Extreme Pressures and Risk of Cavitation in Steeply Sloping Stepped Spillways of Large Dams

Abstract

Stepped spillways have been increasingly used to handle flood releases from large dams associated with hydropower plants, and it is important to evaluate the fluctuating pressure field on the steps. Hydraulic model investigations were conducted on three 53° (1V:0.75H) sloping and relatively large-stepped chutes to characterize the mean, fluctuating, and extreme pressures acting on the most critical regions of the step faces, near their outer edges. The pressure development along the chutes is presented, generally indicating an increase of the modulus of pressure coefficients up to the vicinity of the point of inception of air entrainment, and a decrease further downstream. The extreme pressure coefficients along the spillway are fitted by an empirical formula, and the critical conditions potentially leading to cavitation on prototypes are calculated. The correlation between the cavitation index and the friction factor is also applied for predicting the onset of cavitation on prototypes, and the results are compared with the pressure data-based method. Generally, the results obtained from those methods yield typical values for the cavitation index in the vicinity of the point of inception, varying approximately from 0.8 to 0.6, respectively. In light of these results, maximum unit discharges of about 15–20 m²/s are considered advisable on 53° sloping large-stepped spillways without artificial aeration, for step heights ranging from 0.6 to 1.2 m. For much higher unit discharges, a considerable reach of the spillway may potentially be prone to the risk of cavitation damage.

1. Introduction

Flood-control structures designed to release excess water safely from a reservoir, with the aim of preventing water from spilling over the dam crest, are relevant to major hydropower schemes.

Since the 1980’s, techniques such as roller compacted concrete (RCC) have greatly renewed the interest in using stepped spillways, namely on large dams. Some notable examples of prototype applications can be found elsewhere, namely in [1,2,3]. In particular, various stepped spillways have been incorporated in hydropower schemes, such as the Stagecoach (USA), Petit Saut (French Guayana), Dachaoshan and Shuidong (China), Pedrógão (Portugal), and Dona Francisca, Santa Clara-Jordão, and Anta dams (Brazil). Figure 1a,b illustrates the spillways of the Pedrógão and Dona Francisca dams in operation.

Initially, the unit discharge was frequently limited to some 15 m²/s, such as on the Monskvile (USA) and De Mist Kraal (South Africa) dams, in 1986, Upper Stillwater (USA) and Les Olivettes (France) dams, in 1987, Wolwedans dam (South Africa), in 1990, and La Puebla de Cazalla dam (Spain), in 1991. Since then, larger values have been considered, such as for the Santa Clara-Jordão (unit design discharge q = 27 m²/s), Dona Francisca (q = 32 m²/s), Pedrógão (q = 40 m²/s), and particularly for the Shuidong (q = 100 m²/s) and Dachaoshan (q = 165 m²/s) dams. It should be noted that, in Shuidong and Dachaoshan dams, the spillway configurations were unconventional. In fact, both spillways included flaring pier gates, and, on the latter, a small flip bucket at the downstream end of the piers. According to [4], the objective of such a solution was to enhance self-aeration of the flow and provide enough aeration at the pseudo-bottom (near the step faces). It was also stated by [4] that there would not be enough aeration if the design/check flood discharges of the Dachaoshan chute spillway (165 and 250 m²/s, respectively) flowed onto the steps directly, which could possibly cause some cavitation damage.

To date, major overflow events have occurred on stepped spillways worldwide, and as far as it can be assessed, the performance of stepped spillways on RCC or concrete gravity dams has been satisfactory [2,5,6]. However, on the Dona Francisca dam, some steps presented localized damage after a major flood event in December of 2010 [7]. According to records provided by the Dona Francisca Energética S.A. company, the unit discharge was above 20 m²/s and 15 m²/s for approximately seven and fourteen hours, respectively. In light of the analysis presented in Section 4, the occurrence of cavitation would seem possible for flows of this magnitude on such a type of stepped spillway. It should be noted, however, that on the D. Francisca dam the steps are chamfered, so the previous observation should be carefully interpreted. At Paradise dam, no damage to the stepped chute itself was reported after a succession of major flood events that took place between 2010 and 2013 [6]. In turn, damages were recorded at the basin apron, with pronounced downstream scouring [8,9].

Among the aspects of relevance for the hydraulic design of stepped spillways, the pressure field acting on the step faces has been the subject of in-depth investigation. In [10,11,12], mean pressures were acquired on the step faces, whereas mean, fluctuating, and extreme pressures were obtained in other studies, such as [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. The doctoral dissertations of [17,21,27], in particular, addressed the fluctuating pressure field on the horizontal and vertical step faces of steeply sloping stepped spillways with uncontrolled ogee crests, typical of RCC or concrete gravity dams. Therein, the external region of the horizontal tread, located near the step edge, was found to be influenced by the main stream, whereas the internal part, located around the inner corner, was mainly influenced by the recirculating vortices. Those studies also revealed that the maximum pressure is located on the external region of the horizontal face of the steps, due to the impact of the main flow, whereas the minimum pressure occurs near the outer edge of the vertical face, due to the drag effect caused by the main stream [18]. Empirical models have been developed for predicting the mean and fluctuating pressure distributions along the vertical and horizontal step faces [19,20], or the development of the mean, fluctuating, and extreme pressure coefficients down the chute [21,22,27,29]. Based on the extreme pressures near the outer edge of the vertical face of the steps with 0.1% probability of non-exceedance, cavitation risk was estimated in [21,22,27,32]. In turn, [33] conducted experiments of high velocity flow in a closed conduit, using acoustic emission technology for detecting cavitation characteristics along with high-speed videography, providing additional insight into the flow features that drive the formation of cavitation. The critical cavitation index represented the point of the largest increase in the rate of cavitation activity, a point where performance could be affected. The correlation between the critical cavitation index and the friction factor proposed by [34] for uniformly distributed roughnesses in turbulent boundary layers was found to be applicable in skimming flows. This correlation was applied by [2,35] for predicting the onset of cavitation at the point of inception of skimming flow on 1V:0.75H steeply sloping stepped spillways.

The objective of this study was, firstly, to extend the set of data gathered in previous studies by increasing the maximum unit discharge and the critical flow depth normalized by the step height. Further, by acquiring an extensive set of simultaneous pressure measurements near the outer edges of the vertical and horizontal faces of the steps, embracing the non-aerated and self-aerated flow regions, the aim was to develop robust empirical models for estimating the extreme pressures down the chute. Another goal of the present study was to perform a comparative analysis of the conditions leading to the onset of cavitation and the extension of the spillway potentially prone to cavitation damage, based on the findings of [33], against those derived from the analysis of extreme (minimum) pressures near the outer edge of the vertical face of the steps.

2. Materials and Methods

The experiments were carried out in the framework of [32]. Three relatively large experimental facilities were assembled, two of which (LOH I and LOH II) at the Laboratory of Hydraulic Works of the Institute of Hydraulic Research, Federal University of Rio Grande do Sul (IPH-UFRGS), and the third one at the Laboratory of Experimental Hydraulics (LAHE) of Furnas Centrais Elétricas, also in Brazil.

The stepped spillway physical models used in this research included steps made of metal (LOH I, LOH II) and smooth concrete (LAHE) (Figure 2). The height of the stepped chutes, from crest to toe (H) was 2.45 m (LOH I), 4.60 m (LOH II) and 2.30 m (LAHE), and the respective width (W) was 0.40 m (LOH I), 0.50 m (LOH II) and 1.15 m (LAHE). All stepped chutes had a pseudo-bottom angle from the horizontal θ = 53.13° (i.e., chute slope of 1V:0.75H), with a step height (h) of 0.06 m (LOH I, LOH II) or 0.09 m (LAHE) (

Table 1. Details of the spillway models and location of the pressure taps on the step faces.

Setup	H (m)	W (m)	h (m)	l (m)	NTH (−)	NTV (−)	y (mm)	z (mm)	y/l (−)	z/h (−)
LOH I	2.45	0.40	0.06	0.0450	27	27	4.0	5.0	0.089	0.083
LOH II	4.60	0.50	0.06	0.0450	26	23	8.0	8.0	0.178	0.133
LAHE	2.30	1.15	0.09	0.0675	9	9	5.6	7.5	0.083	0.083

Note: H—height of the chute, from crest to toe; W—chute width; h—step height; l—step length; N_TH—number of pressure taps near the outer edge of the horizontal step faces; N_TV—number of pressure taps near the outer edge of the vertical step faces; y—distance from the external edge to the pressure tap, along the horizontal face of the step; z—distance from the external edge to the pressure tap, along the vertical face of the step (sketch in Figure 3).

). In all models, the uncontrolled spillway crest shapes followed the standard Waterways Experiment Station (WES) developed by the U.S. Army Corps of Engineers. Steps of variable height were not incorporated into the WES profiles. Further information can be found in [32].

Various piezoresistive sensors (Omega PX419, Sitron SP96, and Hytronic TM25, Porto Alegre, Brazil) were used to measure the pressure fluctuations on the step faces with pressure ranges from −0.05 to +0.31 bar (LOH I), −0.35 to +0.35 bar (LOH II), and −0.10 to +0.30 bar (LAHE), with accuracies ranging from 0.08% to ±0.5% FS. The pressure sensors were connected to the taps by means of 0.20-m long, 1.7-mm internal diameter silicone tubes (LOH I, LOH II), and 0.05-m long, 3.95-mm internal diameter copper tubes together with 0.20-m long, 6.35-mm internal diameter silicone tubes (LAHE). The length of the tubes met the recommendations of [36]. The notation used by [17,22] was adopted to define the measuring points (see Figure 3). The fluctuating pressures were acquired at an acquisition rate of 100 Hz, and the acquisition time was set equal to 10 min (60,000 samples per test). The selection of the sampling frequency and time was based on preliminary tests [27,30,32], being of a similar order of magnitude to those adopted by [21,22]. The location of the pressure taps is detailed in Table 1. These correspond to locations nearby those where extreme pressures would be expected, according to the findings of [19,20,22,26,27], among others.

In all models, the discharge was supplied from closed circuits with constant head reservoirs upstream of the chute, after reaching the steady state, and measured by electromagnetic flow meters installed in the supply conduits, with an accuracy of 0.25%. The unit discharges (q) tested ranged from 0.10 to 0.35 m²/s (LOH I), 0.05 to 0.50 m²/s (LOH II) and 0.10 to 0.36 m²/s (LAHE), and the corresponding critical depth normalized by the step height (d_c/h, where d_c = (q²/g)^1/3, g being the gravitational acceleration) varied between 1.68 to 3.87 (LOH I), 1.11 to 4.90 (LOH II) and 1.12 to 2.61 (LAHE). Further details of the test program, including the Reynolds number (Re = q/υ), along with the Froude (Fr_i = U_i/(gd_i)^1/2) and Weber (We_i = U_i/(σ_w/(ρ L_s))^1/2) numbers at the point of inception of air entrainment, are included in

Table 2. Test program.

Setup	q (m2/s)	dc/h (-)	Re (-)	Fri (-)	Wei (-)
LOH I	0.100	1.68	9.9 × 104	5.22	97
	0.125	1.95	1.2 × 105	5.36	105
	0.150	2.20	1.5 × 105	5.47	115
	0.200	2.66	2.0 × 105	5.66	128
	0.250	3.09	2.5 × 105	5.80	141
	0.275	3.29	2.7 × 105	5.87	147
	0.330	3.72	3.3 × 105	5.99	158
	0.350	3.87	3.5 × 105	6.03	163
LOH II	0.054	1.11	5.3 × 104	4.87	75
	0.082	1.47	8.1 × 104	5.11	91
	0.108	1.77	1.1 × 105	5.27	102
	0.150	2.20	1.5 × 105	5.47	115
	0.200	2.66	2.0 × 105	5.66	128
	0.300	3.49	3.0 × 105	5.93	153
	0.400	4.23	4.0 × 105	6.13	171
	0.500	4.90	5.0 × 105	6.29	186
LAHE	0.100	1.12	9.9 × 104	4.87	112
	0.125	1.30	1.2 × 105	5.00	126
	0.150	1.47	1.5 × 105	5.10	134
	0.200	1.78	2.0 × 105	5.28	151
	0.275	2.19	2.7 × 105	5.47	171
	0.330	2.48	3.3 × 105	5.59	185
	0.356	2.61	3.5 × 105	5.64	189

Note: q—unit discharge; d_c/h—relative critical depth, with d_c = (q²/g)^1/3; Re—Reynolds number (Re = q/υ); Fr_i—Froude number at the point of inception (Fr_i = U_i/(gh_i)^1/2); We_i—Weber number at the point of inception, given as We_i = U_i/(σ_w/(ρ L_s))^1/2.

. In the above formulae, υ is the kinematic viscosity of water, U_i and d_i are the mean water velocity and equivalent clear water depth at the point of inception of air entrainment, respectively, σ_w is the surface tension between air and water, ρ is the density of water, and L_s is the distance between the step edges (L_s = h/sinθ).

Considering that both Fr and We increase downstream of the point of inception, practically all of the tested range of Re and We are expected to be exempted from major scale effects with regard to the main macro flow properties, such as air concentration and velocity, characteristic flow depths, and fluctuating pressures on the step faces. According to [11,37], scale effects with respect to the air concentration and velocity were negligible for Re > 3 × 10⁴, whereas [38,39] recommended Re > 10⁵ and We > 100. A more conservative criterion was indicated by [40] for scale effects mitigation based on local air concentration close to smooth chute inverts (Re > 2 × 10⁵). With respect to pressures on the step faces, [21] suggested that scale effects may be neglected for Re > 10⁵.

3. Skimming Flow on Steep Slopes: Main Regions and Flow Properties

Skimming flow down stepped spillways can be divided into various regions. In the non-aerated flow region close to the spillway crest, the boundary layer grows from the spillway floor. Outside the boundary layer the water surface is initially smooth and glassy, but it becomes undulated upstream of the inception of air entrainment. This undulated surface is responsible for the transport of air between the irregular waves, as shown elsewhere [41,42]. When the boundary layer approaches the free-surface, entrainment of air by the multitude of vortices in the turbulent flow begins at the so-called point of inception [1,43,44]. Self-entrainment into the step niches occurs in the vicinity of the point of inception as a consequence of recurring significant surface troughs combined with bubbly flow protruding to the steps [45]. As a result of the highly macroroughness induced turbulence, a rapidly varied air-water flow takes place in a short region further downstream, in contrast with the traditional concept of gradually varied self-aerated flow on smooth chutes [2,41,46]. The location of the section separating the rapidly from the gradually varied flow region was found to occur for L′ ~ 30, irrespective of the relative critical depth [2,41,47]. The normalized distance L′ = (L − L_i)/d_i, where L is the streamwise distance from the crest to the outer edge of the step, d the equivalent clear water depth, and the subscript i refers to the point of inception (Figure 3).

Downstream of the rapidly varied flow region, a trend of a slight increase in the mean air concentration and flow bulking is noticeable along the chute, with a wavy pattern [2,41]. From a practical standpoint, the waviness may be considered negligible, and the flow will become virtually quasi-uniform far downstream of the point of inception. Then, for a given discharge, the main flow properties, such as the mean air concentration, equivalent clear water depth, mean water velocity, and mean pressure distribution on the steps, will practically not vary along the spillway.

Various formulae have been proposed for estimating the location and flow properties at the point of inception in steeply sloping stepped chutes, typical of RCC dams, as well as the main flow properties upstream and downstream of that location, such as [6,39,41,42,44,48,49,50]. Herein, the formulae developed based on the experimental work gathered on a 1V:0.75H sloping chute [51,52,53] were adopted. These include:

Location and equivalent clear water depth at the point of inception [42]:

$$\frac{{\mathrm{L}}_{\mathrm{i}}}{\mathrm{k}}=6.75{\mathrm{F}}_{*}^{0.76}$$

(1)

$$\frac{{\mathrm{d}}_{\mathrm{i}}}{\mathrm{k}}=0.35{\mathrm{F}}_{*}^{0.59}$$

(2)

where k is the step roughness height (k = h cosθ) and ${\mathrm{F}}_{*}$ is a Froude number defined as ${\mathrm{F}}_{*}$ = q/(g sinθ k³)^1/2.

Equivalent clear water depth, upstream of the point of inception [42]:

$$\frac{\mathrm{d}}{{\mathrm{d}}_{\mathrm{i}}}=1.17-0.25\frac{\mathrm{L}}{{\mathrm{L}}_{\mathrm{i}}}+\frac{0.084}{\mathrm{L}/{\mathrm{L}}_{\mathrm{i}}}$$

(3)

Equivalent clear water depth, downstream of the point of inception [41,47]:

$$\frac{\mathrm{d}}{{\mathrm{d}}_{\mathrm{i}}}=\frac{1}{\{1+{\left[21.338-\frac{13.815}{{\left({\mathrm{d}}_{\mathrm{c}}/\mathrm{h}\right)}^2}\right]}^{-1}{{\mathrm{L}}^{\prime }}^{1/2}\}}$$

(4)

$$\frac{\mathrm{d}}{{\mathrm{d}}_{\mathrm{i}}}=0.653+0.347{\mathrm{e}}^{-\frac{{\mathrm{L}}^{\prime }}{27.60}}$$

(5)

where L′ = (L − L_i)/d_i. Equation (4) was applied for d_c/h ≤ 3, whereas Equation (5) was adopted for d_c/h > 3.

The mean water velocity is given as U = q/d.

Air concentration close to the pseudo-bottom, downstream of the point of inception [50,51]:

$${\mathrm{C}}_{\mathrm{b}}=\frac{0.381}{1+{\left(\frac{{\mathrm{L}}^{\prime }}{14.189}\right)}^{-1.232}}$$

(6)

where C_b was obtained at a distance of 0.32 cm from the pseudo-bottom, along its normal.

4. Results and Application

4.1. Pressure Development and Duration of Negative Pressures

In order to evaluate the development of mean and fluctuating pressure along the chute near the outer edges of the horizontal and vertical step faces, the following parameters were analyzed: (1) mean (time-averaged) pressure p_m; (2) root mean square value representative of the pressure fluctuations p′; and (3) extreme pressures characterized by p_x%, x% being the probability of non-exceedance, that is, the probability of occurrence of lower pressure values. Herein, the limits 0.1%, 1.0%, and 5.0% were chosen for the minimum pressures, whereas 95.0%, 99.0%, or 99.9% were adopted for the maximum pressures, in accordance with other studies [18,19,22,27]. These parameters were derived from the pressure measurements acquired simultaneously on the vertical and horizontal faces of the steps, at a frequency of 100 Hz.

Figure 4 shows the streamwise development of the dimensionless mean, fluctuating, and extreme pressures at the LOH II stepped chute, near the outer edges of the vertical and horizontal step faces, for two relative critical depths. Also included in the same figure is the location of the point of inception, based on Equation (1).

The following conclusions may be drawn (also generally valid for the LOH I and LAHE stepped chutes, not shown herein). Near the outer edge of the vertical step faces, the mean pressure is slightly negative or almost null, for both discharges (Figure 4a,b); the fluctuating (p′) and extreme pressures p_95.0%, p_99.0%, p_99.9% tend to increase upstream of the point of inception, whereas the extreme pressures p_5.0%, p_1.0%, p_0.1% tend to decrease (or, in alternative, their respective moduli increase). A decrease in p′, p_95.0%, p_99.0%, p_99.9% (or increase in p_5.0%, p_1.0%, p_0.1%), is noticeable in an initial reach downstream of that location; further downstream, the pressures tend to stabilize, while keeping a typical wavy pattern. Significant negative pressures may occur for p_0.1%, upstream of the point of inception, particularly in its vicinity, reaching −16 γh, for d_c/h = 4.90 (Figure 4b). This corresponds to conditions that could lead (or almost) to cavitation in prototype, for step heights of 0.9 m and 0.6 m, respectively. In turn, near the outer edges of the horizontal step faces, the mean pressure is positive due to the impact of the flow, being of the order of magnitude of the pressure fluctuations; the tendency of the fluctuating and extreme pressures to increase (in modulus) upstream of the point of inception, similar to that observed on the vertical face, is seen for the largest discharge (d_c/h = 4.90), namely for the positive pressures (Figure 4d); shortly downstream of that location, a slightly decreasing trend (in modulus) seem to take place. These results are partially in agreement with those reported by [18], on a similar chute slope (1V:0.8H), along the center of symmetry of the horizontal step faces, relative to p_m, p′, p_5.0%, and p_95.0%. There, it was observed that the mean pressures were positive all along the spillway, and only minimum pressures (p_5.0%) exhibited negative values. It was also found by [18] that both maximum and minimum pressures were located upstream of the point of inception, including for d_c/h = 2.25, which concurs with the results obtained herein for the vertical face of the step, but is apparently in contrast with those gathered on the horizontal face, for a similar relative critical depth (d_c/h = 2.20). However, the definition of the point of inception adopted by [18] was that corresponding to fully aerated flow by visual observation, which is distinct to that used herein. In fact, the point of inception was located at L/k ~ 40 in [18], whereas L/k ~ 33 was obtained in the present study. It is also important to note the distinct location of the pressure taps in both studies (i.e., in the center of the horizontal face versus near the outer step edge), so a direct comparison cannot be made. It is also interesting that significant negative pressures may likewise occur near the outer edge of the horizontal face of the steps, particularly for large unit discharges, where p_0.1% may reach −10 γh (Figure 4d).

Figure 5 shows the streamwise development of the dimensionless mean and fluctuating pressures at the LOH II stepped chute, near the outer edges of the vertical and horizontal step faces, for various relative critical depths. Similar to what was observed by [18] at the center of symmetry of the horizontal face of the steps, a steady wavy pattern is observed down the chute, generally with so-called phase coincidence, i.e., coincidence of local maximum (peak) and minimum (valley) when considering different discharges. Near the outer edge of the vertical face of the steps, the mean pressure oscillates around zero regardless of the relative critical depth, with p_m/(γh) ranging approximately between ±0.25 m for L/k > 70, in the gradually varied flow region. Considerable pressure fluctuations were obtained near the outer edges of the vertical and horizontal step faces, with higher magnitudes on the latter. In general, both the mean pressures and the pressure fluctuations tend to increase with the relative critical depth, as also shown in [18], when considering a similar range of d_c/h values (i.e., 1.41, 1.85 and 2.25). In the present study, the relative differences of p_m and p′ are generally smaller for the largest d_c/h values (3.49 and 4.90).

The percentage of total negative pressure duration with respect to the total time of data acquisition was also investigated herein. Figure 6 presents the streamwise development of the percentage of total time during which the taps located near the outer edges of the vertical and horizontal step faces of the LOH II chute exhibit negative pressures. The results are qualitatively similar to those obtained on the LOH I and LAHE chutes (not shown herein), showing a considerable oscillation down the chute and data scatter, as would be expected, taking into account the broad range of d_c/h values. A slight increasing trend occurs in an initial reach of the spillway (e.g., L′ < 20–40), whereas a plateau seems to occur further downstream. Near the outer edge of the vertical step faces, the percentage of total time during which negative pressures occur is significant, varying from 45 to 80%, for L′ > 40. These values are of the same order of magnitude as those reported by [20] (~50 to 80%), for comparable relative locations of the pressure taps and a comparable range of relative critical depths (i.e., z/h < 0.15 and 1.42 ≤ d_c/h ≤ 2.25). Near the outer edge of the horizontal step faces, the percentage of total time during which negative pressures occur is much smaller, generally ranging from 0 to 20%, and from 10 to 20% for L′ > 40. Even though [20] obtained values up to 20% on the horizontal step faces, considerably smaller values were plotted therein (<5%) for comparable relative locations of the pressure taps. However, it should be noticed that only two locations were selected in [20] (L/k = 69.6 and L/k = 63.5), whereas a broader extension of the chute was analyzed herein (approximately −25 < L′ < 160, or −20 < L/k < 120).

4.2. Mean and Fluctuating Pressures

The mean and fluctuating pressure coefficients are defined as [21,22,28,29] C_p = (p_m/γ)/(U²/2g) and C′_p = (p′/γ)/(U²/2g), respectively. The mean water velocity at each cross section in the non-aerated region was determined from U = q/d, by applying Equation (3), along with Equations (1) and (2), for estimating the equivalent clear water depth; in the self-aerated region, Equations (4) and (5) were used for estimating d, for d_c/h ≤ 3 and for d_c/h > 3, respectively.

Figure 7 shows the development of C_p and C′_p along the LOH II stepped chute for the pressure taps near the outer edge of the vertical and horizontal step faces, for various relative critical depths. Near the outer edge of both step faces, the mean and the fluctuating pressure coefficients exhibit distinct behaviors upstream and downstream of the point of inception, similarly to what was noted by [21,22,28]. Near the outer edge of the vertical step face, C_p is generally very small, with a decreasing trend (or increasing, in modulus) for L′ < 0, whereas an increasing trend (or decreasing, in modulus) occurs in a short reach immediately downstream of the point of inception (Figure 7a). For L′ greater than 20–40, C_p tends to stabilize around a slightly negative value, ranging between 0 and −0.02. The data presents some scatter, which may be explained by the broad range of values of the relative critical depth (1.11 ≤ d_c/h ≤ 4.90). Overall, the influence of d_c/h on C_p is not significant, except for small to moderate relative critical depths, in the vicinity of the point of inception. In turn, C′_p shows a markedly increasing trend for L′ < 0, reaching a peak in the vicinity of the point of inception (Figure 7b); a significant decreasing trend occurs in a short reach immediately downstream of that location, followed by a milder decreasing trend for L′ > 20–40, stabilizing around 0.02. Similarly, as observed for C_p, C′_p is not significantly influenced by d_c/h. In relation to the outer edge of the horizontal face of the steps (Figure 7c,d), both C_p and C′_p display an increasing trend upstream of the point of inception, followed by a decreasing trend downstream of that location, steeply in an initial reach, and more gently further downstream (i.e., approximately L′ > 30). As noted by [22], the presence of air has a cushion effect, reducing the magnitude of the pressures on the step faces. Furthermore, a similar order of magnitude was obtained for C_p and C′_p on the horizontal faces of the steps, which is in line with the results presented by [22].

Interestingly, the separation between these rapidly and gradually varied flow regions downstream of the point of inception seem to occur at a location nearby that previously observed for other flow properties, such as the mean air concentration [41,51].

Figure 8 shows the development of C_p and C′_p along the stepped chutes, using the complete data sets of all facilities (Table 1 and Table 2). The empirical formulae proposed by [22,27] are also plotted in Figure 8. Dashed lines were drawn herein when extending the range of application of the models developed by [22,27], or for the model developed within the scope of this study. A considerable data scatter is noticeable, due to the broad range of values of the relative critical depth (Table 2), as well as due to slightly distinct positions of the pressure taps on the horizontal and vertical faces of the steps (namely for LOH II versus LOH I and LAHE, as indicated in Table 1). These may explain the larger data scatter or slightly lower absolute values of C_p and C′_p for the LOH II model.

The model proposed by [27] generally concurs with the experimental data on indicating increasing or decreasing trends of C_p and C′_p, except for C_p on the vertical face, downstream of the point if inception. However, such a model tends to overestimate C_p and C′_p on the horizontal face of the step upstream of the point of inception (Figure 8c,d). In turn, the model of [22] provides a good qualitative representation of the overall decreasing trend of C_p on the horizontal face, for L′ < 60 (Figure 8c), as well as that of C′_p on both faces of the steps (Figure 8b,d). Considering the significant data scatter, it was not possible to obtain an adequate model that fits all C_p and C′_p results reasonably well, except for C′_p near the outer edge of the vertical step faces (i.e., Equation (7), for 25 ≤ L′ ≤ 50, along with

Table 3. Coefficients of Equation (7) relative to the outer edge of the vertical and horizontal faces of the steps.

Data		a	b	c	d	R2
Full data set (vertical face) −25 ≤ L′ ≤ 50	C′p	0.1313	0.0018	0.0410	0.0020	0.80
	Cp0.1%	−0.6507	−0.0094	0.0421	0.0024	0.80
	Cp1.0%	−0.4210	−0.0072	0.0440	0.0024	0.79
	Cp5.0%	−0.2626	−0.0054	0.0459	0.0023	0.76
Re ≥ 2 × 105; dc/h ≥ 1.78	Cp99.9%	1.1759	0.0084	0.0304	−0.0003	0.46
(horizontal face)	Cp99.0%	0.8956	−0.0012	0.0232	−0.0003	0.58
0 ≤ L′ ≤ 50	Cp95.0%	0.6469	−0.0040	0.0202	−0.0003	0.62

Notes: R²—coefficient of determination. On the horizontal face of the step, Equation (7) essentially provides an indication of the general trend of the pressure coefficient downstream of the point of inception, given the relatively low values of R².

, and C′_p ~ 0.03, for L′ > 50).

4.3. Extreme Pressures

The extreme pressure coefficients are defined as C_px% = (p_x%/γ)/(U²/2g) [21,22,28], where x% is the probability of non-exceedance. The limits 5.0%, 1.0%, 0.1%, and 95.0%, 99.0%, 99.9% were selected for the minimum and maximum pressures, respectively.

Figure 9 shows the development of the most extreme pressures along the LOH I, LOH II, and LAHE stepped chutes, namely, C_p0.1% near the outer edge of the vertical face of the steps, and C_p99.9% near the outer edge of the horizontal face of the steps. Overall, the observed data trends for the distinct stepped chutes are consistent. The influence of the relative critical depth is relatively small on the vertical faces. In turn, the influence of d_c/h is not as negligible on the horizontal faces, taking into account the data scatter. Similarly, as observed for C_p, there is a sharp decreasing trend (increasing in modulus) of C_p0.1% for L′ < 0, whereas an increasing trend (decreasing in modulus) occurs in a short reach immediately downstream of that location (Figure 9a,c,e). For L′ approximately greater than 30, the change in C_p0.1% is gradual, with a tendency to level around negative values from −0.1 to −0.2. For C_p99.9% near the outer edge of the horizontal step faces, the data trend is masked by the relative influence of d_c/h. However, a close look at limited ranges of d_c/h (as shown subsequently for larger values of Re and d_c/h) indicates that the overall trend of C_p99.9% is similar to that previously shown for C′_p, including an increase in C_p up to the vicinity of the point of inception, followed by its decrease downstream of that location, steeply in an initial reach, and mildly further downstream (i.e., L′ > 30). Similar behavior was found for the other corresponding extreme pressure coefficients, i.e., C_p5.0% and C_p1.0%, for the vertical faces, and C_p95.0% and C_p99.0%, for the horizontal faces (not shown herein).

Figure 10 shows the development of C_p0.1%, C_p1.0%, and C_p5.0% near the outer edge of the vertical face of the steps along the stepped chutes. In Figure 10a,c,e, complete data sets were used, whereas in Figure 11b,d,f, only data for Re ≥ 2 × 10⁵ and d_c/h ≥ 1.78 were included (Table 1 and Table 2). The empirical formulae proposed by [22,27] for C_p0.1%, are also plotted, along with a new model developed from the present data set (Equation (7), for −25 ≤ L′≤ 50, along with Table 3). Again, dashed lines were used herein when extending the range of application of the models by [22,27], or for the model developed within the scope of this study. Overall, the development of both C_p5.0% and C_p1.0% is similar to that obtained for C_p0.1%. The model proposed by [27] is able to identify the increasing and decreasing trends of C_p0.1%, C_p1.0%, or C_p5.0%, even though the respective minima are obtained shortly downstream of that derived from Equation (7). The C_p0.1% model presented in [22] for L′ ≥ 0 fits the data well in a reach downstream of the point of inception (i.e., L′ ≤ 15). Further downstream, larger negative values are predicted from the [22] model, in comparison to the data acquired in the present study.

The results obtained for all models considering data satisfying the condition Re ≥ 2 × 10⁵ and d_c/h ≥ 1.78 exhibit a similar trend, which sustains the applicability of the model for high values of d_c/h and Re (Figure 10b,d,f).

$${\mathrm{C}}_{\mathrm{p}}=\frac{\mathrm{a}+{\mathrm{bL}}^{\prime }}{1+{\mathrm{cL}}^{\prime }+{{\mathrm{dL}}^{\prime }}^2}$$

(7)

The development of C_p95.0%, C_p99.0%, and C_p99.9% near the outer edge of the horizontal face of the steps along the chutes is shown in Figure 11. In Figure 11a,c,e, complete data sets were used, whereas in Figure 11b,d,f, only data for Re ≥ 2 × 10⁵ and d_c/h ≥ 1.78 were included (Table 1 and Table 2). The empirical formulae proposed by [27] are also plotted. Overall, the development of both C_p95.0% and C_p99.0% is similar to that obtained for C_p99.9%. The model proposed by [27] overestimates C_p95.0%, C_p99.0%, and C_p99.9% in the vicinity of the point of inception. When considering the model results for Re ≥ 2 × 10⁵ and d_c/h ≥ 1.78, the data scatter is considerably reduced. Near the point of inception, 0.8 ≤ C_p99.9% ≤ 1.4, 0.6 ≤ C_p99.0% ≤ 1.1 and 0.4 ≤ C_p95.0% ≤ 0.8. Hence, the pressure head corresponding to C_p99.0% generally surpasses the mean kinetic head of the flow. Further downstream, in the gradually varied flow region, the extreme pressure coefficients are dampened, due to significant air entrainment and increased flow velocity, approaching C_p99.9% ~ 0.8, C_p99.0% ~ 0.6, and C_p95.0% ~ 0.4. For 0 ≤ L′ ≤ 50, the overall trend of the data is qualitatively well-depicted using Equation (7), with the respective constants included in Table 3. However, the data scatter is considerable, which explains the low values of R².

4.4. Risk of Cavitation: Application to Prototypes

The estimation of the flow conditions that may lead to cavitation on 1V:0.75H steeply sloping stepped spillways was carried out using two distinct approaches. This first one was based on the findings of [54,55], where the pressure with 0.1% probability of non-exceedance was suggested as a representative extreme negative pressure for cavitation tendency verification in macroturbulent flows, based on model–prototype comparisons.

Hence, the empirical model developed herein for predicting C_px% near the outer edge of the vertical step face was used to estimate the critical velocity down the chute [21,22,27,29],

$${\mathrm{U}}_{\mathrm{crit}}=\sqrt{\frac{2{\mathrm{g}\mathrm{p}}_{\mathrm{v}}/\mathsf{\gamma }}{{\mathrm{C}}_{\mathrm{px}\%}}}$$

(8)

where p_v is the vapor pressure (p_v/γ = 0.24 m, for t = 20 °C), and U_crit the critical water velocity taken herein as the velocity which would virtually lead to the occurrence of vapor pressure at prototype, near the outer edge of the vertical face of the step.

Based on the complete set of data, the minimum values of C_p0.1% ranged between −0.45 and −0.9, whereas the application of the empirical model yielded an intermediate value of approximately −0.7, shortly upstream of the point of inception, at L′ = −6 (Figure 10a). This value is identical to that obtained by the model of [27]; however, according to [27], the minimum value is estimated shortly downstream of the point of inception. Also, the minimum value obtained in this experimental study agrees with the conservative value proposed by [22] at the point of inception on a 1V:0.8H sloping chute (C_p0.1% = −0.9).

The cavitation index σ can be estimated from [2,56],

$$\mathsf{\sigma }=\frac{\left(\mathrm{p}-{\mathrm{p}}_{\mathrm{v}}\right)/\mathsf{\gamma }}{{\mathrm{U}}^2/2\mathrm{g}}$$

(9)

where p the pressure at the pseudo-bottom, assuming an hydrostatic pressure distribution (i.e., p = p_atm + γdcosθ, where p_atm is the atmospheric pressure). Considering U = U_crit and d = q/U_crit, the critical cavitation index based on C_p0.1% (σ_{c (Cp0.1%)}) may be calculated from Equation (9). Similarly, predictions of the critical cavitation index based on C_p1.0% (σ_{c (Cp1.0%)}) and C_p5.0% (σ_{c (Cp5.0%)}) were also obtained. In Figure 12, the development of σ_{c (Cp0.1%)}, σ_{c (Cp1.0%)}, and σ_{c (Cp5.0%)} along 1V = 0.75H sloping stepped spillways commonly found on RCC dams is plotted for selected values of the step height and unit discharge. The results show a consistent behavior of the critical cavitation indexes along the chute, rapidly increasing upstream of the point of inception of air entrainment, reaching a peak in the upstream vicinity of that location, and decreasing downstream of such a peak, steeply in an initial reach, and more gradually further downstream. For identical L′, higher values of the critical cavitation index were obtained for decreasing pressure probabilities of non-exceedance, as expected.

Also plotted in Figure 12 is the cavitation index (σ) along the chute obtained from Equation (9), where the mean water velocity and the pressure head at the pseudo-bottom at each cross section in the non-aerated region was determined from Equation (3), along with Equations (1) and (2), with p = p_atm + γdcosθ; in the self-aerated region, Equations (4) and (5) were used for estimating the equivalent clear water depth, for d_c/h ≤ 3 and for d_c/h > 3, respectively. In the chute reach where σ < σ_{c (Cp0.1%)}, it is assumed that there is the risk of occurrence of cavitation. However, provided that the air concentration close to the pseudo-bottom (Equation (6)) is larger than a critical value, cavitation damage is not expected to occur, even if cavitation may take place [57]. Hence, a potential risk of cavitation damage may occur on the reach where σ < σ_{c (Cp0.1%)} and C_b < 0.08, as indicated in Figure 12 as gray shaded zones. Therein one can observe that for a stepped spillway with 0.6 m high steps, σ is slightly lower that σ_{c (Cp0.1%)} in the vicinity of the point of inception for q = 15 m²/s, which corresponds practically to the maximum recommended unit discharge for such a chute based on this method. The influence of the step height on the maximum unit discharge was found to be relatively small. For larger step heights of 0.9 m or 1.2 m, the application of the model would return larger unit discharges of q ~ 17 m²/s and q ~ 19 m²/s, respectively. This result is judged to stem from the fact that the C_p0.1% model does not take into account the influence of d_c/h, which may be considerable near the point of inception. However, the model tends to predict acceptable values of C_p0.1% for large relative critical depths, as expected for design conditions where cavitation may occur. For example, considering the experimental data corresponding to Re > 3 × 10⁵ and d_c/h ≥ 1.78, C_p0.1% varied between −0.79 and −0.67, and U_crit between 15.7 m/s and 17.3 m/s (Equation (8)). Such critical velocities would correspond to maximum (permissible) unit discharges of approximately 13 m²/s and 17 m²/s for h = 0.6 m. With increasing unit discharge (on the same chute), the length of the spillway reach prone to the potential risk of cavitation damage increases, as expected (Figure 12b, grey shaded zone). However, as stated in [33], the use of the unit discharge as a design recommendation seems a bit misguided without knowing the actual conditions of when and if cavitation will form.

The second approach for estimating the flow conditions that lead to the onset of cavitation on 1V:0.75H steeply sloping stepped spillways was based on the correlation between the critical cavitation index and the friction factor proposed for uniformly distributed roughnesses in turbulent boundary layers [33,34,56] (Equation (10)). In [33], the critical cavitation index represented the point of the largest increase in the rate of cavitation activity.

The friction factor in the non-aerated flow region was estimated from Equation (11), which resulted from a reanalysis of closed conduit data for 53° sloping chutes gathered by [13], as per [2,35,51]. Even though Equation (11) was derived from closed conduit air flow, the application of this formula was found to be a good prediction of the non-aerated friction factor on a similar slope, calculated using the Von Karman’s integral momentum equation [21,58].

$${\mathsf{\sigma }}_{\mathrm{c}}=4\mathrm{f}$$

(10)

$$\frac{1}{\sqrt{\mathrm{f}}}=2.20-1.14\log \left(\frac{\mathrm{k}}{\mathrm{d}}\right)$$

(11)

The critical cavitation index estimated from Equations (10) and (11) (σ_{c (f)}) is also plotted in Figure 12 (for the non-aerated flow region), where the equivalent clear water depth at each cross section was determined by applying Equation (3), along with Equations (1) and (2). According to this methodology, the risk of occurrence of cavitation occurs in the chute reach for σ < σ_{c (f)}. The value of σ_{c (f)} at the point of inception varies between 0.55 to 0.60, which is of the same order of magnitude as that found by [33] for the 68° steep slope using laboratory experiments in a specialized reduced ambient pressure chamber (σ_c ~ 0.63); also, it is of the same order of magnitude as that adopted by [59] for stepped spillways up to θ = 55° (σ_c = 0.5), and those obtained by [60] (σ_c = 0.5–0.7). In the latter study, the cavitation index σ was computed according to [56], based on “average hydraulics”, for the limiting discharge of 15 m²/s of [22] on θ = 50° chutes, for distinct step heights of h = 0.3 and 1.2 m. On the other hand, σ_{c (f)} is smaller than σ_{c (Cp0.1%)}, in the vicinity of the point of inception (σ_{c (Cp0.1%)} ~ 0.75–0.8).

As illustrated in Figure 12a,b, cavitation is not expected to occur in the non-aerated region for 1V:75H sloping stepped spillways with 0.6 m high steps for unit discharges up to almost 20 m²/s. More precisely, solving the equation σ = σ_{c (f)} at the point of inception yields q ~ 18 m²/s [2]. The influence of the step height on the maximum unit discharge is relatively small, where approximately 17 m²/s and 15 m²/s were obtained for 0.9 m and 1.2 m high steps, respectively. The relative differences between the maximum recommended unit discharge obtained from σ_{c (Cp0.1%)} in relation to those from σ_{c (f)} were −16%, −2%, and 20% for 0.6, 0.9, and 1.2 high steps, respectively.

For larger unit discharges, the extension of the chute prone to the risk of cavitation damage (red shaded zone in the non-aerated flow region) may be significant, as illustrated in Figure 12, namely for h = 0.9 m and q = 30 m²/s (Figure 12c), and h = 1.2 m and q = 40 m²/s (Figure 12d).

In light of the results obtained herein, it is considered advisable not to adopt design unit discharges considerably larger than 15–20 m²/s in the steeply sloping stepped spillways of large dams, provided that artificial air entrainment is not introduced in the flow (e.g., by piers or aerators). In [61,62,63,64], for example, it is shown that the use of aerators may enable the considerably increase of the maximum (permissible) unit discharge. In addition, for dam spillways of small to moderate heights, larger unit discharges may also be considered adequate, provided that σ > σ_c is assured down the chute on the non-aerated flow region [27,29,65,66]. It should be noted, however, that the values that result from predicting the onset of cavitation are conservative in terms of predicting damage. The severity of damage that may be expected is related both to the intensity of cavitation and time of exposure [56,67].

5. Conclusions

Evaluating the fluctuating pressure field on the step faces, along with the conditions leading to the onset of cavitation and protection against cavitation damage, is of utmost importance for a sound design of steeply sloping stepped spillways on large dams. Limited published guidance is available for estimating the minimum (p_0.1%) and maximum (p_99.9%) extreme pressures near the outer edges of the vertical or horizontal faces of the steps for large relative critical depths (i.e., d_c/h > 3), with a considerable reach of the chute subject to non-aerated flow conditions, as commonly adopted in design practice. These circumstances encouraged a physical model study based on three relatively large-size physical models of stepped spillways, including a vast number of simultaneous instantaneous pressure measurements, along with a broad range of tested unit discharges (up to 0.5 m²/s) and relative critical depths (up to 4.9). In addition, the comparative analysis of the development of cavitation risk on the non-aerated flow region from p_0.1% pressure analysis, and from the correlation of the critical cavitation index with the friction factor, undertaken in this study was, to the best of our knowledge, not available to date.

The following conclusions can be made from this study:

• The pressure development along the chutes indicates a distinct behavior of the pressure field in the non-aerated and self-aerated flow regions, generally with an overall increase of the modulus of the mean, fluctuating, and extreme pressure coefficients up to the vicinity of the point of inception of air entrainment, and a decrease further downstream. This result is in agreement with the findings of [21,22,27,28].
• Shortly downstream of the point of inception, a rapidly varied flow region is noticeable on all pressure coefficients; the downstream end of this region agrees in general with previous findings for other flow properties, such as the mean air concentration (i.e., L′ ~ 30, as per [41,51]).
• The fluctuating and extreme pressure coefficients near the outer edge of the vertical face of the steps along the spillway are fitted by an empirical formula in function of L′ for a broad range of relative critical depths, similarly to the findings of [21,22] for C′_p and C_p0.1%, and by [27] for all related pressure coefficients.
• The empirical formula developed for C_p0.1%, C_p1.0%, and C_p5.0%, based on all data near the outer edge of the vertical face of the steps, is generally valid even if only data that satisfy Re ≥ 2 × 10⁵ and d_c/h ≥ 1.78 are considered; on the other hand, the data scatter for the maximum extreme pressure coefficients C_p95.0%, C_p99.0%, and C_p99.9% near the outer edge of the horizontal face of the steps is considerably reduced for Re ≥ 2 × 10⁵ and d_c/h ≥ 1.78.
• When considering the model results near the outer edge of the horizontal face of the steps, for Re ≥ 2 × 10⁵ and d_c/h ≥ 1.78, C_p99.9% ranges between 0.8 and 1.4 near the point of inception, hence the corresponding pressure head generally surpasses the mean kinetic head of the flow. Further downstream, in the gradually varied flow region, the extreme pressure coefficients are dampened due to the significant air entrainment and increased flow velocity, approaching 0.8 (C_p99.9%), 0.6 (C_p99%), and 0.4 (C_p95%).
• Based on the minimum extreme pressure coefficient analysis (C_p0.1%) applied to prototypes, the critical cavitation index in the vicinity of the point of inception varied typically between 0.75 to 0.80, which is fairly similar to the values obtained from [21] (σ_c ~ 0.8) and larger than those predicted by [27] (σ_c ~ 0.6).
• From the correlation between the cavitation index and the friction factor, the cavitation index in the vicinity of the point of inception at prototypes varied typically between 0.55 and 0.60, hence lower than that predicted from C_p0.1%.

In light of the results obtained in this study, maximum unit discharges of about 15–20 m²/s are considered advisable on 53° (1V:0.75H) sloping large-stepped spillways (without artificial aeration), for step heights ranging from 0.6 to 1.2 m. For significantly larger unit discharges, a considerable reach of the spillway may be prone to the risk of cavitation damage, depending on the intensity of cavitation, time of exposure, and concrete cavitation resistance. However, for small to moderate height dams, larger unit discharges may be considered adequate, as long as the cavitation index is greater than the critical cavitation index in the non-aerated flow region.