# Plausible Differences between the Laboratory and Prototype Behaviors of Spillway Aerator Flows

^{1}

^{2}

^{3}

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## Abstract

**:**

^{6}. Failing to meet this premise would bring about errors for prototype predictions; the error extent depends on both model scale and flow magnitude. In terms of cavity pressure drop, the prototype differs by a factor of less than 10 from its scale model with sufficient air supply. In a model, the air concentration along the chute bottom drops considerably within one to two trajectory lengths. The prototype differs from its model in such a way that the air concentration decay is much slower, with a higher level that is maintained over a longer distance downstream of the impact location. This study is intended to provide insight for laboratory studies and engineering design.

## 1. Introduction

_{atm}+ p

_{g}is local reference pressure, p

_{atm}= the atmospheric pressure, p

_{g}= gauge pressure, p

_{v}= vapor pressure of the water, v = mean reference velocity and ρ = density of the water. This index expresses the relationship between the difference of a local absolute pressure from the vapor pressure of the water and the kinetic energy per volume, which is used to characterize the potential of the flow to cavitate. Cavitation damages are expected at locations where C

_{a}is below 0.2. In addition to flow velocity and depth, the occurrence of the damages also depends on local surface irregularities, material strength, structure elevation and length of spillway operation. Since C

_{a}depends on the local flow velocity and pressure, the maximum flow discharge does not necessarily produce the highest cavitation potential for a specific location.

_{b}= (1–2)% already mitigates the damages and the degree of mitigation is proportional to C

_{b}. If the concentration exceeds C

_{b}= 5%, the damages disappear completely [1,4,5,10,11,12]. Mortenseen [13] shows that, with high-strength concrete (31–55 N/mm

^{2}) for chute construction, C

_{b}= 0.5% would be sufficient to protect the chute from cavitation risk. As a defensive measure, aeration is nowadays the engineering standard for cavitation damage prevention in surface spillways, outlet works and other flood structures.

^{3}/s at the full reservoir water level. The aerator includes an air duct with 13 air vents symmetrically placed about the chute center plane. On each side, air is supplied from an air shaft, between which the chute is 35.0 m wide.

_{b}drops below (6–8)%, another aerator is required to protect the chute [4]. In some circumstances, the flow features even large water depth. As a result, cavitation damages cannot be prevented only by the bottom aerator. Side or lateral aerators are required, usually denoted as 3D aeration [17].

## 2. Theoretical Background

_{a}is also affected by the geometry of the air passage. Figure 3 sketches a typical aerator layout with flow pattern. The functional relationship is expressed as [18]:

_{g}|), V and D = approach flow velocity and depth, α = chute slope with the horizontal, t = chute bottom offset at the aerator, s and θ = deflector height and angle with the chute bottom, l and h = groove length and height, k

_{s}= chute surface roughness, ρ

_{a}= air density, μ = dynamic viscosity of water, σ = surface tension of water, g = gravitational acceleration and f

_{1}= functional symbol.

_{i}= theorem symbol. The equations represent the aerator’s air-water system independent of the choice of dimensionless quantities. To simplify, new non-dimensional groups are introduced by rearrangement.

_{2}= functional symbol. Consequently, F, E and φ are the parameters that govern the aerator two-phase flow. In other words, the cavity air pressure is one of the factors that affects the air flow and ought to be scaled in physical model tests.

_{x}the jet thickness at location x. Based on the equilibrium of the acting forces including Δpdx (per unit width), the trajectory can be expressed as:

_{x}≈ D. With this approximation, the solution of the jet trajectory centerline is obtained.

## 3. Model Scale and Air Demand

_{Q}= λ

^{2.5}, water flow velocity scale λ

_{V}= λ

^{0.5}and water flow pressure λ

_{P}= λ. To address the scale effects of air flow, Pinto and Neidert [22] examined the scale effects of Foz do Areia spillway and found that reasonable results could be obtained from the = 8 and 15 scale models. Volkart and Rutschmann [23] performed experiments in a λ = 18.75 model and made comparisons with the prototype. Tan [24] constructed a λ = 8 model to mitigate the scale effects of air entrainment. It was recommended that a model must be large enough to reduce the scale effects, preferably λ < 10. Morton number, expressed in terms of F, W and R, is also a key variable for addressing the scale effects in modelling air–water flows [7,8,25,26,27,28,29]. If the Froude and Reynolds numbers exceed the minimum values, i.e., W = 110 and R = 1.7 × 10

^{5}, the effects of surface tension and viscous force are negligible [30,31,32,33].

^{3}/s. The aerator offset is at elevation +384.0 m and the offset height is 1.66 m. Model tests were performed in three distinct scales, i.e., λ = 20, 40 and 70, which resulted in significant differences in model flow rates. Upon the dam spillway commissioning, prototype observations of air flows were also made, in which several flow discharges were examined. Together with the field data, Figure 7 plots the model test results as a function of X, in which:

^{3}/s flood discharge, is built with a horseshoe-shaped cross-section and four aerators at varying intervals (Figure 2). Each of the three upstream aerators is composed of both an offset and a deflector. All the offsets are 1.5 m high, and the deflectors are 0.5, 0.4 and 0.2 m high, respectively. The fourth one is shaped in a three-dimensional configuration, featuring sudden-narrowed sidewalls and a convex deflector. From up- to downstream, their protection lengths are 81.0, 87.8, 128.6, and 115.6 m, respectively. An air shaft connects each aerator to the atmosphere.

^{2.5}). After the competition, efforts were made to perform prototype measurements of air flow rate, cavity pressure, air concentration, etc., which provided valuable information to the engineering profession. A comparison is made between the prototype and model results of air demand at the design discharge Q = 3200 m

^{3}/s. The results show that, at aerators No. 2 and 4, the prototype values are 3.4 and 2.3 fold the model ones, respectively. Air demand in the prototype is considerably larger than the upscaled values from the model.

^{6}, thus K = 1, implying that the flow velocity exceeds 7.5 m/s and the air flow rate can be converted from model to prototype in light the Froude law. If 13.6 < Ln(R) < 14.3, i.e., 0.85 × 10

^{6}< R < 1.58 × 10

^{6}, thus K is within the range 1.0–1.86 and direct would give rise to errors. If R becomes even smaller, the flow velocity falls below 4.0 m/s and more significant errors are introduced.

## 4. Threshold Velocity of Approach Flow

^{3}/s. It was measured using a flow meter and an overflow weir. Their relative measurement errors were less than ±3% and ±1%. Air was supplied to the cavity through a lateral duct, with Q

_{a}= 0.0756 m

^{3}/s at maximum. Q

_{a}was determined by a calibrated airflow anemometer, with an inaccuracy within ±(1.5–2)%. A propeller-type water-flow velocimeter measured the flow velocity in the flume, with an inaccuracy level of below ±(2–4)%. Water depths were directly measured with a point gauge and referred to the depth in the direction perpendicular to the flume. For a given cross-section, the obtained water-depth result was also checked against Q and its flow velocity. Measurements of flow velocity and water depth were made in the so-called black-water area upstream from the air cavity. The approach flow velocity and depth, measured at the end of the deflector, amounted to V = 6.6–15.4 m/s and D = 0.050–0.251 m, respectively. F = 4.34–21.90, R = (0.69–1.41) × 10

^{6}and W = 325–490. Data affected by air entrainment from the upper free surface are excluded.

_{1}-X relationship, the flume test results, in which:

_{c}) for air entrainment in free surface flows including the aerator flow. In the literature, three classical theoretical explanations are found.

- (1).
- Based on the relative flow motion between water and air in an open channel, Vojnovich and Schwartz [36] attributed the air entrainment mechanism to surface water fragmentation and concluded that V
_{c}**=**6.92–7.05 m/s. - (2).
- Considering the turbulence intensity of water-air interface, the research work performed by Wood [37] led to V
_{c}= 6.25–7.50 m/s. - (3).
- Based upon the eddy energy approach and in combination with his experimental measurements, Yang [38] showed that V
_{c}= 7.05–7.48 m/s for jet flows.Although some differences exist due to different analytical assumptions and experimental conditions, the commonly used range is V_{c}= 7.00–7.50 m/s.

_{c}= 7.00–7.50 m/s. The formula can be used for estimation of air demand in prototype facilities. For engineering design, the air demand per unit width is often expressed in the form.

_{0}= empirical coefficient and L = air cavity length, referring to the distance from the aerator offset to the reattachment point (maximum flow pressure) on the chute bottom. A partial list based on about 10 prototype spillways is given by Shi [12] and Lian et al. [15]. With exception of a large value at Guri dam (K

_{0}= 0.074), K

_{0}ranges between 0.010 and 0.041, which is still significant. K

_{0}depends obviously on both geometrical layout and flow conditions at the aerator. Note also that the L definition varies in different studies. It may be based on visual observations, air concentration criterion or maximum (stagnation) pressure acting on chute bottom.

_{c}= 6.50–7.50 m/s. A detailed experimental description of aerator water-air flow features is found in Hager et al. [39]. The book written by Shi [12] contains a compilation of detailed flume test data of varying aerator configurations, results from spillway aerator models for different dams and, most of all, invaluable field observation data from quite a few existing dam facilities.

## 5. Pressure Drop in Air Cavity

^{2}/s. Model tests confirmed that the air cavity zone was choked, with Δp ≈ 70 kPa at the maximum flow per unit width 125 m

^{2}/s. Afterwards, the deflector was improved, with s = 0.37 m and θ = 5°, which enlarged the transverse cavity in cross section and allowed for more air flow to reach the chute center. Δp varied thus between 5 and 18 kPa within the same discharge interval. 95 days of spillway operation did not see any cavitation damages. The final aerator design included the 0.37 m deflector and a large bottom offset, with an air gallery supplying air from inside the dam [42].

^{3}/s. Within the discharge range tested, Δp = 6.26 to 15.77 kPa.

## 6. Chute Bottom Air Concentration

_{b}) and its streamwise development is the major issue in aerator design and concerns the prediction of the chute length protected by an aerator. The data from the Jinping-I aerators [15] exemplify a generalized pattern. Figure 14 presents a comparison of C

_{b}between its scale model and prototype. C1, C6, C7, C10 and C11 are prototype monitoring stations, with their approximate locations shown in Figure 2. Points ①–⑧ are model measurement points in the vicinity of their respective prototype locations. ①, ③ and ⑤ are 1.65, 4.20 and 5.74 m upstream, and points ②, ④, ⑥, ⑦ and ⑧ are 2.00, 2.11, 8.59,1.00 and 1.51 m downstream (all in prototype size). In both model and prototype, all the points are placed along the tunnel centerline. Transducers used are of resistance type.

_{b}is several times its model value. Though the points are not at the same positions in the model and the prototype, a direct correlation is not unreasonable. The entrained air detrains much quicker in the model because its flow velocity differs $\sqrt{\lambda}$ times from the prototype value. Stations C6 and C10 are right in the impact area or in the backwater within the cavity. At either station, the C

_{b}level in the model is close to its prototype values and the concentration is high. Probably due to the difference in trajectory length between the model and the prototype, it is intricate to tell the exact locations of C11 and ⑧ relative to the impact position. This is the reason why it is not straightforward to explain the C

_{b}levels at C11 and ⑧.

_{b}and its decay between the models and the prototypes. Hydraulic model tests were performed for the aerators in Liujiaxia spillway (λ = 25) and Ertan (λ = 50), and measurements of C

_{b}were made at several flow discharges [12]. The layout of the discharge tunnel at Ertan is shown in Figure 15. It is 926.0 m long, 13.0 m wide and 13.5 m high. As free-surface flow, the max. discharge per unit width amounts to 288 m

^{2}/s and the flow velocity can be as high as 44–50 m/s. Five aerators exist in the tunnel. The data given here refer to aerator no. 2, at t = 1.4 m, s = 0.5, tanθ = 1:12 and tanα = 7.9%. Figure 16 presents their streamwise C

_{b}distribution from the two models, with x defined in Figure 1 or 2. No model–prototype conversion is made.

_{b}change downstream of the aerator offset can be approximated by

_{b}is characterized by a substantial drop within one to two cavity lengths downstream of the impact location. The chute length needed for C

_{b}to approach zero is also moderately affected by the chute slope. Together with other model test data, it can be stated that, for the same aerator configuration, C

_{b}exhibits a slower decay if the model size becomes larger.

_{b}[12]. The gauge points C1, C2 and C3 are 51.8, 181.3 and 198.8 m from the aerator offset. C2 is right on the deflector of aerator no. 3. Several reservoir levels are tested, combined with varying gate openings. For this aerator, L = 28–32 m. Obviously, over a length of 5L (up to point C2) downstream of the impact location, C

_{b}maintains a high level above 5%.

^{7}and 5.70 × 10

^{7}. Together with the prototype results from the Bpatck spillway aerator, the streamwise C

_{b}variations are plotted in Figure 18 and expressed in the form [12]

_{b}maintains a relatively high level. The model–prototype comparisons made for Jinping-I also confirm this. To use a model for prediction of the protected chute length would therefore give rise to significant underestimation. There are no established criteria to upscale or convert air concentration from model to prototype. In this regard, engineering experiences are probably more useful than laboratory tests. Normally, with a ~30 m/s flow velocity at the aerator, the aerator can protect a chute length of (17–20)L. Attaining more data from prototype observations is crucial for the design of future projects.

## 7. Conclusions

^{6}. If this condition is not met, any attempts to scale up the air flow would lead to errors. The error extent depends on both model scale and flow magnitude. Despite minor differences, the experimental velocity threshold agrees with the three analytical approaches of surface air entrainment.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**An aerator in a 25 m wide single-opening spillway, Sweden. (

**a**) spillway during discharge, looking upstream; (

**b**) aerator with air shaft and air vents. At the aerator, the chute is 35 m wide between the air shafts. There are 13 air vents (each 1.0 m

^{2}in area) connecting the chute with the air duct (5.0 m

^{2}in area) leading to the shafts (each 8.7 m

^{2}in area). At the aerator, the unit flow discharge amounts to ~43 m

^{2}/s at the full reservoir water stage.

**Figure 2.**Aerators in Jinping-I flood discharge tunnel with free-surface flow (adapted from [15]). At the design reservoir water level, Q = 3200 m

^{3}/s at the 100% gate opening.

**Figure 5.**Two examples of formation of enclosed air cavity subjected to sub-atmospheric pressure. (

**a**) Air pocket behind a horizontal sharp-edged weir bounded by side walls; (

**b**) air cavity surrounded by the flow jets over the flip buckets in gated spillway openings [21].

**Figure 6.**Baishan hydropower scheme. (

**a**) dam layout; (

**b**) longitudinal profile of chute spillway with aerator (adapted from [12]).

**Figure 7.**Baishan spillway aerator, β results from model tests of three scales (λ = 20, 40, 70) and prototype measurements (adapted from [12]).

**Figure 10.**Large test rig of chute spillway aerator at IWHR. The flume is 15.0 m long and 0.2 wide, with its angle adjustable between α = 0–49°. (1) water from pump; (2) rotatable pipe joint; (3) flow meter; (4) control valve; (5) water supply pipe; (6) water supply in flume bottom; (7) test flume with aerator; (8) 3D measurement carriage; (9) tailwater; (10) observation window; (11) horizontal rail; (12) horizontally moving wheel; (13) expansion pipe section; (14) vertically moving wheel; (15) water intake to flume; (16) counterweight (not shown); (17) vertical rail; (18) hoister; (19) safety brake; (20) lifting cradle; (21, 22) lifting wire and guide wheel.

**Figure 11.**Comparisons of air demand (β

_{1}-X) among IWHR high-velocity flume tests, and model and prototype data from the Fengjiashan and Baishan aerators (adapted from [12]).

**Figure 12.**Model–prototype comparisons of air demand (β

_{1}-X) for Foz do Areia spillway aerator (based on the data in [22]).

**Figure 13.**Jinping-I aerators, pressure drop in the air cavities (Δp) from the prototype measurements (based on the data in [15]).

**Figure 14.**Jinping-I tunnel aerators, model–prototype comparisons of chute bottom air concentration C

_{b}(adapted from [15]). Points ①–⑧ are model measurement points in the vicinity of their respective prototype locations.

**Figure 15.**Layout of Ertan flood discharge tunnel, with five aerators (three are shown here). C1, C2 and C3 are prototype measurement points of bottom air concentration.

**Figure 16.**Aerator model tests, streamwise change of air concentration C

_{b}downstream of the impact location. (

**a**) Liujiaxia in scale λ = 25; (

**b**) Ertan in scale λ = 50 (based on the data in [12]).

**Figure 17.**Prototype measurement results of C

_{b}at aerator no. 2 in Ertan flood discharge tunnel (based on the data in [12]).

**Figure 18.**Prototype measurement results from the Fengjiashan and Bpatck aerators, streamwise change of bottom air concentration C

_{b}downstream of the impact location (based on the data in [12]).

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Yang, J.; Li, S.; Lin, C.
Plausible Differences between the Laboratory and Prototype Behaviors of Spillway Aerator Flows. *Water* **2022**, *14*, 3264.
https://doi.org/10.3390/w14203264

**AMA Style**

Yang J, Li S, Lin C.
Plausible Differences between the Laboratory and Prototype Behaviors of Spillway Aerator Flows. *Water*. 2022; 14(20):3264.
https://doi.org/10.3390/w14203264

**Chicago/Turabian Style**

Yang, James, Shicheng Li, and Chang Lin.
2022. "Plausible Differences between the Laboratory and Prototype Behaviors of Spillway Aerator Flows" *Water* 14, no. 20: 3264.
https://doi.org/10.3390/w14203264