Dearcolleagues!MynameisNN.IwillpresenttheteamofRussiawiththeproblemFirehose.
Theproblemsays:Considerahosewithawaterjetcomingfromitsnozzle.Releasethehoseandobserveitssubsequentmotion.Weareproposedtodeterminetheparametersthataffectthismotion.
Thephenomenonofsocalledgarden-hoseinstabilitywaswidelyinvestigatedbyPaïdoussis et al.Asyoucansee,hisexperimentaldevicewasenoughlarge and complicated.Ourequipmentismuchmoresimple and small.
Let’sbeginwithourfirstobservations.
The hose is straight anddoesnotoscillate whenagardenerholds it nearthenozzle. Let us increasethelengthbetweenthehandfixationandthefreenozzle. Atcertainpositionthesystembecomesunstableandsmallrandomperturbationsgrowintolateraloscillations with significant amplitude.
Hereyoucanseetheoscillatinghose.Itslengthisabout15cm,anditsoscillatoryfrequencyisabout10timespersecond.
Letmeintroduceastandard theoreticalmodelofthisphenomenon.
Themodeltakesintoaccountthreeforces:elasticforce,centrifugalforceandCoriolisforce.Gravityandviscosityareneglected.
Theelasticforcemanifestsitselfwhenthehoseisbent.Tensileand compression stressestrytostraightenthehose,thuscreatingabendingmoment.
Thecentrifugalaccelerationappearswhenthefluidmovesalongthecurvedsegmentofthehose.Itiscalculatedthisway.
TheCoriolisaccelerationappearswhenthefluidmovesalongtherotatedsegmentofthehose.Thefirstsourceofthisaccelerationisthatthesegmentchangesitsdirection,untilthefluidrunsalongit.Thesecondsourceisthatoneendofthesegmentmovesrelativetotheother,whichgivesitsowncontributionintothechangeofthefluidvelocity.
Themovementofthehoseisdescribedbythisequation.WetookitfromPaïdoussis’ book.Infact,thisequationrepresentsNewton’ssecondlawappliedtoasmallsegmentofthehose.Theleftpartisaproductofmassandacceleration.Therightpartisasumofactingforces.
Thehoseinstabilitydevelopsasaresultofinternalfluidmotion.NoticethatonlytwolasttermsinthisequationdependonthefluidvelocityU.TheCoriolisforceisproportionaltothisvelocityandthecentrifugalforceisproportionaltoitssquare.Therefore,forsufficientlysmallfluidvelocitytheCoriolisforcedominates,andthesystemissubjectedtoflow-induceddamping.Forsufficientlylargefluidvelocitythecentrifugalforcebecomesdominant,andthesystemlosesitsstabilityduetoitsaction.
Introducingthedimensionlessparameterstheequationtakesthefollowingform.Noticethatnowthemotionofthehoseisdeterminedbytheonlyparameterβ,whichistheratioofthefluidmasstothefullhosemass.Sothedimensionlesscriticalvelocityandfrequencyarethefunctionsofthisparameterβ.
ThesefunctionswerenumericallycalculatedbyGregoryandPaïdoussis.Theredpointscorrespondtotheparametersofourexperimentalsetup.
Thetheorypredictsthatthecriticalvelocityisinverselyproportionaltothelengthofthehose,andthecriticalfrequencyisinverselyproportionaltothesquareofthislength. We check thesepredictionsinourexperiment.
Inourfirstexperimentthehosewassituatedintheair.
Wesetthecertainflowrate,measureditandcalculatedthejetvelocity.Thenwefoundthecriticallengthofthehose.Finally,wemeasuredtheoscillatoryfrequency.
Therearethemainparametersofthehoseinoursetup. The mass ratio is about 0.22.
Thisgraphshowshowthecriticalvelocitydependsonthehoselength.Theredlinerepresentsthetheoreticalprediction.Theagreementbetweenthetheoryandtheexperimentis reasonable.
Thereareanalogousresultsfortheoscillatoryfrequency.Theagreementisgoodagain.
Inoursecondexperimentthehosewassubmergedinwater.
Wenoticedthatitscriticalfrequencydecreasedsignificantly.
Criticalvelocitiesintheairandinthewaterarenotsignificantlydifferent:thevalueofthisvelocitydeterminedbythebalanceofforces,whichremainsthesame.
Incontrast,theoscillatoryfrequencyunderthewaterisseveraltimeslessthanintheair.Thisisbecausethe inertia of the hoseunderthewater is increasedbytheaddedmassofsurroundingwater.
Theaddedmassappearsbecauseanacceleratingbodymustmovesome connected volumeoffluidasitmovesthroughit.Onecanshowthatacylinderhasanaddedmassequaltothemassoffluidinitsvolume.
Letmesummarizeourresults.
Firstofall,thegarden-hoseinstabilityiscausedbytheinterplayofcentrifugalandCoriolisforcesgeneratedbytheflowofwaterthroughthehose.
Self-sustaininghoseoscillationsappearwithacertaincriticalfluidvelocity.
Theoscillatoryfrequencyofthehosedecreaseswithincreasingofitsmass.Whenthehoseissubmergedinwater,itsmass is virtually increasedbytheaddedmassof surrounding water.
Theseareourreferences.
Thankyouforyourattention!