Research Article Studies of Two-Phase Flow at a Chute Aerator with Experiments and CFD Modelling

12 Pages·2016·2.79 MB·English

Hindawi Publishing Corporation
Modelling and Simulation in Engineering
Volume 2016, Article ID 4729128, 11 pages
http://dx.doi.org/10.1155/2016/4729128
Research Article
Studies of Two-Phase Flow at a Chute Aerator
with Experiments and CFD Modelling
PenghuaTeng,1JamesYang,1,2andMichaelPfister3,4
1HydraulicEngineering,RoyalInstituteofTechnology(KTH),10044Stockholm,Sweden
2FluidMechanics,VattenfallR&D,81470A¨lvkarleby,Sweden
3CivilEngineering,HauteEcoled’Inge´nierieetd’ArchitecturedeFribourg(HEIA-FR),1705Fribourg,Switzerland
4LaboratoryofHydraulicConstructions,EcolePolytechniqueFe´de´raledeLausanne(EPFL),1015Lausanne,Switzerland
CorrespondenceshouldbeaddressedtoPenghuaTeng;[email protected]
Received17May2016;Accepted25July2016
AcademicEditor:ShengKaiYu
Copyright©2016PenghuaTengetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Thechuteaeratorofaspillwayisastructureinsuchasensethatairis,intheintenseemulsification,entrainedintothehigh-
velocitywaterflow.Correctlypredictingtheairentrainmentandtwo-phaseflowpatternattheaeratorwouldcontributetoreliable
spillwayoperation.Basedonexperimentaldata,2Dnumericalsimulationsarepreformedtopredictstreamwiseairconcentrations
intheaeratedflow,inwhichatwo-fluidmodelisused.Dependingontheairbubblesize,relativelygoodagreementisseenwith
theexperimentsintheaircavityzone.Thesimulationsgiverisetohigherairconcentrationdownstreamofthecavity,whichis
presumablyduetounderestimationoftheinterfacialforcesinthetwo-fluidmodel.
1.Introduction Formanyyears,physicalmodelshavebeenthemajortool
to study the characteristics of the aerated flow. Computa-
Spillwaysareimportanthydraulicstructuresfordamsafety. tionalFluidDynamics(CFD)hasemergedasanimportant
If the water flow velocity exceeds, for example, 20m/s and alternativeinmultiphaseflowmodelling.Bothmethodsare
thecavitationindexdropsbelowacertainlimit,damagemay undoubtedly complementary to each other. With CFD, it is
occur due to cavitation in the chute bottom, which affects possibletoobtain,indetail,air-waterflowfieldsoftheaerated
the safety of the spillway [1]. Hence, protecting spillways flowsoastounderstandtheeffectsofgoverningparameters
from cavitation damage is a primary goal of engineering necessaryforaprojectinquestion.
design. The use of aerators is probably the only economic The Volume of Fluid (VOF) method is an interface
countermeasureforthepurpose.Anaeratorentrainsairinto tracking scheme addressing the topological changes of the
high-speed flow, alleviates the negative pressure near the air-water interface in free-surface flows [10]. To describe
chutebottom,andthusavoidstheriskofcavitation. its hydraulic performance, the VOF model is often used to
Drivenbyengineeringpractice,researchershaveinvesti- simulatetheaeratedflowofaspillway[11–14].
gatedaeratorsbothinthelaboratoryandthroughprototype In an air-water flow, exchange behaviors between the
observations[2–8].KramerandHager[9]examined,through dispersedairphaseandcontinuouswaterphaseaffectforces
experiments,flowvelocity,airconcentration,andairbubble between the phases. Hence, the correct modelling of forces
sizedistributions;theyconcludedthatthebubblerisevelocity and turbulence is of prime importance for capturing the
in chute flows depends on the Froude number. Pfister and physics. The two-fluid model differs from the VOF model
Hager[7,8]analyzedtheeffectsofgeometricalparameterson insuchawaythatthemomentumandcontinuityequations
streamwisedistributionsofairconcentrationdownstreamof aresolvedforeachphase.Furthermore,theinteractionforce
aerators. between phases, the drag force, the virtual mass force, and
2 ModellingandSimulationinEngineering
1
2 3
5 9
6
8
4
7
10
Figure1:Chutemodelconfiguration[7](withpermissionfromASCE).(1)flowmeter,(2)slidevalve,(3)jet-box,(4)approachflowzone,
(5)aerator,(6)airduct,(7)measurementrange,(8)automaticpositionsystem,(9)fiber-opticalprobe,and(10)cavitypiezometer.
the turbulent dispersion force are modelled in the momen- 𝐹=𝑉0/(𝑔ℎ0)0.5 =7.52,where𝑉0 isthemeanapproachflow
tumequations. velocityand𝑔istheaccelerationofgravity.
The two-fluid model was used to simulate the complex The air supply to the air cavity below the jet occurs
hydrodynamics of air-water flow in industrial applications through a lateral duct on each side of the chute; its mass
[15–17].Zhang[18]carriedoutthree-dimensional(3D)mod- fluxismeasuredwithathermoelectricairflowanemometer.
elling with the model. He focused on evaluations of such To measure the spatial distribution of air concentration
parametersasthediameterofairbubble,wallfunction,and (denotedas𝐶),adual-tipfiber-opticalprobeisadoptedwith
interphaseexchangemodels.Zhangetal.[19]performedtwo- asamplingfrequencyof1MHz.Itsmeasurementisbasedon
dimensional(2D)simulationsusingthemodel,inwhichthe differentrefractionindicesbetweenthesapphiretipandthe
turbulencedispersionforcewasincludedinthemomentum surroundingphase.Ifthetipsareintheairphase,thelight
equations. They concluded that the inclusion of the turbu- onthetipsisreflectedanddetected.Otherwise,itis“lost”in
lencedispersionforcegavebetterresultsofairconcentration thewaterphase.The𝐶valueateachpointisobtainedfrom
intheflow,whichagreedwellwiththeexperimentaldata[20]. a period of typically 20s. The measurement points have a
Physical model tests of an aerator were performed at streamwisespacingof0.2m;theintervalperpendiculartothe
the Laboratory of Hydraulics, Hydrology, and Glaciology chutebottomis2to5mm.
(VAW), ETH, Zurich [7, 8]. Based on its configurations, Fortheapproachflow,theWeberandReynoldsnumber
CFD modelling is performed using the two-fluid model. isdefinedasWe=𝑉0/(𝜎/𝑔ℎ0)andRe=(𝑉0ℎ0)/𝜇𝑤,where𝜎
The simulations, time dependent and in 2D, examine the isthesurfacetensionand𝜇𝑤isthekinematicwaterviscosity.
transport of air in the flow. Included in the study are Iftheyexceedtheminimumvalues,thatis,We=110andRe
evaluationsofaircavitylength,air-entrainmentrate,andair =1.7×105,theeffectsofsurfacetensionandviscousforceare
concentrationdistributions.Theeffectofbubblediameter,a negligible[21–25].Inthetests,We=236andRe=1.3×106,
dominatingfactorinthetwo-fluidmodel,isalsoconsidered bothexceedingthelimits.
on the momentum exchange between air and water. With
regardtotheexperimentalresults,thepurposeofthestudyis 3.NumericalModel
toevaluatethesuitabilityofthetwo-fluidmodelinsolution
of the two-phase flow at the aerator and to learn about the Anumericalmodelwassetuptosimulatethewater-airflow
air-waterfeaturesoftheflow. attheaerator.Thecomputationsarebasedonthetwo-fluid
modelinANASYS15[26–28].
2.PhysicalModel
3.1. Two-Fluid Model. The two-fluid model is based on the
Euler-Euler approach. Both separate and interacting phases
Theexperimentalsetofdatawasobtainedfromahydraulic
areallowedtobemodelled.Inthemodel,thecontinuityand
modeltestofanaeratorconductedinaflume0.3mwideand
momentumequationsareformulatedforeachphase.
6.0mlongatVAW,ETH,Zurich(Figure1)[7].Oneaerator
Thecontinuityequationforeachphaseis
configuration,withoutoffsetbutwithadeflector,isselected
for the numerical modelling (Figure2). The chute bottom 𝜕 ⇀󳨀
angleisset hereinto𝛼=30∘ withthehorizontalplane;the 𝜕𝑡(𝛼𝑖𝜌𝑖)+∇⋅(𝛼𝑖𝜌𝑖V𝑖)=0, (1)
defectorangleis𝜑=8.13∘ withthechutebottom.Thechute
lengthupstreamoftheaeratoris2.0m.Theheightofdefector wheresubscript𝑖 = 𝑎and𝑤,referringtotheairandwater
(𝑠) is 0.0133m. The water depth of the approach flow is ℎ0 ⇀p󳨀hase,𝛼𝑖isthephasevolumefraction,𝜌𝑖isthephasedensity,
= 0.084m. The approach flow Froude number is defined as V𝑖isthephasevelocity,and𝑡isthetime.
ModellingandSimulationinEngineering 3
Ahppzroonaech flow z
0
V
0 Jet zone
o z
u
S Fzaorn-fieeld
𝜑
R
𝛼 x
Figure2:Aeratorconfiguration.
Themomentumbalanceequationforeachphaseis If bubbles accelerate relative to the water, virtual mass
effects occur. 𝑀V𝑚,𝑤 is due to the inertia of the water mass
𝜕 (𝛼𝜌⇀󳨀V )+∇⋅(𝛼𝜌⇀󳨀V⇀󳨀V ) encounteredbytheacceleratingbubbles[30],definedas
𝜕𝑡 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
(2) ⇀󳨀 ⇀󳨀
𝑑V 𝑑V
=−𝛼∇𝑝+∇⋅⇀󳨀𝜏 +𝛼𝜌𝑔+𝐺+𝑀, 𝑀 =0.5𝛼 𝜌 ( 𝑤 − 𝑎). (8)
𝑖 𝑖 𝑖 𝑖 V𝑚,𝑤 𝑎 𝑤 𝑑𝑡 𝑑𝑡
where𝑝isthewaterpressure,𝜏𝑖 istheshearstress,𝐺isthe The virtual mass effect is significant when the secondary-
interphase force, and 𝑀 is the interfacial force between air
phasedensityismuchsmallerthantheprimary-phaseden-
andwaterphase.
sity.
𝐺dependsonwater-airfriction,pressure,cohesion,and 𝑀𝑡𝑑,𝑤actsasaturbulentdiffusionindispersedflowsand
othereffectsandiswrittenasasimpleinteractiontermofthe
isbasedondeBertodano[31]:
followingform:
⇀󳨀 ⇀󳨀 𝑀𝑡𝑑,𝑤 =−𝑀𝑡𝑑,𝑎 =𝐾𝑡𝑑𝜌𝑤𝑘𝑤∇𝛼𝑎, (9)
𝐺𝑎𝑤 =𝐾𝑎𝑤(V𝑎− V𝑤), (3)
where 𝑘𝑤 is the water turbulent kinetic energy per unit of
where𝐾𝑎𝑤 =𝐾𝑤𝑎 =interphasemomentumexchangecoeffi- massand𝐾𝑡𝑑istheturbulencedispersioncoefficient.
cientbetweenthewaterandtheairphase. 𝑀𝑙,𝑤actsonairbubblesmainlyduetovelocitygradients
𝑀consistsofseveralindependentforces: inthewaterflowfield.FromDrewandLahey[30],itiswritten
as
𝑀=𝑀𝑑,𝑤+𝑀𝑙,𝑤+𝑀V𝑚,𝑤+𝑀𝑡𝑑,𝑤, (4) ⇀󳨀 ⇀󳨀 ⇀󳨀
𝑀𝑙,𝑤 =−𝐾𝑙𝜌𝑤𝛼𝑎(V𝑤− V𝑎)×(∇× V𝑤), (10)
where𝑀𝑑,𝑤 isthedragforce,𝑀𝑙,𝑤 istheliftforce,𝑀V𝑚,𝑤 is
thevirtualmassforce,and𝑀𝑡𝑑,𝑤istheturbulencedispersion where𝐾𝑙istheliftcoefficient.
force. In the two-fluid model, the air phase is assumed to
form bubbles. The viscous stress creates bubble skin drag; 3.2. Grid, Boundary Conditions, and Grid Independence.
the pressure distribution around the bubble creates form The chute included in the simulations comprised 2.0 and
drag.Thelatterbecomessignificantwhentherelativebubble 5.06mupstreamanddownstreamoftheaerator.Theupper
Reynoldsnumber(Re𝑟)increases.𝑀𝑑,𝑤iswrittenas boundaryofthedomainisparalleltothechutebottomandat
adistanceof1.2mfromit.Theheightofthetransversalgroove
𝑀 = 𝐾𝐷Re𝑟, (5) is𝐻=0.03m;theductwidthis𝐵=0.06m.Thebottomofthe
𝑑,𝑤 24 ductisopentoatmosphere,allowingairtobesuckedintothe
jetcavity.
where𝐾𝐷isthedragcoefficient.Re𝑟isobtainedfrom AquadrilateralmeshisgeneratedintheGambitsoftware.
𝜌𝑤󵄨󵄨󵄨󵄨󵄨⇀󳨀V𝑎−⇀󳨀V𝑤󵄨󵄨󵄨󵄨󵄨𝐷 ThFigeurgee3o.mThetericbaolunsidzaeryancdonnduitmioenrsicaarlegdreidfisneadreinshFoigwunrei4n.
Re𝑟 = 𝜇 , (6) The upstream boundary consists of both water and air in
𝑤
height.Avelocityboundaryissettothewaterphasebelow;
whereDisthebubblediameter.𝐾𝐷 isbasedontheSchiller apressureinlet(theatmosphericpressure)isgiventotheair
andNaumannmodel[29]: phaseabovethewater.Boththetopandtheductbottomare
specifiedaspressureinlet;thedownstreamboundaryissetas
{{24(1+0.15Re𝑟0.687) pressureoutlet.Theotherboundariesaretreatedaswall.
𝐾𝐷 ={{ Re𝑟 Re𝑟 ≤1000 (7) usinSgimthuelasotiftownsaraereFlpueernfotrinmAedNiSnYS2D15.aZndhatnimg[e18d]epcoemndpeanrteldy
{0.44 Re𝑟 >1000. the Standard 𝑘-𝜀 and Realizable 𝑘-𝜀 models for aerated
4 ModellingandSimulationinEngineering
m
2
1.
2
m
5.06m B H
(a) Entiredomain (b) Localrefinementattheaerator
Figure3:Numericalgrid.
Pressure inlet 1.5
Velocity inlet
S1
S2 1.0
S3
S4
S5 Z
S6
S7
0.5
Pressure inlet Pressure
outlet
Figure4:Numericalboundaryconditions.
0
0 0.5 1.0
flow.Theresultshaveshownthattherearesmalldifferences
between them. However, the Realizable k-𝜀 model is more C
suitable to represent features of turbulent free-surface flow
Medium grid
[32].Hence,itischoseninthesimulations.Theparameters Fine grid
chosen for use in the two-fluid model are summarized in Coarse grid
Table1.
Figure5:Checkofgridindependence(atlocation𝑥=0.211m).
To guarantee the numerical quality, a check of grid
independence is necessary. Three quadrilateral grids are
examined, with the number of cells being about 9900
(coarse),33000(medium),and66000(fine),respectively.The thechutebottom.Themainissuesofconcernformodelling,
minimumcellsizeis1mmattheaerator.Thevariable𝑧𝑢refers eitherphysicalornumerical,includetheairdemand,cavity
to the position corresponding to 𝐶 = 0.9 in the upper edge length,andairconcentrationinthecavityzone,attheendof
of the nappe (Figure2). For the grid independence check, thecavityandinthefarfieldofthejet.
the 𝐶 values are compared as a function of 𝑍 = 𝑧/𝑧𝑢. The Acoordinatesystem(𝑥,𝑧)isdefined.The𝑥-directionis
comparison for location 𝑥 = 0.211m is shown in Figure5. onthechutebottom,whilethe𝑧-direction,perpendicularto
Theresultsindicatethatthemediumsizegridissufficientto thechutebottom,isonthedownstreamfaceofthedeflector
modeltheaeratorflow. (Figure2). Several cross sections, denoted as S1–S7 and all
perpendicular to the chute bottom, are used to describe
the flow (Table2). The distance between two neighboring
4.ResultsandDiscussions
locationsis0.2m.
The flow field is usually divided into three zones, that
is, approach flow zone, air cavity zone, and far-field zone 4.1. Selection of Bubble Diameter. The air phase exists pre-
(Figure2).Thelattertwozonesaredividedbythereattach- sumptively in the form of bubbles. The air-water exchange
ment point (R), defined as the intersection of 𝐶 = 0.9 with behavior is a dominating issue in modelling aerator flow
ModellingandSimulationinEngineering 5
Table1:Summaryofparametersinthetwo-fluidmodel.
Turbulencedispersion
Dragforcemodel Turbulencemodel Virtualmassforcemodel Liftforcemodel
forcemodel
SchillerandNaumann Realizable𝑘-𝜀 DrewandLahey deBertodano DrewandLahey
Table2:Locationsofcrosssections. Table3:Comparisonof𝐿’sbetweenexperimentsandsimulations.
S1 S2 S3 S4 S5 S6 S7 Simulations
Experiments
𝑥(m) 0.211 0.411 0.611 0.811 1.011 1.211 1.411 𝐷=1mm 𝐷=2mm 𝐷=3mm 𝐷=4mm
𝐿(m) 1.120 1.175 1.161 1.152 1.155
andisinfluencedbythebubbledimensions.Inthetwo-fluid
model,thisbehaviorisdescribedby𝐾𝑎𝑤:
agreementwiththeexperimentalone;themaximaldifference
𝜌 𝑀 is5%for𝐷=1mm.
𝐾𝑎𝑤 = 𝑎6𝜏𝑑,𝑤𝐷𝐴, (11) Astheblack-watercorestretchestotheendofthecavity,
𝑎
theairsupplytotheaeratorisequaltotheairentrainedinto
where𝐴istheinterfacialarea,𝜏𝑎=(𝜌𝑎𝐷2)/(18𝜇𝑎),and𝜇𝑎is t𝛽hereflsuolwtsfcroormrestphoenldoiwngertonathppeedisffuerrfeancet.𝐷Tavballeu4es.shWowiths tthhee
theairkinematicviscosity. increaseof𝐷,the𝛽valuebecomessmaller.𝐷=1mmgives
ChansonandToombes[33]carriedoutanexperimentin thelargestdifference;𝐷=4mmisclosertotheexperimental
anopenchanneltostudyair-waterflowproperties.Theyindi-
result,withanerrorof9%.Allthesimulationsoverestimate
cated that the probabilityof air bubble chord sizes between the air demand. 𝐷 is a parameter that does affect the air
0and4mmintheflowregionismorethan65%.Takahashi
entrainment.
et al. [34] conducted a study of interactions between free-
surface and cavity recirculation in a stepped channel. The
resultsshowedthatthemajorityofbubblediametersisbelow 4.3.Cavity-ZoneAirConcentration. Airisentrainedintothe
5mm.Chenetal.[35]studiedbubblediameterdistributions waterbystrongemulsificationinthecavityzone,particularly
in an aerated flow by means of physical model tests. They atitsendnearthereattachmentzone.Iftheaeratorgeometry
showed that the bubble sizes are commonly in the range of isgiven,thestateofairentrainmentgovernstheaircapacity
0.5–3mm in the vicinity of the chute bottom and 3–5mm of the aerator, which forms an initial condition for the
closetothefreesurface. streamwisetransportoftheairinthewater.Figure6shows
In the VAW laboratory tests, no systematical measure- the𝐶distributionsasafunctionof𝑍forthefourlocations
mentsweremadeoftheairbubblesizes.However,fragmen- S1, S2, S3, and S4 (Figure4). Each diagram compares the
tarytestsdemonstratedthattheywereusuallybelow4mm. distributions between the experiments and the simulations.
Withreferencetotheabovementionedobservations,𝐷=1,2, The𝐷valuesvaryfrom1to4mm.
3,and4mmareselectedtoexaminetheireffectsontheflow Fortheupperedgeofthenappe,thefour𝐷valueslead
characteristics. toalmostidenticalresultsatS1anddifferhoweverfromthe
experimental result; the range of the so-called black-water
zoneisunderestimated.Asthestreamwisedistanceincreases
4.2. Cavity Length and Air Demand. When the water flows
from the aerator, the simulated 𝐶 values corresponding to
over the aerator, then the deflector separates the flow from
the larger diameters decrease and approach gradually the
thebottomandacavityregionbeneaththenappeofthejetis
experimentalones.AtS4,theresultof𝐷=4mmisingood
generated,inwhichairissuckedin.Thecavitylengthandthe
agreementwiththeexperimentaldata.
airsupplyaretwocharacteristicparametersofinterest.Due
Fortheloweredgeofthenappe,thenumericalandtest
tothehighvelocityandturbulence,awell-definedinterface
results are more close to each other along the cavity. Some
betweentheairandwaterdoesnotexistforthelowernappe
smalldifferencesareseenfor𝐷=1mminthebeginningofthe
surface. The cavity length, denoted as L (m), refers to the
distance from the aerator to the reattachment point 𝑅 on cavity(e.g.,atS1)and𝐷=4mmtowardsdownstream(e.g.,
atS3andS4).
thechutebottom(Figure2).Todescribetheair-entrainment
capacity of an aerator, an air-entrainment coefficient (𝛽) is
definedastheratioofairdischarge(𝑄𝑎,m3/s)towaterflow 4.4.LowerEdgeConcentrationSimilarity. Tofurtherexamine
discharge(𝑄𝑤,m3/s). theairconcentrationofthelowernappeedge,acharacteristic
Other conditions being given, the bubble diameter 𝐷 is thickness, called the air-entrainment thickness 𝛿, is usually
a primary parameter in the two-fluid model. In literature usedtodescribeitsdevelopment[20,36].Foragivencross
review,limitedinformationhasbeenfoundofitseffectson section,itreferstothedifferencebetween𝐶=30%and60%;
the cavity length and the air demand. Table3 compares the thatis,𝛿=𝑧30−𝑧60.Anormalizedthickness,𝜂,isthendefined
averagedresultsof𝐿fromtheexperimentsandthenumerical as𝜂 = (𝑧−𝑧60)/𝛿.Forthefourlocations,Figure7compares
simulations. The simulated 𝐿 values are in relatively good thechangesofthe𝐶valuesasafunctionof𝜂.
6 ModellingandSimulationinEngineering
Table4:Comparisonof𝛽’sbetweenexperimentsandsimulations.
Simulations
Experiments
𝐷=1mm 𝐷=2mm 𝐷=3mm 𝐷=4mm
𝑄𝑎(m3/s) 0.0393 0.0629 0.0550 0.0460 0.0430
𝛽 0.228 0.365 0.319 0.267 0.249
1.5 1.5
S1(x=0.211m) S2(x=0.411m)
1.0 1.0
Z Z
0.5 0.5
0 0
0 0.5 1.0 0 0.5 1.0
C C
Experiments D=3mm Experiments D=3mm
D=1mm D=4mm D=1mm D=4mm
D=2mm D=2mm
(a) CrosssectionS1 (b) CrosssectionS2
1.5 1.5
S4(x=0.811m)
S3(x=0.611m)
1.0 1.0
Z Z
0.5 0.5
0 0
0 0.5 1.0 0 0.5 1.0
C C
Experiments D=3mm Experiments D=3mm
D=1mm D=4mm D=1mm D=4mm
D=2mm D=2mm
(c) CrosssectionS3 (d) CrosssectionS4
Figure6:𝐶distributions.
ModellingandSimulationinEngineering 7
4 4
S1(x=0.211m) S2(x=0.411m)
2 2
𝜂 0 𝜂 0
−2 −2
−4 −4
0 0.5 1.0 0 0.5 1.0
C C
Experiments D=3mm Experiments D=3mm
D=1mm D=4mm D=1mm D=4mm
D=2mm D=2mm
(a) CrosssectionS1 (b) CrosssectionS2
4 4
S4(x=0.811m)
S3(x=0.611m)
2 2
𝜂 𝜂
0 0
−2 −2
−4 −4
0 0.5 1.0 0 0.5 1.0
C C
Experiments D=3mm Experiments D=3mm
D=1mm D=4mm D=1mm D=4mm
D=2mm D=2mm
(c) CrosssectionS3 (d) CrosssectionS4
Figure7:Similarityof𝐶distributions.
Figure8 is a plot of all the experimental 𝐶 values for 4.5.CavityEnd. Theaircontentatendofthecavityevolves
the lower edge. The 𝐶 distributionsare similar for different from the air entrainment along the lower edge of the jet;
locationsandcanbeassembledintotheexpressionsuggested at the same time, it reflects also the air contribution from
byHager[37]: the backflow upstream of the reattachment point 𝑅 and its
𝐶(𝜂)=𝑚1exp(−𝑚2(𝜂+𝑚3)2), (12) inteSraeccttiioonnwSi5thisthloecjeatt’esdloawtetrheedgene.d of the cavity. Figure9
where𝑚1,𝑚2,and𝑚3areconstants,𝑚1=1.04,𝑚2=0.16,and compares,alongthewholecrosssection,the𝐶distributions
𝑚3=1.87. betweentheexperimentsandthemodelling.Figure10shows
8 ModellingandSimulationinEngineering
4 4
S5(x=1.011m)
2
2
𝜂 0
𝜂 0
−2
−2
−4
0 0.5 1.0
−4 C
0 0.5 1.0
C
Experiments D=3mm
D=1mm D=4mm
S1 S4
D=2mm
S2 Equation (12)
S3 Figure10:Similarityof𝐶distributionatS5.
Figure8:Similarityof𝐶distributionsforexperiments.
thehighairconcentrationregionclosetothechutebottom
1.5 edgeisoverestimatedinthesimulations.
S5(x=1.011m)
4.6. Far-Field Evolution. It is the aerated air bubbles close
tothebottomthatprotectthechutebottomfromcavitation
damage. It is therefore of practical significance to examine
thestreamwisedevelopmentdownstreamofthereattachment
1.0
point𝑅.Itprovidesinformationwithrespecttotheeffective
lengthofthechuteprovidedbytheaeratorinquestion.
Z Figure11showsthe𝐶profilesattwotypicalcrosssections
(S6andS7)inthefarfiled,thatis,thechangeof𝐶with𝑍.The
𝐶distributionsbehaveinasimilarmannerfortheexamined
0.5 𝐷range.Forboththeupperedgeandlowerboundaryofthe
flow, 𝐶 is overestimated; all the simulations lead to a much
larger𝐶valueclosetothechutebottom.Theaverage𝐶value
ateachlocationiswellreproduced.
4.7. Discussions. As seen from (11), 𝐷 is a parameter that
0
0 0.5 1.0 affectsthediffusionbetweentheairandwater;themomen-
C tum exchange becomes more intense with augment in 𝐷.
In the simulations, 𝐷 is a constant value in the entire
Experiments D=3mm
computational domain. In actual situations, it may vary in
D=1mm D=4mm
D=2mm the different flow regions. In the approach flow, with the
increaseofthesurfaceturbulence,airstartstobeentrained
Figure9:𝐶distributionatS5. from the free surface at a location upstream of the aerator.
Compared with the flow in the cavity zone, the turbulence
intensityintheflowislow[6].Asaresult,theCdistribution
thesimilarityprofileof𝐶,thatis,thechangeof𝐶asafunction oftheapproachflowmightstillbeoverestimatedevenwith
of𝜂. thesmallestdiametersimulated(𝐷=1mm).
From the experiments, one can see that the black- Previous model tests showed that the roughness of the
water zone becomes small at S5 and is about to disappear surfacewaterincreasesalongthecavityzone[9].Thisimplies
downstreamofit.Thethicknessoftheblackwaterisrelatively thattheexchangeofmomentumbetweenthewaterandair
wellproduced;itspredictedpositionishoweverlower.Again, becomesstronger;alargerbubblediametershouldbeusedto
ModellingandSimulationinEngineering 9
1.5 1.5
S7(x=1.411m)
S6(x=1.211m)
1.0 1.0
Z Z
0.5 0.5
0 0
0 0.5 1.0 0 0.5 1.0
C C
Experiments D=3mm Experiments D=3mm
D=1mm D=4mm D=1mm D=4mm
D=2mm D=2mm
(a) CrosssectionS6 (b) CrosssectionS7
Figure11:𝐶distributionatS6andS7.
describetheprocess.Thisisthereasonwhygoodagreement bubblediameteraffectstheamountofairentrainedintothe
isobtainedfortheupperedgewith𝐷=4mm.Forthelower flow; the air flow rate from the 4mm simulations is close
trajectory, the simulations show only small deviations from to the measured one. Along the cavity zone, the surface-
the experiments. The turbulence intensity is high, but not waterroughnessincreases;asomewhatlargerbubblesizeis
as high as that for the upper edge. The result is somewhat suitable to describe the exchange between the two phases.
insensitiveto𝐷,butsmaller𝐷valuesthan4mmgivebetter Forthelowerjet trajectory,thesimulatedairconcentration
resultsforthemajoritypartofthecavity. is relatively nonsensitive to the bubble size. However, sizes
Inthefar-fieldzone,the𝐶valuesnearthechutebottom smallerthan4mmleadtoresultsclosertotheexperiments.
areoverestimatedinthesimulations.KramerandHager[9] The air concentration in the far field is overestimated; the
studiedtheairtransportprocessinchuteflows.Theypointed experiments showed lower air contents. The contributing
out that air detrained after the reattachment point 𝑅 and reason is presumably the underestimation of the interfacial
the interfacial force between bubbles and water dominated forces in the two-fluid model. Besides, the use of a single
theprocessofdetrainmentinthezone.Thisimpliesthatthe bubble diameter is not sufficient to represent the complex
effectsoftheinterfacialforcesareunderestimatedinthetwo- water-air exchange in the aerator flow but seems to be
fluidmodel. adequateasafirststep.
Notations
5.Conclusions
𝐴: Interfacialarea(m2)
To model the air-water flow at an aerator is a challenging 𝑎𝑖: Constants(−)
issue,especiallyifthewaterflowvelocityishigh,whichisthe 𝐵: Groovewidth(m)
caseinmostspillwayinstallations.Basedontheexperimental 𝐶: Localairconcentration(−)
data of ETH, Zurich, numerical simulations are performed 𝐷: Bubblediameter(mm)
toreproducethecharacteristicsoftheaeratedflow.TheCFD 𝐹: Froudenumber(−)
model is the two-fluid model in ANSYS 15; the parameters 𝐺: Interphaseforce(N)
of interest include air concentration, cavity length, and air- 𝑔: Accelerationofgravity(m/s2)
entrainmentrate.Thebubblediameteraffectsthemomentum 𝐻: Groovedepth(m)
exchangebetweenthewaterandairandarangebetween1and ℎ0: Depthofapproachflow(m)
4mmisinvestigated. 𝐾𝑎𝑤: Interphasemomentumexchange
Intermsofthecavitylength,theexperimentsandCFD coefficient(−)
give similar results irrespective of the bubble size. The 𝐾𝐷: Dragcoefficient(−)
10 ModellingandSimulationinEngineering
𝐾𝑡𝑑: Turbulencedispersioncoefficient (CTH), and Uppsala University (UU) (http://www.svc.nu).
(−) The authors are indebted to Ms. Sara Sandberg of SVC for
𝐾𝑙: Liftcoefficient(−) projectcoordinations.
𝑘𝑤: Waterturbulentkineticenergy(−)
𝐿: Cavitylength(m)
References
𝑀: Interfacialforce(N)
𝑀𝑑,𝑤: Dragforce(N) [1] H. Falvey, Cavitation in Chutes and Spillways, Engineering
𝑀𝑙,𝑤: Liftforce(N) Monograph 42, United States Department of the Interior,
𝑀𝑡𝑑,𝑤: Turbulencedispersionforce(N) BureauofReclamation,Denver,Colo,USA,1990.
𝑀V𝑚,𝑤: Virtualmassforce(N) [2] P.VolkartandP.Rutschmann,“Rapidflowinspillwaychutes
𝑚1,𝑚2,and𝑚3: Constants(−) with and without deflectors a model-prototype comparison,”
𝑝: Waterpressure(N/m2) in Proceedings of the Symposium on Scale Effects in Modelling
𝑄𝑎: Airdischarge(m3/s) HydraulicResearch,EsslingenamNeckar,Germany,1984.
𝑄𝑤: Waterdischarge(m3/s) [3] H. Chanson, “Flow downstream of an aerator—aerator spac-
Re: Reynoldsnumber(−) ing,”JournalofHydraulicResearch,vol.27,no.4,pp.519–536,
Re𝑟: RelativeReynoldsnumber(−) 1989.
𝑠: Deflectorheight(m) [4] P.RutschmannandW.H.Hager,“Airentrainmentbyspillway
𝑡: Time(s) aerators,”JournalofHydraulicEngineering,vol.116,no.6,pp.
⇀⇀󳨀󳨀𝑉V0𝑖:: VAeplporcoitaychofflpohwasveel𝑖o(cmit/ys)(m/s) [5] 7[MP6.h5A–.D7..8K2do¨,isk1s9pe9irnt0aa.tri,oAni]r,-eDnetpraairntmmeennttionfhCigihvislpEenedgifnreeeersiunrgfa,cMeifldodwles
⇀󳨀V𝑎: Velocityofairbubble(m/s) EastTechnicalUniversity,Ankara,Turkey,1996.
V𝑤: Velocityofwater(m/s) [6] K.Kramer,Developmentofaeratedchuteflow[Ph.D.disserta-
We: Webernumber(−)
tion],VAW,ETHZurich,Zurich,Switzerland,2004.
𝑥: 𝑥-coordinate(m)
[7] M. Pfister and W. H. Hager, “Chute aerators. I: air transport
𝑍: Normalized𝑧-coordinate(−)
characteristics,”JournalofHydraulicEngineering,vol.136,no.
𝑧: 𝑧-coordinate(m)
6,pp.352–359,2010.
𝑧30: 𝑧(m)at𝐶=30% [8] M. Pfister and W. H. Hager, “Chute Aerators. II: hydraulic
𝑧60: 𝑧(m)at𝐶=60% design,” Journal of Hydraulic Engineering, vol. 136, no. 6, pp.
𝛼: Chutebottomangle(∘) 360–367,2010.
𝛽: Air-entrainmentcoefficient(−)
[9] K. Kramer and W. H. Hager, “Air transport in chute flows,”
𝜑: Deflectorangle(∘) InternationalJournalofMultiphaseFlow,vol.31,no.10-11,pp.
𝜎: Surfacetension(N/m) 1181–1197,2005.
𝛼𝑖: Volumefractionofphase𝑖(−) [10] C.W.HirtandB.D.Nichols,“Volumeoffluid(VOF)method
𝛼𝑎: Airvolumefraction(−) forthedynamicsoffreeboundaries,”JournalofComputational
𝜌𝑖: Densityofphase𝑖(kg/m3) Physics,vol.39,no.1,pp.201–225,1981.
𝜌𝑤: Waterdensity(kg/m3) [11] D. K. H. Ho, K. M. Boyes, and S. M. Donohoo, “Investiga-
𝜌𝑎: Airdensity(kg/m3) tionofspillwaybehaviorunderincreasedmaximumfloodby
𝜇𝑤: Kinematicwaterviscosity(m2/s) computationalfluiddynamicstechnique,”inProceedingsofthe
𝜇𝑎: Kinematicairviscosity(m2/s) Conferenceonthe14thAustralasianFluidMechanics,pp.577–
𝜏𝑖: Shearstressofphase𝑖(Pa) 580,AdelaideUniversity,Adelaide,Australia,December2001.
𝛿: Air-entrainmentthickness(m) [12] M.C.AydinandM.Ozturk,“Verificationandvalidationofa
𝜂: Normalizedair-entrainment computationalfluiddynamics(CFD)modelforairentrainment
thickness(−). atspillwayaerators,”CanadianJournalofCivilEngineering,vol.
36,no.5,pp.826–836,2009.
[13] J.-M. Zhang, J.-G. Chen, W.-L. Xu, Y.-R. Wang, and G.-J.
CompetingInterests Li,“Three-dimensionalnumericalsimulationofaeratedflows
downstreamsuddenfallaeratorexpansion-inatunnel,”Journal
Nopotentialconflictofinterestswasreportedbytheauthors. ofHydrodynamics,vol.23,no.1,pp.71–80,2011.
[14] V. Jothiprakash, V. V. Bhosekar, and P. B. Deolalikar, “Flow
characteristics of orifice spillway aerator: numerical model
Acknowledgments studies,”ISHJournalofHydraulicEngineering,vol.21,no.2,pp.
216–230,2015.
The numerical modelling presented here was carried out
[15] R. F. Mudde and O. Simonin, “Two- and three-dimensional
as part of a Ph.D. Project “Hydraulic Design of Chute simulations of a bubble plume using a two-fluid model,”
Spillway Aerators,” funded by Swedish Hydropower Centre Chemical Engineering Science, vol. 54, no. 21, pp. 5061–5069,
(SVC). SVC has been established by the Swedish Energy 1999.
Agency, Energiforsk, and Swedish National Grid together [16] S.M.Monahan,V.S.Vitankar,andR.O.Fox,“CFDpredictions
with Lulea˚ University of Technology (LTU), Royal Institute forflow-regimetransitionsinbubblecolumns,”AIChEJournal,
of Technology (KTH), Chalmers University of Technology vol.51,no.7,pp.1897–1923,2005.