A New Type of Pre-Aeration Stepped Spillway

Abstract

Aiming to increase energy dissipation and prevent the cavitation potential of a traditional stepped spillway (TSS) at large unit discharges, a kind of pre-aeration stepped spillway, called a hydraulic-jump-stepped spillway (HJSS), is introduced in this paper. Unlike a TSS, a basin added upstream of the stepped chute in the HJSS plays a vital role in the hydraulic performance owing to the formation of a hydraulic jump in the basin. This paper presents experimental research on the hydraulic performance of the HJSS in comparison to a TSS with the same chute slope (θ = 39.3°) for a wide range of unit discharges, including the flow pattern, energy dissipation, pre-aeration effect, and maximum splash height. The results showed that the HJSS corresponded to a large energy dissipation rate, the air was effectively entrained at the inlet of the stepped chute, and there was an observation of splash formation in the foregoing and downstream steps. Under large unit discharges, the HJSS maintained an energy dissipation rate exceeding 80%. Additionally, at the inlet, the air concentrations reached 4.5% on the bottom and 11.2% on the sidewall. The findings of this research could be used as a general guideline for stepped spillway design with large unit discharges.

1. Introduction

The stepped spillway has been a long-lasting successful operation for over 3000 years [1,2,3,4]. During the past four decades, stepped spillways have been widely used because of the emergence of roller-compacted concrete (RCC) technology and their high energy dissipation efficiency [5]. Three flow regimes are observed successively for the water flowing over a stepped spillway with an increase in discharge, i.e., nappe flow, transition flow, and skimming flow [6]. Most prototypes of stepped spillways operate at large discharges, so skimming flow is predominant [7]. In skimming flow, the step cavity is filled with recirculation fluid. The step edge dominates the turbulent boundary layer, and air entrainment at the free surface occurs when the turbulent boundary layer reaches the free surface. According to the inception point of air entrainment, the flow along the stepped spillway can be classified into non-aerated and aerated flow regions. For both regions, because of the lowest measured pressure on the vertical riser [8] and the gradually increasing flow velocity [9] along the stepped spillway, the steps are prone to cavitation damage [10]. However, the air concentration in the aerated flow region is large, and the air content close to the pseudobottom and sidewall plays a vital role in mitigating and eliminating cavitation damage. For comparison, the non-aerated flow region enlarges as the unit discharge increases, and cavitation damage is likely to be present in this region, especially for large unit discharges. Therefore, a stepped spillway should be generally designed for a small unit discharge, i.e., a unit discharge q < 30 m²/s [11]. In addition, an early stepped spillway has a relatively small height, and the risk of cavitation is low due to the small flow velocity, even at the downstream end of the stepped spillway [12].

Based on this situation, many previous studies have focused on improving the designed maximum unit discharge of the stepped spillway, mainly including two methods. One approach is to change the geometry of the steps or cavities (e.g., inclined steps, wedge-shaped steps, steps with sills or baffles, trapezoidal or rounded steps, and trapezoidal or rounded cavities) to make the inception of air entrainment moving upstream [13,14,15,16]. However, the non-aerated flow region cannot be avoided in the skimming flow, and the length of this region is related to an increasing unit discharge. The other is to add aeration apparatuses to pre-aerate the non-aerated flow region, providing safety against cavitation damage. In typical scenarios, high-velocity spillway flow from large unit discharges can lead to cavitation erosion. Consequently, an aerator is frequently used to artificially introduce air into the flow, thereby mitigating cavitation [17]. In addressing the issue of non-aerated flow regions in stepped spillways, the most commonly employed solution is the addition of a deflector (e.g., ramp or offset). When a deflector is positioned upstream of the first step, it effectively diverts the flow away from the surface of the stepped spillway [11,18,19]. In the cavity formed below the deflecting flow, the air is supplied into the flow due to the local negative pressure [20]. However, the non-aerated flow region still exists along the sidewall, and its length increases with the unit discharge [21]. Currently, stepped spillways with flaring gate piers are applied frequently in many practical projects. The aerated flow is produced through the piers, and a large number of prototypes have experienced operation during large floods (up to 90 m²/s). But the deflecting flow caused by piers may impact the step face at a high velocity under large unit discharges. In addition to flaring gate piers, aeration apparatuses such as labyrinth weirs [22], abrupt contraction aerators [23], and ski-jump aeration basins [24] have been proposed by many researchers. Accordingly, a kind of pre-aeration stepped spillway called the hydraulic-jump-stepped spillway (HJSS) was developed by [25]. Despite that, some attention has also been paid to jet flow formation and air entrainment in the literature [26,27,28]. The operating mechanism, the flow regime characteristics, the pre-aeration effect, the energy dissipation, and the splash height due to the narrow width are of great importance to further clarify.

Figure 1 presents a schematic definition of the HJSS, primarily consisting of a weir, a basin, and a stepped chute. Different from a traditional stepped spillway (TSS), the basin in the HJSS, as a new element, begins from the toe of the weir and ends at the upstream face of the first step of the stepped chute. The function of the weir differs significantly from that of a traditional spillway weir. It not only ensures a smooth transition between the inflow and spillway flow but also introduces a certain drop in elevation, allowing for a specific flow velocity as water enters the basin. Consequently, due to the obstruction created by the basin end sill (i.e., the first step of the stepped chute), a hydraulic jump may occur when the flow changes from a supercritical flow to a subcritical one. The hydraulic jump is characterized by highly turbulent flow and air entrainment. Macro-scale vortices form in the jump roller and interact with the free surface, resulting in the entrainment of air bubbles and the creation of splashes and droplets within the two-phase flow region [29,30].

Therefore, the HJSS structure is designed to utilize the hydraulic jump to provide a relatively stable aeration effect, ensuring that pre-aeration of the stepped spillway occurs as the flow passes over the basin. Traditionally, the basin is positioned downstream of the TSS to dissipate energy. However, in the HJSS structure, the basin functions not only as an energy dissipator but also as an aerator. To further validate this new design, the objective of this paper is to analyze two physical models (i.e., TSS and HJSS) with similar step geometries on the hydraulic characteristics, including flow regimes, energy dissipation rate, pre-aeration effect, and splash height.

2. Experimental Setup and Methodology

2.1. Experimental Setup

The experimental setup consisted of a large feeding basin, a pump, an approach conduit, a stepped spillway model, and a flow return system (Figure 2). Both physical models of the HJSS and TSS included an inflow channel, a stepped chute, and an outflow channel.

The main difference between these two physical models is the configuration of the inlet of the stepped chute (Figure 1). The inlet of the stepped chute in the HJSS additionally adopts a weir and a basin as compared to the TSS. The weir crest was a modification of the Waterways Experiment Station (WES) with a profile of y = 1.1255x^1.85. The height between the weir crest and the basin bottom was 63.00 cm, the slope of the overflow surface was 1 V:0.75 H, and the toe of the weir was connected to the basin bottom by an ogee curve with a radius of 27.50 cm. The length l_a and height h_a of the basin were 122.5 cm and 18.0 cm, respectively. A reverse step was placed in the aeration basin to stabilize the outflow of the basin. The reverse step height h_s = 13.5 cm and its length and width were the same as that of the single downstream step.

Both the stepped chutes in the HJSS and the TSS were made up of 24 steps and 40 steps with the same step geometry, respectively. The step length a was 11.00 cm, the step height b was 9.00 cm, the step width B was 27.50 cm, and the slope angle θ was 39.3°. In order to avoid the water-wing in the foregoing steps at small unit discharges [31], the first step height of the traditional stepped chute was set as 4.0 cm. Both the inflow and outflow channels of these two models were long enough to provide a steady flow and eliminate the effect of air entrainment on the tailwater flow, respectively. The total height of the two models from the weir crest (i.e., the HJSS with a WES crest and the TSS with a broad crest) to the outflow channel bottom (H_dam) was 2.61 m and 3.55 m, respectively.

2.2. Experimental Methodology

Visual observations of these two physical models were conducted for dimensionless unit discharges 0.56 ≤ h_c/b ≤ 2.82 for the HJSS and 0.36 ≤ h_c/b ≤ 1.73 for the TSS, equaling Reynolds numbers of 1.41 × 10⁵ ≤ Re ≤ 1.59 × 10⁶ and 7.14 × 10⁴ ≤ Re ≤ 7.62 × 10⁵, respectively. The model discharge was measured by a standard V-notch weir with a reading accuracy of 1%. In the experiments, the scale effects are negligible if the Reynolds number is Re ≥ 3 × 10⁴ [32]. Hence, the Reynolds numbers of both physical models were large enough to avoid significant scale effects as identified in air–water flows along the stepped spillway.

The energy dissipation rate is defined as η = ΔH/H_max, with ΔH = H_max − H_res, where H_max = the upstream total head above the outflow channel and H_res = the residual head at the last step edge, calculated by the air–water flow properties (i.e., the equivalent clear water flow depth h_w1 and flow velocity v_w1). Figure 3 shows the sketch of the measurement of flow depth h_w1 at the last edge. Section 1-1 is located at the edge of the last step, and Section 2-2 is selected around more than 3–4 times the step length away from the last step of the outflow channel, where the air is expected to be entirely removed from the flow [33].

In this situation, the flow at Section 2-2 can be assumed to be clear water, the flow depth h_w2 at Section 2-2 can be measured by a point gauge with a 0.1 mm accuracy, and the average flow velocity v_w2 = q/h_w2. Neglecting the energy losses between Sections 1-1 and 2-2 and considering the energy equation and continuity equation, we have the following:

$$h_{\mathrm{w}1}\cos \theta +{\displaystyle \frac{{\alpha }_1{v}_{\mathrm{w}1}^2}{2g}}=h_{\mathrm{w}2}+{\displaystyle \frac{{\alpha }_2{v}_{\mathrm{w}2}^2}{2g}},$$

(1)

$$v_{\mathrm{w}1}{h}_{\mathrm{w}1}=v_{\mathrm{w}2}{h}_{\mathrm{w}2},$$

(2)

The combination of Equations (1) and (2) would yield h_w1 and v_w1. And the residual head at the last step H_res can be calculated with h_w1 and v_w1 [15,34].

The pre-aeration of the stepped spillway is characterized by the air concentration at the inlet of the stepped chute. The measurement of air concentration was placed at the edge of the first step of the stepped chute, both for the bottom of the center line and the sidewall of 0.05 m above the bottom. The air concentration was measured by a CQ6-2005 aeration apparatus [35,36,37] with a sampling rate of 1020 Hz, a period of 10 s, and an error of ±0.3%. The splash heights, defined as the vertical distance above the pseudobottom to the wetted splash line on the sidewall, were measured with a steel ruler and validated by photos at each step edge during skimming flow conditions [38].

3. Results and Discussion

3.1. Flow Pattern

Figure 4 shows the flow patterns observed in the HJSS and TSS as unit discharges increase. At small unit discharges (h_c/b ≤ 0.76), the flow pattern of the stepped chute in the HJSS was nappe flow (Figure 4a), while the TSS exhibited a nappe flow at dimensionless discharges of h_c/b ≤ 0.64, consistent with previous findings on a slope of 40° by [39]. The basin in the HJSS was observed to have a small impact on the downstream stepped chute flow, a phenomenon also seen in the transition flow (Figure 4b).

As the unit discharge increased, a skimming flow was observed in both the HJSS (0.99 ≤ h_c/b ≤ 2.82) and TSS (0.86 ≤ h_c/b ≤ 1.73), as seen in Figure 4c. However, the difference in the inlet of the stepped chute between these two models for the disappearance of the cavity in the skimming flow may be related to the inlet of the stepped chute. In the HJSS, the basin end sill (i.e., the first step) resulted in the formation of a deflected flow at larger unit discharges [40]. On one hand, the deflected flow generates a more pronounced increase in water surface elevation compared to the TSS, necessitating additional attention. On the other hand, the cavity formed between the deflected flow and the downstream steps results in a greater value for the onset of skimming flow (i.e., h_c/b = 0.99).

Except for the incipient condition of the skimming flow, the basin in the HJSS plays an important role in the air entrainment of the stepped chute owing to the formation of a hydraulic jump in the basin. At small unit discharges (as shown in Figure 4a,b), the upstream weir flow entered the basin without forming a hydraulic jump, and there was no difference in the inflow conditions of the stepped chute between the HJSS and TSS. However, for large unit discharges (Figure 4c), the upstream weir flow induced a hydraulic jump in the basin, which resulted in pre-aeration of the stepped chute in the HJSS. On the other hand, the hydraulic jump splashed over the end sill of the basin, resulting in a large splash height in the foregoing steps. For the TSS, the stepped chute flow had gradually increased aerated flow depth due to the increase in the turbulent boundary layer.

3.2. Energy Dissipation

The energy dissipation of the stepped spillway is a key parameter, e.g., to design a downstream stilling basin [15]. In this study, the energy dissipation data are reported in Figure 5 in terms of the energy dissipation rate η at the downstream end of the stepped chute. The data are presented as functions of the dimensionless drop in elevation H_dam/h_c between the weir crest and the downstream end. The data of η of the TSS with a slope of 40° are also illustrated in this figure [39].

For both the HJSS and TSS, the energy dissipation rate increased as the dimensionless drop increased. For the TSS, the step size in this study differed from the size in Tian’s data, but the energy dissipation seemed to have the same value. The results revealed that the energy dissipation rate was mainly related to the slope in the TSS, which was the same as the findings of [41,42].

The HJSS had a higher energy dissipation rate as compared to the TSS, and the difference between the value of η in the HJSS and TSS became larger as H_dam/h_c decreased, i.e., a large η corresponded to a large unit discharge for the same drop. As observed in Figure 5, the energy dissipation rate η exhibited values of 0.96 and 0.88 for the HJSS and TSS, respectively, at a relatively small unit discharge with H_dam/h_c being approximately 51. Notably, with an increase in the unit discharge and H_dam/h_c approaching 18, η for the HJSS remained a relatively high value of 0.84. In contrast, η decreased significantly to 0.68 for the TSS, indicating an improvement in the energy dissipation rate of 23.5% for the HJSS. The literature review indicates that when the ratio of H_dam/h_c exceeds 8, the rate of energy dissipation can be represented by the equation η = m(H_dam/h_c)ⁿ, where m and n are variables associated with the chute slope [43]. The correlation is supported by measured data from the TSS, the HJSS, and Tian’s data.

$$\eta =0.477{\left(H_{\mathrm{dam}}/h_c\right)}^{0.150} \mathrm{for} \mathrm{TSS}, 22.84\le h_{\mathrm{dam}}/h_{\mathrm{c}}\le 110.78,R^2=0.94$$

(3)

$$\eta =0.552{\left(H_{\mathrm{dam}}/h_c\right)}^{0.138} \mathrm{for} \mathrm{HJSS}, 10.28\le h_{\mathrm{dam}}/h_{\mathrm{c}}\le 51.79,R^2=0.95$$

(4)

$$\eta =0.383{\left(H_{\mathrm{dam}}/h_c\right)}^{0.201}, \mathrm{for} \mathrm{Tian’s} \mathrm{data}, 14.08\le h_{\mathrm{dam}}/h_{\mathrm{c}}\le 118.19,R^2=0.90$$

(5)

Furthermore, in addition to the R² (coefficient of determination), Equations (3)–(5) are further evaluated using MSE (mean squared error), RMSE (root mean squared error), and MAE (mean absolute error) to assess the energy dissipation model η = m(H_dam/h_c)ⁿ, with the values presented in

Table 1. Comparison of different error analyses.

Equation	R2	MSE	RMSE	MAE
Equation (3)	0.94	0.0003	0.0002	0.0250
Equation (4)	0.95	0.0157	0.0129	0.1259
Equation (5)	0.90	0.0170	0.0147	0.1580

.

Equations (3)–(5) indicate a general conformity within this range to the aforementioned equation, though Tian’s data exhibit more significant dispersion, particularly at the minimal and maximal values of H_dam/h_c. Nevertheless, this formula still performs commendably in predicting the energy dissipation for stepped spillways with varying configurations. Combined with the geometry of the HJSS, the higher energy dissipation rate of the entire structure was attributed to the hydraulic jump in the basin and the friction factor of the stepped chute. At a large unit discharge, the performance of the hydraulic jump was more obviously associated with high turbulent intensity, and thus the energy dissipation effect was greatly improved.

3.3. Pre-Aeration Effect

According to previous studies, once the flow on the stepped chute becomes aerated, the air concentration downstream of the inception point of air entrainment is larger for preventing cavitation damage [44]. Therefore, the pre-aeration of the stepped chute is crucial, and the air concentration at the inlet of the stepped chute plays a vital role in determining the downstream air concentration. To illustrate this, Figure 6 displays the air concentration on both the bottom and sidewall at the first step (i.e., inlet) of the stepped chute in the HJSS and TSS. To compare the pre-aeration effect of the basin structure in the HJSS under identical inflow conditions, this study also employed the same inflow conditions for the TSS.

Given the existence of a non-zero air concentration, it was evident that the flow at the inlet of the stepped chute was entrained on both the bottom and sidewall, resulting in the absence of a non-aerated region in the HJSS. The air concentration on the sidewall was larger than that on the bottom, and the value increased with the dimensionless unit discharge, except for h_c/b = 0.89 at the sidewall. The efficiency of air entrainment at the inlet can contribute to a relatively large air concentration along the stepped chute, especially for the upstream foregoing steps. In contrast, for the TSS, under identical inflow conditions (0.89 ≤ h_c/b ≤ 2.82), there was no air entrainment at the inlet of the TSS until the 5th, 6th, 8th, 10th, and 12th step, where air entrainment was observed. For comparison, the height between the first step and the inception point of the air entrainment (H_i) of the TSS could be calculated as in this study, which showed a good agreement with the literature [43,45]:

$${\displaystyle \frac{H_{\mathrm{i}}}{k}}=4.98F_{\ast }^{0.55}, \mathrm{with}{R}^2=0.98$$

(6)

where k is the roughness height perpendicular to the pseudobottom (k = bcosθ) and F_* is the roughness Froude number ($F_{\ast }=q/\sqrt{g\sin \theta k^3}$). The results indicated the existence of the non-aerated region in the TSS, especially for large unit discharges. For instance, Equation (6) indicated that air entrainment would not occur until the height H_i extended to 1.81 m in the case of q reaching 0.401 m²/s in the TSS model. On the other hand, in the HJSS model, the air concentration at the inlet of the stepped chute was 4.5% and 11.2% on both the bottom and sidewall, respectively.

3.4. Maximum Splash Height

Strong air–water interactions occur in the aerated flow region downstream of the inception point of air entrainment for the stepped spillway. Excessive splashing over the sidewall of the stepped chute can cause scouring and increase the risk of failure [38,46]. Therefore, the design of a vertical sidewall on a stepped spillway chute is determined by the maximum splash height, particularly for the local region. In the skimming flow, the maximum splash height downstream of the free-surface inception point, y_sp, is the distance above the step edge to the observed splash line on the sidewall. In the HJSS, excessive splash formation was observed in both the upstream and downstream steps, attributed to jet formation caused by the basin end sill and the increased air concentration along the stepped chute [25]. In the TSS, the inception point of air entrainment shifts downstream as the unit discharge increases (Equation (4)), resulting in splash occurring exclusively at the downstream steps. The splash phenomena in both the TSS and HJSS are illustrated in Figure 4c.

The maximum splash height of both the HJSS and TSS are presented in Figure 7, and the value in the HJSS should be considered additionally because of the existence of the deflected flow. It was evident that the maximum y_sp of the HJSS was slightly larger than the value of the TSS, no matter the foregoing steps and downstream steps in the range of b/h_c. In the cases of splash flow not induced by deflected flow, there exists a linear relationship between splash flow and inflow conditions for both the TSS and HJSS. A simplified linear regression of both data sets yielded Equations (7) and (8) with coefficients of determination R² = 0.93 and R² = 0.89, respectively.

$${\displaystyle \frac{y_{sp}}{h_c}}=2.78{\displaystyle \frac{b}{h_c}}+1.67 \mathrm{for} \mathrm{TSS},0.38\le b/h_c\le 1.01$$

(7)

$${\displaystyle \frac{y_{sp}}{h_c}}=1.13{\displaystyle \frac{b}{h_c}}+2.49 \mathrm{for} \mathrm{HJSS},0.58\le b/h_c\le 1.16$$

(8)

From Equations (7) and (8), it is apparent that for aerated flow in the skimming flow region, the maximum splash height of both the TSS and HJSS increased approximately linearly with the dimensionless unit discharge, which corresponded to the findings by [38]. In comparison, under small unit discharges (i.e., for large b/h_c), the difference in the dimensionless maximum splash height between the TSS and HJSS is considerable, whereas this difference diminishes under high unit discharges. Considering the splash occurring near the basin end sill (deflected flow) in the HJSS, the dimensionless maximum splash height displayed substantial variability, oscillating between 3.15 and 4.58. Therefore, it is advisable to elevate the sidewall in the area adjacent to the basin end sill in practical engineering applications.

Overall, although the basin upstream of the stepped chute induces this deflected flow, particularly near the stepped chute inlet, the resulting aerated flow may mitigate the risk of cavitation damage in the non-aerated flow region. Future research on the HJSS will involve a more comprehensive analysis or modeling of the turbulence dynamics and air–water interactions, aiming to elucidate the mechanisms by which the HJSS outperforms the TSS, especially at a large unit discharge.

4. Conclusions

A kind of pre-aeration stepped spillway, called the hydraulic-jump-stepped spillway (HJSS), was introduced in this paper, which aims to achieve energy dissipation and prevent cavitation potential. The hydraulic performance of this design, including the flow pattern, energy dissipation, pre-aeration effect, and maximum splash height, was demonstrated for a wide range of unit discharges. Compared with a traditional stepped spillway (TSS) with the same stepped chute slope, the HJSS effectively addresses the challenges associated with large unit discharges, even when the stepped chute inlet experiences significant splash height due to deflected flow. The HJSS achieves a higher energy dissipation rate, maintaining over 80% efficiency, which represents a 23.5% improvement over the TSS. Additionally, the HJSS provides a superior pre-aeration effect, ensuring consistent aeration at the inlet of the stepped chute. At large unit discharges, the air concentration at the inlet reached 4.5% on the bottom and 11.2% on the sidewall, whereas the TSS recorded an air concentration of 0. These findings could help improve the safety and effectiveness of stepped spillway chutes in hydraulic engineering applications. Research is ongoing to conduct more comprehensive analysis or modeling of the turbulence dynamics and air–water interactions in the HJSS compared to the TSS.