Performance and Design of a Stepped Spillway Aerator

Abstract

Stepped spillways are frequently limited to specific discharges under around 30 m²/s due to concerns about potential cavitation damages. A small air concentration can prevent such damages and the design of bottom aerators is well established for smooth chutes. The purpose of this study is to systematically investigate the performance of a deflector aerator at the beginning of stepped chutes. Six parameters (chute angle, step height, approach flow depth, approach flow Froude number, deflector angle and deflector height) are varied in a physical model. The spatial air concentration distribution downstream of the aerator, the cavity sub-pressure, water discharge and air discharges are measured. The results describe the commonly used air entrainment coefficient, the jet length, as well as the average and bottom air concentration development to design an aerator. The lowest bottom air concentration measured in all tests is higher than the air concentration recommended in literature to protect against cavitation damages. And, unlike smooth chutes, there appears to be no significant air detrainment downstream of the jet impact. One deflector aerator seems therefore sufficient to provide protection of a stepped spillway.

1. Introduction

1.1. Stepped Spillways

Stepped spillways are commonly used, owing to their efficient energy dissipation and the emergence of roller compacted concrete (RCC) dams in the 1980s. Steep step spillways (chute angle φ ≈ 50°) are found on the downstream face of RCC or gravity dams, while moderate slope stepped spillways (φ ≈ 30°) are found on embankments dams.

Stepped spillways have been widely studied in the past decades in terms of flow regime, energy dissipation, self-aeration and pressure distribution. General summaries of researches are given in [1,2,3]. One major drawback of stepped spillways is that the specific discharge q has been limited to q ≤ 30 m³/s/m [4,5,6,7] to avoid cavitation damages upstream of the inception point.

1.2. Cavitation

Cavitation corresponds to the formation of vapor cavities in water due to a reduction of the local pressure below the vapor pressure. The low pressure usually results from local flow separation of a high velocity flow along the boundary. Vapor bubbles are swept downstream until they reach a zone with pressures high enough for their collapse. A shock wave is formed during the collapse, as well as a microjet if the collapse happens close to a boundary. The resulting pitting harms the boundary.

The inception of cavitation can be assessed with the cavitation index σ = (p + p_a − p_v)/(ρu²/2), where p is the local pressure, p_a the atmospheric pressure, p_v the vapor pressure, ρ the fluid density and u the average flow velocity. Due to local velocities and pressure fluctuations, cavitation inception does not start at σ = 0 but at a critical cavitation index σ_c, which depends on the geometric chute configuration and surface irregularity.

Multiple studies summarized in [8] determined σ_c for various geometric singularities. The critical cavitation index below which damages are likely is σ_c = 0.2 on smooth chutes with strict construction tolerances [8,9,10].

Stepped chutes are probably more prone to cavitation due to steps protruding into the flow. Using acoustic measurements, [11] showed a large increase in cavitation activity for 0.30 ≤ σ ≤ 0.65 for two stepped chute angles of φ = 21.8° and φ = 68.2°. They suggested that the critical cavitation index σ_c is related to the Darcy friction factor f with σ_c = 4f. A consecutive research [12], showed that after a few hours of operation at σ = 0.30, consistent damages were observed on steps for φ = 21.8°, while lesser damages were observed for φ = 68.2°. Using pressure measurements for φ = 51.3°, [6] showed that the lowest pressures occur just upstream of the inception point on the vertical face close to the step edge.

For spillways, flow aeration is the most economical measure to avoid damages. A few researches [13,14,15] showed that an average air concentration of 5–8% eliminated damages. However, the local air concentration close to the samples investigated for damages remains unknown in these studies.

1.3. Chute Aerators

On smooth chutes, aerators artificially entrain air into high velocity flow to limit cavitation damages. Many researches of the 1980s [16,17,18,19,20,21,22,23] focused on the air entrainment coefficient β = q_A/q—the ratio of specific air discharge q_A to specific water discharge q—and showed that it is mainly related to the jet length L, the approach flow Froude number F_o and the air cavity sub-pressure Δp. Further research investigated the stream-wise air transport and air detrainment to determine the bottom air concentration C_b close to the aerator [24] and far downstream [25].

Little research was performed on stepped chutes. Two geometrical configurations were investigated for limited sets of parameters: the deflector aerator commonly used on stepped chutes [26], and a novel the step aerator [27,28]. The results showed a significantly better performance of the deflector aerator compared to the step aerator. Alternatives to aerate stepped chutes include lateral contraction of the flow with flaring gate piers [29], abrupt lateral contraction [30], and macro-roughness elements on the stepped chute that displace the inception point upstream [31]. The latter seems however unlikely to be an effective solution for high specific discharges.

A few prototype stepped spillways with various aerator designs are reported: Dachaoshan dam (China) [32,33], Wadi Dayqah dam (Oman) [34,35], Boguchany dam (Russia) [36], and enlarged Cotter RCC dam (Australia) [37]. They were all designed for q ≥ 30 m²/s and included physical model tests.

1.4. Objectives

Potential cavitation damages currently limit stepped spillway designs for high specific discharges (q ≤ 30 m³/s/m). Since research on aerators in the past decades solved a similar issue on smooth chutes, some preliminary studies of aerators on stepped chutes were carried out with promising results [26,27,28].

The objective of the present study was to investigate the performance of a deflector aerator at the beginning of stepped chutes to prevent cavitation damages and therefore allow high specific discharges. The deflector aerator was studied on a physical model [38]. The influence of all relevant parameters was for the first time systematically varied to study the aerator performance and its effect on the downstream flow features [38,39].

2. Experimental Setup

2.1. Physical Model

The herein described study was performed in the frame of the work cited as [38]. A 0.5 m wide prismatic channel at the Laboratory of Hydraulic Constructions (LCH) of EPFL was used (Figure 1a) [38]. One sidewall was made of acrylic to allow flow observation and it had an adjustable bottom angle φ (Figure 1b). The stream-wise channel length was 6 m for a chute angle φ = 50° and 8 m for φ = 30°.

The discharge was supplied to the channel via a jet-box [40]. It generated the transition between pressurized to free surface flow. Guiding walls inside the jet-box ensured a homogeneous flow distribution. Unlike an uncontrolled ogee, the jet-box independently set the approach flow depth h_o and Froude number F_o. The jet-box opening height h_jb could be varied in the range 0.015 ≤ h_jb ≤ 0.113 m, and the maximum discharge provided by the pump if the jet-box was fully opened was Q = 0.245 m³/s, corresponding to a unit discharge of q = 0.490 m²/s.

The channel was equipped with steps made of folded aluminum sheets with a step height s = 0.06 m. The fold radius was 0.0055 m for φ = 50° and 0.006 m for φ = 30°. A 0.47 m long smooth section was located between the jet-box (protruding 0.28 m in the channel) and the first step. In total, 67 steps were installed for φ = 50° and 60 steps for φ = 30°. Rectangular wood inserts with a 0.001 m chamfer could be placed in each aluminum step to have a step height s = 0.030 m, which doubled the step count.

The aerator consisted of an aerator-box, as well as an interchangeable deflector and air duct (Figure 2). The aerator-box was impervious and had a sufficiently large cross-section to limit air head losses and therefore a sub-pressure in the cavity under the jet created by the deflector. The air slot below the deflector had a width of 0.48 m and a height of 0.02 m. The air duct was connected perpendicularly to the aerator-box to measure the air discharge. Two ducts were used: a 1 m long one with a 0.103 m internal diameter (for tests with low Froude numbers of F_o = 3.2) and a 1.8 m long one with a 0.153 m internal diameter (for all other tests).

2.2. Instrumentation

The water discharge was measured with an electromagnetic flowmeter (ABB COPA-XE, Zürich, Switzerland; Q_max = 0.667 m³/s; accuracy of ±0.005 Q_max for Q > 0.047 m³/s). The approach flow depths were measured with a point gauge at x = −0.20 m. Two piezoresistive pressure transducers (Baumer ED701 J20.633, Frauenfeld, Switzerland; p_max = 10 kPa; accuracy of ±0.001 p_max) were installed on the bottom at x = −0.27 m to estimate flow turbulence. The sub-pressure in the cavity below the jet Δp/(ρg) was measured using a U-shaped water manometer placed in the center of the channel at the exit of the air slot. A thermoelectric anemometer (Schiltknecht ThermoAir64, Gossau, Switzerland; u_max = 5 m/s; accuracy of 0.005 u_max + 0.015 u) was placed in the center of the duct cross-section and at 60% of the length from the intake to calculate the air discharge assuming a logarithmic velocity profile.

The local air concentration C was measured during 20 s with a phase-detection fiber-optical probe using an 80 μm prism at the tip (RBI Instrumentation, Meylan, France). Probes of the same manufacturer were used by [31,41,42,43,44,45]. The measurement error was found to be below 5% by [41]. The probe was compared to a phase-detection conductivity probe and showed a good agreement in the air concentrations measured [46]. Air concentration profiles were measured at the edge of the steps, starting from step 1 and with a regular stream-wise interval of 3 steps (steps 1, 4, 7, 10…) for the steps of s = 0.06 m. For the steps of s = 0.03 m, the same coordinates x was kept (steps 2, 8, 14, 20…). A total of 25 points were systematically measured for each profile. The depth-wise interval ∆z was kept constant for all profiles of the same test and varied in the range 0.004 ≤ ∆z ≤ 0.01 m. The number of profiles measured were 19 for φ = 50° and 18 for φ = 30°.

2.3. Test Program

The tests were designed to study the influence of the chute geometry—chute angle φ and step height s—the influence of the approach flow conditions—Froude number F_o and flow depth h_o—as well as the deflector geometry—deflector angle α and deflector height t (Figure 1b). A total of 18 reference tests without the aerator and 44 aerator tests were performed (

Table 1. Test numbers and their parameters: deflector angle α, deflector height t, chute angle φ, step height s, Froude number F_o and flow depth h_o.

α	t	φ	s	Fo (-)	Fo (-)	Fo (-)	Fo (-)	Fo (-)
(°)	(m)	(°)	(m)	3.2	5.5	5.5	5.5	7.5
				ho (m)	ho (m)	ho (m)	ho (m)	ho (m)
				0.075	0.052	0.075	0.092	0.075
-	-	50	0.06	17	15	13	14	16
-	-	50	0.03	10	9	8	7	6
-	-	30	0.06	41	40	38	39	42
-	-	30	0.03	56		54		55
5.71	0.015	50	0.06	29	33	32	30	31
9.46	0.015	50	0.06	27	25	24	26	23
9.46	0.015	50	0.03			12
9.46	0.015	30	0.06			47
9.46	0.015	30	0.03			58
9.46	0.030	50	0.06	18	21	20	19	2
9.46	0.030	50	0.03	3	2	1	4	5
9.46	0.030	30	0.06	52	46	44	45	43
9.46	0.030	30	0.03	62		57		58
14.04	0.015	30	0.06			48
14.04	0.030	50	0.06	28	34	35	36	37
14.04	0.030	50	0.03			11
14.04	0.030	30	0.06			49
14.04	0.030	30	0.03			60
14.04	0.045	30	0.06	53		50		51
14.04	0.045	30	0.03			61

).

In addition, two reference tests and two aerator tests with a metal grid (0.008 m square holes every 0.01 m for a thickness of 0.0015 m, placed from x = −0.48 to x = −0.19 m) were performed to study the influence of the approach flow turbulence. Finally, two aerator tests were performed with pressurized air injected in the water supply pipe upstream of the jet-box to study the effect of approach flow pre-aeration. These 6 tests were performed with the chute φ = 30° and s = 0.03 m, the approach flow F_o = 5.5 or 7.5 and h_o = 0.075 m, as well as with the deflector α = 9.46° and t = 0.03 m.

2.4. Scale Effects

As usual for free surface flows, a Froude similitude was adopted to conserve the ratio between inertia and gravity forces. For two-phase flows, forces due to turbulence and surface tension should also be considered. They are respectively described by the Reynolds number R(L_ref) = uL_ref/ν and the Weber number W(L_ref) = u/(σ_w/ρ/L_ref)^0.5, where L_ref is a reference length, ν is the kinematic viscosity, ρ the water density and σ_w the surface tension between air and water. The Froude similitude leads to air bubbles being too large in the model, which results in a higher detrainment rate and lower transport capacity compared to a prototype [47]. The water drops of the disintegrating jet are also too large, although the jet dispersion is not affected [48].

Various smooth chute aerators, stepped spillways and other hydraulic structures studies have provided recommendations on the Reynolds number R, the Weber number W or the model scale factor λ to limit scale effects [21,47,48,49,50,51,52,53,54,55,56]. The recommendations depend on the characteristic being studied: air entrainment coefficient, average air concentration, air concentration profile, velocity profile, air bubble properties, etc. In addition, [57] highlighted that a turbulent velocity of 0.25 to 0.3 m/s is required to break the surface tension and to initiate the aeration process.

The present study mainly focused on the average and bottom air concentrations. For these characteristics, the recommendations are R_o = R(h_o) ≥ 10⁵ and W_o = W(h_o) ≥ 110–140. The tests performed were in the ranges of 2.02 × 10⁵ ≤ R_o ≤ 4.86 × 10⁵ and 87 ≤ W_o ≤ 209. The Reynolds number recommendations was respected for all tests, while the two approach flow conditions with the lowest discharges (F_o = 3.2 with h_o = 0.075 and F_o = 5.5 with h_o = 0.052) might have been affected by scale effects based on the Weber number recommendations. However, the turbulence generated by the aerator and the steps might lower the scale effects limit observed in previous studies.

3. Results

3.1. Reference Tests without Aerator

The inception point was defined where the bottom air concentration reached C_b = 0.01. The inception position x_i, inception mixture depth h_i, inception equivalent blackwater flow depth h_wi and inception average air concentration C_ai were linearly interpolated between the two profiles adjacent to C_b = 0.01.

The mixture flow depth h_i at the inception point showed a good agreement with literature [4,6,58,59] (Figure 3a). The two tests slightly below the general trend are the tests with F_o = 7.5 and s = 0.06 m. At this Froude number, the flow was decelerating when reaching the steps (increasing h_w). With the step height s = 0.06 m, the inception point occurred during the deceleration, while it occurred after the deceleration phase for s = 0.03 m. These two tests are unusual cases for stepped spillway and were not considered for the inception point flow depth analysis. The following relationship was obtained for 30° ≤ φ ≤ 50° (r² = 0.993)

$$\frac{h_i}{k}=0.384·{\mathrm{F}}_{\mathrm{k}}^{0.614}$$

(1)

where k = s·cosφ is the step roughness and F_k = q/(gk³ sin φ)^0.5 the step roughness Froude number.

The equivalent blackwater flow depth at the inception point h_wi gave slightly smaller depths than [60], which might be explained by a different method to locate the inception point (Figure 3b). The following relationship was obtained for 30° ≤ φ ≤ 50° (r² = 0.988)

$$\frac{h_{wi}}{k}=0.287·{\mathrm{F}}_{\mathrm{k}}^{0.626}$$

(2)

The average air concentration at the inception point C_ai decreased with an increasing chute angle φ with C_ai = 0.241 for φ = 30° and C_ai = 0.201 for φ = 50°. The influence of the chute angle φ was previously found by [4] and slightly lower values were obtained in the present study. The results agreed well with C_ai = 0.20 found by [58,60] for φ = 53.1°.

The flow depths at the end of the channel agreed well with the uniform mixture depth h_90u and uniform equivalent blackwater flow depth h_wu suggested by [61].

The uniform flow average air concentration C_au appeared to converge to the C_au values of smooth chutes as suggested by [4,62,63]. It signifies that C_au only depended on the chute angle φ. The influence of the step height s on C_au found in some relationships [4,64] might therefore be the consequence of insufficient flow development due to limited channel lengths. The C_au of [65] was used to normalize C_a in aerator tests results.

The uniform bottom air concentration C_bu also appeared to depend on φ. A value of C_bu = 0.17 is suggested for φ = 30°, and C_bu = 0.32 for φ = 50°. These values were used to normalize C_b in aerator tests results. Compared to smooth chutes [66], C_bu was close for φ = 30° (C_bu = 0.18), but there was an important difference for φ = 50° (C_bu = 0.50). The difference either indicates that steps have an influence or that uniform flow was not fully reached.

Based on the development of C_a and C_b presented in [38], a length of at least X = (x − x_i)/h_c ≥ 20 appeared to be required to reach quasi-uniform flow conditions.

3.2. Presentation of a Typical Aerator Test

The air concentration distribution obtained in a typical aerator test is presented in Figure 4a.

The deflector is shown with a black triangle at x ≤ 0. Above, the black line is the flow depth measured with the point gauge. Below, the 0.02 m high air slot is visible in the vertical face of the first step. The deflector combined with the aeration underneath allowed for the generation of a free jet. The wavy pattern of the jet surfaces particularly visible for 0 ≤ x ≤ 0.5 m results only from contour interpolation artifacts. The jet was formed of a blackwater core (0 ≤ C ≤ 0.05) with a decreasing thickness in the stream-wise direction as both jet surfaces became gradually aerated. For long and thin jets, the blackwater core ended within the jet, whereas for short and thick jets, the blackwater core ended due to the jet impact on the steps. An air cavity was formed under the jet and allowed for the aeration of the lower jet surface.

The jet length L was defined via the impact of the lower jet surface trajectory on the pseudo-bottom [38]. A roller was located directly upstream of the jet impact. It was multiple steps high, and higher than that observed on smooth chutes [67,68]. The roller included water ejected from the jet by flow turbulence and the resulting nappe flow on the steps. After the jet impact, spray was formed, which increased the water surface z₉₀ with a maximum at x = 3.15 m in this example. Spray returned to the flow close to the channel end.

The average air concentration C_a steadily increased along the jet (Figure 4b), and a local maximum was attained at x ≈ 1.4 m. The ensuing decrease was due to the inclusion of the roller in the profile and to the jet impact where air was compressed and detrained. A second maximum at x ≈ 3.15 m was linked to spray. A gradual convergence towards the uniform average air concentration C_au was then observed although quasi-uniform flow conditions were not exactly reached for this test.

The bottom air concentration was by definition C_b = 1 in the air cavity (Figure 4b). However, there was generally a small number of drops below the jet that slightly decreased C_b. Upstream of the jet impact, the roller decreased C_b and a minimum often occurred in the roller (here at x = 1.5 m). After the jet impact, there were various trends that will be described further below. Unlike what was observed for smooth chute aerators, C_b did not rapidly decrease towards values below a few percent. The minimum air concentration C_min was located in the center of the jet (blackwater core) at the beginning, and was then located at the pseudo-bottom after the jet impact as indicated by the overlap of C_b and C_min.

Finally, Figure 4c shows a photo of the jet. At takeoff (x = 0), both the upper and lower jet surface became white, indicating the onset of aeration. The blackwater core can be seen until step 8. The sidewall locally slowed down the jet and affected the local aeration. The consequence was that the jet length and blackwater core extended further than it is visible in Figure 4a. After the jet impact, spray began as indicated by a rough surface.

3.3. Flow Zones

To identify flow zones, the stream-wise coordinate x was normalized with the jet length L as used by [48] for smooth chutes. The quasi-uniform flow values identified above in reference tests were used to normalize flow properties (Figure 5). The behavior and zones of stepped chutes were similar to smooth chutes.

In the approach flow (x/L ≤ 0), three groups of normalized flow depths z₉₀/h_90u can be distinguished and reflect the three approach flow Froude numbers.

In zone I, the jet zone where 0 ≤ x/L ≤ 1, the flow upper surface elevation was highly affected by the deflector geometry. The average air concentration C_a gradually increased until a maximum was reached at x/L ≈ 0.8. The inclusion of the roller in the air concentration profiles then decreased C_a although the air entrainment continued at the surface. A minimum appeared at the jet impact (x/L ≈ 1) due to flow compression and air detrainment around the jet impact. The bottom air concentration was C_b ≈ 1 in the jet zone for x/L < 0.6 due to the air cavity. The roller was then encountered, which very rapidly decreased C_b. At the jet impact, C_b was around the quasi-uniform bottom air concentration C_bu.

In zone II, the spray and reattachment zone where 1 ≤ x/L ≤ 3, the flow upper surface elevation showed fluctuations due to the jet impact and spray. The average air concentration C_a rapidly increased directly after the jet impact. A majority of tests reached a second C_a maximum in this zone. A decrease of the bottom air concentration C_b below C_bu was observed following the jet impact. Some tests showed a C_b minimum and a maximum before reaching C_bu while others directly stabilized around C_bu. Tests with F_o = 3.2 had a Weber number W_o ≈ 88 and showed a different trend in C_b likely due to scale effects.

In zone III, the far-field zone where x/L ≥ 3, the flow characteristics stabilized and tended towards the quasi-uniform values observed in chutes without an aerator.

3.4. Air Entrainment Coefficient

The air entrainment coefficient β = q_A/q—the ratio of specific air discharge q_A to specific water discharge q—is commonly expressed as a function of the relative jet length L/h_o. Since the jet length L is itself a function of the chute, flow and deflector characteristics, the air entrainment coefficient can be empirically derived from these characteristics. The following relationship best described the air entrainment coefficient for stepped chute aerators (r² = 0.906, Figure 6)

$$\beta =0.013·{\mathrm{F}}_{\mathrm{o}}·{\left(1+\sin \phi \right)}^{1.5}·{\left(1+\tan \alpha \right)}^3-0.096,$$

(3)

where F_o is the approach Froude number, φ the chute angle and α the deflector angle.

The approach flow Froude number F_o and deflector angle α were used by [24] for smooth chutes. The addition of the chute angle φ for stepped chutes indicated that the process of air entrainment depended on φ with the presence of steps. The relative deflector height t/h_o had only a marginal an effect on β, and the relative step height s/h_o had an even smaller effect.

No satisfactory relationship providing β for both smooth and stepped chutes could be obtained due to the small influence of s/h_o, which is the only characteristic that differed in the two chute types. Ref. [69] showed that air detrainment is related to the jet impact angle γ on the chute bottom. On stepped chutes, the jet impacted the horizontal face of the steps whereas on smooth chutes the jet impacts a surface at an angle of φ. This likely important difference cannot be considered with s/h_o. A different process of air entrainment, detrainment and circulation in the cavity might occur, especially since the roller seemed to extend further upstream on the stepped chute.

Equation (3) shows that for a deflector with α = 10° air entrainment might start at F_o = 2.0 for φ = 50° and at F_o = 2.5 for φ = 30°. However, these values were not tested on the model.

3.5. Jet Characteristics

A common approach to express the relative jet length L/h_o as a function of the basic parameters was investigated and the following relationship was obtained for a small sub-pressure (i.e., Δp/(ρgh_o) ≤ 0.1, r² = 0.979, Figure 7a)

$$\frac{L}{h_o}=0.5·{\left(1+\sin \phi \right)}^2·{\left(1+\tan \alpha \right)}^4·{\mathrm{F}}_{\mathrm{o}}^{1.4}·{\left(t/h_o\right)}^{0.4}$$

(4)

which is valid for 0° ≤ φ ≤ 50°, 3.2 ≤ F_o ≤ 10.4, 5.7° ≤ α ≤ 20°, 0.10 ≤ t/h_o ≤ 1.65 and Δp/(ρgh_o) ≤ 0.1. An alternative method based on jet trajectories using a ballistic parabola was presented in [38].

The relative jet blackwater core length L_bwc/h_o for the present study was shorter than the estimation obtained from [70] (Figure 7b). All tests with F_o = 3.2 and all tests with h_o = 0.052 m except one (test 34 with the steepest deflector) still had a blackwater core at the jet impact and therefore no L_bwc could be calculated. The results with F_o = 5.5 showed that h_o still had an influence on L_bwc/h_o, which was likely an effect of a different approach flow turbulence for each h_o. The slope of the [70] relationship could be adapted with satisfactory results to (r² = 0.798)

$$\frac{L_{bwc}}{h_o}=49·{\mathrm{F}}_{\mathrm{o}}^{-1}·{\left(1+\tan \alpha \right)}^{-0.5}·\left(1+\sin \phi \right)$$

(5)

3.6. Development of Air Concentrations Downstream of the Aerator

3.6.1. Average Air Concentration

The development of the average air concentration C_a downstream of the aerator showed common characteristics with three distinct extrema (Figure 5b). The first was a maximum shortly before the jet impact. This point will be referred further as the jet maximum with its streamwise coordinate x₁ and average air concentration C_a1. It was quickly followed by an impact minimum close to the jet impact (x₂ and C_a2). The average air concentration increased again in the spray zone to a second maximum, the spray maximum (x₃ and C_a3). Finally, the average air concentration tended towards the quasi-uniform average air concentration C_au.

The position of the jet maximum was on average x₁/L = 0.78. The C_a development in the jet is a function of the blackwater core length L_bwc (Figure 8a). The data agreed with [70] with slightly lower values. An adapted relationship was found for x/L_bwc < 3

$$C_a=\tanh \left(0.32{\left(x/L_{bwc}\right)}^{0.7}\right)$$

(6)

The jet maximum average air concentration C_a1 can be determined by substituting x₁ = 0.78 L and L_bwc from Equation (5) in Equation (6).

The position of the impact minimum was in average x₂/L = 1.03. The minimum average air concentration C_a2 was related to the relative jet length L/h_o and could be better expressed directly from the base parameters (r² = 0.863, Figure 8b)

$$C_{a2}=0.812·{\left(1+\sin \phi \right)}^{1.1}·{\left(1+\tan \alpha \right)}^{2.5}·{\mathrm{F}}_{\mathrm{o}}^{1.1}·{\left(t/h_o\right)}^{0.15}$$

(7)

On smooth chutes, the air detrainment at the jet impact was related to the jet impact angle γ [69,71]. On stepped chutes, there was no correlation of the air detrainment with γ and no relationship could be derived.

Unlike the two previous extrema, the spray maximum streamwise coordinate x₃ varied for each test. A few tests were excluded of the analysis because the jet was too long and the spray maximum was not reached before the end of the channel, or the spray maximum was not well defined. The coordinate x₃/L was somewhat correlated with the relative jet length L/h_o, with x₃/L decreasing for long L/h_o. The following relationship was obtained for 1.5 ≤ x₃/L ≤ 3 (r² = 0.958, Figure 9a)

$$\frac{x_3}{L}=1+8.26·{\left(1+\sin \phi \right)}^{-1}·{\left(1+\tan \alpha \right)}^{-5}·{\mathrm{F}}_{\mathrm{o}}^{-0.75}·{\left(t/h_o\right)}^{-0.55}·{\left(s/h_o\right)}^{0.3}$$

(8)

The relationships starts at x₃/L = 1 which is the jet impact point. The four first parameters also influence the jet length L; thus, the negative exponents indicate that x₃ increases proportionally less than L when one of the four parameters increases. A larger relative step height s/h_o increases x₃/L which suggests a dampening of the deflection at the jet impact by high steps.

The spray maximum average air concentration C_a3 can be obtained from (r² = 0.876, Figure 9b)

$$C_{a3}=0.491·{\sin }^{0.7}\phi ·\left(1+\tan \alpha \right)·{\mathrm{F}}_{\mathrm{o}}^{0.25}·{\left(t/h_o\right)}^{0.15}$$

(9)

3.6.2. Bottom Air Concentration

Trends in the bottom air concentration C_b were less distinct than for C_a (Figure 5c). The bottom air concentration decreased along 0.5 ≤ x/L ≤ 1 due to the roller. Some tests showed a roller minimum. This location was not considered critical for cavitation since C_b > 0.1 and the flow velocities were small. After the jet impact at x/L = 1, C_b showed a rapid but continuous decrease similar to observations on smooth chute aerators, until a minimum was reached (x₄ and C_b4). From the minimum, C_b increased to a maximum (x₅ and C_b5), which was more pronounced for φ = 50° than for φ = 30°. The maximum was not always distinct, especially for φ = 30°. After the maximum, C_b tended towards the quasi-uniform bottom air concentration C_bu.

Similarly to x₃, the streamwise coordinate of the minimum x₄/L was correlated with L/h_o. The minimum x₄ was close to the jet impact for long L/h_o, and further away for shorter L/h_o. A relationship for x₄ was obtained with the four parameters characterizing the jet length (r² = 0.841, Figure 10a)

$$\frac{x_4}{L}=3.22·{\left(1+\sin \phi \right)}^{-1}·{\left(1+\tan \alpha \right)}^{-1.75}·{\mathrm{F}}_{\mathrm{o}}^{-0.17}·{\left(t/h_o\right)}^{-0.23}$$

(10)

The minimum bottom air concentration C_b4 was obtained for 1 ≤ x₄/L ≤ 2 from (r² = 0.924, Figure 10b)

$$C_{b4}=0.184·{\sin }^{1.4}\phi ·\left(1+\tan \alpha \right)·{\mathrm{F}}_{\mathrm{o}}^{0.3}·{\left(t/h_o\right)}^{0.15}·{\left(s/h_o\right)}^{-0.3}$$

(11)

Tests with W_o < 100 showed measured values as low as half the estimation of Equation (11) indicating scale effects that are discussed further below.

As for x₃ and x₄, the streamwise coordinate of the minimum x₅/L was correlated with L/h_o. Many tests with φ = 30° did not show a distinct maximum and were therefore excluded. A relationship for x₅ was obtained for 1.4 ≤ x₅/L ≤ 3.5 (r² = 0.845, Figure 11a)

$$\frac{x_5}{L}=53.8·{\left(1+\sin \phi \right)}^{-2}·{\left(1+\tan \alpha \right)}^{-4}·{\mathrm{F}}_{\mathrm{o}}^{-1}·{\left(t/h_o\right)}^{-0.3}·{\left(s/h_o\right)}^{0.3}$$

(12)

The relative deflector height t/h_o and relative step height s/h_o have opposite exponents and can be combined to s/t.

The maximum bottom air concentration C_b5 was obtained from (r² = 0.782, Figure 11b)

$$C_{b5}=0.502·{\sin }^{1.5}\phi ·{\mathrm{F}}_{\mathrm{o}}^{0.2}·{\left(t/h_o\right)}^{0.25}·{\left(s/h_o\right)}^{-0.1}$$

(13)

Quasi-uniform flow was considered to be attained at x/L ≥ 4.

3.6.3. Comparison with Preliminary Tests

The present study tests are compared in terms of bottom air concentration with the tests of [26] in Figure 12. The difference between the two datasets is explained by a partial aeration of the flow in the tests by [26]. In the latter, the aerator only had a local effect close to the bottom. This performance difference can be explained by a smaller relative deflector height 0.04 ≤ t/h_o ≤ 0.24 in [26] compared to 0.16 ≤ t/h_o ≤ 0.60 for the present study. [57] showed that a deflector generates turbulence in the flow. A small deflector likely created less turbulence than a larger one, and therefore less air is entrained. It also shortened the relative jet length L/h_o, with values ranging from 0.5 ≤ L/h_o ≤ 10.3 for [26] compared to 6.3 ≤ L/h_o ≤ 44 for the present study, which further reduced air entrainment. Due to the partial aeration, the air was gradually detrained as observed for smooth chutes [24,25] until the entire depth became fully aerated from the surface (inception point). These results demonstrate minimal deflector dimension requirements to obtain the relatively high C_b observed in the present study. It is recommended to respect α ≥ 5.71° for deflector angle, t/h_o ≥ 0.16 for the relative deflector height and L/h_o ≥ 8 for the jet length.

3.6.4. Scale Effects

Tests with F_o = 3.2 (W_o = 88 and R_o = 2.05 × 10⁵) showed a significant underestimation of the minimum bottom air concentration C_b4 due to scale effects (Figure 10). However, they showed no clear evidence of scale effects for the maximum bottom air concentration C_b5 (some tests did not show a maximum) or the general development of the average air concentration C_a. Tests with the same unit discharge q = 0.205 m³/s but with a higher Froude number F_o = 5.5 (R_o = 2.05 × 10⁵ and W_o = 106) showed no clear evidence of scale effects for C_a or C_b. Therefore, values of W_o ≥ 100 and R_o ≥ 2 × 10⁵ are suggested to limit scale effects. It highlights that a Reynolds number limit is not sufficient for F_o < 5 as explained by [72].

3.6.5. Cavitation Damages Protection

Excluding tests with scale effects, a minimum bottom air concentration C_b = 0.09 was observed for φ = 30° and C_b = 0.16 for φ = 50°. These bottom air concentrations were higher than the air concentrations observed by [13,14,15] to limit cavitation damages. For the configurations tested, a stepped spillway was fully protected against cavitation damages downstream of one single aerator.

3.7. Effect of Turbulence and Pre-Aeration

In addition to the six parameters varied, literature shows that turbulence has an influence on air entrainment [51,57]. Having steps upstream of the aerator instead of the smooth bottom used in the model would increase turbulence. To qualitatively evaluate the influence of increased turbulence on the aerator performance, a grid was attached to increase the bottom roughness. Holes in the grid were cut around the pressure transducers. It reduced the average pressure upstream of the deflector by 15 to 35% (helping dissipate the remaining jet-box pressure) and increased root mean square of the pressure fluctuations by 40 to 60% compared to tests without the grid.

In special circumstances, for example if another aerator upstream is present or for small discharges, the approach flow can be partially aerated. Two tests were performed to evaluate the resulting aerator performance. Pressurized air was injected 15 m upstream of the jet-box to ensure a homogeneous distribution within the flow. An air concentration profile was measured at x = 0 m without a deflector and the jet-box opening height was iterated until the equivalent blackwater depth reached h_w = 0.075 m. For both tests, a unit air discharge of 0.080 m²/s was measured at x = 0 m. The deflector was then installed, and the test started.

Compared visually to the standard test (Figure 13), the grid test showed an increase in the jet spread revealed by a shorter blackwater core and an impact of the lower jet surface on the pseudo-bottom closer to the aerator. Less spray was observed for the grid test downstream of the jet impact. The pre-aerated test showed exclusively whitewater. No distinct difference could be observed on the upper jet surface, while a slightly larger spread was visible for the lower jet surface. This spread was lower than that of the grid test. As for the grid test, less spray was observed downstream of the jet impact as compared to the standard test.

Both the grid and pre-aerated tests showed a decrease in the jet length L as well as an increase in the air entrainment coefficient β and air cavity sub-pressure Δp/(ρgh_o) as compared to the standard tests (

Table 2. Jet characteristics of standard, grid and pre-aerated aerator tests.

Test	Type	Fo (-)	L (m)	β (-)	Δp/(ρgho) (-)
57	standard	5.49	1.22	0.112	0.00
64	grid	5.52	1.00	0.216	0.04
67	pre-aerated	5.54	0.95	0.149	0.01
58	standard	7.51	1.93	0.207	0.07
65	grid	7.29	1.56	0.297	0.16
68	pre-aerated	7.47	1.66	0.244	0.09

). This was due to a higher turbulence that enhanced the jet spread.

The average air concentration C_a was significantly increased in the jet for the grid and pre-aerated tests as compared to the standard tests, which lead to a higher jet maximum C_a1 (Figure 14a). The impact minimum C_a2 was, however, not proportionally increased. The spray maximum C_a3 was reduced for the pre-aerated and grid tests, suggesting a dampening of the jet impact and the resulting spray by a higher air concentration in the jet. In the far-field zone, the grid tests were similar to the standard tests, whereas the pre-aerated flow tests had a higher C_a.

The bottom air concentration C_b was decreased at the minimum C_b4 and maximum C_b5 compared to the standard tests (Figure 14b). For the pre-aerated tests, C_b was increased at the minimum C_b4 and there was no clear trend for the maximum C_b5 (decrease for F_o = 5.5 and increase for F_o = 7.5).

In the far-field zone, the grid tests did not show a significant difference as compared to the standard tests while the pre-aerated tests appeared to converge faster to quasi-uniform flow conditions.

4. Design Example

An example is given to demonstrate the application of the results obtained. An aerator is planned on a stepped chute with a unit design discharge q_D = 60 m²/s, a chute angle φ = 50° and a step height s = 1.5 m. The spillway includes first a smooth standard ogee crest [73] (Figure 15). The ogee design head is H_D = 9.09 m and the natural air self-inception point would be located at a distance L_i = 70.8 m from the crest according to [60].

The tangency point is located at x_cr = 14.97 m and z_cr = 7.89 m, which is far upstream of the inception point. Based on [73], at the tangency point the flow depth is h_wr = 3.49 m and the bottom pressure p_r/(ρg) = 0.50 m. This leads to a mean flow velocity u_r = q/h_r = 17.2 m/s, a Froude number F_r = 2.94 and a head H_r = 7.67 m. Considering an atmospheric pressure p_atm = 101 kPa and a vapor pressure p_v = 1.23 kPa, the cavitation index is σ_r = 0.71.

The bottom pressure is reduced on the ogee due to convex curvilinear flow but is assumed to be hydrostatic downstream of the tangency point. This leads to a discontinuity at the tangency point: the bottom pressure increases from p_r/(ρg) = 0.50 m to 2.43 m, the flow depth from h_r = 3.49 m to 3.79 m, and the cavitation index from σ_r = 0.71 to 0.98.

The first two steps are filled to have a smooth bottom and accelerate the flow to provoke a better aerator performance as well as flow conditions inside the parameter range of the present study. The drawdown curve is calculated until the second filled step edge (x_co = 18.90 m and z_co = 10.89 m) using a Strickler coefficient K = 70 m^1/3/s. The approach flow values obtained are F_o = 3.13 and h_o = 3.13 m.

A deflector with an angle α = 9.46° (slope of 1:6) and height t = 0.65 m (relative height t/h_o = 0.19) is chosen. The deflector takes the length of two steps and its downstream end is aligned with a step edge (x_ct = 22.80 m and z_ct = 13.89 m), resulting in the configuration shown in Figure 15.

The aerator parameters as well as the jet characteristics and the air concentration extrema obtained with the relationships presented in this article are shown in

Table 3. Design example characteristics compared to test 27 using a scale factor λ = 45.

Characteristic	Equation	Example	Test 27	Unit
Chute angle φ		50	50	°
Step height s		1.5	2.7	m
Approach flow Froude number Fo		3.13	3.16	-
Approach flow depth ho		3.35	3.38	m
Deflector angle α		9.46	9.46	°
Deflector height t		0.65	0.68	m
Unit discharge q		60	61.3	m2/s
Relative step height s/ho		0.46	0.80	-
Relative deflector height t/ho		0.19	0.20	-
Air entrainment coefficient β	(3)	0.056	0.050	-
Jet length L	(4)	24.8	28.3	m
Blackwater core length Lbwc	(5)	85.8 1	-	m
Jet maximum coordinate x1		19.3	14.4	m
Jet maximum average air concentration Ca1	(6)	0.11	0.14	-
Impact minimum coordinate x2		25.5	25.0	m
Impact minimum average air concentration Ca2	(7)	0.14	0.13	-
Spray maximum coordinate x3	(8)	68.9	78.0	m
Spray maximum average air concentration Ca3	(9)	0.49	0.52	-
Minimum coordinate x4	(10)	41.4	- 2	m
Minimum bottom air concentration Cb4	(11)	0.21	- 2	-
Maximum coordinate x5	(12)	94.8	- 2	m
Minimum bottom air concentration Cb5	(13)	0.30	- 2	-
Quasi-uniform flow conditions xu		99.1	-	m

¹ Theoretical value as the jet impact ends the blackwater core. ² No values due to scale effects.

. The results are compared with test 27 using a geometrical scale factor λ = 45. The six base parameters are nearly identical except for the step height s, which only has a minor overall influence. The calculated jet length is shorter than the scaled test 27. The C_a development is well described by the three extrema (Figure 16a) whereas the C_b development cannot be compared due to scale effects of the model test (R_o = 2.03 × 10⁵ and W_o = 87, Figure 16b). Quasi-uniform flow is reached 91.2 m after the aerator or 114 m downstream from the crest. Finally, it is important to design the air supply system in a way to limit the cavity sub-pressure to Δp/(ρgh_o) ≤ 0.1 to obtain the aerator performance described above, and to include an efficient drainage system.

For lower specific discharges than q_D, the aerator approach Froude number F_o would increase, leading to a higher air entrainment coefficient β and a similar jet length L. The average air concentration extrema C_a1, C_a2 and C_a3 would increase. The minimum bottom air concentration C_b4 would be approximately stable as the increase of the F_o and t/h_o terms is compensated by a decrease of the s/h_o term. The maximum bottom air concentration C_b5 would increase.

Without the aerator and filled steps, the critical cavitation index σ_c = 0.70 of stepped chutes would be reached shortly downstream of the third step edge considering a friction factor f = 0.1 for the stepped bottom drawdown.

5. Conclusions

The performance of a stepped spillway aerator using a deflector was for the first time systematically investigated by varying the governing parameters on a large-scale physical model. The following conclusions were drawn from aerator tests:

• Three zones can be identified as a function of the jet length L: (i) jet zone for 0 ≤ x/L ≤ 1, (ii) spray and reattachment zone for 1 ≤ x/L ≤ 3 and (iii) far-field zone for x/L ≥ 3. They are similar to the zones observed on smooth chutes;
• The air entrainment coefficient β is mainly influenced by the chute angle φ, the approach flow Froude number F_o and the deflector angle α. For φ = 50° and α = 10°, air entrainment may start at values of F_o = 2.0;
• Relationships are presented for the jet length L;
• Relationships are presented for the three identified average air concentration C_a extrema and two bottom air concentration C_b extrema, which allow to characterize the flow development downstream of the aerator. Unlike smooth chutes, there appears to be no significant air detrainment in the far-field zone as the flow rapidly (x/L ≥ 4) converges to quasi-uniform conditions. One aerator seems thus sufficient to aerate the entire chute;
• The lowest C_b measured at the step edges in all tests is higher than the air concentration recommended by literature to protect against cavitation damages. A deflector aerator is sufficient to provide protection of a stepped spillway when the parameter ranges given below are respected;
• Increased turbulence in the approach flow increases β and has a minor effect on C_a and C_b;
• Scale effects are observed on the bottom air concentration C_b for tests with F_o = 3.2 and W_o ≈ 88, while no distinct effects are observed for the tests with F_o = 5.5 and W_o ≈ 106. An approach flow Weber number W_o ≥ 100 is therefore recommended to limit scale effects.

Due to potential cavitation damages on stepped spillways, an aerator is required for high specific discharges (q ≥ 30 m³/s/m) as one of the most efficient mitigation measures. The results of the present study allow for the design of a stepped spillways aerator and an example was provided. The results of the study apply within the following parameter ranges: 30° ≤ φ ≤ 50°, 0.32 ≤ s/h_o ≤ 1.16, 3.2 ≤ F_o ≤ 7.5, 5.7° ≤ α ≤ 14°, 0.16 ≤ t/h_o ≤ 0.60, Δp/(ρgh_o) ≤ 0.1 and L/h_o ≥ 8.