Abstract
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.
HIGHLIGHTS
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
INTRODUCTION
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.
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 AND METHODOLOGY
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.
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.
Experimental cases and geometric parameters of the aeration basin and the critical condition for the onset of the jet flow
Cases . | la (10−2 m) . | ha (10−2 m) . | hs (10−2 m) . | (hc/b)cri . | (Frb)cri . | Re (104) . | We (102) . |
---|---|---|---|---|---|---|---|
M111 | 87.5 | 18.0 | 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 | 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 | 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 | 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 | 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 | 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 |
Cases . | la (10−2 m) . | ha (10−2 m) . | hs (10−2 m) . | (hc/b)cri . | (Frb)cri . | Re (104) . | We (102) . |
---|---|---|---|---|---|---|---|
M111 | 87.5 | 18.0 | 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 | 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 | 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 | 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 | 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 | 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 |
RESULTS AND DISCUSSION
Flow patterns
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.
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.
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.
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.
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.
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.
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.
Flow through the first step with different heights of reverse steps at hc/b= 2.13: (a) case M213; (b) case M214.
Flow through the first step with different heights of reverse steps at hc/b= 2.13: (a) case M213; (b) case M214.
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).
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).
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.
Critical conditions of the HJSS with different reverse step heights in the aeration basin.
Critical conditions of the HJSS with different reverse step heights in the aeration basin.
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.
Variation of (hc/b)cri with (la/b)0.78(ha/b) − 0.90(7.00 +hs/b)2.98.
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.
CONCLUSION
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.
ACKNOWLEDGEMENTS
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).
DECLARATION OF COMPETING INTEREST
The authors declare no conflicts of interest.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.