to be nonexistent. Compared with the flow pattern for case M211, for an identical inflow condition of hc/b = 0.76 and 1.17, the flow patterns with and without a reverse step were similar. However, when hc/b increased to 1.37 and 1.60, the flow through the first step without a reverse step exhibited a local jet flow.

Furthermore, the height of the reverse step also had a significant effect on the flow pattern, as shown in Figure 5. For case M213, the flow began to detach from the top of the first step when hc/b equaled 2.13 (Figure 5(a)), whereas the water surface profile for case M214 seemed to be smooth for the conditions of the higher reverse step height (Figure 5(b)). The reverse step, as an effective method to smooth the water surface profile, changed the angle of the streamline at the entrance section with a horizontal flat step (Tajabadi et al. 2018). The commonly used method of smoothing the water surface profile involves adding an inserted apparatus, e.g., the step edge arrangements in the stepped spillway proposed by Pfister et al. (2006b) as shown in Figure 6. Obviously, the flow through the first step had to separate in the HJSS. There was a critical condition for the onset of the jet flow when hc/b increased to some extent, regardless of the geometry of the aeration basin.

Figure 5

Flow through the first step with different heights of reverse steps at hc/b= 2.13: (a) case M213; (b) case M214.

Figure 5

Flow through the first step with different heights of reverse steps at hc/b= 2.13: (a) case M213; (b) case M214.

Close modal
Figure 6

Local flow patterns over stepped spillways with different step edge arrangements in foregoing steps for n = (a) 0, (b) 2, (c) 5 and hc/b = 0.59 (Pfister et al. 2006b).

Figure 6

Local flow patterns over stepped spillways with different step edge arrangements in foregoing steps for n = (a) 0, (b) 2, (c) 5 and hc/b = 0.59 (Pfister et al. 2006b).

Close modal

Critical conditions of jet flow

As mentioned above, if the unit discharge increased to some extent, the flow through the first step of the HJSS would exhibit a local jet flow. There was a critical hydraulic condition (hc/b)cri required for the formation of the jet flow, and the jet flow occurred once (hc/b) > (hc/b)cri. Figure 7 shows the critical condition (hc/b)cri with different reverse step heights in the aeration basin.

Figure 7

Critical conditions of the HJSS with different reverse step heights in the aeration basin.

Figure 7

Critical conditions of the HJSS with different reverse step heights in the aeration basin.

Close modal

Clearly, all critical conditions (hc/b)cri for the jet flow formation increased with the relative height of the reverse step (hs/b) in the aeration basin regardless of the relative length (la/b) or relative height (ha/b) of the aeration basin, and their trend had a roughly similar growth rate. As shown in this figure, the value of (hc/b)cri increased with increasing la/b and decreasing ha/b.

Utilizing the multiple linear regression method and based on the experimental data, we can obtain Equation (2) by considering the geometric parameters of the aeration basin, as shown in Figure 8.

(2)
where 9.7 ≤ la/b ≤ 13.6, 2.0 ≤ ha/b ≤ 3.0 and 0 ≤ hs/b ≤ 1.5. Equation (2) reflects not only the relationship between the critical conditions and geometric parameters of the aeration basin (i.e., (hc/b)cri increased with increasing la/b and hs/b, but decreased with increasing ha/b), but also the magnitude of the effect of these parameters. The relative reverse step height hs/b had a greater effect on the critical conditions (hc/b)cri than the other geometric parameters of the aeration basin. The reverse step proved useful for smoothing the water surface profile.
Figure 8

Variation of (hc/b)cri with (la/b)0.78(ha/b) − 0.90(7.00 +hs/b)2.98.

Figure 8

Variation of (hc/b)cri with (la/b)0.78(ha/b) − 0.90(7.00 +hs/b)2.98.

Close modal
To integrate the influence of the geometric parameters of the aeration basin into the inflow of the stepped spillways, Figure 9 summarizes the experimental results regarding the onset of the jet flow as a function of the Froude number at the brink of the first step (Frb). The critical condition (hc/b)cri can be expressed as:
(3)
Figure 9

Variation of (hc/b)cri with Frb.

Figure 9

Variation of (hc/b)cri with Frb.

Close modal
As seen in Figure 9, for the HJSS, with the increasing Frb, (hc/b)cri increased, and the relationship appeared to be linear. For the conventional stepped spillway, the hydraulic conditions required for the formation of the jet flow were defined as follows (Chanson 1996):
(4)
where θb is the initial angle of the streamlines in the horizontal direction, and the jet flow occurred for hc/b < (hc/b)cri. Because the stepped spillway of the HJSS was similar to the conventional stepped spillway for the flat crest, by substituting θb = 0 into Equation (4), a linear form between (hc/b)cri and (Frb) can be obtained in Equation (5) as follows (Figure 9):
(5)

The data for this study and those of Chanson (1996) both had the linear relation. This indicated that the risk of flow deflection at the first step for the HJSS could be reduced by increasing the step height, contrary to the conventional stepped spillway at a given unit discharge. The difference between the two curves was larger as Frb increased, which further reveals the appearance of the jet flow at larger unit discharge in the HJSS, whereas for small unit discharges in the conventional stepped spillway, both Equations (3) and (5) reveal that the jet flow could be attributed to the Froude number Fr at the entrance of the stepped spillway.

Although the aeration basin upstream of the stepped spillway increased the risk of flow deflection for large unit discharges, the resultant aerated flow might reduce the risk of cavitation damage of the non-aerated flow region of the stepped spillway (Wu et al. 2018; Zhou et al. 2021). In future research on the HJSS, it will be important to focus on the air concentration, velocity profiles, and other hydraulic performances by numerical simulations.

For the hydraulic-jump-stepped spillway we proposed, the jet flow near the entrance of the stepped spillway with a large unit discharge was experimentally investigated. The critical condition for the formation of the jet flow increased with the increase in the length of the aeration basin and the height of the reverse step in the aeration basin, but the critical condition decreased with the increase in the height of the aeration basin. The height of the reverse step height was more effective in lowering the risk of jet flow. By integrating the influence of the geometric parameters of the aeration basin into the inflow condition of the stepped spillway, a linear relationship could be obtained, i.e., the occurrence of jet flow in the hydraulic-jump-stepped spillway could be attributed to the Froude number at the entrance of the stepped spillway. Compared with conventional stepped spillways, the hydraulic-jump-stepped spillway could effectively extend the practical application in large unit discharges with the knowledge of jet flow conditions. Research is ongoing on the hydraulic performance of a hydraulic-jump-stepped spillway with numerical methods.

This research is supported by the National Natural Science Foundation of China (Grant No. 51479057), the Joint Funds of the Zhejiang Provincial Natural Science Foundation of China (Grant No. LZJWZ22E090004), the Open Funding of the Key Laboratory for Technology in Rural Water Management of Zhejiang Province (Project No. ZJWEU-RWM-20200103B), the Science and Technology Project of Zhejiang Water Resources Department (RA1904, RB2119, RC2142 and RC2143) and the General Research Project of Zhejiang Provincial Department of Education (Y202045107).

The authors declare no conflicts of interest.

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

Amromin
E.
2018
Prediction of cavitation inception within regions of flow separation
.
Journal of Fluids Engineering
140
(
1
),
011103
.
https://doi.org/10.1115/1.4037505
.
Biethman
B.
,
Ettema
R.
,
Thornton
C.
,
Hogan
T.
&
Lan
Y.
2021
Air entrained in flow along a steep-stepped spillway: data and insights from a hydraulic model
.
Journal of Hydraulic Engineering
147
(
5
),
05021001
.
https://doi.org/10.1061/(ASCE)HY.1943-7900.0001880
.
Boes
R. M.
&
Hager
W. H.
2003a
Two-phase flow characteristics of stepped spillways
.
Journal of Hydraulic Engineering
129
(
9
),
661
670
.
https://doi.org/10.1061/(ASCE)0733-9429(2003)129:9(661)
.
Boes
R. M.
&
Hager
W. H.
2003b
Hydraulic design of stepped spillways
.
Journal of Hydraulic Engineering
129
(
9
),
671
679
.
https://doi.org/10.1061/(ASCE)0733-9429(2003)129:9(671)
.
Chamani
M. R.
&
Rajaratnam
N.
1999
Onset of skimming flow on stepped spillways
.
Journal of Hydraulic Engineering
125
(
9
),
969
971
.
https://doi.org/10.1061/(ASCE)0733-9429(1999)125:9(969)
.
Chanson
H.
1996
Prediction of the transition nappe/skimming flow on a stepped channel
.
Journal of Hydraulic Research
34
(
3
),
421
429
.
https://doi.org/10.1080/00221689609498490
.
Chanson
H.
2015
Discussion of ‘Cavitation potential of flow on stepped spillways’ by K. Warren Frizell, Floriana M. Renna, and Jorge Matos
.
Journal of Hydraulic Engineering
141
(
5
),
07014025
.
https://doi.org/10.1061/(ASCE)HY.1943-7900.0000808
.
Dong
Z. S.
,
Wang
J. X.
,
Vetsch
D. F.
,
Boes
R. M.
&
Tan
G. M.
2019
Numerical simulation of air–water two-phase flow on stepped spillways behind X-shaped flaring gate piers under very high unit discharge
.
Water
11
(
10
),
1956
.
https://doi.org/10.3390/w11101956
.
Han
Y.
,
Fen
R. L.
,
Tian
J. N.
&
Li
B. L.
2006
Water-wing on steep slope of stepped spillways (in Chinese)
.
Journal of Hydroelectric Engineering
25
(
1
),
114
118
.
Hunt
S. L.
&
Kadavy
K. C.
2017
Estimated splash and training wall height requirements for stepped chutes applied to embankment dams
.
Journal of Hydraulic Engineering
143
(
11
),
06017018
.
https://doi.org/10.1061/(ASCE)HY.1943-7900.0001373
.
Pfister
M.
,
Hager
W. H.
&
Minor
H.-E.
2006a
Bottom aeration of stepped spillways
.
Journal of Hydraulic Engineering
132
(
8
),
850
853
.
https://doi.org/10.1061/(ASCE)0733-9429(2006)132:8(850)
.
Pfister
M.
,
Hager
W. H.
&
Minor
H.-E.
2006b
Stepped chutes: pre-aeration and spray reduction
.
International Journal of Multiphase Flow
32
(
2
),
269
284
.
https://doi.org/10.1016/j.ijmultiphaseflow.2005.10.004
.
Qian
S. T.
,
Wu
J. H.
&
Ma
F.
2016
Hydraulic performance of ski-jump-step energy dissipater
.
Journal of Hydraulic Engineering
142
(
10
),
05016004
.
https://doi.org/10.1061/(ASCE)HY.1943-7900.0001178
.
Renna
F. M.
&
Fratino
U.
2010
Nappe flow over horizontal stepped chutes
.
Journal of Hydraulic Research
48
(
5
),
583
590
.
https://doi.org/10.1080/00221686.2010.507016
.
Silvestri
A.
,
Erpicum
S.
,
Archambeau
P.
,
Dewals
B.
&
Pirotton
M.
2013
Stepped spillway downstream of a piano key weir – critical length for uniform flow
. In:
Proceedings of the International Workshop on Hydraulic Design of Low-Head Structures
(D. B. Bung & S. Pagliara, eds), Bundesanstalt für Wasserbau, Karlsruhe
,
Germany
, pp.
99
107
.
Stojnic
I.
,
Pfister
M.
,
Matos
J.
&
Schleiss
A. J.
2021
Effect of 30-degree sloping smooth and stepped chute approach flow on the performance of a classical stilling basin
.
Journal of Hydraulic Engineering
147
(
2
),
04020097
.
https://doi.org/10.1061/(ASCE)HY.1943-7900.0001840
.
Tajabadi
F.
,
Jabbari
E.
&
Sarkardeh
H.
2018
Effect of the end sill angle on the hydrodynamic parameters of a stilling basin
.
The European Physical Journal Plus
133
(
1
),
10
.
https://doi.org/10.1140/epjp/i2018-11837-y
.
Terrier
S.
,
Pfister
M.
&
Schleiss
A. J.
2022
Performance and design of a stepped spillway aerator
.
Water
14
(
2
),
153
.
https://doi.org/10.3390/w14020153
.
Tullis
B. P.
,
Jorgensen
T. J.
&
Crookston
B. M.
2021
Effects of a labyrinth weir with outlet ramps on downstream steep-stepped chute sidewall height requirements
.
Journal of Irrigation and Drainage Engineering
147
(
12
),
04021057
.
https://doi.org/10.1061/(ASCE)IR.1943-4774.0001635
.
Wang
J. B.
&
Chen
H. C.
2009
Experimental study of elimination of vortices along guide wall of bank spillway
. In:
Advances in Water Resources and Hydraulic Engineering
(C. K. Zhang & H. W. Tang, eds), Springer
,
Berlin, Germany
, pp.
2059
2063
.
Wehrmeister
E.
&
Ota
J. J.
2021
Separation in overflow spillways: a computational analysis
.
Journal of Hydraulic Research
60
(
2
),
357
362
.
https://doi.org/10.1080/00221686.2021.1908438
.
Wu
J. H.
,
Zhou
Y.
&
Ma
F.
2018
Air entrainment of hydraulic jump aeration basin
.
Journal of Hydrodynamics
30
(
5
),
962
965
.
https://doi.org/10.1007/s42241-018-0088-4
.
Yu
D.
,
Lee
J. H. W.
&
Wong
C. K. C.
2008
Stormwater overflow in stepped channel
.
Journal of Hydro-Environment Research
2
(
2
),
119
128
.
https://doi.org/10.1016/j.jher.2008.05.004
.
Zare
H. K.
&
Doering
J. C.
2012
Energy dissipation and flow characteristics of baffles and sills on stepped spillways
.
Journal of Hydraulic Research
50
(
2
),
192
199
.
https://doi.org/10.1080/00221686.2012.659840
.
Zhou
Y.
,
Wu
J. H.
,
Ma
F.
&
Hu
J. Y.
2020
Uniform flow and energy dissipation of hydraulic-jump-stepped spillways
.
Water Supply
20
(
4
),
1546
1553
.
https://doi.org/10.2166/ws.2020.056
.
Zhou
Y.
,
Wu
J. H.
,
Ma
F.
&
Qian
S. T.
2021
Experimental investigation of the hydraulic performance of a hydraulic-jump-stepped spillway
.
KSCE Journal of Civil Engineering
25
(
10
),
3758
3765
.
https://doi.org/10.1007/s12205-021-1709-y
.
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