Explicit calculation of natural aeration using a Volume-of-Fluid model

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ORE Open Research Exeter
TITLE
Explicit calculation of natural aeration using a Volume-of-Fluid model
AUTHORS
Lopes, P; Tabor, G; Carvalho, RF; et al.
JOURNAL
Applied Mathematical Modelling
DEPOSITED IN ORE
20 March 2017
This version available at
http://hdl.handle.net/10871/26694
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How to cite this paper: Lopes, P., Tabor, G., Carvalho, R.F., and Leandro, J., 2016. "Explicit calculation of
natural aeration using a Volume-of-Fluid model". Applied Mathematical Modelling, 40 (17–18), pp.
7504–7515, DOI: 10.1016/j.apm.2016.03.033.
Explicit calculation of natural aeration using a Volume-of-Fluid
model
Pedro Lopesa,b,∗, Gavin Taborc, Rita F. Carvalhoa,b, Jorge Leandroa,b
aDepartmentofCivilEngineering,UniversityofCoimbra,Portugal
bMARE-MarineandEnvironmentalSciencesCentre,UniversityofCoimbra,Portugal
cCollegeofEngineering,MathematicalandPhysicalSciences,UniversityofExeter,UnitedKingdom
Abstract
Accurate prediction of the air-entrainment process in air-water two-phase turbulent flows is one of the most computationally
challenging subjects under current investigation in hydraulic engineering. An ideal numerical model for air-entrainment needs
to be accurate and fast in the definition of a macroscopic interface and simultaneously precise enough to take into account the
formationofbubblesthroughthefree-surface,theirtransportandtheirnaturalinteractions: bubble-bubbleandbubble-fluid. The
problemismademorecomplexbythestrongcouplingbetweenmeshandsolutionexhibitedbyinterfacecapturingschemeswhich
arecommonlyusedforsuchproblems. Thispaperexaminesnumericalandmodellingaspectsoftheentrainmentprocessfortwo
canonicalcases; the2Ddambreakand3Dcircularplungingjetcases. WestartbyinvestigatingthecapacitiesofaVolume-of-
Fluidbasedmodeltodetectthefree-surfaceandpredictthevelocitiesinsidethewaterphase,examiningtheeffectofcoarsening
andrefiningthemeshonthepredictionoftheinterfacelocation. Areformulatedexplicittermisusedtodetectbubbleformation
andair-entrainmentatthefree-surface,withouttheneedofacalibrationprocessandadaptedtoruntogetherwithVolume-of-Fluid
models. Theresultsobtainedwiththisnewapproacharefurthercomparedwithsimilarcasesintheliteratureintermsofbubble
formationandfree-surfacewave’samplitude.Thecorrectdefinitionofthefree-surfacewasfoundtobestronglydependentonthe
meshrefinementinawaythathasverysignificantimplicationsforthedevelopmentofair-entrainmentmodelling.
Keywords: Air-entrainment,Air-watertwo-phaseflow,Sub-gridexplicitmodel,Volume-of-Fluid
1. Introduction
1
Air-entrainmentoccursinmostturbulentfree-surfaceflowsinnatureresultinginadispersedtwo-phaseflowbelow
2
the surface with a complex turbulent mixture structure, where compressibility and density are important physical
3
propertiesaffectingtheair-entrainmentandtransportcharacteristics. Theaccuratepredictionofair-entrainmentisa
4
veryambitiousgoalformostComputationalFluidDynamics(CFD).Theair-waterinterfaceisveryunstableandthe
5
length scales of turbulence range from those influenced by the bubbles and surface tension (order from microns to
6
∗Correspondingauthorat:Hydraulic,WaterResourcesandEnvironmentLaboratory,RoomS.H.4.6,DepartmentofCivilEngineering-Polo2
UniversityofCoimbra,RuaLu´ısReisSantos,3030-788Coimbra,Portugal;Tel.:[phone].
Emailaddresses:[email protected](PedroLopes),[email protected](GavinTabor),[email protected]
(RitaF.Carvalho),[email protected](JorgeLeandro)
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Lopes et al. / Applied Mathematical Modelling 40 (17-18) 7504-7515 7505
millimetres) to those of the mean flow (order of meters). The simulations should be able to work on a coarse grid
1
inordertosimulatethemeanflowbehaviour,thefree-surfacepositionandallthesimilarlargelengthscales(LLS),
2
whilst at the same time being sufficiently detailed to model the transport of the bubbles of gas within the flowing
3
fluid and simulate important phenomena as bubble formation, breakup, coalescence and collision, which take place
4
onsmalllengthscales(SLS).
5
In CFD, Volume-of-Fluid (VOF) [16] and Level-Set (LS) [32] based models are typically used to capture the
6
interfacebetweentwofluids,solvingasinglesetofNavier-Stokesequationsrepresentingbothairandwater,together
7
with an indicator function propagated by an advection equation. These interface models are widely used for many
8
free-surface flows with macroscopic interfaces and LLS systems, as in vertical plunging jets [17, 24, 28]. Direct
9
NumericalSimulationscouldsimulateallaspectsoftheentrainmentprocessrightdowntothesubsequentdynamics
10
ofthebubbles;howeverthiswouldbeincrediblytime-consumingand,insomecases,theapplicationofVOFmethod
11
todispersedphasescouldleadtoanon-physicalinterpretationofbubblesordroplets[7]. Instead,forarealisticmesh
12
resolutiontherepresentationoftheentrainedair(SLSsystem)canbeaccomplishedbytheinclusionofanEulerian
13
dispersedphasemodelwithasecondsetofequationsrepresentingthedynamicsofthebubbles. Differentapproaches
14
thatfollowthisideacanbefoundintheliterature. ThemodelofCerneetal.[7]blendstheVOFinterfacetracking
15
with a two-fluid model formulation [11]. In this model, in zones where the phase separation are clear and where
16
just one fluid is present, the two-fluid model is switched off and a single set of Navier-Stokes equations together
17
with the VOF method is solved instead. A criterion based on the local dispersion of the interface is used to switch
18
betweenthetwoformulations, however, theaccuracyoftheresultshasexhibitastrongdependenceonthevalueof
19
this threshold which goes against the essence of a blending model. To overcome the dependency of the dispersion
20
threshold,YanandChe[39]introducedaunifiedsolutionframeworkforcouplingVOFwithatwo-fluidmodel. The
21
idea was to blend the two formulations ensuring the conservation of all three phases; i.e. when the LLS of air are
22
present(phase2),theinterfacetrackingmodelisactivated;thenwhentheLLSareabsentfromthecell,thetwo-phase
23
modelisusedtosolveforthelocalcharacteristicsofthefluids(phase1forwaterandphase3forbubbles). Themodel
24
showntobeefficientforthesimulationofarisingbubbleandaswarmofbubblesinaverticalpipe. However,volume
25
fractionconservationwhenthethreephasesarepresentisnotguaranteed,withparticularlossesoccurringforphase2.
26
Wardle and Weller [36] introduced in OpenFOAM(cid:13)R a hybrid formulation based on the combination of an Eulerian
27
multifluidframework(tosolvetheSLS)withaninterfacecapturingmethodusingVOF(tosolvetheLLS),alongwith
28
aswitchingfunctionbasedontheworkofCerneetal.[7]. ShonibareandWardle[33]extendedthishybridmodelto
29
dealwithvariablebubblesizeusingthereducedpopulationbalancemethodandappliedittoaverticalplungingjet.A
30
similarconceptualapproachtothelastwasimplementedbyMarschallandHinrichsen[25]inOpenFOAM(cid:13)R forsolely
31
two-phaseflows. Ha¨nschetal.[14]extendedtheinhomogeneousMultipleSizeGroup(MUSIG)-approachbyadding
32
a continuous gas phase in order to solve simultaneously in the same domain, dispersed and continuous gas phases.
33
Thetransitionbetweenthetwowasmodelledbythe”clusteringmethod”thatutilisesanadditionalinterfacialforce
34
appliedtotheEulerianmultifluidframework. Thesolutionswereverifiedforthecaseofplungingjets. Yetanother
35
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Lopes et al. / Applied Mathematical Modelling 40 (17-18) 7504-7515 7506
type of subgrid models combine an interface model (VOF/LS) with the two-phase flow formulation by including a
1
sourcetermtodetecttheairformationatthefree-surface[15,22,23]. Sourcetermsattheinterfacerelatetherateof
2
bubbleformationtosurfaceflowpropertiessuchaslocalturbulenceandthesizeofinterfacewaves.
3
A typical experiment in which air-entrainment has been observed and extensively studied is the plunging jet
4
[1, 2, 4, 8, 10, 18, 29, 31]. Plunging jets are efficient mechanisms to dissipate energy, and in doing so produce
5
and transport significant quantities of bubbles through the body of the water. Practical examples of plunging jets
6
include dam spillways and plunge pool stilling basins, waste-water treatment, oxygenation of bioreactors and river
7
re-oxygenation. Air-entrainmentinthepooldependsuponthejetimpact,thephysicalpropertiesofairandwater,the
8
jetdiameter,thefreefallingdistancebetweenthejetandthepool,andthejetturbulence. Attheintersectionpointof
9
thejetandthewater,free-surfaceinstabilitiesarethereasonforair-entrainmentwhenthejetimpactvelocityexceedsa
10
characteristicvelocityoronsetvelocity[8]. Slightlyabovetheonsetvelocity,theairentrainsintheformofindividual
11
bubbles and small pockets. At the impact point, the free-surface is observed to assume a shape which balances the
12
forces between both sides of the interface. With increased liquid velocity at the jet impact zone, the local stress
13
is increased, and small cavities are formed and pulled bellow the free surface. Deeper into the pool, these cavities
14
decreaseinradiusbutcontinuebalancingthesurfaceforces. Acriticalconditionisreachedwhereeither(a)asteady
15
balanceofforcescannolongerbemaintainedor(b)theinferiorendofthecavityorthefree-surfaceisdisruptedby
16
disturbancesatthefreejetstreamand/oronthepool. Theresultisthesequentialgenerationofsmallbubblesorair
17
pocketsbelowthefreesurface. Inthiscase,therateofair-entrainmentisverysmallandchallengingtomeasurewith
18
commonintrusiveprobes. Athigherjetvelocities,asubstantialairpocketorsheathisformedattheperipheryofthe
19
jetbelowthefreesurface; thisairpocketisunstableanditsbreakupformstheentrainedairbubbleswhicharethen
20
thesubjectoftransportwithinthebodyofthewater[4].
21
This paper proposes a reformulated explicit term for bubble formation that is independent of calibrating factors
22
by using a renewed formula for surface wave’s amplitude. The explicit term was included in the well validated
23
interFoam VOF solver from the OpenFOAM(cid:13)R toolkit to accurately predict the interface position [19, 20, 21]. An
24
interfacelocationcoefficientisnecessaryforthiscoupling. Thisstudyrepresentsastartingpointforthedevelopment
25
ofaVOF-basedmodelwithamodelledclosuretorepresenttheentrainedair. Section2developstheequationsbehind
26
thenumericalmodelalongwiththeinterfacelocationtechniqueandtheair-entrainmentterm. Section3presentsthe
27
resultsandtheirdiscussionfortwocanonicaltestcases;a2Ddambreakcaseanda3Dplungingjet. Finally,section4
28
presentsconclusions.
29
2. NumericalModel
30
2.1. Generalconcepts
31
InterfacecapturingmodelssuchastheVOFmodel(implementedintheOpenFOAM(cid:13)R toolkitasinterFoam)rep-
32
resent the free-surface dynamics of the two phases in terms of single phase-weighted velocity and pressure fields,
33
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Lopes et al. / Applied Mathematical Modelling 40 (17-18) 7504-7515 7507
governed by a single set of mass and momentum equations. For a system of isothermal, incompressible and im-
1
miscibletwo-phaseflow,theReynoldsaveragedequationsformassconservationandmomentumarewrittenintheir
2
conservativeform:
3
∇·U=0 (1)
∂ρU
+∇·(ρUU)=−∇p∗+g·x∇ρ+∇·τ+f (2)
∂t
whereρ(kgm−3)isthefluidlocaldensity,g(m s−2)thegravitationalacceleration,U(m s−1)thevelocityvector,
4
τ (Pa) the shear stress tensor, p∗ (Pa) a modified pressure adopted by removing the hydrostatic pressure (ρg·x)
5
fromthetotalpressureandf(kgm−2s−2)thevolumetricsurfacetensionforce. Itisimportanttonotethatininterface
6
capturingmodels,thevelocityvectorUactsasasharedvelocityofthetwofluids,i.e. U=U =U ,ratherthanin
7 f1 f2
mixturemodelswhereU ,k={1,2}canassumedifferentmagnitudes.
8 fk
ThedecompositionoftheviscousstresstermisgivenbytheStokes’sstressconstitutiveequation(Eq.3)where
9
µ(kgm−1s−2)isthedynamicviscosity.
10
∇·τ=∇·[µ(∇U+(∇U)T)]=∇·(µ∇U)+(∇U)·∇µ (3)
Togetherwiththepreviousequations,interFoamusestheVOFtechnique[16]tocapturetheinterfacebetweenthe
11
twofluidsbysolvingatransport/advectionequation(Eq.4). Basically,ateachcellofthedomainwedefineanalpha
12
(α) value representing the fraction of the volume of the fluid in that cell [6, 34]. Cells completely filled with fluid
13
1 (f1) will be represented by α = 1 and cells filled with fluid 2 (f2), by a value α = 0. The interface is localised
14
to the cells where α is intermediate between these two values. The advection equation also includes an interfacial
15
compressivetermtoconfinethisinterfaceregionintoassmallaregionofspaceaspossible(thelasttermonthel.h.s.
16
ofEq.4)[3,37],ratherthanusinginterfacereconstructionschemes[30].
17
∂α
+∇·(αU)+∇·[U α(1−α)]=0 (4)
∂t c
Thetermα(1−α)ensuresthatthecompressivetermiscalculatedjustattheinterfacialcellsofthedomain,while
18
C isabinarycoefficientthatactivates(C = 1)ordeactivates(C = 0)theinterfacesharpeningterm. Thesource
19 α α α
termalsoincludesthecompressivevelocity(U )thatactsasavelocityperpendiculartotheinterfaceandiswrittenas:
20 c
∇α
U =C |U| (5)
c α |∇α|
Itisworthemphasisingthattheinterfaceismerelybeinglocalisedwithinthevolumeofspaceforwhich0<α<1.
21
Forsimplicityitisoftentakenthattheinterfaceisrepresentedinpost-processingbytheisosurfaceα=0.5,butstrictly
22
speakingthisissimplyanassumption,onewhichwewishtoexploreinthepresentwork.
23
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Lopes et al. / Applied Mathematical Modelling 40 (17-18) 7504-7515 7508
Whenαassumesvaluesbetween0and1,thephysicalpropertiesofthetwo-fluidmixturearedefinedasaweighted
1
averageoftwofluidproperties:
2
ρ=αρ +(1−α)ρ (6)
f1 f2
µ=αµ +(1−α)µ (7)
f1 f2
ThevolumetricsurfaceforcefunctionisexplicitlyestimatedbytheContinuumSurfaceForce(CSF)model(Eq.8)
3
developedbyBrackbilletal.[5]whereσ(kg s−2)isthesurfacetensionandκ(m−1)isthesurfacecurvaturecalculated
4
asκ=∇·(∇α/|∇α|).
5
f=σκ∇α (8)
Theturbulentkineticenergy(k)andrateofenergydissipation(ε)arecalculatedusingtherealizablek−εturbulence
6
model,withtheeffectivedynamicviscosity(µ)beinggivenbyasumofmolecularviscosity(ν)andturbulentviscosity
7
(ν)(µ=ρ(ν +ν)). Thechoiceoftherealizablek−εisbasedonitsknownabilitytoaccuratelypredictthespreading
8 t t
rateofbothplanarandroundjets[13,40]. Thisturbulentmodelisalsosuperiortothestandardk−εmodelforthe
9
simulationofflowsinvolvingrotation,boundarylayersunderstrongpressuregradients,separationandrecirculation.
10
The realizable k −ε also requires less computational time than Re-Normalisation Group (RNG) k −ε which was
11
derivedtodealwiththeswirleffectonturbulence[13].
12
2.2. Bubbleformation
13
The air-entrainment process is described by the inclusion of a function E (m−3s−1) which describes the rate of
14 g
bubblegenerationatthefree-surface.Itiscalculatedattheendofeachtime-stepanddoesnotchangeanyconvergence
15
processwithinthestandardsolver. TheformulationofthisvariablefollowscloselytheworkofMaetal.[23], who
16
representitas:
17
(cid:42) (cid:43)
∂U a
E = n (9)
g ∂n φ V
ent g
whereU isthenormalvelocitycomponenttothefree-surface,a(m)istheamplitudeofthecavitiesformedatthe
18 n
free-surface(Fig.1),V (m3)isthevolumeofaspherecalculatedassuminganaveragedbubblediameterandφ (m)
19 g ent
istheinterfacethickness, givenbyφ = 0.05L, where L (m)isacharacteristiclineardimension(equaltothepipe
20 ent
diameterforpipefloworthehydraulicdiameterwhendealingwithriversystems). Thesymbols(cid:104)(cid:105)areusedtoturn
21
thenormalderivativezeroifitsvalueislessthanzero:
22

(cid:104)f(x)(cid:105)=f(x) , f(x)>0 (10)
0 , f(x)≤0
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Lopes et al. / Applied Mathematical Modelling 40 (17-18) 7504-7515 7509
A rough way to predict the volume fraction occupied by the bubble phase at each cell can be done by E =
1 g,frac
E V ∆t, where ∆t (s) is the computational time step. This neglects the advection and diffusion terms of the void
2 g g
fractionconservationandtransportatthefree-surface,however,whenthevelocitiesaresmall,itcanprovideagood
3
firstimpressionofvoidfractionmagnitude.
4
InthisworkandaccordingtoErvineandFalvey[12],theamplitudeofthesurfacedisturbances(a)isconsidered
5
as having the same order of magnitude as the radius of the turbulent eddies at the free-surface (l(cid:48)). From here, we
6
obtain:
7
k3/2
a=l(cid:48)/2= C3/4 (11)
µ 2ε
whereC isaturbulencemodelconstantwhichinthek−εmodeltheoryassumesthevalue0.09[27,35]. Similar
8 µ
criteriaisusedbyHirt[15]tocalculatesurfacedisturbancesatfree-surfaceintheCFDcommercialcodeFlow-3D(cid:13)R.
9
InthemodelofMaetal.[23],theamplitudeofthesurfacedisturbances(a)iscalculatedasa =C k/g,whereC
10 ent ent
isaconstantthatneedstobecalibratedforeachcase.
11
idealized free-surface
water with vorticity
to trap air bubbles
air a
water σ r
σ
l' Un
Figure1.Free-surfacewithphysicalparametersused.AdaptedfromErvineandFalvey[12].
2.3. Interfacelocation
12
Thelocationoftheair-entrainmentinourVOFmodeldoesnotfollowthesamecriteriaasinworkfromMaetal.
13
[23](whichusesLSmethods). InLS,theinterfacecaneitherbelocatedbyassumingacertainthresholdforinterface
14
thickness,orbylimitingthevelocitytosomethresholdvalue. However,inVOFmodels,theinterfacialstructurecan
15
only be detected by the change of volume fraction or by its gradient, as for example in the the algebraic equation
16
formulatedbyHa¨nschetal.[14]. Inthiswork, E iscalculatedusingasimilarcriterionasusedforthecompressive
17 g
term,i.e.bymultiplyingthelastbythefunctionφ =4α(1−α).Thiswillworkasamaskingfunction,whichreturns
18 FS
0whencalculatedincellsawayfromthefree-surfaceand1attheinterface. Fromthefactthatthiscoefficientuses
19
αtodetectthefree-surface, itisfairlyessentialkeeptheinterfaceconfinedtoasmallnumberofcellspossible, and
20
fromthisweseetheneedofameshdependencestudy.
21
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Lopes et al. / Applied Mathematical Modelling 40 (17-18) 7504-7515 7510
3. Results: Examplecases
1
3.1. 2DDam-break
2
Theexamplepresentedhereisthebenchmarking2Ddam-breakcasewithoutobstacle. Thiscaseisusedbymany
3
other authors to show the capabilities of free-surface models and of interest for this work as it can exhibit several
4
possiblezonesofnaturalaerationoftheflowandtheinterfacelocation. Astaticwatercolumnisinitiallyheldagainst
5
theleftwallbyaverticalgateasshowninFig.2. Onremovingthegate,thewatercolumncollapsesandtheresulting
6
sloshhitstherightwallbeforereturningintheformofasmallpropagatingwave.
7
atmosphere
n=0.146 m
wall
2n
water air
n
4n
Figure2.2Ddam-breakdomain.
Numerical simulations are performed in four different meshes: 16 × 16, 32 × 32, 46 × 46 and 64 × 64 cells.
8
The fluid properties were chosen to represent the physical values of water and air at 15oC. Surface tension is set
9
to 0.072 kg s−2, the averaged bubble radius (r ) is 0.002 m and φ = 0.05n = 0.0071 m, where n stands for the
10 b ent
watercolumnwidth(Fig.2). Thedomainisboundedbywallswiththeexceptionofthetoponethatrepresentsthe
11
atmosphere. The pressure at the walls is given by Neumann boundary conditions (BC), whilst the velocity is set to
12
no-slip(Dirichet-BC).
13
Figure3presentssnapshotsofthefree-surfacepositionfort = 0.1, 0.3, 0.7and1.0 sforfinermesh. Figure3a
14
shows the φ function and Fig. 3b the values of the terms E and E . Figure 4 demonstrates the influence of
15 FS g g,frac
changingthemeshsizeonthefree-surfacepositionfort = 0.8 s. Again,Fig.4ashowstheφ functionandFig.4b
16 FS
thevaluesofthetermsE andE .
17 g g,frac
Thefunctionusedtodetectthefree-surfacehasdemonstratedexcellentaccuracythroughoutthesimulationtime
18
even when a large cavity is formed inside the fluid (Fig. 3a4). One second after the collapse of the water column
19
(Fig.3b1),duetothefrictionwiththebottomwall,airisforcedtoenterfromthebottom,wherethetermE assumes
20 g
highvalues. Highvaluesofair-entrainmentcanalsobeseeninthezonewherethewatercollideswiththerightwall
21
(Fig.3b2)andonitswaybackintheareawhereasmallcavityofthebreakingwaveisformed(Fig.3b3). Themesh
22
influencestudyshowalargedifferencebetweenmeshes16×16and32×32cellsbothforφ and E values. The
23 FS g
free-surfaceinthelastthreefinermesheshaveasimilarshape,whereasthemaximumvaluesofE areontheorderof
24 g
4×107inthemeshwith32×32cells(Fig.4b2)andoftheorderof5×107for46×46and64×64cells(Fig.4b3and
25
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Lopes et al. / Applied Mathematical Modelling 40 (17-18) 7504-7515 7511
(a1) (a2) (a3) (a4) φFS
1.00
0.75
0.50
0.25
0.00
(b1) (b2) (b3) (b4) Eg(x107 ) Eg,frac
7.00 1.00
6.00 0.75
4.00 0.50
2.00 0.25
0.00 0.00
Figure3.Dam-breakprocessforthetimestepst=0.1,0.3,0.7and1.0susingthefinestmesh(64×64cells).Topimagesarephotographsofthe
experiment,middlefigures(a)showthefunctiontodetectthefree-surfaceposition(φFS)andbottomfigures(b)showthebubbleformationterm
(Eg)anditsvolumefractiononthecells(Eg,frac).
φ
(a1) (a2) (a3) (a4) FS
1.00
0.75
0.50
0.25
0.00
(b1) (b2) (b3) (b4) Eg(x107 ) Eg,frac
7.00 1.00
6.00 0.75
4.00 0.50
2.00 0.25
0.00 0.00
Figure4.Meshinfluencestudyofdam-breakcaseatt=0.8s.Fromthelefttorightthemeshesare:16×16,32×32,46×46and64×64cells.Top
figures(a)showthefunctiontodetectthefree-surfaceposition(φFS)andbottomfigures(b)showthebubbleformationterm(Eg)anditsvolume
fractiononthecells(Eg,frac).
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Lopes et al. / Applied Mathematical Modelling 40 (17-18) 7504-7515 7512
Fig.4b4). Thisshowsthat,evenwiththedecreasingofthemeshsize,thevaluesof E arequasimeshindependent,
1 g
althoughitsspaciallocationissensibletothegridsize.
2
3.2. 3Dverticalplungingjet
3
The 3D computational domain represents a part of the experimental apparatus used by Chanson and Manasseh
4
[9],schematicallyrepresentedinFig.5. Thisexperimentalworkprovidesextensivedataonvoidfractionandbubble
5
countrateinsidethewater,whichdespitenotbeingrelevantinthiswork,willbeusefultocomparetheresultswhena
6
fullair-entrainmentmodeliscompleted. Thesimulationofa3Ddomainratherthana2Ddomainallowsustobetter
7
representalltheflowstructuresacrosstheinterfacewhicharemanifestly3Dstructures. Also,comparisonstothe3D
8
modelofMaetal.[22]intermsofE valuesandlaterforvoidfractionscanbedoneclearlyandwithoutconstraints
9 g
(notethatMaetal.[22]appliedtheirmodeltothesamecaseintheirstudy).
10
Aplungingjetofcleanwaterisejectedfromad =0.025mdiameternozzle,intoapoolofstationarywater. The
11 0
distancebetweenthewatersurfaceandthenozzleisconstantandequalto x − x = 0.1 m. Theimpactvelocityof
12 1 0
thejetatthepoolisU = 3.5 m s−1. Velocityprofilesareobtainedatdifferenthorizontalplanesfrom0.8d (m)to
13 1 1
10.0d (m) below the initial pool free-surface, where d is the jet diameter at the impact zone (d ≈ 0.024 m) and
14 1 1 1
r the jet radius (r ≈ 0.012 m). To investigate the impact of mesh resolution on the solution, three different grids
15 1 1
withdifferentmeshresolutionsweregeneratedandtheircharacteristicssummarizedinTable1. Inordertodecrease
16
thecomputationaltime,insteadofusingspatiallyuniformmeshes,threezoneswithdifferentgridsizesarecreatedas
17
definedinFig.5.
18
celllength,∆x(m) N.Cells N.Proc. Time(s)
jetandsurface(Z1) centre-bottom(Z2) lateral(Z3)
Coarsegrid(G0.005) 0.005 0.005 0.005 84192 16 1963
Mediumgrid(G0.0025) 0.0025 0.0025 0.005 521124 16 39891
Finegrid(G0.00125) 0.00125 0.0025 0.005 2104460 16 281073
Finnestgrid(G0.0008) 0.0008-0.00125* 0.0025 0.005 2458835 32 284242
Table1.Gridcharacteristicstodemonstratethemeshdependency.ZonesZ1,Z2andZ3arerepresentedinFig.5.*Inthiscase,cellswith0.0008m
edgeswereplacedjustontheintersectionbetweenjetandpool.
TheinitialconditionsaresetasrepresentedinFig.5. Thepoolisfilledwithstationarywater(U = 0 m s−1)toa
19
constantdepth.Acylindricalcolumnofwaterwithafixedinletvelocityisusedtorepresentthejet.Inordertogetfast
20
convergenceofthesolutionandcomputationalstabilitytheairvelocityinthevicinityofthejetissetequaltothejet
21
velocity. Fourdifferentboundaryconditions(BC)areusedinthissimulationasshowninFig.5. Thevelocityatthe
22
(cid:113)
inletiscalculatedbasedonthejetimpactvelocityusingBernoulli’sprinciple,U = U2−2g(x −x )=3.21m s−1.
23 0 1 1 0
TheatmospherejustallowstheairtoleavethedomainbysettingU asdependentonpressurewithtotalpressureset
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as zero. The lateral boundaries have Neumann-BC for α, p equal to hydrostatic pressure and non-slip Dirichet-BC
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forU. Forthebottomboundarythehydrostaticpressureissettoρ g(x −x )=3915.2 PaandNeumann-BCare
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