Lower nappe aeration in smooth channels: experimental data and numerical simulation

17 Pages·2010·0.83 MB·English

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AnaisdaAcademiaBrasileiradeCiências(2010)82(2): 521-537
(AnnalsoftheBrazilianAcademyofSciences)
ISSN0001-3765
www.scielo.br/aabc
Lower nappe aeration in smooth channels: experimental data
and numerical simulation
EUDES J. ARANTES1, RODRIGO M. PORTO2, JOHN S. GULLIVER3,
,
ALBERTO C.M. LIMA4 and HARRY E. SCHULZ25
1UTFPR,CampusCampoMourão,CaixaPostal271,87301-005CampoMourão,PR,Brasil
2DepartamentodeHidráulicaeSaneamento/EESC/USP,
Av. TrabalhadorSão-carlense,400,13566-590SãoCarlos,SP,Brasil
3DepartmentofCivilEngineering,UniversityofMinnesota,2ThirdAvenueS.E,55455Minneapolis,MN,USA
4UniversidadedaAmazônia,Av. AlcindoCancela,287,Umarizal,66060-000Belém,PA,Brasil
5NúcleodeEngenhariaTérmicaeFluidos/EESC/USP,
Av. TrabalhadorSão-carlense,400,13566-590SãoCarlos,SP,Brasil
ManuscriptreceivedonSeptember14,2008;acceptedforpublicationonSeptember30,2009
ABSTRACT
Bed aerators designed to increase air void ratio are used to prevent cavitation and related damages in
spillways. Air entrained in spillway discharges also increases the dissolved oxygen concentration of the
water,whichcanbeimportantforthedownstreamfishery. Thisstudyconsidersresultsfromasystematic
series of measurements along the jet formed by a bed aerator, involving concentration profiles, pressure
profiles, velocity fields and corresponding air discharges. The experimental results are, then, compared,
withresultsofcomputationalfluiddynamics(CFD)simulationswiththeaimofpredictingtheairdischarge
numerically. Comparisonswithjetlengthsandtheairentrainmentcoefficientsfromtheliteraturearealso
made. It is shown that numerical predictive tools furnish air discharges comparable to measured values.
However,ifmoredetailedpredictionsaredesired,verificationexperimentsarestillnecessary.
Keywords: spillwayaerators,airentrainment,air-waterflows,multiphaseflows.
INTRODUCTION
Bottomaeratorsareatechniqueusedtopreventcavitationerosiononspillwaysandtoenhancetheoxygen
contentofthewater. Airventedthroughthebottomaeratorsisentrainedintotheflowingwater,increasing
thecompressibilityoftheair-watermixtureandloweringthevelocityofpressurewaves. Whenimplosion
ofcavitationbubblesoccurs,thehighercompressibilityofthesurroundingfluiddampenstheimpactofthe
pressure waves. Additionally, the bubbles increase the contact area between air and water, improving the
oxygendissolutionintothewaterandtheDOcontentdownstreamofthespillway.
Correspondenceto: Prof. HarryEdmarSchulz
E-mails: [e-mail];[e-mail];[e-mail]
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Experimentalinvestigationsonspillwayairentrainmentbybottomaeratorshaveresultedinempirical
designequations. SchwarzandNutt(1963)presentedatheoreticalequationforthejetlengthformedafter
the ramp. Pan et al. (1980) and Pinto et al. (1982) related the air discharge to geometrical parameters of
the jet. Additionally, Tan (1984) and Rutschmann and Hager (1990) explained the dependence of the air
dischargeonthejetlength. ThejetlengthpredictionsobtainedbyTan(1984)areclosetothoseofSchwarz
andNutt(1963).
InBrazil,thestudiesonaeratedspillwayswereintensifiedduringtheconstructionofthehydropower
damsinthe1970and1980decades. ThefirstrelevantconclusionsforspillwayswerepresentedbyPintoet
al. (1982),whileBorsari(1986)andFuentes(1992)furnishedreviewsofimportantstudiesandprocedures.
These aerator studies helped in the establishment of locally adopted procedures. Some early studies,
like Volkart (1980) and Wood (1985), added important conceptual contributions for the understanding of
aeratedflows. Practicalapplicationsoftheresearchresults,however,requiremoredetailedmeasurements
andthereviewofexistentresults. KökpinarandGögüs(2002)conductedanextensiveexperimentalstudy
and furnished correlations not only for the jet length, but also for the air entrainment in lower and upper
nappes, and a graphical presentation of the effect of ramp heights and bed slopes. The redistribution of
flowvelocityintheaerationzonewasconsideredbyToombesandChanson(2005),whilethedetailsofthe
geometryoftheair-waterinterfacewereusedtoproposetheconceptof“entrappedair”byWilhelmsand
Gulliver(2005). AsimilarentrappedairconceptwasusedbyLimaetal. (2008)toexplainmeasurements
ofairvoidratiosinlowernappeaeration.
This paper seeks to compare CFD simulations with a detailed experimental study of the lower nappe
of a jet generated by an aerator in a laboratory chute. Measurements of velocity fields, pressure and air
concentration(voidratio)profileswillbecomparedtoCFDresults. TheCFDsimulationsofairdischarges
were comparable to the measured values. It is also shown that, if a more detailed description is needed,
experimentsarestillnecessary.
EXPERIMENTALMETHODS
The experiments were conduced in a chute built in the Laboratory of Environmental Hydraulics of the
.
SchoolofEngineeringatSãoCarlos,Brazil. Thechutehadaslopeof14 5◦,anusefullengthof5.0m,with
arectangularcrosssection0.20mwideand0.50mhigh. Thebedaeratorwascomposedofarampwitha
.
lengthof23.0cm,afinalheighttr = 4 0cmandanangleof10◦ relativetothechute. Thechamberunder
thejethadadepthof12.0cm,alengthof18.0cmandawidthof20.0cm. Theairdischargewasmeasured
in the air supply tube, which had a diameter of 71.65mm. Air velocities were obtained from pressure
measurements,withamicromanometerhavingonesideopenedtotheatmosphereandtheotherfixedina
pre-calibratedpositioninthetube. TheconcentrationmeasurementsintheflowweremadewithaCesium
137probe,asshowninFigure1. Calibrationwasmadewiththechannel a)fullwithwaterand b)empty.
TheCesium137radiationwasprojectedperpendicularlytotheflowinthechannelandacounterregistered
the remaining radiation after passing the mixture of air and water and the glass walls. The concentration
measurements thus consider the entire width of the channel. Air concentration profiles were obtained by
positioningtheprobeandradiationcounteralongtheverticalofeverystudiedcrosssection. Eachobtained
concentrationisahorizontalmeanvaluetransversetotheflowdirection.
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BED AERATION IN SMOOTH CHANNELS: EXPERIMENTS AND NUMERICAL SIMULATION 523
Fig. 1–EquipmentusedfortheconcentrationmeasurementswithCesium137probe.
An electromagnetic flow meter was used to measure the water discharges, which were checked with
a rectangular weir located at the outlet of the channel. The measurement of velocity fields in the jet was
performed for nine runs, using a mirror inside of the flow. Velocity fields were measured using particle
imagevelocimetry(PIV).ThelightsourcewasacoppergasLaser,withameanpowerof20Wandpulses
at 10 kHz. The pulse power ranged from 60 to 140 kW. Generated wavelengths were 510.6 nm (green)
and578.2nm(yellow). ACCDcamerawitharesolutionof1024pixels 1024pixelswasusedtorecord
×
the images. After capturing and storing the images in the computer, PIV software was applied to each
image to obtain velocity vector fields by using auto-correlation calculations. 234 images were taken for
each run. Figure 2 shows all cross sections used in the present study. Sections 1, 2 and 3 were used to
obtain the approach flow information (velocity and water depth). Sections 4 through 8 were used for the
concentrationmeasurements. AdditionalsectionsSIX andSX wereusedforvelocitymeasurementsinthe
jetcore. Forvelocitymeasurementsinsections1and2,thelasersheetwasintroducedintheflowfromthe
bottomofthechannel.
Fig. 2–Measurementsectionsforthepresentstudy. Distancesareincentimeters.
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For velocity measurements in the jet core, a mirror was positioned downstream in the jet before the
upper and lower white water regions came together, as shown in Figure 3, and the laser sheet was intro-
ducedintotheflowfromthesideofthechannel. Detaileddescriptionsofthechuteandthemeasurement
equipmentmaybefoundinCarvalho(1997)andLima(2004).
Fig. 3–Experimentalarrangementforthevelocitymeasurementsinthejetcoreusingamirror.
SIMULATIONMETHODS
In this study, the inhomogeneous multiphase model was applied with the liquid and the gaseous phases
considered. Thereisonesolutionfieldforeachseparatephase,andthefluidsinteractviainterphasetransfer
terms. Forexample,thetwophasesmayhaveseparatedvelocityandtemperaturefields,buttherewillbea
tendencyforthesetocometoanequilibriumthroughinterphasedragandheattransferterms(CFX2004).
Thefollowingequationsforinhomogeneousmultiphaseflowwereusedtosimulatetheairandwater
flows,andairuptake:
ContinuityEquations:
∂ Np
ρ ρ 0
∂ ra a +∇ × ra a ∪a = SMSa + ab (1)
t
(cid:0) (cid:1) (cid:0) (cid:1) Xb=1
MomentumEquations:
∂
ρ ρ
∂ ra a ∪a +∇ × ra a ∪a ×∪a =
t
(cid:0) (cid:1) (cid:0) (cid:0) Np (cid:1)(cid:1) (2)
μ T 0 0
−ra∇pa +∇ × ra a ∇ ∪a + ∇ ∪a + a+b ∪b − b+a ∪a +SMA + Ma
(cid:0) (cid:0) (cid:0) (cid:1) (cid:1)(cid:1) Xb−1(cid:0) (cid:1)
where: ra is the volume fraction of each phase (phases indicated by “a”, with “a” assuming values from
ρ −→
1 to Np, the total number of phases), a is the density of phase “a”, Ua is the velocity vector of phase
μ 0
“a”, a is the viscosity of phase “a”, SMSa represents mass sources specified by the user; ab is the mass
flow rate per unit volume from phase “b” to phase “a”. This term only occurs if interphase mass transfer
takesplace; SMa representsmomentumsourcesduetoexternalbodyforces,and Ma representsinterfacial
forces acting on phase “a” due to the presence of other phases. The term 0a+b−→Ub −0b+a−→Ua represents
(cid:0) (cid:1)
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BED AERATION IN SMOOTH CHANNELS: EXPERIMENTS AND NUMERICAL SIMULATION 525
momentumtransferinducedbyinterphasemasstransfer. Thevolumefractionsofthephasessumtounity,
thatis:
Np
ra = 1 (3)
Xa=1
Thereare4NP +1equationstodescribethefluiddynamicsforthe5NP unknownsra,Ua||,||Va,Wa,
−→
(componentsofthevelocityvectorialfield Ua),and Pa (thepressurefieldofeachphase). Theadditional
NP − 1 equations needed to close the system of equations were supplied by assuming that all phases
sharedthesamepressurefield, P,thatis:
,...,
Pa = P forall a = 1 NP (4)
φ
Ascalarvariable inphase“a”hasthecorrespondingtransportequation:
∂ μ
ρ φ ρ φ ρ (φ) ta φ (φ) (φ)
∂ ra a a +∇ × ra a ∪a a −∇ × ra aDa + ∇ a = Sa +Ta (5)
t (cid:18) (cid:18) Scta(cid:19) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
(φ) φ (φ) (φ)
where Da is the Diffusivity of a, Sa represents an external source in phase “a”, and Ta represents
φ
sourcesof a duetointerphasetransfers. DetailsofthemultiphasemodelmaybefoundinCFX(2004).
Toinitializeasimulation,arelativelygrossmeshwasfirstusedtolocatetheair-waterinterface. Then,
theregionsoftheupperandlowerfreesurfaceswererefined,allowingadetailedlocationoftheinterface
for the definitive run. The final mesh of each case presented a higher concentration of cells in the upper
andlowernappesofthejet,asshowninFigure4. DetailsaredescribedbyArantes(2007). TableIshows
the main characteristics of the grids and the simulated fluids (air and water). The number of nodes and
elementsofthegridsaremeanvaluesforallsimulatedconditions.
Fig. 4–Exampleofrefinedmeshusedforthenumericalcalculations.
The model allows the adjustment of constants and boundary conditions in order to calibrate the pre-
dictions with measured results. Different combinations were tested, but they in general reproduced only
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526 EUDES J. ARANTES et al.
TABLEI
TheDomain,FluidsandSimulationcharacteristics,
initialandboundaryconditions.
Domain(afteradaptation)
NumberofNodes 550000
≈
NumberofElements 2750000
≈
Fluid: Water
Temperature 25◦C
DynamicViscosity 8.899*10−4kg.m−1.s−1
Density 998kg.m−3
SurfaceTensionCoef. 0.0732N.m−1
Fluid: Air
Temperature 25◦C
DynamicViscosity 1.831*10−5kg.m−1.s−1
Density 1.185kg.m−3
Simulation
Timestep 0.1s
SimulationTime 10s
Processor Processor3.2GHz
Characteristics (Opteron64bits)
CPUProcessingtime 1*105s
≈
a portion of the observed results. The CFX standard constants and conditions, on the other hand, led to
acceptablepredictionsforthewholesetofexperiments,beingthususedinthisstudy.
Theconditionsintheinletwereimposedas:
– Pressure: hydrostaticpressure,
– Turbulenceintensity: fixedat5%ofthewatervelocity(standardforCFX),
– Turbulence model: SSG Reynolds Stresses using CFX standard parameters. The model allows the
simulationofnon-isotropicsituations,andtheuseofthelawofthewall.
– Roughnessforthelawofthewall: 1.0mm.
Thecalibrationwasprimarilybasedonthepredictionsofthejetlengths.
RESULTSANDDISCUSSION
Fourteenrunswereconductedinthechute. TableIIprovidestheexperimentalconditions. Thejetlength
of run 2 is not available because the jet was longer than the channel. Velocity field measurements using
PIVwereperformedintheninerunsindicatedby(*). Resultsofallfourteenrunswereusedtocheckthe
reproducibilityofthedata,whiletheresultsoftheninerunswithvelocitydatawereusedtocomparewith
CFDpredictions.
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BED AERATION IN SMOOTH CHANNELS: EXPERIMENTS AND NUMERICAL SIMULATION 527
TABLEII
Experimentalconditions.
Opening
Depth Water Velocity Froude Jet Airflow Air Water
ofthe
Run atS2 flowrate atS2 Number Length rate Temp. Temp.
floodgate
h (cm) Qw (l/s) Vw (m/s) Fr L (m) Qair (l/s) (◦C) (◦C)
H (cm)
1* 3.37 45.77 6.79 11.81 2.08 23.06 27.5 24.2
3
2* 3.51 63.93 9.11 15.52 NA 27.82 26.0 25.1
3* 5.26 47.65 4.53 6.31 1.08 12.66 24.0 22.0
4 5.50 58.85 5.35 7.28 1.38 18.37 24.7 24.0
6
5* 5.40 64.37 5.96 8.19 1.48 20.17 25.5 24.1
6* 5.85 92.05 7.87 10.39 2.28 29.49 23.0 24.1
7 6.00 44.44 3.70 4.82 0.78 9.92 29.0 28.0
8* 7.05 64.38 4.57 5.50 0.98 14.26 28.0 22.5
9
9* 8.31 98.20 5.91 6.55 1.48 22.92 21.4 21.0
10 8.54 119.86 7.02 7.67 2.08 31.69 29.0 22.0
11 6.92 46.58 3.37 4.09 0.68 11.28 23.8 21.8
12* 7.94 64.38 4.05 4.59 0.88 15.52 25.8 23.0
11
13* 9.10 94.64 5.20 5.50 1.18 20.28 25.8 22.1
14 10.16 128.76 6.34 6.35 1.58 27.45 21.4 21.0
COMPARISON OF MEASUREMENTS WITH LITERATURE
Verificationofthejetlength L
Jet lengths are commonly used to quantify the air uptake by the water flows. The jet lengths of Table II
werecomparedwiththeoreticalpredictionsofSchwarzandNutt(1963)andTan(1984),givenrespectively
byequations6and7,andplottedinFigure5. Alsoshownarethepredictionsoftheempiricalcorrelation
ofKökpinarandGögüs(2002),givenbyequation8. Thetheoreticalequations6and7wereobtainedwith
α
norestrictionsto . Equation8wasadjustedfromexperimentaldata,beingthusrestrictedbytherangeof
α
tested during the experiments. A systematic underestimation of equation 8 is observed, which may be
< α < .
duetoextrapolationsofitslimitsofapplication. Forexample,therampslopeislimitedto0 9 45◦,
α
butinthisstudy 10◦.
=
Thegoodagreementbetweenmeasuredandliteratureresultsistakenasanindicationthattheexperi-
mentaldataforjetlengthcanbeusedtocomparewithCFDsimulations.
α( α θ) θ
L 1+ tan 2 tr +ts cos tr +ts θ
Fr 1 1 2 tan (6)
h = cosθ (cid:26) +(cid:20) + h (αFr)2(cid:21)(cid:27)+ h
Lh = gs2ihnθt2+ (V chosθ)t, t = g coVsθsi+nαρ1wgPh 1+tvuu1+2(tr +ts)g(cid:16)c(oVssθin+αρ)1w2gPh(cid:17) (7)
(cid:16) (cid:17)  
 
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528 EUDES J. ARANTES et al.
. .
L . ( α)0.22 1.75 tr +ts 044 ( θ) Aa −0087
0 28 1 Fr 1 tan (8)
= + +
h (cid:18) h (cid:19) (cid:20) Aw(cid:21)
α θ α
Intheaboveequations, istherampslope, isthechute slope, isthetakeoffangle(whichisthe
α α
angle followed by the liquid when it leaves the ramp. is equal to because no inertial effects altered
the observed direction of the flow after leaving the ramp), tr is the ramp height, ts is the step height (zero
ρ 1
in the present experiments), t is the time, w is the water density, g is the gravity acceleration, P is the
/
relative pressure under the jet (taken positive), h, V, Fr and L are defined in Table I, and Aair Aw is the
ratiobetweentheairsupplypipeentrancearea, Aa,andtheareaofthewaterflowbeforetheaerator, Aw.
Figure 5: Predicted and measured jet lengths.
Fig. 5–Predictedandmeasuredjetlengths.
β /
Verificationoftheairentrainmentcoefficient asafunctionof L h
/ /
Pinto et al. (1992) proposed that the ratio between Qair and Qw||, ||Qair Qw, is proportional do L h,
with a coefficient of proportionality between 0.023 and 0.033. Rutschmann and Hager (1990) found a
β
coefficient of proportionality of 0.030, for the maximum value of (zero relative pressure in the cavity
β / /
underthejump). Theauthorsshowedthatthestraightlineof vs. L h interceptedthe L h axisaround
/ β . ( / )
L h 5forafirstsetofdata,sothattheequation 0 030 L h 5 wasproposed. Forasecondsetof
= = −
β . / /
data,thesimpleproportionality 0 030L hwasfollowed. Theauthorsmeasured L huptoaround45.
=
β / / <
Chanson(1991)suggestedthat and L h followlineartrendswithlowerslopesfor L h 20andhigher
/ > /
slopesfor L h 20. The L h valuesrangeduptoaround25. KökpinarandGögüs(2002)presentedthe
empiricalcorrelation
β . ( / )0.82 / ( θ 0.24
= 0 0189 L h Aa A2 1+tan (9)
β β( / ) (cid:2)(cid:0) (cid:1) (cid:1)(cid:3)(θ) / /
in which the coefficient of = L h depends on the channel slope and the ratio Aair Aw. L h
β /
ranged up to 30. Equation 9 shows a nonlinear dependence between and L h, and that the coefficient
< θ < .
maybelowerthan0.0189. Theequationisvalidfor0 51 3◦.
β / β /
Figure6shows against L h forthedataoftheseexperiments. Aconstant seemstooccurfor L h
/ <
between10and15. However,asnomeasurementsweremadefor L h 10,thisapparentconstancymay
beonlythescatterofthedata. Figure6alsoshowstheregioncontainingthelowernappedataofLow(1986)
θ .
andanalyzedbyChanson(1991),obtainedforachuteslope 52 33◦;andtheequationsofRutschmann
=
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BED AERATION IN SMOOTH CHANNELS: EXPERIMENTS AND NUMERICAL SIMULATION 529
θ . θ . β β( / )
and Hager (1990), for 51 3◦ and 34 4◦. The dependence of the function L h on
= = =
geometricalcharacteristics,suchastheslopeofthespillway,isclearlyvisible. Applyingequation9forthe
(θ , )
conditionsofthisstudy 14 5◦ ledtogoodpredictionsofthemeasureddata,withdeviationsoccurring
=
/
forthelowerrangeof L h. Thesedeviationsmaybeduetotheextrapolationsoftheapplicationlimitsof
α
equation9orbiasinthemeasurements. Asalreadymentioned,theusedrampslope 10◦ extrapolates
=
< α < . ( ) . < < .
the 0 9 45◦ limits. Also the experimental range of Froude numbers Fr , 4 08 Fr 15 5,
. < < .
extrapolatestheapplicationlimitsofequation4,givenas5 56 Fr 10 00. However,equation9shows
thatthedatafollowedthegeneralexpectedbehaviorfortheappliedexperimentalconditions,indicatingthat
theexperimentaldataforair/waterflowratiocanbeusedtocomparewithCFDsimulations.
β / θ
Fig. 6–Measured andL hvalues,togetherwithdataandcorrelationsfromotherauthors. istheangleofthespillwayineach
experiment. ThepresentsetofdataapproximatestothegeneraltrendofthecorrelationofKökpinarandGögüs(2002).
β
Verificationoftheairentrainmentcoefficient asafunctionof Fr
β
Figure 7 shows the data of the present study (white circles) and those of Kökpinar and Gögüs (2002)
(graycircles)againsttheFroudenumber. ThedataofKökpinarandGögüswereobtainedforflowsovera
/
stepheighttsof5cm, while, forthepresentdata, ts = 0. Moreover, their tr h valuesrangedfrom0.0to
0.4,whileinthepresentstudytheyrangedfrom0.4to1.2. Thedifferentexperimentalconditionsgenerated
β
cloudsofpointswithsimilartrends,butseparatedinthegraph vs. Fr,asshowninFigure7.
β
Fig. 7–Measured againsttheFroudenumber.
β
Toconsiderthedifferentparametersinacorrelationfor ,adimensionalanalysisleadsto
β , tr , ts
f Fr (10)
=
(cid:18) h h(cid:19)
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530 EUDES J. ARANTES et al.
( )/
The combined effect of tr and ts is represented here through the ratio tr +k ∙ts h, where k is an
adjustedconstant. Amultipleregressionanalysisfurnishedtheresult
.
β . 1.044 tr +3ts −0460
0 0278Fr (11)
=
(cid:18) h (cid:19)
Predicted and measured data are presented in Figure 8, showing that the present data and those from
literature are complementary and form a single cloud of points. The agreement between present and
literature data (Figs. 5, 6 and 8) allowed the use of the corresponding experimental results of air concen-
trationprofiles,pressureprofiles,velocityprofilesandairdischargestocomparetotheCFDpredictions.
β β
Fig. 8–Calculated obtainedusingequation6againstmeasured .
COMPARISON OF CFD PREDICTIONS WITH EXPERIMENTS
Jetlength
TableIIIshowssixpredictionsofthejetlength. Asmentioned,theturbulenceconditionswereadjustedin
theinletofthenumericaldomain,buttheCFXstandardparametersoftheturbulencemodelfurnishedthe
bestpredictionsforthesetofjetlengths. Theseconditionsweresimilarforallruns. Thesimulatedlengths
wereintherangeof0.81to1.08timesthemeasuredlengthforthesixruns. Theserunswereusedtoobtain
numericalairdischarges.
TABLEIII
Simulatedruns,showingpredictedandmeasured
lengthsofthejet.
Measured Predicted
Lpredicted
Run Length Length
Lmeasured
(m) (m)
3 1.08 1.17 1.08
5 1.48 1.18 0.81
8 0.98 1.02 1.04
9 1.48 1.32 0.89
12 0.88 0.88 1.00
13 1.18 1.21 1.03
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