8
Chute Aerators: Steep Deflectors and Cavity Subpressure Michael Pfister, D.Sc. 1 Abstract: Chute aerators are applied to high-velocity spillways to entrain air into the flow so that cavitation damage is avoided. Air entrain- ment occurs locally at the aerator, whereas further downstream the flow is deaerated. This process is relevant because it defines the influence range of an aerator. A preliminary study investigated the effect of the aerator geometry and of the approach flow conditions on the streamwise bottom and average air concentration characteristics. Two aspects were excluded, namely, the effect of (1) steep deflectors, which operate more efficiently regarding air entrainment yet with simultaneously poor flow features; and (2) cavity subpressure effect on the streamwise air concentration field. A cavity subpressure reduces, in particular, the streamwise bottom air concentrations or it provokes aerator choking so that the cavitation protection is not ensured. Physical model tests indicate that optimum aerator performance results at deflector angles around 10°, i.e., a slope of 15 relative to the chute bottom with acceptable shock wave formation, spray generation, and jet height. DOI: 10.1061/ (ASCE)HY.1943-7900.0000436. © 2011 American Society of Civil Engineers. CE Database subject headings: Aeration; Cavitation; Deflection; Spillways; Velocity. Author keywords: Aerator; Cavitation; Cavity; Chute; Deflector; Spillway; Subpressure. Introduction Chute aerators are common devices on high-head spillways entraining air to avoid cavitation damages. The aerators separate the flow from the chute bottom to generate a free jet disintegra- ting and entrapping air. The air is then transported downstream of the aerator, thereby avoiding cavitation damage. Three basic aerator geometries are commonly applied: (1) deflector, (2) offset, and (3) a combination of both. The air entrained into the flow is primarily provided by an air supply system connected with the free atmosphere that typically consists of two lateral ducts and a transverse groove, which allows for transverse distribution of the air below the jet. A historical review on the chute aerator development is provided by Hager and Pfister (2009). Extensive model experiments were conducted by Koschitzky (1987), Rutschmann (1988), Chanson (1988), Rutschmann and Hager (1990), and Kökpınar and Göğüş (2002), among others. These studies primarily focused on the aer- ator air entrainment coefficient as the ratio of entrained air to water discharge β ¼ Q A =Q, in that Q A = air supply discharge through ducts and Q = water discharge. However, this global coefficient neither specifies the precise air distribution in the flow nor its detrainment rate. As cavitation damages occur along the chute boundaries, the associated air concentration is of prime interest. Chanson (1989a) proposed two differential equations to derive the air concentration and velocity profiles downstream of aerators on the basis of the flow depth and the streamwise average air concentration. He demonstrated an analogy between developing chute flow and aerator flow. Kramer et al. (2006) illustrated air concentration profiles downstream of chute aerators for a specific test. Their data analysis focused on the air concentration develop- ment in the far field of aerators primarily to study the self-aeration process and to localize the position of a second aerator if the chute bottom air concentration falls below a threshold value. However, the gap in the near field of chute aerators remained open. Pfister and Hager (2010a, b) describe the air concentration field downstream of chute aerators regarding the relevant hydraulic and geometrical parameters, providing predictions of the streamwise average (subscript a) C a and bottom (subscript b) C b air concen- tration development. No direct correlation between β and C a or C b was proposed because of weak data correlation. This first study is considered as a basis for the results presented in this paper. In par- ticular, two restrictions limit the basic investigation: 1. Common prototype aerators typically include deflector angles in the rage of 5.7 to 11.3°, i.e., 110 to 15(DeFazio and Wei 1983; Marcano and Castillejo 1984; Pinto 1984, 1989; Koschitzky 1987; Wood 1991). Steeper deflectors are com- monly not applied because they are reported to generate shock waves, spray, and high jets requiring high sidewalls. However, steep deflectors operate more efficiently than do flat deflectors with the optimum performance for the steepest tested aerator angle at 11.3°. Additional tests were conducted herein to compare the aerator efficiency of common deflector angles of 0° and 5.7° (110), 8.1° (17), 11.3° (15) with excessive steep angles of 18.4° (13) and 26.6° (12). 2. Subpressures in the air supply system limit the aerator per- formance. They control the equilibrium between air demand of the flow and the capacity of the air supply system. Notable subpressures provoke poor flow features that include (1) insuf- ficient protection from cavitation along the spillway due to reduced air entrainment and (2) pulsations in the air supply system. Ultimately, aerators may even become cavitation- generating devices. The effect of cavity subpressure on β was investigated, e.g., by Tan (1984), Chanson (1988), and Rutschmann (1988). The relationship between cavity subpres- sure and the streamwise air concentration field was illustrated by Chanson (1989b) who found a small correlation. In parallel, the deaeration process at the cavity end was described as a 1 Laboratory of Hydraulic Constructions (LCH), Ecole Polytechnique Fédérale de Lausanne (EPFL), ENAC IIC, CH-1015 Lausanne, Switzer- land; formerly, Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH Zurich. E-mail: [email protected] Note. This manuscript was submitted on May 5, 2010; approved on April 8, 2011; published online on September 15, 2011. Discussion period open until March 1, 2012; separate discussions must be submitted for in- dividual papers. This paper is part of the Journal of Hydraulic Engineer- ing, Vol. 137, No. 10, October 1, 2011. ©ASCE, ISSN 0733-9429/2011/ 10-12081215/$25.00. 1208 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / OCTOBER 2011 J. Hydraul. Eng. 2011.137:1208-1215. Downloaded from ascelibrary.org by UNIVERSIDADE FEDERAL DE PARANA on 09/02/13. Copyright ASCE. For personal use only; all rights reserved.

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Page 1: Chute Aerators: Steep Deflectors and Cavity Subpressure

Chute Aerators: Steep Deflectors and Cavity SubpressureMichael Pfister, D.Sc.1

Abstract: Chute aerators are applied to high-velocity spillways to entrain air into the flow so that cavitation damage is avoided. Air entrain-ment occurs locally at the aerator, whereas further downstream the flow is deaerated. This process is relevant because it defines the influencerange of an aerator. A preliminary study investigated the effect of the aerator geometry and of the approach flow conditions on the streamwisebottom and average air concentration characteristics. Two aspects were excluded, namely, the effect of (1) steep deflectors, which operatemore efficiently regarding air entrainment yet with simultaneously poor flow features; and (2) cavity subpressure effect on the streamwise airconcentration field. A cavity subpressure reduces, in particular, the streamwise bottom air concentrations or it provokes aerator choking sothat the cavitation protection is not ensured. Physical model tests indicate that optimum aerator performance results at deflector angles around10°, i.e., a slope of 1∶5 relative to the chute bottom with acceptable shock wave formation, spray generation, and jet height. DOI: 10.1061/(ASCE)HY.1943-7900.0000436. © 2011 American Society of Civil Engineers.

CE Database subject headings: Aeration; Cavitation; Deflection; Spillways; Velocity.

Author keywords: Aerator; Cavitation; Cavity; Chute; Deflector; Spillway; Subpressure.

Introduction

Chute aerators are common devices on high-head spillwaysentraining air to avoid cavitation damages. The aerators separatethe flow from the chute bottom to generate a free jet disintegra-ting and entrapping air. The air is then transported downstreamof the aerator, thereby avoiding cavitation damage. Three basicaerator geometries are commonly applied: (1) deflector, (2) offset,and (3) a combination of both. The air entrained into the flow isprimarily provided by an air supply system connected with thefree atmosphere that typically consists of two lateral ducts and atransverse groove, which allows for transverse distribution of theair below the jet.

A historical review on the chute aerator development is providedby Hager and Pfister (2009). Extensive model experiments wereconducted by Koschitzky (1987), Rutschmann (1988), Chanson(1988), Rutschmann and Hager (1990), and Kökpınar and Göğüş(2002), among others. These studies primarily focused on the aer-ator air entrainment coefficient as the ratio of entrained air to waterdischarge β ¼ QA=Q, in that QA = air supply discharge throughducts and Q = water discharge. However, this global coefficientneither specifies the precise air distribution in the flow nor itsdetrainment rate. As cavitation damages occur along the chuteboundaries, the associated air concentration is of prime interest.Chanson (1989a) proposed two differential equations to derivethe air concentration and velocity profiles downstream of aeratorson the basis of the flow depth and the streamwise average airconcentration. He demonstrated an analogy between developingchute flow and aerator flow. Kramer et al. (2006) illustrated air

concentration profiles downstream of chute aerators for a specifictest. Their data analysis focused on the air concentration develop-ment in the far field of aerators primarily to study the self-aerationprocess and to localize the position of a second aerator if the chutebottom air concentration falls below a threshold value. However,the gap in the near field of chute aerators remained open.

Pfister and Hager (2010a, b) describe the air concentration fielddownstream of chute aerators regarding the relevant hydraulic andgeometrical parameters, providing predictions of the streamwiseaverage (subscript a) Ca and bottom (subscript b) Cb air concen-tration development. No direct correlation between β and Ca or Cb

was proposed because of weak data correlation. This first study isconsidered as a basis for the results presented in this paper. In par-ticular, two restrictions limit the basic investigation:1. Common prototype aerators typically include deflector angles

in the rage of 5.7 to 11.3°, i.e., 1∶10 to 1∶5 (DeFazio andWei 1983; Marcano and Castillejo 1984; Pinto 1984, 1989;Koschitzky 1987; Wood 1991). Steeper deflectors are com-monly not applied because they are reported to generate shockwaves, spray, and high jets requiring high sidewalls. However,steep deflectors operate more efficiently than do flat deflectorswith the optimum performance for the steepest tested aeratorangle at 11.3°. Additional tests were conducted herein tocompare the aerator efficiency of common deflector anglesof 0° and 5.7° (1∶10), 8.1° (1∶7), 11.3° (1∶5) with excessivesteep angles of 18.4° (1∶3) and 26.6° (1∶2).

2. Subpressures in the air supply system limit the aerator per-formance. They control the equilibrium between air demandof the flow and the capacity of the air supply system. Notablesubpressures provoke poor flow features that include (1) insuf-ficient protection from cavitation along the spillway due toreduced air entrainment and (2) pulsations in the air supplysystem. Ultimately, aerators may even become cavitation-generating devices. The effect of cavity subpressure on βwas investigated, e.g., by Tan (1984), Chanson (1988), andRutschmann (1988). The relationship between cavity subpres-sure and the streamwise air concentration field was illustratedby Chanson (1989b) who found a small correlation. In parallel,the deaeration process at the cavity end was described as a

1Laboratory of Hydraulic Constructions (LCH), Ecole PolytechniqueFédérale de Lausanne (EPFL), ENAC IIC, CH-1015 Lausanne, Switzer-land; formerly, Laboratory of Hydraulics, Hydrology and Glaciology(VAW), ETH Zurich. E-mail: [email protected]

Note. This manuscript was submitted on May 5, 2010; approved onApril 8, 2011; published online on September 15, 2011. Discussion periodopen until March 1, 2012; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Hydraulic Engineer-ing, Vol. 137, No. 10, October 1, 2011. ©ASCE, ISSN 0733-9429/2011/10-1208–1215/$25.00.

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Page 2: Chute Aerators: Steep Deflectors and Cavity Subpressure

function of the jet impact angle, which correlates with thecavity subpressure (Chanson 1994).

These two essential aspects are addressed herein. An impression ofstreamwise air concentration profiles is given in Fig. 1. These pro-files were measured in the hydraulic model as described. The caseof Fig. 1(a) is considered as a reference (test no. 139) with condi-tions according to Table 1, i.e., with a standard deflector and with-out cavity subpressures, whereas Fig. 1(b) shows the same test butwith a very steep deflector (α ¼ 26:6°) and Fig. 1(c) with a relevantsubpressure for otherwise identical conditions. The profiles weremeasured at the sections indicated in the legend, giving the unnor-malized distance to the deflector lip with an ordinate Z adjusting theflow between its upper and lower boundary at the 90% air concen-tration isoline or the chute bottom. It is visible in Fig. 1(b) that asteep deflector generates a longer jet than a flat defector becauseC ¼ 0:9 at Z ¼ 0 up to approximately 1.7 m, in contrast to 0.75 min Fig. 1(a). The steep deflector further generates higher concen-trations at the last profile (3.61 m). Fig. 1(c) shows that a chokedaerator with large cavity subpressures inhibit the generation of afree jet so that the cavity is filled with water. Therefore, the firsttwo profiles at 0.01 and 0.41 m lack a clear trend at Z < 0:5.

Hydraulic Model

Additional experiments focusing on steep deflectors and on cavitysubpressures were conducted in a 0.300-m-wide by 6-m-long sec-tional chute model at the Laboratory of Hydraulics, Hydrology andGlaciology (VAW), ETH Zurich. The constant chute bottom anglewas φ ¼ 12° relative to the horizontal. Furthermore, the constantoffset height was s ¼ 23 mm. Flows of variable approach (sub-script o), flow depths ho, and Froude numbers Fo ¼ Vo=ðghoÞ0:5were generated with a jet-box, where Vo = approach flow velocityand g = gravitational acceleration. Other parameters affecting theair entrainment at aerators were systematically varied and includethe deflector height t, the deflector angle α, and the cavity subpres-sure head hs (Table 1, Fig. 2). A summary table showing the basicexperimental characteristic is given by Pfister and Hager (2010a).Their experiments were all conducted with approximately atmos-pheric pressure in the cavity.

The discharge was measured with an electromagnetic flowmeter(Krohne, Germany), the flow depths upstream of the aerator weremeasured with a point gauge, and the pressure head in the air cavitywas measured with a U-shaped manometer. Thus, three pressuretapings were transversely installed within the air cavity onto thevirtual chute bottom level at x ¼ 60 mm, combined and adaptedto a single manometer. The coordinate x is defined in the stream-wise direction along the chute bottom starting at the foot of the

Fig. 1. Aerator generated streamwise air concentration profiles of Test (a) 139 (Table 1) as reference, (b) 133 with an extraordinary steep deflector,and (c) 153 with a considerable cavity subpressure, for otherwise identical conditions

Table 1. Test Program for Additional Experiments, s ¼ 23 mm andϕ ¼ 12°

Test number Fo [-] ho [m] α [°] t [m] P [-] Series

(123) 7.6 0.065 11.3 0.013 0.03 A

(124) 8.8 0.066 11.3 0.013 0.06 B

(125) 7.3 0.086 11.3 0.013 0.03 C

127 6.0 0.065 18.4 0.013 0.03

128 7.5 0.065 18.4 0.013 0.05 A

129 9.0 0.065 18.4 0.013 0.04 B

130 8.8 0.051 18.4 0.013 0.05

131 7.2 0.086 18.4 0.013 0.03 C

132 5.9 0.066 26.6 0.013 0.04

133 7.4 0.066 26.6 0.013 0.06 A

134 9.0 0.065 26.6 0.013 0.08 B

135 7.3 0.086 26.6 0.013 0.03 C

136 7.4 0.066 8.1 0.013 0.05 A

137 9.0 0.065 8.1 0.013 0.06 B

138 7.3 0.086 8.1 0.013 0.02 C

139 7.5 0.065 5.7 0.013 0.03 A

140 8.9 0.066 5.7 0.013 0.05 B

141 7.3 0.086 5.7 0.013 0.02 C

142 7.5 0.065 5.7 0.013 0.31

143 9.1 0.065 5.7 0.013 0.54

144 7.6 0.084 5.7 0.013 0.44

145 6.1 0.065 18.4 0.013 0.17

146 7.5 0.065 18.4 0.013 0.20

147 9.0 0.065 18.4 0.013 0.45

148 9.0 0.065 11.3 0.013 0.34

149 9.0 0.053 11.3 0.013 0.13

150 7.5 0.065 5.7 0.027 0.42

151 7.5 0.065 5.7 0.007 0.32

152 7.5 0.065 0.0 0.000 0.54

153 7.5 0.065 5.7 0.013 2.08

154 9.0 0.065 5.7 0.013 0.31

155 7.2 0.086 5.7 0.013 0.23

156 6.0 0.065 18.4 0.013 1.74

157 7.5 0.065 18.4 0.013 2.00

158 8.8 0.066 18.4 0.013 0.24

159 9.0 0.065 11.3 0.013 3.08

160 9.1 0.053 11.3 0.013 0.19

161 7.5 0.065 5.7 0.027 0.28

162 7.5 0.065 5.7 0.007 3.08

163 7.5 0.065 0.0 0.000 0.11 A

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offset. The coordinate perpendicular to x is z starting at x ¼ 0. Theeffective transverse groove exceeds the chute bottom to guaranteefree air flow below the jet. Thereby, choking of the pressure tapingswas avoided, and they were protected from splash water. Theair entrained by the aerator was supplied through a lateral ductand measured with a thermoelectric anemometer (Schiltknecht,Switzerland). The two-dimensional (2D) air concentration distribu-tion of the flow was measured by using a fiber-optical twin probe(RBI Instrumentation, France). The measurement principle takesthe different refraction indexes between the sapphire tips and thesurrounding air or water phase into account. Light supplied tothe tip is reflected and detected if the tip is positioned in the airphase, whereas it is lost in the water phase. This probe measuresthe totally transported air, not distinguishing between entrapped orentrained air in contrast to other arrangements, as for instancecapacitive probes. The local air concentration C of the two-phaseair water flow resulted from an acquisition period of 20 s. The jetlength L was visually detected as distance between x ¼ 0 and thereattachment point P of the lower jet trajectory defined along theisoconcentration line of C ¼ 0:90.

In addition to the data set derived for the basic study (tests 34 to126 from Pfister and Hager 2010a, b), two new test series wereconducted: (1) variation of α up to 26.6° (tests no. 127 to 141),and (2) effect of hs (tests 142 to 163). The parameters φ ¼ 12°and s ¼ 23 mm were kept constant for these tests because theireffect was previously investigated. The remaining parameters, Fo ;ho; α; t; and hs were varied to define their effect and to validateprevious relationships. The test program is given in Table 1.

Subpressures were generated by partially or fully closing the airduct supplying air to the cavity. If the corresponding throttle valvewas fully opened, then pressures around atmospheric values, i.e.,hs ≈ 0, resulted in the cavity, whereas for a fully closed valve hswas up to 3:1ho. For large subpressures, the air cavity below the jetis filled with water, i.e., choked so that no air pressure measurementwas possible. For these tests, the water subpressures within thecavity were measured in a way that no air was entrained by theinstrumentation.

Scale effects relating to the hydraulic model tests were discussedby Pfister and Hager (2010a) and found small as limitations relativeto the approach flow Weber number W ¼ Vo=ðσ=ρhoÞ0:5 and theapproach flow Reynolds number Ro ¼ ðVohoÞ=υ were respectedwith ρ = water density; σ = surface tension; and υ = kinematic waterviscosity. Typically, these limitations exclude tests with approxi-mately Wo < 140 and Ro < 2 · 105. Tests no. 127 to 163 rangedwithin 143 ≤ Wo ≤ 232 and 2:8 · 105 ≤ Ro ≤ 5:2 · 105.

Effect of Steep Deflector

General

Three test series A, B, and C with five or six different α valuesand otherwise identical conditions were conducted (Table 1) forapproximate atmospheric air cavity pressure. Fig. 3 shows theeffect of α on the three characteristic parameters: (a) L; (b) β;and (c) hj with hj = maximum jet height along coordinate zperpendicular to the chute bottom (Fig. 2). Both L and hj werenormalized with ho, a constant within a test series.

It is observed from Fig. 3(a) that L=ho increases primarily forα < 20°, beyond which the increase is reduced. The maximum re-sults theoretically for α ¼ 45° following the point-mass parabola,whereas the take-off angle differs from α for a real fluid (Helleret al. 2005). The figure also includes data of Rutschmann(1988) for 4:4° ≤ α ≤ 7:4° with the increasing trend. Fig. 3(b)shows that β increases if α < 12° but remains almost constantfor higher values. The increase of β for small α is confirmed bythe combined data set of Tan (1984) and Low (1986) related tothe Clyde spillway investigation, and the data of Rutschmann(1988). For steep deflectors, L increases with α, typically associ-ated with steep impact angles at point P. These are known to be lessefficient regarding air entrainment (Wood 1988; Ervine et al. 1995;Chanson 1988, 1994; Attari and Zarrati 1997). Fig. 3(c) shows thathj=ho increases with α (Pfister and Hager 2009) so that higher side-walls including sufficient freeboard are necessary. The deflectortested in series B, for instance, generated a jet height of almost4ho. Consequently, the aerator efficiency regarding β increaseswith α, although this trend reduces for steep α. Simultaneously,

Fig. 2. Definition sketch with investigated parameters

Fig. 3. Effect of deflector angle α on (a) L=ho, (b) β, and (c) hj=ho for otherwise identical test conditions (Table 1)

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the required sidewall freeboard along the chute increases, andshock waves as well as spray occur.

Jet Length and Air Entrainment Coefficient

The maximum (subscript M) jet length for nearly atmosphericcavity pressure, i.e., P≈ 0 is (Pfister and Hager 2010a)

LMho

¼ 0:77Foð1þ sinφÞ1:5� ffiffiffiffiffiffiffiffiffiffi

sþ tho

rþ Fo tanα

�ð1Þ

Eq. (1) was derived for 0° ≤ α ≤ 11:3° and was validated with otherdata, including prototype measurements in Pfister and Hager(2010a). A comparison of LM with steeper deflectors as shownin Fig. 4(a) indicates that Eq. (1) may be applied up toα ¼ 18:4°. The coefficient of determination is R2 ¼ 0:95 for theadditional tests if using Eq. (1) such that its basic limitationmay be extended to 0° ≤ α ≤ 18:4°. In contrast, R2 ¼ 0:88 onlyfor α ¼ 26:6° because the values of LM are overestimated withEq. (1), i.e., the increase of LM with α reduces for steep deflectors[Fig. 3(a)].

The maximum air entrainment coefficient β for P≈ 0 is (Pfisterand Hager 2010b)

βM ¼ 0:0028F2oð1þ Fo tanαÞ � 0:1 ð2Þ

This equation was limited to deflectors with 0° ≤ α ≤ 11:3° andwas validated with other data, including prototype measurements inPfister and Hager (2010b). For α ¼ 18:4° results R2 ¼ 0:93 be-tween Eq. (2) and the additional tests such that the above limitationmay be expanded to 0° ≤ α ≤ 18:4°. For steep deflectors withα ¼ 26:6°, Eq. (2) overestimates βM for identical reasons, as statedpreviously [Fig. 4(b)].

Air Concentration Development

An additional and more significant indication of the effect of α onthe aerator performance is the resulting impact on the air concen-trations Cb and Ca downstream of the aerator. These were derived

for the test series A to C. In Fig. 5 relating to series A, the coor-dinate x is normalized with L0 = jet length of Test 163 withoutdeflector, i.e., with α ¼ t ¼ 0 as offset aerator. Accordingly, thereattachment point P of this test is located at x=L0 ¼ 1. The jetlengths become longer for α > 0 than for α ¼ 0 [Fig. 3(a)], suchthat P migrates in a streamwise direction.

The Cbðx=L0Þ profiles in Fig. 5(a) further indicate that steepdeflectors result in higher air concentrations than flat deflectorsat the same location x=L0. An almost identical concentration profileresults for α ¼ 11:3°, 18.4° and 26.6°. An aerator with a steepdeflector of α > 11:3° therefore entrains marginally more air atx=L ¼ 0 than if α ≤ 11:3° [Fig. 3(b)] but the additional air isdetrained rapidly away from the chute bottom. Steep deflectorsdo not significantly increase the bottom air concentration as com-pared to the standard deflectors with 11.3°, e.g., 1∶5. This is relevantbecause Cb is the most significant parameter relating to cavitationprotection.

The streamwise development of average air concentrationCaðx=L0Þ is shown in Fig. 5(b). Values of Ca result from integrationof the air local concentration profiles CðzÞ over the flow depthbetween the chute bottom or the lower jet trajectory and the freesurface at h90 ¼ hðC ¼ 0:9Þ as (Straub and Anderson 1958)

Ca ¼1h90

Zh90

0CðzÞ dz ð3Þ

Steep deflectors values increase up toCa ¼ 0:6 and include localflow phenomena as the jet zone, as well as the reattachment andspray zone (Pfister and Hager 2010a). Representative values of Caoccur downstream of these zones at x=L ¼ 3 as shown by Pfister andHager (2010b). Although this point was not reached in all tests ofseries A, a comparison of Ca at x=L0 ≈ 13 indicates that for the α ¼11:3° result is large values similar to those of steeper deflectors.

The previously described observation also applies for theseries B and C. To conclude, for the present test setup with φ ¼ 12°,

Fig. 4. Comparison between predicted and measured values of (a) LM[m] and (b) βM [-], both for various α

Fig. 5. Test series A (a) Cbðx=L0Þ and (b) Caðx=L0Þ for various α butotherwise identical conditions

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a deflector with α ¼ 11:3° (1∶5) generates almost maximum airconcentrations in the streamwise direction with limited shock waveheight, spray, and jet height thereby not requiring excessively highsidewalls. Steeper deflectors entrain slightly more air at the aeratoryet without affecting the streamwise air concentration.

Effect of Cavity Subpressure

Definition

The air supply system connects the cavity with the free atmosphere.It consists of typically two lateral air ducts connected with thetransverse cavity, i.e., the air space below the jet. This cavity istypically generated by the offset or the deflector of an aerator(Fig. 6), optionally combined with a lowered chute bottom down-stream of the aerator. This setup prevails especially for wide spill-ways to avoid extensive cavity subpressure and to assure uniformcavity air flow (Marcano and Castillejo 1984).

The air entrained at an aerator causes energy loses in the supplysystem (Rutschmann et al.1986), which may be derived from theBernoulli equation. Losses reduce the air flow, limiting the air en-trainment capacity of the aerator. As a consequence, a steady stateemerges between the air demand of the aerator and the air supplycapacity. For large subpressures this equilibrium results in small airentrainment rates with a poor aerator performance. A thoroughdesign of the air supply system with large cross sections and anaerodynamic shape is at least as important as the aerator design.Furthermore, the air supply system should self-drain to avoidchoking.

Experiments with notable cavity subpressure were conducted toinvestigate its effect on L and β by comparing results with Tan(1984), Chanson (1988), and Rutschmann (1988) and to investigateits effect on the streamwise developments of Ca and Cb. The cavity

subpressure head hs, i.e., the subpressure in the transverse air cavitybelow the jet (Fig. 2) is normalized with ho as

P ¼ hsho

ð4Þ

The basic tests included small subpressures with values of0:01 ≤ P ≤ 0:08, whereas additional tests (142 to 163) includelarge subpressures up to P ¼ 3:08.

Jet Length and Air Entrainment Coefficient

As proposed by Rutschmann and Hager (1990), the effects of P onL and β are normalized with their maximum values βM and LMunder small cavity subpressures P≈ 0. These maxima representthe optimum aerator performance without restrictions from theair supply system. As soon as the free air flow is limited, a sub-pressure P > 0 occurs and these values are reduced. Values ofLM for P≈ 0 follow from Eq. (1), whereas these of βM for P≈ 0follow from Eq. (2). An exponential relationship between L=LMand P was derived, considering the present tests 143 to 163,the data sets S1 and S2 provided by Rutschmann (1988), andthe data of Tan (1984). Test series with deflector aerators and mini-mum initial P < 0:1 were considered for Rutschmann and Tan.This minimum initial P serves as comparative value P≈ 0 to deriveLM . It then follows ðR2 ¼ 0:92Þ

LLM

¼ expð�0:85PÞ; for 0 < P ≤ 3 ð5Þ

Fig. 7(a) compares the test data with Eq. (5). Note that even a smallP drastically reduces L. For large P > 1, the cavity typically col-lapses and chokes as observed in the model. Then, a two-phase air-water flow occurs in the residual cavity rotating with high intensity.Taking the lower shear zone as virtual jet trajectory, an approximateL may be derived for these tests.

To derive β=βM as a function of P, tests 143 to 163, the data ofTan (1984), test series S1 and S2 of Rutschmann (1988), and those

Fig. 6. Chute aerator and air supply system of Kárahnjúkar Dam spill-way in Iceland that consist of a 1∶10 deflector with t ¼ 0:15 m ands ¼ 0:40 m (image by author) Fig. 7. (a) L=LMðPÞ and (b) β=βM ½ðP=FoÞ0:5� ∈

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of Chanson (1988) were considered. A data analysis indicatedthat beside P, Fo affects the reduction of β. It was found that thisreduction is smaller for large than for small Fo under otherwiseidentical conditions, following the linear approximation

ββM

≅ �6ðP=FoÞ0:5 þ 1:3; for 0:5 ≤ β=βM ≤ 1 ð6Þ

A coefficient of determination was not derived because the datascatter is large, especially for large ðP=FoÞ0:5. Eq. (6) is limitedto approximately β=βM ≥ 0:5, below which the values are partiallyunderestimated. Fig. 7(b) compares all data considered with Eq. (6).For β=βM ¼ 0, i.e., a fully closed air supply system, P ¼ Fo=21results theoretically from Eq. (6). Because of the data scatter closeto β=βM ¼ 0 with a trend to higher values ðP=FoÞ0:5 as compared toEq. (6), the choking limitation may be set to P≈ Fo=10. This alsocorrelates with the value derived from the data of Chanson (1995)as P ¼ Fo=9:4. He links the choking with intense spray along thecavity trajectory and also mentions P and Fo as relevant parameters.The effect of the subpressure is relevant for P > Fo=400 fromβ=βM ¼ 1 in Eq. (6).

To ensure optimum aerator performance, the subpressurecoefficient should be as small as possible, at least P ≤ 0:1 forthe relevant discharges with a potential of cavitation. Then, sub-pressure hardly affects the air entrainment and Eqs. (1) and (2)may be applied, as the model tests indicate. For 0:1 < P < 1,the air entrainment is reduced by the cavity subpressure, whereasthe aerator will choke if approximately P > 1. Chanson (1995) listsP values during cavity choking with an average of P ¼ 0:96.

Average Air Concentration Characteristics

Because of local flow phenomena downstream of an aerator, thebasic investigation with P≈ 0 considered Cað3LÞM at x=L ¼ 3 asthe relevant air concentration to determine the aerator efficiency.For the present tests, the location x=LM ¼ 3 was partially notreached because of long jets for P≈ 0. Accordingly, no referencevalue was derived. Furthermore, the determination of L or 3L,respectively, for choked cavities was partially inaccurate. There-fore, the effect of P on Cað3LÞ is not on the basis of a large dataset. The data trend, indicates that Caðx=L > 1Þ slightly increasesunder aerator presence for high values of P.

Bottom Air Concentration Characteristics

The air concentration CðzÞ measured closest to the chute bottomwas considered as Cb with z ¼ 2 to 3 mm. The values along 0 ≤x=L ≤ 1 are Cb ¼ 1 as jet flow occurs. For the data analysisCbðx=LÞ-values were considered along the reattachment and sprayzone 1 ≤ x=L ≤ 3. For deflector-aerators, the relevant parameter is(Pfister and Hager 2010b)

f D ¼�xL� 1

�FnoðtanαÞm

�ho

sþ ho

��0:2ð7Þ

with

m ¼ 0:5� ð1:5 sinφÞ3 ð8Þand

n ¼ �1� ð1:5 sinφÞ3 ð9ÞThe additional tests for P > 0 indicate that Cb reduces as P in-creases, primarily because L overproportionally decreases with P[Fig. 7(a)]. The additional tests were also normalized with f D ina semilogarithmic plot [Fig. 8(a)]. Tests with small P essentiallycollapse, whereas those with P > 1:5 lie above the other points.The effect of reduced L is obviously not correctly contained in

Eq. (7), such that a corresponding term respecting P was added.The effect of the cavity subpressure was added to f D as

f DP ¼ f D

�1

1þ P

�0:7

ð10Þ

For P ¼ 0, ½1=ð1þ PÞ�0:7 ¼ 1, whereas for P ¼ 1 the term is 0.62:Cavity subpressure therefore reduces both L and Cb.

The basic test data were fitted as Cb ¼ 1� tanhð4:8f 0:25D Þ,provided 0:01 ≤ P ≤ 0:08, as is typical for prototype aerators.Inserting the cavity subpressure effect ½1=ð1þ PÞ�0:7 in the preced-ing expression results in (R2 ¼ 0:81)

Cb ¼ 1� tanh

�4:8

�xL� 1

�0:25

F0:25no ðtanαÞ0:25m

×

�ho

sþ ho

��0:05�

11þ P

�0:18

�for 1 ≤ x=L ≤ 3 ð11Þ

Fig. 8(b) shows the basic and additional test data as Cbðf DPÞ.All data essentially collapse, such that f DP instead of f D includesthe cavity subpressure effect. Eq. (11) applies for 1 ≤ x=L ≤ 3if 5:8 ≤ Fo ≤ 10:4; 12° ≤ φ ≤ 50°; 0:1 ≤ ðt þ sÞ=ho ≤ 2:1; 0° ≤α ≤ 11:3°; and particularly 0 < P ≤ 3.

Aerator Choking

Aerators generate small streamwise air concentrations even if theair supply system is closed (β ¼ 0) or the cavity is choked (P > 1),i.e., filled with recirculating water. Then the average and bottomair concentrations beyond x=L ¼ 1 are often higher than at theapproach flow section. However, these concentrations are smalland insufficient to prevent cavitation damage downstream of theaerator. This air entrainment is related to the turbulence productionby the deflector (Ervine et al. 1995), generating a rougher flowsurface, and entraining more air from the free surface (Fig. 9).

Fig. 8. Bottom air concentration Cb versus (a) f D for present tests and(b) f DP for basic and present tests

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Page 7: Chute Aerators: Steep Deflectors and Cavity Subpressure

A choked aerator is dangerous because it provokes local cavitationdamages.

The following example illustrates the effect of large P on theaerator performance and its impact on the chute. The Guri Damin Venezuela was heightened from 100 to 150 m so that the spill-way slope had to be increased. As chute aerators were required,model and prototype experiments were conducted (Zagustin et al.1982; Marcano and Castillejo 1984; Marcano and Patiño 1989).The provisional aerator during the constructional stage II ofChute 3 was inserted in 1982. It consisted of a deflector witht ¼ 0:14 m without transverse groove over the spillway widthof 48 m. After 190 h of operation at a specific discharge of41 m2=s, cavitation damage occurred. With ho ¼ 2:0 m andhs ¼ 6:0 m (Marcano and Patiño 1989) a value of P ¼ 3 occurred.An inspection indicated an irregular deflector lip and height. Theaerator was then repaired and equipped with pressure tapingsmounted at the vertical deflector face. Tests demonstrated thatthe cavity zone was choked, the lowest subpressure was hs ≅ 7 mfor maximum discharge, and an intermittent noise with regular ex-plosions was reported (Marcano and Patiño 1989). The improveddeflector with t ¼ 0:37 m and α ¼ 5° increased the transversecavity cross section allowing for air flow to the center of thechute and resulted in P ≤ 0:5 for discharges up to 100 m2=s(Table 2). For this specific discharge, no cavitation damages wereobserved even after an operational period of 95 days (Marcano andPatiño 1989). The final stage aerator includes, in addition to thedeflector, a large offset with an air gallery feeding the aerator frominside the dam (Castillejo and Marcano 1988).

Conclusions

The herein presented hydraulic model tests complete a basic inves-tigation on chute aerators (Pfister and Hager 2010a, b). In the frameof this basic investigation, the relevant parameters deflector heightand angle, offset height, chute bottom angle, approach flow Froudenumber, and depth were systematically varied. However, the steep-est deflector tested had the best performance regarding air entrain-ment and streamwise maximum air concentrations. To generalizethis result, steeper deflectors were tested herein, for which theair entrainment slightly increases, whereas the effect on the stream-wise air concentration is small. In parallel, the jet height increasesand large spray and shock waves result. The optimum aerator

performance results for a deflector angle of some 10°, i.e., a slopeof 1∶5. Flatter deflectors entrain less air and generate smallerstreamwise air concentrations whereas steeper deflectors generatesimilar air concentrations but notable shock waves and sprayrequiring higher chute sidewalls.

Subpressures in the air cavity below the jet were excluded in thebasic investigation although their effect is relevant. These occur ifthe air supply system has insufficient cross sections or if the trans-verse cavity air flow is constrained. Then, the aerator performanceis poor because less air is entrained into the flow. Both the jet lengthand the air entrainment coefficient decrease with increasing sub-pressure following the herein-derived equations. The related dataset was completed and compared with measurements of other stud-ies. The streamwise bottom air concentration decreases particularlywith increasing subpressure. A term accounting for subpressurewas added to the equation presented in the basic study. It is dem-onstrated that an aerator chokes if the cavity subpressure head is ofthe order of the approach flow depth. Subpressure heads belowsome 10% of the approach flow depth result in a small effecton the flow. An example finally illustrates the effect of a chokedair cavity suggesting the importance of a thorough design of the airsupply system.

Acknowledgments

Professor Dr. Willi H. Hager, Laboratory of Hydraulics, Hydrologyand Glaciology (VAW), ETH Zurich, supported the writer withvaluable advice. Professor Dr. Arturo Marcano, Electrificaciondel Caroni C.A. (EDELCA), Venezuela, kindly provided infor-mation related to the Guri Dam aerator damage. Thanks also toMr. Alexander Schmid who conducted measurements related tocavity subpressure.

Notation

The following symbols are used in this paper:C = air concentration;F = Froude number;f D = streamwise normalization function;f DP = streamwise normalization function considering P;g = acceleration of gravity;h = flow depth;hj = maximum jet height;hs = cavity subpressure head;L = jet length;P = cavity subpressure coefficient;Q = water discharge;QA = air discharge;R = correlation coefficient;R = Reynolds number;s = offset height;t = deflector height;V = flow velocity;W = Weber number;x = streamwise coordinate;z = coordinate perpendicular to chute bottom;α = deflector angle;β = air entrainment coefficient;υ = viscosity of water;ρ = density of water;σ = surface tension of water; andφ = chute bottom angle.

Fig. 9. Choked flow conditions for Test 162 with P ¼ 3:08, with roughand aerated flow surface

Table 2. Guri Dam Aerator Characteristics Derived from Data of Marcanoand Patiño (1989)

Provisional aerator Improved aerator

q ½m2=s� ho [m] hs [m] P [-] hs [m] P [-]

18.8 1.0 4.0 4.0 0.5 0.5

42.7 2.0 6.0 3.0 0.8 0.4

62.5 2.5 6.2 2.5 1.2 0.5

84.4 3.0 6.5 2.2 1.5 0.5

100.0 3.5 6.6 1.9 1.8 0.5

125.0 4.5 6.7 1.5

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Subscripts

a = average;b = bottom;M = maximum, i.e., P≈ 0; ando = approach flow.

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