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Modelling flow-induced vibrations of gates in hydraulic structures

Erdbrink, C.D.

Publication date2014

Link to publication

Citation for published version (APA):Erdbrink, C. D. (2014). Modelling flow-induced vibrations of gates in hydraulic structures.

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5 Physicalexperiment3

5.1 Preface

Thischapterandthenextareconnectedbyacommonaim:toinvestigatethedynamicsofanewgatelay‐out.ThephysicalmodeldiscussedinthischapterwaspreparedandexecutedatDeltaresinDelft,TheNetherlands.Amoredetaileddescriptionoftheexperimentisreportedin Erdbrink (2012a). Chapter 6 continues the study of the same gate shape by numericalphysics‐based simulation of its FIV response. This will give an idea of the feasibility ofapplying fundamental numerical models as assessment tools for gate designs, as acomplementary toolnext tophysicalmodelling. In thepresent chapter, the set‐upand theresults of the physical scale model experiment are given. Conclusions and a morecomprehensivediscussionareincludedinChapter6.

5.2 Introduction

AslaidoutinChapter2,thedynamicresponseofahydraulicgateduetoitsinteractionwiththeflowstronglydependsondetailsofthegatebottomgeometry.Numerousexperimentalstudies of flow‐induced vibrations (FIV) of gates have previously looked into thecharacteristics of gate shapes (Hardwick 1974, Vrijer 1979, Kolkman 1984). The gainedinsightinexcitationmechanismshasresultedinwidespreadrulesofthumbforunfavourabledesigns that should be avoided as well as favourable design features (e.g. Thang 1990,Naudascher and Rockwell, 1994). However, fundamental knowledge and practicalexperience have not culminated in one ideal universal shape – partly because thesurroundingstructureisanimportantfactor.Experimentalandnumericalmodelsareincapableofcapturingalldegreesoffreedom(d.o.f.)experienced by real‐life gates (mass‐vibration mode in cross‐flow and in‐flow direction,bending, torsion). Streamwise (horizontal) vibrations are usually studied separately (e.g.Jongeling 1988) and sometimes in combination with the cross‐flow mode (Billeter andStaubli,2000).Inthisstudyweconsiderthemostfrequentlyencounteredandinvestigatedmodeforavertical‐liftgate:oned.o.f.inthecross‐flowdirection.Theemergenceandseverityof flow‐relateddynamic forcesonthegatearerelatedto flowinstabilitiesandbodymotioneffects(Section2.4).Fluctuationsoftheseparatedflow’sshearlayer may incite a mechanism called Impinging Leading Edge Vibrations (ILEV) for gateswithasharpupstreamedge.Ifthegatebottomhasanextendinglipinstreamwisedirection,the shear layer separated from the upstream edgemay reattach to the gate bottom in anunstableway, givingdynamic excitation. In a differentmechanism, periodic forces are theresult of initially small gate movements. This self‐exciting process is called Movement‐InducedExcitation(MIE). 

3Thischapterisbasedonandusestextandcontentfrom“Reducingcross‐flowvibrationsofunderflowgates: experiments and numerical studies” by C.D. Erdbrink, V.V. Krzhizhanovskaya, P.M.A. Sloot,currentlyunderreviewattheJournalofFluidsandStructures.

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Previous investigations have proved that most severe vibrations of underflow gates insubmerged flow occur at small gate openings and are predominantly caused by ILEV andMIEmechanisms(Hardwick,1974;ThangandNaudascher,1986aand1986b).Thecurrentinvestigation therefore focuses on small gate openings and does not look at the distinctlydifferentmechanismofnoiseexcitation.OthernotableexperimentalstudiesareKapurandReynolds(1967),NaudascherandRockwell(1980),Thang(1990),Kanneetal.(1991),Ishii(1992)andGomesetal.(2001).Onlythelaststudyalsoincludesnumericalmodelling.Assuming that adding structuraldampingoravoiding critical gateopeningsareunfeasibleoptions, the shape of the gate bottom is the decisive factor determining the tendency tovibrate.Iftheflowpassesthegatewhileremainingattached,orifthereisafixedseparationpoint and a stable reattachment, or if the shear layer is kept away from thebottom in allcircumstances,thentheILEVmechanismmaybeavoided.Athin,sharp‐edgedgeometrywithseparation from the trailing edge is favourable (e.g. the rightmost profile in Figure 2 inSection 2.2.2), because potential shear layer instabilities occur downstream from the gateand a small bottom area inhibits the occurrence of large lift forces on the gate, thusminimizing theriskofMIEvibrations.Butsuchadesign isoftennotpossibledue tootherdesignconstraints.Atthestartofthisstudyanumberofnewideasforattenuationmeaureswereidentified4:(i). Counter‐balance the vibrating gate mass by adding an extra elastically mounted

weighttothegatethatissetintomotiontocompensatethegatemovement.(ii).Tondl(1998)hasmadeseveralanalyticalstudiesintothequenchingofself‐excited

vibrations by varying the stiffness of the support (this is called parametricexcitation).

(iii).Influencethehydrodynamicgatepressuresactivelybyinjectingorsuckingwaterthroughapump‐regulatedsystemoftubesflowingoutthroughthegatebottom.Thisshouldworktodisturbtheexcitations.

(iv).Make holes (or shafts or slots) in the gate bottom profile so that an intentionalleakage flowcandevelop, againwith the goal of influencing thebottompressuresbeneficially.

(v).Adjustthebottomgeometryinsuchawaythattheprofileactsasa(semi‐)aerofoil.Thiswouldenableactive ‘steering’ofthegatebyusing(rotationsof)theprofiletoincrease or reduce the steady lift force. This acts as an aid for gate lifting andloweringunderaheaddifferenceandmayhelptoavoidcriticalgateopeningswherevibrationsoccur.

As far as known from literature studies, none of these measures have been investigatedbeforeor tested for theirachievability foruse inhydraulicstructures.The investigationathand chooses measure (iv) as the central new idea. An unfavourable thick flat‐bottomrectangulargateistakenasareferencegateandanewdesignwithleakageflowthroughtwoopeningsinthebottomsectionisinvestigatedasapotentialwayofimprovingthevibrationproperties,seeFigure5.1.

4 Ideas (i) and (iii)‐(v) originate from discussions at Deltares with Tom Jongeling, who helpedenormouslyinthedesignstageoftheexperiment.

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Figure 5.1. Streamwise cross‐section of gate configuration showing ventilated gate design (detail ofbottomelementontheright).Dimensionsinmillimetres;notdrawntoscale.

5.3 Experimentalset‐up

The experiment of a gate section was performed in a 1 m wide and about 90 m longlaboratoryflume.Astraight,verticallyplacedunderflowgate issuspendedinasteel framethatisfixedtotheflume.Figure5.2containsdrawingsoftheplacementofthegateandtheframe in the flume. The dimensions of the gate are 1100x600x50 mm (height x width xthickness); it is a stiffplateand thusacts asa linearmass‐springoscillatorbodywithonedegreeoffreedominthecross‐flowverticaldirection.Topreventmeasurementequipmentand frame parts from influencing the flow, the flume was locally narrowed to 0.5 m byconstructing side walls out of waterproof film‐coated plywood and a sloped ramp of thesamematerialaroundthegate.Inthesectionclosesttothegate,thewallsweremadeoutoftransparentPerspexplastictoallowvisualinspection.Theflowdirectlyupstreamfromthegatewasattachedtothewallsandhaslowturbulenceintensity.

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Figure5.3showsthegate’sfrontviewandsuspension.AsmentionedintheIntroduction,twogate types were tested. The flat rectangular‐shaped bottom (with smooth surface, sharpedgesandwithoutextendinglip)willbecalled‘originalgate’.Theadaptedgatediffersfromthe original gate in that it has five horizontal slots in the upstream face of the bottomsection, as shown in Figure 5.3, and five identical slots in the bottom face of the bottomsection, as shown in Figure 5.4. This gatewill be called ‘ventilated gate’. The slots on theupstreamsideactasinflowopeningsforleakageflowandtheslotsonthedownwardfacingsideactasoutflowopenings.Thedimensionsoftheslotswerechosensuchthattheeffectofthe leakagewould be distinctly perceptible, butwithout compromising the rigidity of thegate.

Figure5.3.Frontview fromupstreamside,showing inflowventilationslots inthebottomsectionofthegate and the three supporting springs.Measurement frame is left out for clarity. Dimensions are inmillimetres.Onlythebottomelementisdrawntoscale.The gate is supported in vertical directionby three springs, one spring in the centerwithadjustablestiffnessandtwosidespringsof lowerstiffness.Priortoeachmeasurement,thegate was set to the desired height and the main central spring was set to the desiredstiffness.Thenthetensioninthetwosidespringswasadjusted(symmetrically)bychangingthelengthofthechainsconnectingthesidespringswiththeframe,seeFigure5.3.Thiswasdoneinsuchawaythatthetwolowstiffnesssidespringscarriedmostofthestaticloadsinvertical direction. The dynamic loads were mostly carried by the stiff central spring. Itsadjustablestiffnessenabledacontrolledvariationofthenaturalfrequency.Forthetwoweakside springs, linear coil springs (Alcomex TR‐1540)were used; for the stiffmain spring adouble leaf springwas custom‐builtwith high yield strength steel and a high elastic limit(Armco17‐7PH,hardeningconditionTH1050).Thetwobendingbladesofthemainspringhavedimensions600mmx30mmx4mm(lxbxt).Thebending lengthL (<600mm)canbeadjustedbymovableblockswithclamps,thusvaryingstructural stiffness, seeFigure5.3.Linearityof thisspringwasconfirmedbystatic

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loadingtestsfordifferentbendinglengths.Themainspringisinstalledinparallelwithtwosidesprings,eachwithconstantkside=0.57N/mm.TherelationbetweenbendinglengthandtotalstiffnessofthethreespringsisderivedfromconstitutiverelationsplusHookeslaw:

28

1.14, (5.1)

withktotalthecombinedspringstiffnessinN/mm,EthemodulusofelasticityinN/mm2,bthewidth and t the thickness of the leaf spring blades in mm and L the length between theclamps in mm. This formula was calibrated with free vibration tests in air to increaseaccuracybetweenchosenLandachievedktotalanddrynaturalfrequencyf0.Themaingatebodyconsistsofarigidsteelgridfilledwithwater‐resistingfoamandcoveredwiththinplasticplatesonbothsides.ThebottomelementiscarvedoutofPVCmaterialandisscrewedontothemainbody,thusformingonestiffmasswithit.Inordertoreduceweight,thebottomelement ishollow; itcontains fivechambers(seeFigure5.4).Thetotalmassofthe original gatewas 17.2 kg. The bottom element of the ventilated gate is lighter due toremoval of material, but the openings allow more water into the cavities of the bottomelementsothatthetotalmassofthemodifiedgatewasonlyslightlyhigher:17.3kg.Thesevaluesexcludeaddedmassduetowaterdisplacementduringoscillation.

Figure5.4.Planviewofgatewithcrosssectionthroughbottomelement,revealingtheoutflowslotsinthebottomfaceofthegate.Thesupportingframeisleftoutforclarity.Measurementsinmillimetres.Onlythegateisdrawntoscale.AplanviewofthegateissketchedinFigure5.4.Thewaterflowsbetweentheperspexwalls,inthefigurefrombottomtotop.Thespacebetweentheperspexwallsandtheflumewallsisfilledwithstillwaterat thedownstreamwater level.Thesidesealsconsistof thinverticalstripsthatarecarefullyinstalledsuchthatsideleakageisminimizedandatthesametimenocontact is made with the gate. Observations during the experiment indicated that theinevitablesidewaysleakagewasonlysignificantatrelativelyhighhydraulicheads,althoughthe leakage appeared to be only a small fraction of the total gate discharge. It was also

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observed that thedistancebetweengateand sidewallson thedownstreamsidehad littleimpact on the underflow discharge. Figure 5.4 contains a gate cross‐section through thebottomsection,suchthattheoutflowopeningsoftheventilatedgatebecomevisible.Thesefive openings are identical to the five openings on the upstream face of the gate. All slotswerecutperpendicularlytothefaces,seeFigure5.1.Fivehorizontalsupports, threehingedsteelrodsin longitudinaldirectionandtwoincrossdirection, enableverticalmovementof thegatewhile fixing thegatepositionhorizontally.Sixforcemeterswereinstalled:verticallyoneforeachspringandhorizontallyoneforeachlongitudinalsupport.Onlythemainverticalsupportforceisusedintheanalysisofthegateresponse.Thesamplefrequencywas200Hz.Thelengthoftheanalyseddatafileswas90sonaverageandhadaminimumof60s,yieldingafrequencyresolutionofatleast0.0017Hz.Recording started after reaching equilibrium water levels and horizontal support forces.Figure5.5showsasamplefromananalysedmeasurementsignalofthemeanleafspring.Thesignal analysis was done in MATLAB and consisted of a fast Fourier transforms analysis(FFT).First, the timeaverageof theremainingsteadycomponentof themainspring forcewas subtracted from the signal. Then a slidingwindowwas used to calculate themovingaverage and the amplitude envelopes, which were smoothed with a simple triangularsmoothing function. The Hilbert transform was used to find the envelopes. For eachmeasurement signal, the representative vibration amplitudewas determined as themeandifferencebetweentheenvelopesandthemovingaverage.

Figure5.5.Anexampleexcerptfromameasuredsignalofthemainspring,withcomputedmovingaverageandenvelopesadded.Furthermore,thedischargeandthewaterlevelsontheupstream(h1)andthedownstreamside(h2)ofthegateweremeasuredusingresistance‐typewaterlevelmeters.ThelocationsofthewaterlevelmetersareshowninFigure5.2.SeeErdbrink(2012a)formoredetailsontheexperimentalset‐up.

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5.4 Definitions

The motion equation in vertical z‐direction for partly submerged bodies includeshydrodynamiccoefficients(cf.Sections2.3‐2.4):

. (5.2)

Here,m ismass, c is damping, k is stiffness,F is the excitation force and the subscriptwindicates the added coefficients. The excitation is a time‐dependent hydraulic forcewhichcanbewrittenas

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, (5.3)

withWthecross‐flowwidthofthegatesectiononwhichFworks,F0astationaryreferencefluid force and CF is a periodically varying force coefficient. In this chapter not thedisplacementzbuttheresponseforceFzismeasured,theamplitudeofwhichisdenoted .Furthermore, for the pressure on the gate bottom boundary pbound we use the pressurecoefficient CP defined logically as pbound divided by ρg∆h. The two‐dimensional dischargeformulaforanunderflowgatesectioninsubmergedflowis

2 ∆ , (5.4)

with q the discharge per unit width in m3/s/m or m2/s, CD the dimensionless dischargecoefficientforsubmergedflow,atheliftingheightorgateopening,Utheflowvelocityinthevena contracta, estimatedwithBernoulli’s formula, inm/s and∆ .The reducedvelocityVrisusedasdimensionlessdescriptivequantityfortheflow‐inducedvibrations.Itisdefinedhereas

2 ∆

, (5.5)

where fz is the dominant response frequency, the numerator represents the characteristicflow velocity and the gate thickness in flow direction D (see Figure 5.1) is taken ascharacteristic lengthscale.To find theaddedmassmw, it ishereestimatedexperimentallyfromfreevibrationtests inairandstillwater. It follows fromtheratioof thewetanddrynaturalundampedfrequencies.Forf0,waterwehave

,12

, (5.6)

andtheratioisthus

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, 1 ⁄

1 ⁄. (5.7)

becausetheaddedrigidity(hydraulicstiffness)kwduetobuoyancyonthesubmergedpartofthe gate is found via Hooke’s law: , . The error made by neglectingdampinghereislessthan2%.Numerically,mwnearawallinstillwaterwascomputedinapotentialflowmodelwithafinitedifferencemethodbyKolkman(1988),andstudiedmanytimessincetheninthecontextofgates(e.g.Anamietal.,2012).AveryuniversalapproachistheobservationinKolkman(1984)thatthetotalkineticenergyofthefluidcanbeexpressedas

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. (5.8)

Summingthevelocitymagnitudeoverallcomputationalnodesshouldyieldavalueformw,assumingthattheobjectvelocityisknown.

5.5 Measurementconditionsandvariationofparameters

Themeasurementprocedurewas tovaryVr anda/D,whichwasdonemostlybychangingsuspensionstiffnessandgateheightandoccasionallybychangingthepumpdischargeandheightoftheoutflowweiroftheflume.Thepresentstudydealswiththecriticalgateopeningrange for cross‐flowvibrations:98%of themeasurements lie in the interval0.48≤a/D ≤1.50. During the experiment the stiffness was varied between and 19.3 N/mm and 967N/mm,correspondingtoarangeinachievedundampednaturalfrequencyinairof5.33Hz< < 37.7 Hz for the original gate with closed bottom section. The gate submergence

⁄ ,withh2 thewaterdepthmeasureddownstream fromthegatewas in therange 4.2 < Cs < 6.2, for 98% of the data. Thismeans flow conditionswere close to fullysubmergedwithminor freesurface fluctuations.DischargecoefficientCD isestimated fromthemeasuredpumpdischarge to beon average0.80with standarddeviation0.11 for theoriginalgateandonaverage0.83withstandarddeviation0.10fortheventilatedgate.Theachieved Vr ranges were 1.2 < Vr < 11.6 for the original gate and 1.8 < Vr < 9.5 for themodifiedgate.TheReynoldsnumberdefinedasRe=UD/ν,againwithU=(2g∆h)0.5,wasinthe range 3.2·104 < Re < 1.3·105. The mass ratio, defined as / , iscomputedfortheoriginalgateas12.3≤mr≤12.7,andfortheventilatedgateas12.3≤mr≤12.8.Dampingwasmonitoredthroughouttheexperimentin32freevibrationtestsinstillwater,wherethegatewassetinmotionbyamanualtaponthetop.Foreachtest,thelogarithmicdecrementofdamping wasdeterminedoverthefirsttenperiods.Atsmalldampinglevels,we have 2 . This formula is used to compute values of the damping ratio . For theoriginal gate, was on average 0.013 and had a standard deviation of 0.0065. For theventilatedgate, theaveragevalueof was0.020withastandarddeviationof0.0061.Thedampingvaluescontainnotrendsrelatedtowaterdepthorstiffnessanddidnotchangeovertime during the experiment. However, there is some inherent variability caused by themanualexcitationforcenotbeingthesameeachtime.Also,itmakesadifferencewhichpart

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ofthedecayingfreevibrationisusedfordeterminingδ.Theventilatedgateshowedasmallnumber of deviant values of higher damping at small openings a/D < 0.7, these are notincludedinthegivenranges.Thehighestrecordeddampingratiointhisparticularsituationwas =0.093.Apossibleexplanation is that flowresistance inside thechambersbetweenthe inflow and outflow slots is more pronounced at small gate openings due to higherrelativevelocities.Furthermore,eventhoughdampingwasnothigheratlargerwaterdepthsin still water, the observed leakage through the side seals at high hydraulic heads (thatexistedforasmallpartofthemeasurements)mayhavehadasignificanteffectondampingduetoskinplatefriction.Thiswasnotquantifiable,sincethefreevibrationtestswereonlyfeasibleinstagnantwater.ThedimensionlessScrutonnumbercombines themassanddamping ratiosand is, for lowdamping,definedbySc 2 4 .Itwasfoundthatfortheoriginalgate1.0≤Sc≤2.3andfortheventilatedgate0.9≤Sc≤3.7.ThevariationintheScrutonnumberisalmostcompletelyduetothediscussedvariationindamping.TheoutliersfortheventilatedgateatsmallopeningsarenotincludedintheScranges.Table5.1givesadditionalinformationonmeasurementconditions.Table5.1.Flumesettingsduringmeasurements.

min max unit

pumpdischargeQpump 13.0 50.0 l/sgateopeninga 22.5 100 mmupstreamwaterdepthh1 0.297 0.656 mdownstreamwaterdepthh2 0.120 0.352 mheaddifference∆h 0.063 0.355 m

Parameter ranges given in Table 1 are based on the combined data set for both gates,consistingof145measurementsfortheoriginalgateplus85measurementsforthemodifiedgate.Theachievedunderflowdischargewassomewhatlowerthanthepumpdischargeasaresult of the sideways leakage through the seals. An observed variation in the pumpdischargeof+/‐0.5l/swasoflittleinfluencesincethefrequencyofthisvariationwasverysmallcomparedtosignalrecordinglength.

5.6 Resultsofphysicalexperiment

The focus of the experimental data analysis is on determining how dominant forceamplitudes in cross‐flowdirection changewithVr for both gate types. Absolutemaximumforceamplitudesdependonstructuraldampingofaparticularset‐upandareoflessinterest;thereforeresponseamplitudesarepresentedfordifferentsettingsrelativetothestationaryhydrodynamicforceF0.Figure 5.6 shows the dimensionless dynamic force response of the closed gate and theventilated gate. Judging from the plot, the response may be divided into three differentregions: 2 <Vr<3 (relatively high stiffness), 3 <Vr<8.5 (medium stiffness) andVr > 8.5(relatively low stiffness). Response values of 5 represent significant, regularoscillations with response frequencies in the range 4.6 Hz < fresp,z < 20.2 Hz. Verticaldisplacementamplitudesestimatedfromtheforceamplitudeas ̂ ⁄ wereoverall less

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than0.1D.ThestrongestrecordedforceamplitudeinthehighstiffnessregionwasfoundatVr=2.54.ThemaximumresponseintherelativelylowstiffnessvibrationregionoccurredatVr =10.16.The excitationmechanismsassociatedwith these two regions arediscussed insection4.1.

Figure 5.6. Overview of vibration characteristics of the original rectangular gate (left) and the newventilatedgate (right): reducedvelocityVrversusdimensionlessvibrationamplitude ⁄ of themainverticalforcemeter;aisthegateopening,gatethicknessDis50mm.Theresultsshowthatthevibrationsfoundat lowVroccurquitesuddenly.Therearesteepincreases in force amplitude around Vr = 2 and Vr = 3.0–3.5; most significantly for gateopeningslessthanorequaltoD.Testsatgateopeningssmallerthan0.5Dwerehinderedbytheriskofthegatehittingtheflumebottom.Althoughthegateopeningwasnotvariedovera

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largerange, thedataseemstoshowthatthe forceresponseatVr≈2.5occursatasmallergateopeningthanfortheresponsemaximumatVr≈10.Zoominginonthedenselysampledareaof relativelyhigh frequencyresponseat2<Vr<3gives further insights (Figure5.7).Themeasurementsweredonebymakingseriesofabouttendatapointsofdifferentstiffnesssettings,keepinggateopeninganddischargeconstant.

Figure5.7.Themostsignificantvibrationsat1.5<Vr<4.Measurementsseriesrepresentconstantgateopening a/D and head difference ∆h for varying suspension stiffness. Gate submergence relative todownstreamlevelisdenotedCs.Thesolidtrianglesmarkthegatewithslots,allothersymbolsdenotetheoriginalgate.Responseswith ⁄ 0.25notbelongingtotheseseriesareleftoutforclarity.Theseresultsreconfirmthatinthisreducedvelocityregionsignificantcross‐flowvibrationsoccuratgateopenings intherange0.5D≤a≤D.For1.5<Vr<4,withthestrongestforceamplitudesarounda/D=0.5.FromFigure5.7itappearsthatforheaddifferences∆h≈1.5D–1.75D, vibrationmaximadecreasewith increasinggateopening.Theserieswithhighheaddifference∆h=283mm≈5.7Ddiffersfromtherestbyitshigherhydraulicheadandhigherstiffness settings. Because the still water damping tests proved that damping is notsignificantly higher for high spring stiffness settings, the most likely explanation of thesmallerresponseofthisseriesisanincreaseddampingassociatedwiththehigherhydraulichead.Thisispossiblyrelatedtotheobservedleakagethroughthesidesealsathighhydraulichead.Thisfindingdoesnotinfluencethecomparisonsbetweenthetwogatetypes.However,thereportedhigherdamping for theventilatedgateata/D<0.7, seeSection2.3, suggeststhattheresponsesoftheventilatedgateseriesata/D=0.52areunderestimated.Despite the smaller data set size of the new gate design, the overall effect of applyingventilation holes seems clear. For very similar conditions at 1.5 < Vr < 3.5, the forceamplitudeofthevibrationisafactor3lessfortherectangulargatewithaddedholes.For3.5<Vr <9.5, theholedgate showsavery lowresponse for similar conditions (openingsandheaddifference)astheoriginalgate.FromtheavailabledatapointsatVr>8.5,thepictureoftheeffectofaddingholesisincomplete.MoredataisneededforVr≥10.Theavailabledatasuggeststhattheventilationsucceedsinhavingamitigatingeffectonthevibrationsinthatrangeaswell.

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TheventilatedgateshowsanoverallshifttowardslowerachievedVr‐valuescomparedtotheoriginal gate. The maximum force response of the new gate lies at Vr = 2.19. Also, thismaximum is found at a higher gate opening than themaximum of the closed gate (0.74Dagainst0.48D).Thisistheneteffectofthealteredgatedesign:aslightlydifferentmassandadditionaldischargecausedbythejetofrelativelyhighvelocitythroughthegate.

5.7 Comparisonwithotherexperimentalresults

The measurement data of the original rectangular gate resembles results from previousexperimentalresearch.Itisimportanttorealisethatmostotherstudiesmeasureresponseindisplacement insteadof force. In thehigh stiffnessatVr<5,whichhasbeen studiedmostfrequently, thereducedvelocityatwhichhighestamplitude is found liesclose toVr =2.75foundbyHardwick(1974)andVr,nat=2.5 foundbyThangandNaudascher (1986a,b).ThereducedvelocitybasedonthenaturalfrequencyVr,natissomewhatlowerthantheVrusedinthis study.Thegateopeningsofmaximumamplitudewerea/D =0.67anda/D=0.65 forthesetwostudies,slightlyhigherthanwhatisfoundhere.BilleterandStaubli(2000)reportmaximumforcecoefficientsinz‐directionatVr=2.75andVr=3.0.TheexperimentsofVrijer(1979)coveredvibrationsintherange10<Vr<80.Theirmodeldimensionsweresmaller(gatewidthof10mm).Acomparisonbasedonthesmalloverlapofless than ten data points suggests that the lower bound of vibrations in the low stiffnessregioninthepresentstudyisataslightlylowerVrvalue(Vr=9.0‐9.5)thaninVrijer(1979)whereamplitudesstarttoincreaseforVr>10.

5.8 Summary

Thisexperimentalstudyinvestigatesanewwaytoreducecross‐flowvibrationsofhydraulicgateswith underflow. A rectangular gate section placed in a flumewas given freedom tovibrate in the vertical direction. Horizontal slots in the gate bottom enabled leakage flowthroughthegatetoentertheareadirectlyunderthegatewhichisknowntoplayakeyrolein most excitation mechanisms. For submerged discharge conditions with small gateopeningstheverticaldynamicsupportforcewasmeasuredinthereducedvelocityrange1.5< Vr < 10.5 for a gate with and without ventilation slots. The leakage flow significantlyreducedvibrations.Thisattenuationwasmostprofoundinthehighstiffnessregionat2<Vr<3.5.

5.9 Photographsfromtheexperiment

Figures5.8‐5.12showphotographstakenduringthephysicalexperiment.

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Figure5.8.Sideviewofmeasurementframeinflume.Flowfromlefttoright.

Figure5.9.Detailofthemeasurementinstallation:gatebody,frameandmainleafspring.

lowstiffnesssidespringtobalancestaticforces

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Figure5.10.Originalclosedflat‐bottomgateprofile(top)andmodifiedgateprofilewithventilationholes(bottom).

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Figure5.11.Close‐upofgatebottomsectionwithoriginalbottomprofileinstalled.Flowfromlefttoright.

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Figure5.12.Adjustingthestiffnessoftheleafspring.