Experimental study of submerged hydraulic jumps generated downstream of rectangular plunging jets

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International Journal of Multiphase Flow

journalhomepage:www.elsevier.com/locate/ijmulflow

Experimental study of submerged hydraulic jumps generated downstream of rectangular plunging jets

José M. Carrillo

^∗

, Francisca Marco , Luis G. Castillo , Juan T. García

Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos e Ingeniería de Minas, Universidad Politécnica de Cartagena, Paseo Alfonso XIII, 52 - 30203 Cartagena, Spain

a rt i c l e i nf o

Article history:

Received 10 June 2020 Revised 11 January 2021 Accepted 23 January 2021 Available online 27 January 2021 Keywords:

Air-water flow Back-flushing Pitot tube Falling jet

Optical fiber probe Overtopping

Submerged hydraulic jump

a b s t r a c t

Thisexperimentalstudyanalyzessubmergedhydraulicjumpsgeneratedina1.05-mwidestillingbasin thatreceivesoverflowrectangularimpingingjets,withtheverticaldistancebetweenthesharp-crested weirandthebottomoftheplungepoolbeing2.20m.Fivesubmergedhydraulicjumpsdownstreamof non-developednappejetswereanalyzed,withverticalfallingdistancestobreak-uplengthratiosbetween 0.76and0.78.Theimpingementvelocityrangewasfrom6.12to6.33m/s,whilethewaterdepth/im- pingementjetthicknessratioswerebetween8.45and15.00.Thevelocityfieldandairentrappedwere measuredinseveralcross sections,withthe farthestsectionsbeinglocated1.0mfromthestagnation point.Theentrappedair (voidfraction and phasechangedetection)behavior measuredwithaphase- detectionprobewasclassifiedasafunctionofthe distancetothestagnationpoint /impingementjet thicknessratioand/orthemeanvoidfractionoftheverticalprofiles.Themaximumvoidfractionswere observednearthefreesurface.However,thehigherphasechangedetectionfrequencies(upto130-140 Hz)wereobtainednearthebottomforlocalvoidfractionssmallerthan0.15-0.20.Thenon-dimensional velocitydistributionseemsto besimilar topreviouspublishedstudies, extendingtheirvalidity range.

Furthermore,the freehydraulicjumpsformula hasbeenadaptedtoestimatetheair rateintheshear stresslayerandthegrowthofthecharacteristiclengthintheplungepool.

© 2021TheAuthor(s).PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Inrecentdecades,improvementsintheknowledgeofthetime- series and changes in dam safety regulations have made it nec- essary to re-evaluate the spillways and outlets capacity of large dams, and their operational scenarios around the world. Several studies (e.g., FEMA, 2013, 2014) have demonstrated that many weirs may have an insufficient flow discharge capacity, thereby increasing thepossibility of beingovertopped inextreme events.

Duringextremeconditions,theoverflowmaybeconsideredasan additionaloperationalstrategy,althoughattheriskofraisingnew hydrodynamicscenariosforthedamsthatmaycauselocalerosion andscouratthetoeofthedam.

The type of stilling basin is a technical-economic choice be- tweenalinedshallowplungepooloran unprotecteddeepplunge pool. For safety operations, the designers require data regarding the magnitudeandfrequencyofthehydrodynamicactionsatthe

∗Corresponding author.

E-mail address: [email protected] (J.M. Carrillo).

bottomoftheplungepool.Theimpingementjetcharacteristicsand thewatercushiondepthsdefinethebottomrequirementstoresist scouratthetoeofthedam(Annandale,2006).

Severalstudies havefocused on the airentrainment,free sur- face perturbations and break-up length of turbulent jets (e.g., Horeni, 1956; Ervine and Falvey, 1987; Ervine et al., 1997; Castillo, 2006). In addition, the characteristics ofthe near region flow of turbulent jets entering water cushions have been stud- ied by several authors (e.g., Albertson et al., 1950; Ervine and Falvey, 1987; Ervine et al., 1997; Bollaert and Schleiss, 2003; Chanson et al., 2004; Manso et al., 2007; Bertola et al., 2018; Xuetal.,2018).

A better understanding of the flow patternin submerged hy- draulic jumps ofplunge poolsis essential forthe designers asit may affect their designs. The entrapped air in the plunge pool may play an important role in hydraulic structure safety opera- tions.Thiscreatesagreatincrementintheflowdepthofthemix- ture. The entrained air may also affectthe behavior ofthe still- ing basin, influencing the characteristics of the vortices. Accord- ingtoDuarteetal.(2015),twoopposingeffectsmaybegenerated

https://doi.org/10.1016/j.ijmultiphaseflow.2021.103579

0301-9322/© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Fig. 1. Definition sketch of a submerged hydraulic jump downstream of a nappe jet.

byairentrainment.Duetoitslowerapparentdensity,aeratedjets havelower momentumthan similarclearwaterjets, contributing tohavinglowerpressuresonthebottomoftheplungepool.How- ever,theairbubbleslowertheshearstressesinthepool,allowing theaeratedjetstoflowwithahighervelocity.Theinfluenceofair entrainmentisabalancebetweenthesetwoeffects.

Ahydraulicjumpisalocalnon-uniformtransitionfromsuper- criticaltosubcriticalflowregime.The hydraulicjumpisadiscon- tinuity inthefree surfacecharacterizedby asteep upwardslope, the development ofeddies atdifferent scales, surfacewaves and undulationofthefree surface,spray,energydissipationandlarge amounts ofairentrainment.Due totheir highenergydissipation, theymaybedesignedasenergydissipaters,whoseefficiencyisdi- rectly relatedtothesupercritical Froudenumber.Areviewofthe generalknowledgeinfreehydraulicjumpsandequipmenttomea- suretheflowpropertiesmaybefoundinValeroetal.(2018).

According to Wood(1991),the voidfractionis oneof thekey parameterstobeanalyzedinhydraulicjumps.Theentrainedairin stillingbasinsmayaffectthebehaviorandthesafetyoperationof hydraulic structures(e.g.,toprotecthydraulicstructuresfromcav- itationdamage,todesigntherequiredwallelevationofthestilling basin toavoidoverflows duetobulkingand/orundulationsofthe freesurface,amongothers).

In freehydraulic jumps,the airentersmainly atthe jumptoe into afree shearlayer,characterized byintensive turbulencepro- duction,predominantlyinvorticeswithaxesperpendiculartothe flow direction. A better knowledge of mixing, diffusion, transfer processes and air/watergas exchanges dependson the ability to measurethesetwo-phaseturbulentflows(Murzyn,2010).

When the tailwater levelis higherthan theconjugated orse- quent water depth, for the same inlet Froude number, the hy- draulic jumpisdrowned out.In submergedhydraulic jumps gen- erated downstream of gates, the air entrainment and the turbu- lenceintensities aresmallerthanfortheir freejumpcounterparts (Rajaratnam, 1965), as well as showing relatively gentle surface profiles (Longetal.,1991).Long etal.(1990)consideredthatsub- merged hydraulic jumps may be treated asa transition between free hydraulic jumps and wall jets, and may be treated as wall jets under an adverse pressure gradient. The authors also dis- tinguished three regions in submerged hydraulic jumps: the de- veloping, the fully developed, and the recovering regions. Later, Peiqingetal.(1998)distinguishedthreeregions inthemainflow

directionofplunge pools:region Ilocated betweenthe impinge- ment point ofthe free fallingjet andthe distancefromthe bot- tom whereits influencedoesnot affectthe characteristics ofthe submerged jet;region IIlocated nearthe stagnationpoint,where theflow experiencessuch considerabledeflectionthatthe veloci- tiesdecreaserapidlyandthepressuresincreasesharply;regionIII inwhichtheflowbehaviorissimilartothatofasubmerged wall jet,themainflowdiffusesrapidlyalongthebottomoftheplunge poolandalargerecirculatingregionisformedatthesurface.

Fig.1showsasketchofasubmergedhydraulicjumpgenerated inaplungepool.The figureincludesthevelocity andairconcen- tration distributionsbasedonthe resultsobtainedandtheobser- vationsmadebyseveralauthorsinhydraulicjumps.

Withregardtotheplungepoolaeration,thereareatleastthree main mechanismsasreportedby Ervine (1998).The first mecha- nism considers the surface disturbances of the falling jet on the aerationprocess,withtheminimumjetvelocitytoentrainairde- pendingonthejetturbulencelevel(Ervineetal.,1980).Inthesec- ondmechanism,athinairboundarylayerisdraggedalongbythe jet; the air flow maybe ableto enter the receiving water when a gap isformed between the jet andthe recirculating flow. This secondmechanismisusuallyonlythoughttobesignificantforjet velocitiesbeyond5m/s.Thethirdmechanismisafree-surfaceaer- ation which contributes to the overall aeration rate. This can be presentinthefallingjet,particularlyathighvelocities(>6m/s), andcan alsooccurduetotheviolentdisturbancesonthesurface oftheplungepool.

Usingphase detectionprobes, severalstudies havefocused on the void fraction or air concentration analysis in free hydraulic jumps (e.g., Chanson, 1995; Murzyn and Chanson, 2007; 2009; Murzyn,2010).Twodifferentregionsmaybedistinguished:anair- watershearlayernearthebottom,characterizedbysmallerbubble sizes(typicallyinmillimeters)andhighaircontent,andarecircu- lation regionlocated in the upperpart ofthe vertical profiles. A strong unsteady recirculation appears in the second region, with bubbles of differentsizes, and the presence ofair-water packets.

Nearthefreesurface, afoamstructurelayermaybeobserved.To thebestofourknowledge,thecurrentunderstandingoftwo-phase flows inthesubmerged hydraulicjump offreefallingrectangular jetsisscarce(e.g.,Castilloetal., 2017; Carrilloetal.,2020a). The presentwork aims to improve the studies inthe far field of the plungepool (namely fullydeveloped submerged wall jet region).

Fig. 2. Detail of the plunge pool in the experimental facility with the optical fiber probe.

This study experimentally analyzes submerged hydraulic jumps generateddownstream ofrectangularfallingjets, withfallingdis- tances from the weir crest to the bottom of the plunge pool of 2.20m.Anopticalfiberprobeandaback-flushingPitottubehave beenemployedtoanalyzetheentrappedairandthevelocityfield, respectively. Void fraction rates, free surface undulation pattern, phase-change detection frequency, mean velocities near the bot- tom,andSautermeanbubblediameterhavebeenanalyzedinthe submerged hydraulic jumpsgenerateddownstreamofthe stagna- tionpointofrectangularfreefallingjets.

2. Materialsandmethods 2.1. Experimentalsetup

Theexperimentswereconductedinanexperimentalfacilitylo- cated in the Hydraulic Laboratory of the Universidad Politécnica deCartagena(Spain).Therelativelylargeinfrastructuremaybedi- videdintotwoparts:

A 4.00 m long and1.05 m wide inlet channel that ends ina 0.85mrectangularsharp-crestedweirwithaweirheightof0.325 m.Thechannelhasalabyrinthtoincreasetheflowtrajectory,two gravelfiltersanda0.30mhoneycombpanelfordissipatingenergy andstraightening the streamlinesbefore theweir. The weircrest elevation was located at vertical distance P_l = 2.20 m fromthe bottomofthestillingbasin.

A1.05mwide,3.00mlong,and1.60mhighstillingbasin,built in methacrylate (Fig. 2). The plungepool enablesdifferent water cushiondepthstobeanalyzedbymodifyingtheheightofthewall locatedinthedownstreamside.Furtherinformationontheexper- imentaldeviceisavailableinCarrillo(2014).

The flows were measured with an Endress + Hauser electro- magnetic flowmeter installed in the supply line. The device had an accuracy of±0.5%.The upstream energyheadoverthe sharp- crested weirwasestimatedwitha pointgauge, withanaccuracy of±0.5mm.

Anoptical fiberprobeandamodifiedPitottubewere usedto measure the entrapped air and the velocity field in the center- line of the stilling basin. The submerged hydraulic jumps gener- ateddownstreamofthestagnationpointoftheimpingementjets were measured in ten cross sections, spaced 0.10 m apart. The stagnationpointwasconsideredbyusingthetrajectoryofthecen- tral nappe proposed by Scimemi (1930).According to our previ- ousfindings, thisexpressionseems tobea goodestimatorofthe maximumpressuredistributionpoint measuredon thebottomof theplungepool,withdifferencesof±1 cmforq >0.037m³/s/m (Carrillo, 2014; Castilloet al., 2014). In each section, the vertical profilewasmeasured.Theprobes weremountedonatrolleythat enabled longitudinal, transversal and vertical displacements. The desired location in the three directions was measured by rules, withanestimatedaccuracyof±1mm.

The flow conditions were selected to analyze non-developed jets(verticaldistancebetweentheenergyheadovertheweircrest and at the end of the plunge pool to the break-up length ra- tio H/L_b < 1.0), andeffective water cushion depths (water depth at the end of the plunge pool to impingement jet thickness ra- tio Y/B_j > 5.5according to Castillo etal., 2015). Tofacilitate the analysis of the water cushion ratio influence (Y/B_j from 8.45 to 15.00), thecasesselected hadasimilar impingement jetvelocity, H/L_b ratioandangleof theimpingement jet

θ

j,estimatedby the Scimemi(1930)formula, andmeanvoid fractionatthe impinge- mentconditionC_mean,j.

Fivedifferent submerged hydraulic jumps were studiedin the stilling basin, with specific flow rates q from 0.099 to 0.073 m³/s/m, andequivalent impingement Froude number F_r^∗, that is, calculatedwithgravitationaljetthicknessBgandgravitationalve- locityV_j,from15.51to17.13.Thegravitationaldatawereobtained withthe Bernoulliformula, neglecting energy lossesandwithout considering air entrainment. Tables 1 and2 summarizesome of themostrepresentativeparametersinrectangularfreefallingjets, wherehistheenergyheadovertheweircrest,andR_eandW_eare, respectively,theReynoldsandWebernumbersobtainedupstream ofthesharp-crestedweir.

Table 1

Experimental flow conditions: dimensional parameters.

q (m ³ /s/m) h (m) H (m) Y (m) V j (m/s) B g (m) B j (m) L b (m) θj ( °) 0.099 0.144 2.044 0.290 6.33 0.017 0.034 2.69 78.4 0.089 0.135 2.001 0.324 6.26 0.016 0.032 2.61 78.7 0.085 0.130 1.979 0.341 6.23 0.015 0.030 2.57 78.9 0.079 0.124 1.938 0.376 6.16 0.014 0.028 2.51 79.0 0.073 0.118 1.907 0.401 6.11 0.013 0.027 2.45 79.2

Table 2

Experimental flow conditions: non-dimensional parameters.

q (m ³ /s/m) Y/B j (-) H/L b (-) C mean,j (-) F _r^∗ (-) R e (-) W e (-) R e0.2 W e0.6 (-) 0.099 8.45 0.76 0.54 15.50 161463 2794 1286 0.089 10.25 0.77 0.55 15.80 146565 2456 1168 0.085 11.28 0.77 0.55 16.24 138498 2277 1103 0.079 13.20 0.77 0.54 16.62 129021 2072 1028 0.073 15.00 0.78 0.56 17.11 119771 1876 954

FollowingCastillo(2006)andCastilloetal.(2015),theimpinge- mentjetthicknesswasestimatedas:

B_j=Bg+2

ξ

⁼



^q

2gH+4

ϕ 

h



_√

2H− 2



h



(1)

where

ξ

^{isthe lateraljet spreadthicknessdue tothe turbulence}

effect and

ϕ

= K_ϕTu,withTu beingthe turbulence intensityand K_ϕ =1.24fornappeflowcase.

The break-up length L_b defines the distance from where the jet becomes fully developed, which means that the non-aerated jet core disappears (see L_b in Fig. 1). It was calculated as per (Castilloetal.,2015):

Lb

B_iF_i² = K



K_ϕTuF_i²



0.82 (2) withB_i,F_iandT_ubeingthejetthickness, theFroudeNumber,and theturbulentintensityattheissuanceconditions,respectively,and K≈ 0.85adimensionlesscoefficient.

For prototypespecific flows (q >>0.25m³/s/m),a mean tur- bulence indexTu ~ 1.2%maybeconsidered(Castillo,2006).How- ever, for laboratory specific flow (q < 0.25 m³/s/m), the tur- bulence intensity at issuance conditions may be estimated as (Castilloetal.,2015):

T_u^∗= q^0.43

IC (3)

IC=14.95g^0.50

K^1.22C_d^0.19 (4)

withICbeingtheinitialconditions,gthegravitationalacceleration, andC_d thedischargecoefficient.

Possible scale effects may affect the initial conditions of the jet in the weir. To reduce the scale effects in rectangular weirs, Ranga RajuandAsawa(1977)estimatedthattheeffectsofviscos- ityandsurfacetensionvanishforR⁰_e^.2W_e^0.6 >900orheadof11.0 cmforwaterat20°C,inwhichRe=



gh³/

ν

^andWe=

ρ

^gh2/

σ

^are

the ReynoldsandWeber numbers, and

ν

^,

ρ

^and

σ

^{are the kine-}

matic viscosity, the waterdensity andthe surfacetension ofthe water,respectively.

AlthoughthevaluesshowninTables1and2fulfiltherecom- mendations ofRanga Raju and Asawa (1977),the air bubblesize is not correctlyscalable. Forthat reason, the air-waterflow phe- nomenonmustbescaledwithparticularcaution(Chanson,2009).

2.2. Opticalfiberprobe

Toregisterthelocalvoidfraction,anRBI-Instrumentsphasede- tectionchangedevicewasused.Theequipmentconsistedofanop-

ticalprobe that endsin a Descartes’ prism. Considering that the waterandthe airhavedifferentrefractionindexes,1.33and1.00, respectively,theintensityofthereflectedlight maybe relatedby theFresnelformula. Anoptoelectronic modulemeasuresthelight intensity reflected or diffracted by the prism depending on the fluid incontactwiththeprismandconvertsthe signal intovolt- age.Theriseandfallofthevoltagearethearrivalanddepartureof thephasechangeatthesensortip,respectively.Thus,opticalfiber equipmentcanbe usedtoestimate thelocalphasechangeofthe air-watermixture(RBI-Instruments,2012).

Murzyn et al. (2005) considered that the minimum bubble sized detected by the device may be in the order of the optical fiberdiameter(40

μ

^{m)inslowmovingflows.However,thesmall-}

estsizebecomesafunctionofthesignalsamplingfrequency(10⁶ Hz)athighvelocities.

StutzandReboud(1997a,1997b)foundthattherelativeuncer- taintyofthevoidfractionmeasuredwithopticalprobescanbees- timatedatapproximately15%ofthemeasuredvalue,andthesen- sitivitytoathresholdvariationislessthan1%.

Foranaccurateestimationoftheairentrappedintheflow,the statisticalcountofthenumberofairbubblesincontactwiththe tipsofthe probeshould be considered (Stutz, 1996). Hence,it is necessarytohavelargeenoughmeasurementsforaccurateresults.

In a studyof steppedspillways, Boesand Hager(2003) consider thattheaccuracyofthevoidfractioninair-waterflowsisrelated tothevariation oftheair-waterphase, insteadofthesampledu- ration t.Moreover, in steppedspillways, André etal. (2005)per- formedasensitivityanalysisoftheprobebehaviortotestthemin- imumduration oftheir registers. The authorsconsiderthat mea- surementsof60smaybeconsideredagoodbalancebetweenthe duration of the sampling sequence andaccuracy in order to ob- tain quasi-stationaryvaluesthat are statisticallyrepresentative of the void fraction.To err on the side of caution,previous studies insubmerged hydraulic jumps haveconsidered a sampletime of 90s(Carrilloetal.,2020b),obtainingarelativeuncertaintyofless than 1% of the mean value of the void fraction in this type of tests.

Ervine et al. (1997) proposed that the dominant frequencies in plunge poolsmay divide into two cases related withthe im- pingementjet velocity V_j.The lowest frequencies arelinked with largescaleeddieswithdimensionsintheorderoftheplungepool depth.Thosefrequenciesarecharacterizedbyarecirculatingveloc- ityVr ≈ 0.035V_j,anda Strouhalnumberofthedominanteddies S=fY/V_j≈ 0.01,wherefisthedominantfluctuationfrequency.The eddysizescontainedwithhalfofthe shearlayerwidthare char- acterizedbytheStrouhalnumberintheshearlayer Ss =fsY/V_j ≈ 0.25.

0 2000 4000 6000 8000 10000

0 15 30 45 60 75 90

segnahcesahpforebmunevitalumuC

t (s)

0.1 0.12 0.14 0.16 0.18 0.2

0 15 30 45 60 75 90

Cair(-)

t (s)

Void fraction Mean value

±1% relative uncertainty

Fig. 3. Evolution of the phase changes and the air concentration measured by the optical fiber probe during a test.

Fig. 3 shows that, although the phase change detection fre- quencydoesnotseemtobeaffectedbythesampletime(theslope ofthecumulative numberofphasechangesissimilar throughout theentireregister),morethan60sseemtobenecessarytoobtain arelativeuncertainlybelow1%inthevoidfractionmeasurements.

Takingthoseideasintoaccount,thefarfielddownstreamofthe impingement jet point ofthestillingbasin wasanalyzed.A sam- ple duration of90s wassetinall the experimentalcampaignto reduce the relative uncertaintyof themeasurements, considering theexpectedlarge-scaleeddiesintheplungepool.Thestatistically representative values ofthe voidfraction were validatedthrough repeatabilitytests.

Measurements were obtained withthe probe located perpen- dicular to the bottom. In the vertical position, the sensitive tip forms a45° angle withthe horizontalanditwasorientedtothe upstream side. This design may reduce the bias effect in three- dimensional flowstructures whenonlyone tipisbeingused.Ac- cording to Boes and Hager (1998), the size of the probe tip is rather small,so that it allows the measurement of tiny bubbles of aboutthe probe tip diameterandthe disturbance tothe flow issmall.However,astheflowmaynotbeorientedinasingledi- rectioninahydraulicjump,thisequipmentwasusedonlytomea- surevoidfractions.Averticaldefinitionof0.0067mwasusednear the bottom andnearthefree surfaceto capturethewall jet and thefreesurfaceundulation.Inthemiddlepartoftheverticalpro- files,theverticalresolutionwasincrementedupto0.020m.Into- tal, morethan 1600 pointswere obtainedwiththe opticalprobe device. These resultsallowed several parameters to be obtained:

thelocalvoidfractionC_air,thephasechangedetectionfrequencyF, andthewaterdepthswherethelocalvoidfractionwas90%(y₉₀), amongothers.

2.3. Back-flushingpitottube

The time-averaged velocity field wasmeasured by a modified Pitottubetoavoidairentrainmentinthepipelines.Thetipofthe tube waspositioned inthe samepositions asthe opticalfiber in the stilling basin. The Pitottube has a diameterhole of 2.3mm to measure the total pressurehead atthe tip of thetube, anda ringofportslocatedinthecircumferenceofthetubetoobtainthe staticpressurehead.The externaldiameterofthePitottubeis12 mm.TotalandstaticpressureheadswererecordedwithGEDruck model UNIK 5000 pressure sensors, with a measuring range be- tween-200and+800mbarandanaccuracyof±0.04%oftheirfull

scale.The accuracyoftheinstantaneous pressuredata was±0.01 mafterthestaticcalibration.Signalswererecordedfor90sat20 Hz.

To avoid the entrainment of airinto the pneumatic pipelines ofthePitottube,a back-flushingsystemwasmounted.Following MatosandFrizell(2000)andMatosetal.(2002),theback-flushing wasforcedbyareservoirofconstantheadwhichfedboththetotal andthestaticpressure ports. Needle valveswere usedto control theback-flushingflow.Toreducetheperturbationofthemeasure- ments by theback-flushing system, the flow rate waslimited to almostzero.

Wood(1983)consideredthefollowingexpressiontoobtainthe meanvelocity fromtheback-flushingPitottube inair-waterflow conditions:

V=



2g



^P

ρ

w

(

¹−

λ

^Cair

)

⁽⁵⁾

whereVisthetime-averagedvelocity, gthegravitationalacceler- ation,࢞P thedifferencebetweenthetotal pressureheadandthe staticpressurehead,

ρ

w thewaterdensity,C_airthelocalvoidfrac- tion, and

λ

the tapping coefficient which accounts for the non- homogeneous behavior of the air-waterflow near the stagnation pointofthePitottube.

Consideringthat the recirculation andbackward flow may af- fectthemeasurements ofthePitottube(MatosandFrizell,2000; Matosetal.,2002),thelocalmeanvelocityanalysiswaslimitedto thewalljetregionnearthebottomoftheplungepool.

3. Resultsanddiscussion 3.1. Voidfractionintheplungepool

ThelocalvoidfractionC_airmaybeestimatedwithphasechange equipment relating the sum of the time the probe tip is in air (tGi) andthe total time duration t ofthe measurement.An op- ticalfiber probe wasused to obtain the void fraction profiles in submerged hydraulicjumps ofnappejets.Ineach submerged hy- draulicjump,crosssectionsfromX=0.10mto1.00mwereana- lyzed.Atleast30pointsweremeasuredpercrosssection,obtain- ingmorethan1600registers.

Data of the different cross sections have been classified as a functionoftwodifferentdimensionlessparameters:theX/B_jratio, withXbeingthedistanceoftheverticalprofile tothestagnation

Table 3

Minimum and maximum values of X/B j and C ^mean per submerged hydraulic jump.

q (m ³ /s/m) Y/B j (-) H/L b (-) X/B j,min (-) X/B j,max (-) C meanmin (-) C mean,max (-)

0.099 8.45 0.76 2.9 29.4 0.17 0.24

0.089 10.25 0.77 3.1 31.3 0.12 0.21

0.085 11.28 0.77 3.6 35.7 0.09 0.20

0.079 13.20 0.77 3.6 35.7 0.09 0.20

0.073 15.00 0.78 3.7 37.0 0.08 0.20

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

y/y90

C_air(-)

3 < X/B^j≤ 10 10 < X/B^j≤ 20 20 < X/B^j≤ 30 30 < X/B^j< 37

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

y/y90

Cair(-)

0.08 < Cmean ≤ 0.13 0.13 < Cmean ≤ 0.19 0.19 < Cmean < 0.24

Fig. 4. Local void fraction profiles in the plunge pool.

point andB_j theimpingementjetthickness, andtheaveragecon- centrationCmeanineachcrosssection,definedas:

Cmean= 1 y90

y90



0

Cairdy (6)

where y₉₀ is the vertical distance from the bottom at which C_air=90%.

Table 3 details the range of the X/B_j ratio and the Cmean ob- tained per submerged hydraulic jump. Regarding the X/B_j ratio, all theflow conditionshaveverticalprofilesbetween3-4and30.

However, thehighestspecific flow,withthebiggest impingement jet thickness, hasthe smallestmaximumX/B_j ratio.TheC_mean ex- treme values obtained in each submerged hydraulic jump show greater variability, withfew overlaps betweenthe higherspecific flow(withthesmallerY/B_jratio),andtheotherspecificflows.

Fig.4showsthevoidfractionprofiles(C_air=tGi/t)ofthefive submergedhydraulicjumpsanalyzed.They-axishasbeennormal- ized bythey90value. Thedatahavebeengroupedintofoursub- classesasafunction oftheX/B_jratio.Different behaviorsmaybe consideredintheverticalprofiles.Alltheprofilesshowtheirmin- imum values of C_air nearthe bottom (y/y₉₀ < 0.10). The vertical profilesclosertothestagnationpointshowafirstmaximumpeak around 0.15<y/y₉₀< 0.30,withthelargestvoidfractionsreach- ingvaluesofaround20-25%.Thoselocalpeaksseemtoberelated withtheairbubblesandtheairpocketsentrapmentmechanisms observed by Ervine(1998).As Chanson (1995)reported,theseair pockets are broken up intovery thinairbubbles asthey are en- trained in the turbulent shear region, characterized by large air content.Astheflowmovesdownstreamfromthestagnationpoint, the bubblesare diffusedintoregions oflowershear stresses,and themaximumpeakofvoidfractiondetectednearthebottomtends todecrease.For0.40<y/y₉₀<0.70,theC_air valuetendstoacon- stantvalueinthevertical(^dCair/(^y/y₉₀)≈ 0)^{,thatreducesfrom8-}

13%to 5-7% as the distancefrom the stagnationpoint increases.

Nearthefreesurface(y/y₉₀>0.75-0.80),theairentrappedvalues tend to increase rapidly from0.15-0.20 to 0.9. Thismay indicate theunsteadyflowdepthgeneratedbyintenseturbulenceandvor- ticityinthereceivingpoolasErvine(1998)suggested.

The data can also be classified by considering the mean void fractioninthevertical profile.Fig.4 alsoshowsthevoidfraction datainadimensionlesswayasa functionoftheaverageconcen- trationinthevertical.Threedifferentsub-classescanbeidentified.

The vertical profiles with the maximum Cmean values (from 0.19 to0.24) show afirst maximumpeak neartothe bottom(around 0.15< y/y₉₀ < 0.30),with thelargest voidfractionvalues reach- ing around 20-25%. Thissub-class wasmainly obtainedwith the cross sections measured by the maximum specific flow and the minimum water depth, and for the cross sections located closer to thestagnation point inthe other specific flows (see theCmean

rangeinTable3).Themaximumpeaknearthebottomisnot de- tectedbytheopticalprobeequipmentintheothersub-classes.The datalocated atgreaterX/B_j ratios fromthestagnationpoint tend tocoincide withthesmaller C_mean values(see Fig.4). For0.20<

y/y₉₀<0.80(for0.30<y/y₉₀<0.70inthemaximumCmeanvalues sub-class),theC_air value tends toa constantvalue inthe vertical thatreducesfrom8-13% to5-7%astheaverageconcentration de- creases.Near the free surface, thevoid fraction increasesrapidly independentlyofthesub-class.

Takinginto account theanalytical distribution ofvoidfraction considered byChanson andBrattberg(2000) andby Murzynand Chanson (2009) in the air-water shear layer of free hydraulic jumps,the formulacan beadapted to analyzethe distributionof thevoid fractionmeasured nearthe stagnationpoint inthe sub- mergedhydraulicjumpsas:

C_air=Cmaxe

−

( (

^y−yCmax

)

^/Bj

)

²

4D∗

(

^X/Bj

)

validfory < y_Cmin (7)

Fig. 5. Vertical distribution of void fraction at the beginning of the submerged hy- draulic jump for the minimum and maximum specific flows.

whereCmaxisthemaximumvoidfractionintheshearlayer,Y_Cmax the verticaldistancefromthebottomto themaximumvoidfrac- tion C_max, D^∗ a dimensionless turbulent diffusivity, and y_C

min the verticaldistancefromthebottomtothelocalminimumvoidfrac- tion at the boundary between the turbulent shear flow andthe recirculationregions(seeFig.1).

The dimensionlessdiffusivityD^∗= Dt/(V_jB_j),withDt beingthe turbulent diffusivityassumedconstant ata givenlongitudinalpo- sition,andV_jandB_jare thejet thicknessandvelocity attheim- pingement point, respectively. D^∗ deduced frombest-fit data per verticalprofile,rangedfrom0.2to0.4,wasintheorderofthatre- portedbyBertolaetal.(2018)forasimilarimpingementReynolds number inplanar plunging waterjets, tending toincrease asthe impingementReynoldsnumbertendstoreduce.

Fig.5showsthedetailoftheshearstresslayermeasuredwith the minimum and maximum specific flows (q = 0.099 m³/s/m, Y/B_j =8.45, andq = 0.073m³/s/m, Y/B_j = 15.00,respectively) in thecrosssectionslocatedbetween0.10and0.30mfromthestag- nationpoint.AlthoughtheformulaproposedbyMurzynandChan- son (2009)was obtainedfor partially-developed hydraulic jumps downstream of a gate, the adjustments show reasonably good agreementonceEq.(7)hasbeenmodifiedforrectangularimpinge- mentjets.

3.2. Freesurfaceundulationpattern

The watersurfacealonga submergedhydraulic fluctuates,and the free surface is usually treated as an averaged depth profile (Longetal.,1990).Anotherrecurrenttreatmentofthefreesurface inair-waterflowsistousetheequivalentwaterheightheq,defined astheequivalentwaterdepththattherewouldbeintheabsence ofentrainedair.Ineachcrosssection,theequivalentwaterheight canbeobtainedas:

heq=

(

^{1− Cmean}

)

^{y90 (8)}

Fig.6showstheequivalentwaterdepth,heq,asafunctionofY, withYbeingthey₉₀valueatcrosssectionX=1.0moftheircorre- spondingsubmerged hydraulicjumps,andthemeanvoidfraction Cmean along theplunge pool.Ingeneral, all theCmean valuestend todecreaseastheXdistanceincreasesfromthe stagnationpoint, whiletheheq/Yratiotendstoincrease,from0.63-0.80neartheim- pingementpoint to0.83-0.92atX = 1.0m. Thosevaluesconfirm theexpectedreductionsinmeanairconcentrationandmeanflow

Fig. 6. Variation of the free surface undulation pattern and mean void fraction.

velocity as the cross sectionsare located further away from the impingementpoint.

Severalstudies(e.g.,Lopes,2011;Carrilloetal.,2019)analyzed thevariation oftheunsteadyflow depthata givenlocationwith conductivity probe measurements. Following the same approach, Fig.6alsoshowsthe(y₉₀-y₂₀)/Yratio,withy₂₀ beingthevertical distancefromthebottomatwhichC_air =20%.The y20 valuewas selected taking into account the smaller void fraction above the dC_air/(^y/y₉₀)≈ 0 region observed inFig. 4, inwhich all the void fraction profiles show C_air ≤ 0.20, occurring for y/y₉₀ ratios be- tween 0.4and0.7.Taking thesharper changeindC_air/(^y/y₉₀)^into account,the(y₉₀-y₂₀)/Yratiowasconsideredasanestimatortoan- alyze the free surface undulationin the plunge pool. In general, thedistancebetweeny₂₀andy₉₀tendstodecreaseslightlyasthe flow moves downstream. This indicates that there is a small re- duction in the free surface undulation, namely waves, along the stillingbasin that seems to be somehowrelatedto the C_mean re- duction. The results registered with the higher specific flow and thesmallerwatercushiondepth(Y/B_j=8.45case)tendtoobtaina constant(y₉₀-y₂₀)/Yratioaround0.20alongthedifferentcrosssec- tions.Thisdifferentbehaviorseemstoberelatedwiththesmaller submergenceratioofthishydraulicjump.

3.3. Phasechangefrequency

Severalauthors (e.g., Murzyn et al., 2005; Murzyn andChan- son,2007) haveanalyzedthephasechangedetectionfrequencyF infreehydraulicjumps.However,therearefewreferencestosub- mergedhydraulicjumpsdownstreamofrectangularimpingements freefallingjetsflows.

Fig.7showsthephasechangefrequencydetectedbytheoptical probeasafunctionoftheX/B_j ratio.Thesub-classes identifiedin theprevioussectioncanbeusedtodefinedifferentbehaviors.The localbubblecount ratehas beenplottedina dimensionlessway, enablingcomparisonwiththosedistributions previously obtained byseveralauthorsinfreehydraulicjumps(e.g.,ChansonandBrat- tberg,2000;Murzynetal.,2005; ChachereauandChanson,2011; Chanson andChachereau, 2013). As those authors indicated, the maximum change detection frequency may be linked with the largeturbulent shearstressesnearthe bottomoftheplungepool thatbreakuptheentrappedbubblesintofinerparticles(typically millimetricinsize).Thisphenomenonresultedinahighernumber ofbubblesimpactingontheopticalfibertip.Theresultsshowthe characteristicprofilereportedbyChansonandBrattberg(2000)in freehydraulicjumps,whichisatriangularprofileintheturbulent

Fig. 7. Dimensionless change detection frequency profiles in the submerged hy- draulic jump.

shear region,an abruptchangeofslope attheupperedgeofthe shearregionandaflattershapeinthere-circulationregion.

The profiles closest to the beginning of the submerged hy- draulic jump show a clear maximum peak around 0.10 < y/y₉₀

< 0.30, with the maximum reaching values of around 130-140 Hz for a X/B_j ratio between 3 and 10. The measured maximum frequency values were similar to those obtained by Murzyn and Chanson (2007), who measured maximum frequencies of around 120 Hz with a Froude number of 8.30 in free hydraulic jumps.

The maximum frequency peak is related with the entrapped air in the plunge pool entered by the air-water turbulent jet that generates high irregularities around the impingement point (Ervineetal.,1980).

As the flow movesdownstream, the maximum peak detected by theprobenearthebottomtends todecrease.For0.50< y/y90

<0.80,thedetectedfrequencytendstoaconstantvalueinalmost all the distances(Fbetween10 and50), withthemaximum val- ues for the cross sections located at X/B_j ratios between 20 and 30, while the closest sections to the stagnation point show the smallest values. Near the free surface (y/y₉₀ > 0.90),the change phase frequency tends to reduce rapidly. This sharpreduction in frequencyisconsistentwiththefree surfaceundulationobserved, whichisassociatedwithlargertimescales.

ConsideringtheCmeanvalues,themaximumpeakobservednear the bottommaybe identifiedwiththemaximum valuesofC_mean observedinFig.4(between0.19and0.24).Themaximumfrequen- cies near the bottom tend to reduce asthe Cmean inthe vertical profiles reduces.In the intermediate part ofthe vertical profiles, the minimumfrequencies tend tocorrespond withtheminimum Cmean values (between0.08and0.13),while thevaluesregistered intheothertwosub-classesconsideredinFig.4tendtobemixed.

The frequency profiles can also be analyzed if they are nor- malized asa function of the maximum frequency Fmax obtained ineachverticalprofile(Fig.8).Differentregionsareobserved.The higher F/F_max valuesare obtainedforlowlocal concentrationval- ues (C_air < 0.20).Those pointsmainlycorrespond tothe C_air and F peaks observed nearthe bottom inthe previous figures. How- ever,incrosssectionswithsmallC_meanvaluesand/or locatedfur- ther away from thestagnation point, therange offrequencies in theverticalissmallerthanneartheimpingementregion.

For 0.20 < C_air < 0.80, the F/Fmax values are around 0.30 for 3 <X/B_j< 20.However,thedataare scatteredintheprofileslo- catedfurtherfromthestagnationpoint.Nearthefreesurface(C_air

Fig. 8. Normalized frequency profiles as a function of the maximum frequency.

>0.80),theF/Fmaxratiotendstorapidlyreducetozero(largertime scalesregion).

3.4. Distributionofvelocitiesinthedissipationbasin

Using a modified Pitot tube corrected by the local void frac- tion with Eq. (5), the time-average velocity measurements were obtained in the same positions and conditions registered with theopticalprobe.Thisequipmentenablesthequasi-unidirectional flow in the wall jet of the submerged hydraulic jump to be an- alyzed. Near the bottom it may be considered that the stream- linesaremainlyparalleltothebottom,andthePitottubeseemsto be accurateatobtaining thehorizontalmeanvelocity V_x.Outside thewalljet,theflowisnot quasi-unidirectionaland,accordingto Matos andFrizell (2000) andMatos etal. (2002), measurements maybe affectedby theangleofthe velocityvector. Forthat rea- son,thevelocitiesinthebackwardsregionwerenotconsideredin theanalysis.

Previous studies (e.g., Rajaratnam, 1965; Wu and Rajaratnam, 1995)haveobtainedthattheverticalmeanvelocityprofilesinthe forwardflow offreeandsubmerged hydraulic jumps maybe an- alyzed ifthey are normalized witha velocity scale equal to the maximumvelocityV_max atanysection,andwitha lengthscale

δ

l

equaltotheverticaldistanceybetweenthebottomandwherethe localvelocityV =Vmax/2.Thevelocity gradientisnegativeatthis location.

Assumingthisnormalization,Castilloetal.(2017)proposedan adjustment formula based on physical and numerical measure- ments ofa submerged hydraulic jump downstreamof nappejets withfourspecific flowsbetween0.023and0.082m³/s/mfordis- tances from the stagnation point that are greater than 0.4 m.

DeDiosetal.(2017)proposedadifferentadjustmentforimpinge- ment Froudenumbers between3and 5forsubmerged hydraulic jumpsgenerateddownstreamofaverticalsluicegate.

In this work, the length scale

δ

1 and the maximum velocity V_max were obtainedineachvertical profileofthefivesubmerged hydraulicjumps.Intheanalyzedcases,themaximummeanveloc- itieswereobtainedinthevicinityofthebottominthecrosssec- tionslocatedclosesttothestagnationpoint,withmaximumveloc- ityvaluesbetween2.54and4.05m/s.

Fig.9showsthenormalizedvelocityprofilesinthesubmerged hydraulic jump.Althoughthe samesub-classesbased onthe X/B_j ratiohavebeen considered inthefigure, noremarkable behavior changewasfoundamongthedata.Thehighestvelocitieswerereg-

Table 4

Experimental adjustment proposed by several authors.

Authors Experimental adjustment Research R ²

Rajaratnam (1976) V x

V max= e ^−0.693 y

δl 2

Plane turbulent free jet– nozzle 0.94 Lin et al. (2012) V x

V max = 2 . 3 _y δl

0.42 1 − er f 

0 . 886 y δl

 Free hydraulic jump– sluice gate 0.91

De Dios et al. (2017) V x

V max = 2 . 0 _y δl

1/7 1 − er f 

0 . 55 y δl

− 0 . 39 Submerged hydraulic jump– sluice gate 0.96

Castillo et al. (2017) V x

V max= 1 . 48 _y δl

1/7 1 − er f 

0 . 66 y δl

 Submerged hydraulic jump– nappe flow 0.97

New adjustment V x

V max= 1 . 70 _y δl

0.11 1 − er f 

0 . 53 y δl

− 0 . 26 Submerged hydraulic jump– nappe flow 0.98 where er f(u ) = 2

√ π

 u 0 e ^−t2dt

Fig. 9. Dimensionless profiles of horizontal mean velocity in the plunge pool.

istered closeto thebottom. The meanvelocity tends todecrease as theflow separates fromthebottom. The registered valuesare alsoinagreementwiththedifferentdimensionlessvelocity distri- butionsproposed by thestudies listedin Table4,ascan beseen fortheir respective R² valuesforthecurrentdataset.As reported by Long etal. (1990), the profiles of submerged hydraulic jumps aresimilartothoseofaclassicalwalljetfory/

δ

1 uptoabout1.5.

Beyondthat,theVx/Vx,max profilesaremostlyinfluencedbythere- versing flow.Theresultsalsoconcurwiththemodelproposedby Rajaratnam (1965),considering thesubmerged jumpasa walljet underanadversepressuregradient.

TheRajarantam(1976)expressionseemstofollowthetrendas themeasurementpointsmovefromthebottom.However,thisex- pressionisunabletoreproducetheexpectedvelocityreductionfor y/

δ

1 ratios<0.1.TheLinetal.(2012) expressionwasobtainedfor free hydraulicjumps andtends toobtainthemaximumvelocities slightlyfurtherawayfromthebottomwallthantheother formu- lae. Theempirical formulaeobtainedby Castilloetal.(2017) and De Diosetal.(2017)show similarbehavior fory/

δ

1 ratios< 1.A newdimensionlessadjustmenthasbeenproposed,slightlyimprov- ingprevious adjustments.Althoughtheinletflowconditionswere different,theresultsindicatethatthevelocitydistributionnearthe bottom ofsubmergedhydraulic jumpsdownstreamofrectangular free falling jets may be described with relative good agreement withthoseformulaeobtainedindifferentflowconditions.

Following Long etal.(1990),the characteristiclength

δ

l along the submerged hydraulic jumps can be considered foranalyzing the velocity scale decay in submerged jumps. Fig. 10 showsthe

0 1 2 3 4 5 6 7

0 10 20 30 40

δl/Bj

X/Bj

0.08 < Cmean ≤ 0.13 0.13 < Cmean ≤ 0.19 0.19 < Cmean < 0.24 Adjustment (mean) Mean±1.0 Std dev.

Fig. 10. Variation of the characteristic length scale in different sections of the sub- merged hydraulic jump.

length scale values in a dimensionless wayfor the impingement jet thicknessB_j. Data havebeen plottedconsidering theprevious sub-classesofthemeanvoidfractionintheverticalprofiles.

The behavior is similar to that observed by Wu andRajarat- nam(1995)inwalljets,withdatatendingtofallwithinonestan- darddeviationofthemeanvalue.Consideringtheresultsobtained inthistypeofsubmergedhydraulicjumpdownstreamofanappe jet,anadjustmentcanbeobtainedwithR² =0.95,independently oftheCmeanvalue:

δ

l

B_j =0.121X

B_j+1.205 (9)

The slopeis similarto those oftheresults reportedby differ- ent authors. Forinstance, Rajaratnam and Humphries(1984) ob- tainedaslopeof0.07inplanesurfacejets;Longetal.(1990)con- sideredthat submerged jumpsmay betreatedaswall jetsunder anadversepressuregradient.WuandRajaratnam(1995)observed that the free jumps grow atabout the same rate as that of the wall jet (slopeof0.0728) inthe early stage butgrowfaster later on.Inaddition,mostoftheobservationsforsubmergedjumpsare containedwithinone standarddeviationofthewalljet,whilethe datanear theendof thejump show an acceleratedgrowthrate;

RaifordandKhan(2009)observedthatthegrowthrateincircular jetsforlowsubmergenceswas0.071forx/d=24– 35,and0.13– 0.17forfurthersections,withdbeingthenozzlediameter.

Fig. 11. Sauter mean bubble diameter in the submerged hydraulic jump.

3.5. Meanbubblesize

Assumingthatthetime-averagevelocitypresentedintheprevi- oussectionisright,andtakingintoaccountthesimplificationsthat the bubbles can be considered to be spherical in shape, equally distributedintimeandthattheirmovementisone-directional,the meanbubblediametermaybeobtainedbytheSautermeanbubble diameterD_sm (Cliftetal.,1978;RBI-Instrumentation,2012).Thisis themeanbubblediameterwhosevolume/surfaceratioisthesame as that calculated for all the bubbles detected during each mea- surement.TheSautermeanbubblediametercanbedefinedas:

Dsm=3C_airV

2F (10)

whereC_airisthevoidfraction,Vthemeanvelocityofthebubbles obtainedwithEq.(5)(assumingthebubbleshavethesameveloc- ityastheflow),andFthechangephasedetectionfrequency.

Fig.11showstheSautermeanbubblediameterobtainedinthe plungepool.Thenewadjustmentofdimensionlessvelocity distri- bution has alsobeen plotted. As thevalidity ofthe velocity field was limited to the wall jet region, the upper part of the verti- cal profiles hasbeen discarded.The diametersare mostlyaround 2 mm,withmaximumvalues below4mm.Once thedistanceto thebottomislargerthanthecharacteristiclengthscaleofthesub- mergedhydraulicjump(y>

δ

l),thedataseemtoshowascattered behavior with a reduction in the Sauter mean bubble diameter.

Scattereddataare generallythosedataobtainedincross sections located further away fromthe stagnationpoint andwith lessair concentration. The profiles withthelarger valuesof Cmean, hence those closest to thestagnationpoint, obtain amore varied range ofSautermeanbubblediameter,includingmanyofthelargestDsm

values,while theminimum valuesofDsm are mainlyobtained in cross sections with the larger X/B_j ratio values. These diameters concurwiththoseobtainedbyCarrilloetal.(2020a)usingacom- binationofexperimentalandnumericaltechniques.

The phase change size may be classified asa function ofthe bubble chord length distributions, obtainedasthe time the opti- cal probe tipis inairmultipliedby the localmeanvelocity (e.g., Chanson,2007;ChansonandChachereau,2013).Usingthevelocity fieldvalidatedintheprevioussection,Fig.12showstheprobability offinding eachbubblechord length inthefirstsection measured inthewalljet(locatedatX=0.10m),locatedatthesamedimen- sionlessdistancefromthebottom(y/y90=0.08-0.09),forthemax- imum, intermediate and minimum specific flows. Similar values wereobtainedfordifferentdimensional(frequency,numberofde-

Fig. 12. Probability distribution function (PDF) of the bubble chord length at X = 0.10 m downstream of the stagnation point.

tectedbubbles)anddimensionless(X/B_j,y/y₉₀andC_air)parameters.

Inturn,thechoice ofthosepointsallowstheinfluence oftheef- fectivetailwaterrateY/B_j(from8.45to15.00)tobeanalyzed.Con- sideringtheplottedresults,the trendofthebubble chord length distributionseems tobe independent oftheY/B_jvalue. Inall the cases,theprobabilitydistributionfunction(PDF)tendstobehigher forbubble chord lengthsbetween 0and 0.5mm. The datawith thesmallerflow velocity obtainarelatively largermaximumPDF value.Asthebubblechordlengthincreases,aright-skeweddistri- bution isobserved astheprobability of findingeach sub-interval tends to reduce drastically. As several researchers havestated in theirdescriptionofpunctualphasechangeprobes,thisPDFdistri- butionwiththemaximum inthesmallerinterval mayberelated withthedifficultyofpuncturingthebubblesontheirexactcentral axis.Furthermore,largerbubblestendtoan ellipsoidalshape,be- comingmoredistortedandlesssphericalinshapewithincreasing size(Harmathy,1960).

In the three cases considered, around 71-85% of the bubble chord lengths are smaller than 2 mm, and around 85-94% are smaller than 4 mm, with those percentages beinghigher as the localflowvelocityofthetestedpointissmaller.Thoseresultsare alsoinagreementwiththeSautermeanbubblediameterobserved inFig.11.

4. Conclusions

The study of two-phase air-water flows is a challenge in hy- draulicstructuresduetotheundilutedmixtureoftheflow.

Inthepresentstudy,thecharacteristicsoftheair-waterflowof five submerged hydraulic jumps generated downstream of nappe jetshave been analyzed.The equipment (optical fiberprobe and modifiedPitottube)wasusedconsideringtheirlimitations.Hence, thevelocitymeasurementswerelimitedtothewalljetregion.

Thedistancefromthestagnationpointtotheimpingement jet thicknessratioX/B_jandthemeanvoidfractionintheverticalpro- fileCmean seemtobegooddimensionlessparameters foridentify- ingdifferentbehaviorsofthevoidfractiondistributioninvertical profiles. However, the dimensionlessvelocity profiles seem tobe independentofthoseclassifications.

Near the bottom, the local peaks of void fraction and phase detection frequency were obtained close to the stagnation point of the rectangular jet, reaching void fraction values around 20- 25% andwith a frequencyaround 130-140Hz. Foranalyzing the

local void fraction peak, a formula proposed for free hydraulic jumpswasadaptedwithasuitablefit.

The void fraction and the frequency local peaks tend to de- crease as the flow moves downstream from the impingement region. Near the free surface, the local void fraction increases rapidly duetotheundulationobserved inthe freesurfaceofthe plungepool.However, thephasechangefrequencytendstomini- mum valuesinall themeasured crosssections.Thephasechange frequency behavior is similar to those reported by several au- thors infree hydraulic jumps (e.g.,Chanson andBrattberg,2000; Murzynetal.,2005).

The normalizedvelocitiesfollowthe behaviorobservedinfree hydraulicjumps,withthemaximumvelocitiesbeinginthewalljet region. Ourfindingextends thevalidity rangeofprevious studies performedbyseveralresearchers(e.g.,Castilloetal.,2017;DeDios etal.,2017)inthefieldofsubmergedhydraulicjumps.Theveloc- ities tend to decrease asthe flow exits fromthe shearlayer. Di- mensionlessvelocityadjustmentswereproposedforthemeanve- locityandthe growthoftheshear layerforsubmerged hydraulic jumps generateddownstreamofrectangularfree fallingjets, with R² =0.98and0.95,respectively.

The bubble chord length distributions showed a higher prob- ability forvalues ofbubble chord length between0 and0.5mm nearthestagnationpoint,independentlyoftheY/B_jratio.

Data onairentrappedandvelocity fielddownstreamofnappe jetsarescarce.Althoughinthistypeofjetstheflowentersalmost vertically,ingeneraltheformulaepreviously publishedinfreehy- draulic jumps and submerged hydraulic jumps downstream of a gate may be used and/or adapted for submerged hydraulic jump downstreamofnappejets.Thisallowstoconsiderthatourresults mayberepresentativeofthistypeofflowconditions.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingfinan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

José M. Carrillo: Conceptualization, Methodology, Validation, Formalanalysis,Writing-originaldraft,Writing-review&editing, Resources, Supervision,Fundingacquisition. Francisca Marco:In- vestigation,Formalanalysis,Visualization.LuisG.Castillo:Concep- tualization, Methodology, Validation, Resources,Visualization, Su- pervision, Writing-review& editing,Fundingacquisition. JuanT.

García:Conceptualization,Writing-review&editing,Supervision.

Acknowledgments

The researchers express their gratitude for the financial aid received from the "Ministerio de Ciencia, Innovación y Universi- dades" (MCIU), the "Agencia Estatal de Investigación" (AEI) and the "FondoEuropeode DesarrolloRegional"(FEDER), throughthe Project “La aireación del flujo en el vertido en lámina libre por coronación depresas a nivelde prototipoysuefecto encuencos dedisipacióndeenergía”,Ref:RTI2018-095199-B-I00.

The work was partially funded by the "ComunidadAutónoma de la Regiónde Murcia" through the "Programa Regional de Fo- mento de laInvestigaciónCientífica yTécnica(Plan de Actuación 2018) delaFundación Séneca-Agenciade CienciayTecnologíade laRegióndeMurcia”,Project“Análisisdelacapacidaddedescarga de vertederos tipo laberinto yde la disipación de energía aguas abajodelosmismos”,Ref:20879/PI/18.

ThesecondauthorthankstheMinisteriodeEducación,Cultura yDeporteforthefinancialaidreceivedfromtheUniversityTeacher TrainingGrant(FPU),referencenumberFPU16/05658.

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