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Tracking fluid flow in a spinning disk reactor

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Tracking fluid flow in a spinning disk reactor
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Korzhova, Valentina N
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Inclination angle
Mathematical modeling
Spiral wave
Velocity components
Wavelength
Dissertations, Academic -- Computer Science -- Masters -- USF
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bibliography   ( marcgt )
theses   ( marcgt )
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Abstract:
ABSTRACT: The flow of a liquid film over a rapidly rotating horizontal disk has many applications inmedical, industrial, and engineering fields. A specific example is the heat and mass transfer processes between expanded liquid and surrounded dense gas. Diferent wave regimes of a liquid film depend on a flow conditions such as the properties of a liquid, its initial speed,parameters of environment, etc. Therefore, experimental investigation of the film flow over a spinning disk is needed to both validate theoretical predictions and establish methods for fluid flow monitoring.This thesis presents novel video-based algorithms for detection and tracking wave structural data of the liquid film flowing over a spinning disk reactor. The algorithms are based on the spiral model of wave and the quasi-optimal method for estimation of a wave velocity as ill-posed problem. Their performance is compared with results predicted by the fluid dynamics based on the Navier-Stokes equations in the case of thin film.Using experimental video data, the developed models and algorithms allow investigators to estimate the characteristics of wave regimes such as wavelengths, inclination angles, and the radial and azimuthal velocity components of the fluid. The accuracy of estimated characteristics was analyzed. It was shown that average distance between consecutive two waves,their spiral shapes, and the radial velocities of waves confirm the theoretical results and predictions. In particular, computed wavelength is within 1% and a change of the inclination angles is within 2% of the predicted values.
Thesis:
Thesis (M.A.)--University of South Florida, 2006.
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Includes bibliographical references.
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by Valentina N. Korzhova.1
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TrackingFluidFlowinaSpinningDiskReactorbyValentinaN.KorzhovaAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofScienceinComputerScienceDepartmentofComputerScienceandEngineeringCollegeofEngineeringUniversityofSouthFloridaMajorProfessor:DmitryB.Goldgof,Ph.D.RangacharKasturi,Ph.D.SudeepSarkar,Ph.D.DateofApproval:March24,2006Keywords:inclinationangle,mathematicalmodeling,spiralwave,velocitycomponents,wavelengthcCopyright2006,ValentinaN.Korzhova

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DEDICATIONThisthesisisdedicated,especially,tomyparentsand,also,tomyhusband,son,anddaughterfortheirlove,support,guidance,andunderstanding.

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ACKNOWLEDGEMENTSIwouldliketothankDr.DmitryGoldgofforgivingmetheopportunitytoworkwithhimandtheinvaluableacademicguidanceandhoursthathededicatedinthedoingofthisthesis.IalsothankDr.RangacharKasturiandDr.SudeepSarkarforthetimetheytooktoreviewthisthesisandtheirhelpfulcomments.IamgratefulalsotoDr.Sisoevforusefuldiscussionsandhelp.Iwouldliketothankmyfamilyandallofmyfriendswhogavemeinspiration,support,andcomfortthroughoutthistoughroad.

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TABLEOFCONTENTSLISTOFTABLESiiLISTOFFIGURESiiiABSTRACTivCHAPTER1INTRODUCTION11.1OverviewoftheRelatedWork11.2ContributionsofthisThesis31.3LayoutoftheThesis4CHAPTER2THEORYOFFLUIDFLOW62.1GeneralCase62.2MathematicalModeling82.2.1EvolutionEquations82.2.2LinearStabilityAnalysis112.3TheSpiralEquations13CHAPTER3DATAACQUISITIONANDCAMERACALIBRATION153.1DataAcquisition153.2OverviewofCameraCalibration173.3CameraCalibrationAccuracy18CHAPTER4ALGORITHMSANDRESULTS214.1RegimesofFluidFlow214.2VelocityComputationandAccuracyEstimation214.2.1ComputationofRadialVelocityandAccuracyEstimation224.2.2EstimationofAzimuthalVelocityComponent244.3WaveFrontDetection254.3.1EstimationofInclinationAnglesandWavelengths264.3.2CorrespondencetotheMathematicalModel274.3.3Comparison28CHAPTER5CONCLUSIONS315.1Summary315.2FutureResearch31REFERENCES32i

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LISTOFTABLESTable4.1ResultsoftheNumericalSolutionsofSpiralEquations.24Table4.2ExperimentalRadialVelocity.24Table4.3CalculatedResultsofdr dandd2r d2.27Table4.4CalculatedAveragedWaveInclinations.27Table4.5RadiioftheFirstandSecondWavesR1andR2,Respectively.28Table4.6InputDataandModelCoecients.28Table4.7CalculatedAveragedWavelengths.29Table4.8CalculatedAveragedWavelengthOverFiveVideosandTheoret-icallyCalculatedWavelength.30Table4.9InclinationAngleChangesfortheVariousRadii.30ii

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LISTOFFIGURESFigure2.1FluidElementofVolumeV=xyzLocatedatPositionX.6Figure2.2Determinationofand.14Figure3.1ExperimentalSetup.15Figure3.2RotatingDiskCloseup.16Figure3.3PlanarPattern.16Figure4.1RegimesoftheDiskRotationa300rpmandb520rpm.21Figure4.2Block-SchemeofEstimationanInstantaneousVelocity.22Figure4.3Model-BasedSpiralPointsRedontheExperimentalVideoData.25Figure4.4Block-SchemeofWaveFrontDetection.25Figure4.5aEnhancedImagewithSelectedLines.bIntensityDistribu-tionoftheAverageResult.26Figure4.6DetectedPointsofWaves.26Figure4.7aAmplicationFactors.bAxis-symmetricn=0andNon-axis-symmetricn=100Perturbationsat~r=10cm.29Figure4.8DependenceofWavelengthsOverRadii.Asterisksandx-sCor-respondtoExperimentalandTheoretical,Respectively.30iii

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TRACKINGFLUIDFLOWINASPINNINGDISKREACTORValentinaN.KorzhovaABSTRACTTheowofaliquidlmoverarapidlyrotatinghorizontaldiskhasmanyapplicationsinmedical,industrial,andengineeringelds.Aspecicexampleistheheatandmasstransferprocessesbetweenexpandedliquidandsurroundeddensegas.Dierentwaveregimesofaliquidlmdependonaowconditionssuchasthepropertiesofaliquid,itsinitialspeed,parametersofenvironment,etc.Therefore,experimentalinvestigationofthelmowoveraspinningdiskisneededtobothvalidatetheoreticalpredictionsandestablishmethodsforuidowmonitoring.Thisthesispresentsnovelvideo-basedalgorithmsfordetectionandtrackingwavestruc-turaldataoftheliquidlmowingoveraspinningdiskreactor.Thealgorithmsarebasedonthespiralmodelofwaveandthequasi-optimalmethodforestimationofawaveveloc-ityasill-posedproblem.TheirperformanceiscomparedwithresultspredictedbytheuiddynamicsbasedontheNavier-Stokesequationsinthecaseofthinlm.Usingexperimentalvideodata,thedevelopedmodelsandalgorithmsallowinvestigatorstoestimatethecharacteristicsofwaveregimessuchaswavelengths,inclinationangles,andtheradialandazimuthalvelocitycomponentsoftheuid.Theaccuracyofestimatedchar-acteristicswasanalyzed.Itwasshownthataveragedistancebetweenconsecutivetwowaves,theirspiralshapes,andtheradialvelocitiesofwavesconrmthetheoreticalresultsandpre-dictions.Inparticular,computedwavelengthiswithin1%andachangeoftheinclinationanglesiswithin2%ofthepredictedvalues.iv

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CHAPTER1INTRODUCTION1.1OverviewoftheRelatedWorkTheowofaliquidlmoverarapidlyrotatinghorizontaldiskhasmanyapplicationsinchemical,medical,engineeringandbioengineeringelds.Examplesofimportantapplicationsareprocessesofheatormasstransferbetweenexpandedliquidandsurroundedgas,bloodoxygenation,andcoolingdevices.Tointensifythoseprocesses,technologicalperformancehastoadjusthydrodynamicparametersofowwithrelevantchemicalkinetics.So,thecontrolledlmowisused.Controllingtransportratesintheliquidlmoersthepossibilityoftheformationofdierentkindofwaves.Sincevariousregimesoflmowhavestronginuenceonthoseprocesses,itisimportanttocontroltheformationoftheregimes.Theexperimentalinvestigationsofowoveraspinningdiskhaveattemptedtomeasurelocalmaximumormeanofalmthicknesstoobtaininformationaboutsurfacewaves[2].Var-iousmechanical[6,24],electrical[4,29],andoptical[19,48]techniqueswereemployed.Themostpromisedwastheopticalmethodusedtocollectinformationaboutwavesobserved.Intherespectiveexperiments,acamerawasplacedabovethediskandconnectedtoacomputerprovidingwithvideoimaginghardwareandsoftware.Tomeasurelmthicknessoveradiskdomain,calibrationofmechanicalandopticaltoolsandestimationofabsorptioncoecientwereused.All,mechanical,electrical,andopticaltechniquesgaveinsucientinformationforclassifyingwaveregimesandselectingmostecientregimeforspecictechnologicalapplica-tions.Earlyexperimentalinvestigations[6,11]providedsomequalitativeandquantitativeun-derstandingtheeectofow-rateandrotationalspeedontheowcharacteristicsforagivensetofphysicalparameters.Experimentalobservations[2,4,19,20,22,31,42,48]havedemonstratedthatatasmallow-rate,asmoothlmisformed,andatamoderatelyhigher1

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ow-rate,circumferentialwavesmovingfromthediskcentertothediskperipheryareformed.Furtherincreasingowrateleadstotheappearanceofspiralwavesunwindinginthedirectionofrotation[20].Itisshownin[48]thattheinitiallyuniformlmbreaksdownintowell-denedspiralripples,whichthenbreakdownfurtherintoaconfusedassemblyofwavelets.Circum-ferentialandspiralwaveswerefoundtodecayatlargediskradii.Comparisonthoseobservedwavesandtheirassociatedparameterswiththewavesbelongingtotherstfamilyandsecondfamilyofwavesinfallinglmsshowstheirsimilarity.Thewavesinfallinglmswerestudiedexperimentallyintheclassicalwork[17]andtheoreticallyin[3,5].Theoreticalexplanationofexperimentalresultshasreceivedincreasingattentioninrecentpublishedresearch.Therearethreemaindirectionsfortheoreticalinvestigationsofalmowoverarotatingdisk:calculationofwavelessow,analysisofitslinearstability,andnon-linearsimulationsofnite-amplitudewaves.Thewavelesssolutions,asymptoticandnumerical,wereinvestigatedin[14,28,30].ThesesolutionsgeneralizetheNusseltsolutioninseriesexpansioninthepowersoftheEckmannumberortheradius.Steadyowintheframeoftheboundarylayerapproximationwasconsideredin[10,40].Thelinearstabilityanalysiswasexaminedandperformedusingasymptoticmethods[6,27]andthefullNavierStokesequationforniteEckmannumbers[38,39].Inrecentpapers[21,35,36,37]anevolutionsystemofequationstomodelaxis-symmetricnite-amplitudewaveswasderivedandanalyzed;thismodelwasextendedfornon-axis-symmetricowstoexplaintheexperimentalresults.Nevertheless,thereareproblemsthatcouldbetreatedbyparallelapplicationexperimentalandtheoreticalapproaches:sensitivityofwaveregimestoowconditionsandthree-dimensionalstructuresobservedinexperiments.Inthelastdecade,therehasbeensignicantworkinimageprocessingrelatedtothemotionanalysisofnon-rigidobjects[8,16].In[16],theauthorsshowthatFiniteElementModelingshowspromiseformotionanalysisofbiologicalobjects.Mostoftheworkshaveconcentratedonarticulatedandelasticmotion[1,13],butanalysisofuid-likemotionwasalsoattempted[8,26,49,52].Recently,workhasbeguninaneorttocombinepreciseexperimentalsetup,theoreticalderivation,andbasicimageanalysistechniques[43,44,48].Theequationsconstituteasetofphysicalconstraintsthataredierent[16]fromthosecommonlyusedinthestudyofsolid2

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motion,e.g.,therigidityconstraint.Forthefastuid-likemotionintheair,havingturbulentcharacter,detectionofinterfacebetweenuidandairisimportant.Aspecialso-calledparticleimagevelocimetryPIVtechniquewasdeveloped[23]tomeasurethekinematicsofturbulentuidowincontrolledlaboratoryexperiments.GivenatypicalensembleofPIVimages,theaimistocalculatetheinstantaneousinterface,includingtheinstantaneousvelocityonthesurfaceofuidandaircontact,ecientlyandwithareasonabledegreeofaccuracy.Algorithmsusedaretypicallybasedonlter-likemotion.The3Dvectorsofmodelvariablesdescribinguidmotionarevaryinginspace.Inpractice,however,experimentaldata,obtainedforasequenceoftimeinstants,containinformationthat,generallyspeaking,diersfromthemodelvariables.Thepurposeofthisresearchistodevelopasystemofvisualscanning,recording,andtrackingofthelmowoveraspinningdiskwiththeintentiontodetectregimesoftheuidowwithregardtodierentconditionsusingasinglecamerasystem,tocalculateuidowparametersandcharacteristics,andtocomparethemwiththesolutionsoftherelevantmathematicalmodels.Inthisresearch,combinationofdirectvisualizationwithimageanalysissoftware,utilizingresultsandmethodsofmathematicalmodelingissuggested.1.2ContributionsofthisThesisThisthesisdescribesthefollowingnovelresearchcontributions:1.Unlikethemajorityofpreviousworks,estimationofvelocitycomponentsisgivenwithestimatesoftherespectiverelativeerrorsbasedontheso-calledquasi-optimalmethodforthesolutionofill-posedproblems.2.Forpracticalrealizationofthatquasi-optimalmethod,oneneedstoknowestimatesofthesecondandhigherderivativesofthemotion.ThesecondderivativescanbeestimatedduetotheNewton'slawthroughtheratiooftherespectiveforcestothemass.So,thecontributionofthisthesisiseectiveestimationofthesecondderivativesbasedonthemodelofspiralsofwaveswithregardtothreeforcesoftheuidowoveraspinningdisc:centrifugal,frictionoversurfaceofthedisc,andairresistance.3

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3.Thethirdcontributionisexperimentalestimationofwaveinclinations,usingthevideoimageofradiiasthefunctionofazimuthalangle,whichallowsonetousearbitrarystepalongtheangleunderestimationofderivativesofradiioverangles.Oncontrary,usingexperimentalmotionasthefunctionofthetimeforestimationofwaveinclinationsmayrequiretoosmalltime-stepwhichcannotberealizedinpractice.4.Thenovelmodelofspiralsofwavesanditsnumericalrealizationshowrathergoodcoincidencewiththeexperimentalspirals.So,thefourthcontributionisanovelrathersimplemodelincomparisonwiththegeneralmodelofNavier-Stokes,whichcanbeusefulforestimationsofvariousparametersandcharacteristicsofspiralwaves.1.3LayoutoftheThesisChapter2consistsofdescriptionofthegeneraltheoryofuidowbasedontheequationsofNavier-Stokes,includingtheimportantparticularcaseofthethinlmow.Besidesthewell-knownresultsofvariousauthors,especiallyProfessorSisoev,thechaptercontainsanovelequationsofspiralsofwaves,whichcomparedlaterwiththerespectiveexperimentaldataandusesfortheproperestimationsofwavevelocitycomponents.Chapter3describesexperimentalsetupandcameracalibration.Section3.1isdevotedtodataacquisitionfortheexperimentaldiscreactorusedinthisstudyandtocomparisonofthedataandreactorwiththesimilaronesoftheotherauthorsandworks[19,42,48].Sections3.2and3.3describethealgorithmofcameracalibration,andgivestheanalysisofitsaccuracy,whichisusedinChapter4underestimationoferrorsofexperimentalvelocitycomponentsandinclinationangle.Chapter4consistsofdescriptionofexperimentalresultsandalgorithmsforestimatesofparametersofwavespiralssuchastheirvelocitycomponents,inclinationangles,andwave-lengths.ItdescribescomparisonofthoseresultswiththerelatedresultsoftheoreticalmodelsintroducedinChapter2.Itshouldbenotedthatthealgorithmsofthischapterarefocusedonautomaticdataprocessingstartingfromtheinitialvideodataandendingbytheresultsoftheanalysis,includingestimationoferrorsofnumericalresults.Qualitativeandquantita-4

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tivecomparisonshowsrathergoodcoincidencewithin1-2%ofexperimentalandtheoreticalresults.5

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CHAPTER2THEORYOFFLUIDFLOW2.1GeneralCaseThisChapterdescribesthemathematicalmodelsofuidow[9].Consideranidealinvisciduid.Theparametersofuidmechanicsare:theuiddensityx;t;thevelocityvectoreldux;t,andthepressurepx;t;x2Rdisthespatialcoordinateind-dimensionalregionofspace.AninnitesimalelementoftheuidofvalueVlocatedatpositionxattimethasmassm=x;tVandismovingwithvelocityux;tandmomentummux;t.Thenormalforcedirectedintotheinnitesimalvolumeacrossafaceofareanacenteredatxisnx;ta,wherenistheoutwardunitvectornormaltothesurface.Thepressureisthemagnitudeoftheforceperunitarea.ThesedenitionsareillustratedinFigure2.1. Figure2.1FluidElementofVolumeV=xyzLocatedatPositionX.6

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Recallingthedenitionoftheconvectivederivativeandtheequationoftheconservationofmass[9],theEuler'sequationsforanincompressiblehomogeneousuidare:@u @t+uru=1 rp;ru=0;wherer=@ @x;@ @y;@ @z.Ifanexternalforceisappliedtotheuid,thenEuler'sequationsforanincompressibleuidbecome@u @t+uru+1 rp=1 fx;t;ru=0;wherefx;tistheappliedforceperunitvolume.AnincompressibleuidcanbedescribedwiththeNavier-Stokesequationsthatarefourcouplednonlinearpartialdierentialequationsforfourunknownfunctionsthethreecomponentsofuandthepressurep:@u @t+uru+1 rp=u+1 fx;t;ru=0;whereisthekinematics'viscosity.ThediusionofmomentumbetweenneighboringelementsoftheuidisanewingredientintheincompressibleNavier-StokesequationscomparingtotheincompressibleEulerequations.Also,thereisthematterofboundaryconditions[9].Usingthecylindricalsystemcoordinates,thesteadyNavier-Stokesequationsare:u@u @r+w@u @zv2 r=1 @p @r+@2u @z2+@2u @r2+1 r@u @ru2 r;.1whereu;v;warethecomponentsofvelocityinther;;zdirection.Theauthors[30]showthatforathinlm,h=r1,theradialowcanbeobtainedfrom.1tolowestorderbysolvingtheequations2r=@2u @z2;whereistheangularvelocity.7

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Thelasttwoformulaeaboveusedin[19]toestimatethethirdzcoordinateofthewavesinthecaseofthinlmow.2.2MathematicalModelingThissectiondescribesDr.Sisoevderivationoftheevolutionequationsofuidow[34],thenumericalsolutionsofwhichareusedforcomparisonwithexperimentalresultsinChapter4.2.2.1EvolutionEquationsAmodelgivenbelowfollowsfrom[35]withaccountingnon-axis-symmetricterms.Theauthors,G.M.Sisoevetal.,consideralmofincompressibleviscousliquid,ofkine-maticviscosityanddensity,owingoverasoliddiscspinningwithangularvelocity.Astationarycylindricalcoordinatesystem,~r;;~z,alongwiththevelocityeld~ur;~u;~uz,isintroducedtoformulatethemathematicalmodeldescribingthedynamicsofthelmboundedbythefreesurface~z=~handunderlyingsolidsubstrate~z=0.Theowisgovernedbythecontinuityequation,theNavier-Stokesequations,andanappropriatesetofboundaryconditions:no-slipandno{penetrationatthediscsurface,thekinematicboundarycondition,andnormalstressbalancesatthelmsurface.Tomaketheseequationsdimensionless,thefollowingscalingisperformed:~t=Et ;~r=Rcex;~z=Hcz;~ur=~ru E;~u=~r1+v E;~uz=Hcw E;~p=2~r2p;~h=Hch;.2where~pisthepressureand~tisthetime.Inequations.2,E==H2crepresentstheEckmannumber;HcandRcarethecharacteristicscalesforthelmthicknessandradialcoordinate.Hcischosenas:Hc=Qc 22R2c1 3:.38

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TheobservedwaveshaveacharacteristiclengthscalewhichismuchsmallerthanRc,sothefollowingre{scalingsareintroducedx=x ;t=t ;w=w;.4whereisasmallcoecienttobedetermined.InadditiontotheEckmannumberE,thederivedbyDr.Sisoevequationsandboundaryconditionsintermsofthedimensionlessvariablesdenedin.2and.4incorporatetwodimensionlessparameterizingfunctionsWe=2~r2Hc ;"=Hc ~r;whereWeisthelocalWebernumber,denotessurfacetension,and"isthelocalaspectratio.Asshownin[35],abalancebetweenthecapillaryforcesandviscousstresses,whichistypicalforcapillarywavesinviscousliquidlms,isprovidedbyusing=Hc 2R4c1 3:Analysisofexperiments,alsocarriedoutin[35],revealedthatarelation"2=21issatisedinalldataavailablewhenacapillarywavesareobserved.AfteromittingtermsofO"2=2intheproblemstatement,thepressuremaybeeliminatedandaftersubstitutingux;z;t=3qu hz hz2 2h2;vx;z;t=5qv 4h2z hz3 h3+z4 4h4;.5whereowratesinradialandazimuthaldirectionshavebeenintroducedasquhZ0udz;qvhZ0vdz;theapproximatemodelfollowsintheform9

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@h @t+@qu @x+2qu+@qv @#!=0;@qu @t+a11@ @xqu2 h!+"a12@ @#quqv h!+a13qu2 h+a14qv2 h= 4h+22qvb1qu h2+e2xh@ @xe2x@2h @x2+2@2h @#2;@qv @t+a21@ @xquqv h!+a22@ @#qv2 h!+a23quqv h= 4he4x@3h @x2@#+2@3h @#3b2qv h222qu.6withthecoecientsgivenbya11=6 5;a12=17 14;a13=18 5;a14=155 126;b1=3;a21=17 14;a22=155 126;a23=34 7;b2=5 2:.7In2.6,#=Etistheazimuthalanglerelatedtothespinningdisc.Thesimilarityparameter=45E21=1 4528R4cH11c 1 3hasbeenintroduced.Thelmparameteralsoappearsinafallinglmproblem[5].EisheEckmannumberand=45E21.10

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2.2.2LinearStabilityAnalysisWithincreasingradius,capillarywavesareformingonalmsurface.Alocalizedversionofequations.6allowstocomputeparametersofdevelopinglinearwaves.Thelocalizedequationsaresee[34]:@h @t+@qu @x+@qv @#=0;@qu @t+a11@ @xqu2 h!+"a12@ @#quqv h!+a13qu2 h+a14qv2 h= 4"h@3h @x3+2@3h @x@#2b1qu h2+h+22qv#;@qv @t+a21@ @xquqv h!+a22@ @#qv2 h!+a23quqv h= 4h@3h @x2@#+2@3h @#3b2qv h222qu;.8Thesystem.8hastheaxis-symmetricstationarysolution:h=H,qu=Qu,qv=Qv,calculatedfrom4 Ha13Qu2+a14Qv2=Hb1Qu H2+22Qv;a234QuQv H+b2Qv H2+22Qu=0;.9whereQu=1inaccordancewiththescale.3.Tocarryoutthelinearstabilityanalysisofthebasicsolution.9,thesolutionsof.8arerepresentedinthefollowingnormal-modeform:h;qu;qu=H;Qu;Qv+^h;^qu;^qvexpix+n#r!t;.1011

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wherethequantitieswithhatsrepresentsmallperturbations;andnaretherealwavenumbersgiveninradialandazimuthaldirections,and!isthecomplexwavefrequency.Substituting.10into.8inparallelwithitslinearizingleadstothefollowingeigenvalueequationi!3+D3!2+D2!+D1=0:.11Thedependence!;nisfoundnumericallydeterminesthemostgrowingperturbationscomparedwithexperimentaldata.Thecorrespondinginstantlocalinclinationofthespiraltan=1 ~rd~r d#=n .12mightbecomparedwithexperimentalmeasurements.12

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2.3TheSpiralEquationsUsingthetheory[32,33,45,47],thefollowingspiralequationswereutilized:ar=r00t=2r8fr0tc 2sinr02t;as=v0st=8frvstc 2cosv2st;tan=r0t vst=y0x+y x=y0xyx=1 rdr d;0t=r0t rt1 tan=vst rt;dr d=rtr0t vst;d2r d2=r02+rtr00t vstv0str0trt v2st1 0t;xt=rtcos;yt=rtsin;y0x=r0sin+rcos0 r0cos+rsin0;vt=r02t+v2st1 2;r=0;r0=r00;rT=200;vs=0;.13wherearandasareaccelerationsofuidalongradiusandperpendiculartoradius,fisacoecientofuidfriction,andcisresistanceofairtouid.Theformula.13isnotvalidinvicinityofr=0.Therefore,weneedtouseanothermodelfortonthesegment;bforsmallb.Lett=atonthissegment.Then0=0whichwasneededanda=0b=r0b rbtanb.Itisclearthatistheanglebetweendirectionofthespiralandtangent.Tondt,theanglebetweenradiusatthemomenttandtheaxesx,considertherighttriangleABC,wherelegsAB=dr=r0tdtandBC=rd=r0t,angleisoppositetordseeFigure2.2.13

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Figure2.2Determinationofand.Notethatthesecondandthirdformulaefortanabovecanbeusedbyimagealgorithms,sincetheydonotdependonthetimebutonlyongeometricpropertyofthespiral.Theapproximatesolutionforthisnon-linearsystemisfoundusingEuler'snumericalmethodwithdecreasingstepsofcomputationuntilapproximatesolutionsarestabilized.Inthecaseofwaterwehave8f=0:4mm1,c=2mm1,andr00=3:8106 28522:53280mm/s.Here=40radian secisobtainedfromtherespectivegraphforthereactorused,andr0isobtainedfromtheexperiment,inwhich3:8106mm3isvolumeofagallonofwater,285sisthetimeforthewatertorunoutoftherespectivecapacityseeFigure3.1,2:5mmistheradiusofthetube,and3mmisthegapbetweentheendofthetubeandthedisksurface.14

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CHAPTER3DATAACQUISITIONANDCAMERACALIBRATION3.1DataAcquisitionTheexperimentalset-upconsistsofamotor;aluminumat/roundstock;reservoir;tubing;brassadapters;bungedcords;ow-meter;coppertubing;aluminumcontrolbox;switches;returnpump;miscellaneoushardware.Themaincharacteristicofmotorisgivenbytheturntablecalibration.TheimageofthedeviceisshowninFigure3.1. Figure3.1ExperimentalSetup.Measurementswereperformedinthefollowingway.Watercontainedinaplasticcontainerwithanadjustmentvalvefortheowwasdrainedthroughcoppertubingataconstantowstartingrater00,whichcanbechangedintherange0.2-0.8lpmliterperminute.Liquidemergedfromthenozzleasafreejetpouringoutontothecenterofaconstantlyrotatingaluminumdiskwithadiameterof400mm.Therotationalfrequencyofthediskwasmonitoredbyamotorcontrol.Waterleavingtherotatingdiskiscollectedatthebottomreservoirandpickeduptothetopreservoirbyapump.15

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TenvideosweretakenatdierentparametersettingsdierentarrangementsoflightandsettingsofthecamerausingtheportablecamcorderCanonOptura20,capableofcapturingimagesat30fpsframepersecond.Figure3.2showsasampleimageoftheliquidlmthatowsoveradiskrotatingwiththeangularvelocityof520rpmreversesperminuteandtheowrate0.8lpm.ItcanbeseenFigure3.2thelmsurfaceiscoveredbyspiralwaves. Figure3.2RotatingDiskCloseup.Thepatternof810squareswasusedtocalibratethecameraseeFigure3.3.Three Figure3.3PlanarPattern.hundredtwentypointscornersofsquaresontheexperimentalpatternwerechosenintheCartesiansystemcoordinatewiththeorigininthecenterofthedisk.Siximagesofthemodel16

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objectweretakenunderdierentorientationsbymovingthemodelobject.Thetechniquedescribedin[49,51]wasusedforndingtheintrinsicandextrinsicparametersofthecamera.3.2OverviewofCameraCalibrationCameracalibrationisanecessarystepin2Dand3Dcomputervisioninordertoextractmetricinformationfromvideoimages;anditisimportantforaccuracyin2Dand3Drecon-struction.Inparticular,itisacriticaltaskforstereovisionanalysis.Muchworkhasbeendone,startinginthephoto-gram-metriccommunity[46],andmorerecentlyincomputervi-sion[7,12,15,18,41,50,51].Cameracalibrationistheprocessofrelatingtheidealmodelofthecameratotheactualphysicaldeviceanddeterminingthepositionandorientationofthecamerawithrespecttoaworldreferencesystem.Dependingonthemodelused,therearedierentparameterstobedetermined.Forthepinholecameramodeltheparameterstobecalibratedareclassiedintwogroups:1.Internalorintrinsicparameters.Internalgeometricandopticalcharacteristicsofthelensesandtheimagingdevice.2.Externalorextrinsicparameters.Positionandorientationofthecamerainaworldreferencesystem.Therelationshipbetweena2DpointX;Yanditsimageprojectionx;yisgiven[50,51]by266664xy1377775=A[r1;r2;t][X;Y;1]T;A=266664scx0scy001377775.1where[r1;r2;t]aretheextrinsicparameterstherotationsandtranslationthatrelatetheworldcoordinatesystemtothecameracoordinatesystem;Aisthecameraintrinsicmatrix,inwhichcx;cyistheprincipalpoint,s=f=sx=f=sy,fisthefocallengthinthepixels,sisthescalefactor,accuracyofwhichtobeaccountedforanyuncertaintyduetoimperfections17

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intheviewingcamera,sx=syistheeectivesizeofthepixel,andistheskewnessoftheimageaxes.3.3CameraCalibrationAccuracyParametersofthecamerausedintheexperimentsare:sx=1=1:6=0:625mmands=f=sx=512mm.So,assumingthattheabsoluteerror4sofsisnotmorethanhalfofthepixel,wehavethat4s=s<0:4=512=0:1=128<0:001.Consideringlensdistortionofacamerawithcoecientsk1andk2,theidealdistortion-freepixelcoordinatesx;yandthecorrespondingrealobservedordistortedimagecoor-dinatesxd;yd,therelationbetweenthecoordinatesofthedistortedandtheundistortedare:xd=x+kxcx;yd=y+kycy;k=k1r2d+k2r4d;r2d=x2d+y2d:Theserelationsallowustocalculatexandy,afterwhichvaluesXandYarecalculatedfrom.1.AstheresultsX;Y;1]T=r1;r2;t]1A1[x;y;1]T;x=xd+cxk 1+k;y=yd+cyk 1+k:Withregardtothefactthatthemainerrorofthecalibrationmethod[12]usedinourpaperisdeterminedbytheeectofdistortion,thestandardrelativedeviationsforx,yandX,Yareestimated,assumingforthesimplicitythatcx=cy==0.Inthisconnection,theresultin[51],page11isused,thatthestandardrelativedeviationsfortheestimatesofk1andk2donotexceed3-4%.Sinceinthecaseconsideredx=xd 1+k;y=yd 1+k;18

PAGE 26

withregardtothemaintermsassumingkissmall,k0:03,andk1=k2,thevariationsdx=dxd+xddkdy=dyd+yddk;dk=dk1r2d+r4d;dk k=dk1 k1<0:03;.2fromwhere,dx x=dxd xd+kdk k;dy y=dyd yd+kdk k;drd=dr=xdx+ydy x2+y21=2:Usingthefactthatdxd xd;dyd yd<0:3 720,where720ismaximalvalueofthenumberofpixelsand0:3istheupperboundfordxdanddyd,itiseasetoestimatethatdx x<0:3 720+0:030:030:0013;dy y0:0013:.3Itfollowsfrom.3thatinthecaser=p x2+y2,dr rdx x+dy y0:0026:Similarly,inthecaseofR=p X2+Y2withregardtoR=1 sx2+y21 2;ds s<0:001;onehasdR R
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Theseestimatesoftherelativeerrorswillbeusedforexperimentalestimationoftheradialve-locitycomponentandthespiralwaveinclinationofthelmowasanill-posedcomputationalproblemofdierentiation[25].20

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CHAPTER4ALGORITHMSANDRESULTS4.1RegimesofFluidFlowTheuidowoveraspinningdiskof300rpmand520rpmareillustratedinFigure4.1. a bFigure4.1RegimesoftheDiskRotationa300rpmandb520rpm.Fromtheobservationoftheuidowsoveraspinningdiskfollowsthatwavesoftheuidowscreatethespiralstructurefordierentregimes.Thedescriptionoftherespectivespiralsinmoredetailsisgivenlater.4.2VelocityComputationandAccuracyEstimationTheblock-schemeofthevelocitycomponentestimationisgiveninFigure4.2.21

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Figure4.2Block-SchemeofEstimationanInstantaneousVelocity.4.2.1ComputationofRadialVelocityandAccuracyEstimationTodeterminetheradialvelocitycomponent,thesequencesoftheimagesofthelmowsareusedwiththetimedierenceequaledtot=jt2t1jsec.Choosingthesystemofcoordinateatthecenteroftherotationdisc,thechangesoftheradiiatthedierenttimesarecalculated:r=jr2r1j,wherer1isthevalueoftheradiusfromcentertothepointonthewaveatthemomentt1andr2atthemomentt2.Theestimateoftheradialvelocitycomponentr0exp=r jt2t1j.Thisproblemisill-posed,i.e.,forsuchproblems,arbitrarysmallerrorsoftheinitialdatacangive,ingeneral,arbitrarylargeerrorsoftherespectiveresults.Ifrtistheradiusatthetimet,~Rtisitsexperimentalvalue,andjrt~Rtj
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whereM2=maxjr00tj,and0jtj=1 2M22 jtj2=0;jtj=4 RR M21 2;=jr0t~Rt+t~Rt tj1 2M24 M21 2+2 4 M21 22M21 2; M12M2R R1 21 M1;M1=maxjr0tj:.2Theestimatesin.1,.2givethequasi-optimalvalueofstepofdierentiationt,forwhichtherelativeerrorofdierentiationisgivenattheendof.2.Formulae.1and.2areappliedfortheexperimentalestimationoftheradialvelocitycomponentandinclinationangles.Inourcase,wecanestimate=1=2mm,=R=0:003,duetotheresultoftheaccuracycalibration.Fortheproperexperimentalestimationofderivatives,theestimatesofthemaximalabsolutevaluesoftherstandthesecondderivativesarerequired.Thatinformationcanbeobtainedduetothemodelofspiralsoftheowwaves.13.ThoughthatmodeldoesnotregardallconditionsofthelmowastheequationsofNavierStokes,ithastherightrelationsforgoodestimatesoftherequiredmaximalvaluesoftherstandsecondderivativeofthemotion.Indeed,thesecondderivativeisdirectlyproportionaltotherespectiveforceduetotheNewton'slawasshowninequationsofspiralofwaveswithregardtothreemainforces.Theresultsofnumericalsolutionofequations.13arecontainedinTable4.1.Usingthistable,thevalueM2andM1canbeestimatedas:M2 R=5063=200=25,M1=781.Therefore,using.2,thequasi-optimalt:003=251 21=100sandtherespectiveerrorforestimationoftheradialvelocitywithregardingtorandomizing M10:065.Thus,therelativeerrorforestimationofradialvelocityis6:5%.TheradiiandexperimentalradialvelocitycomponentforonesequenceofimagesaregiveninTable4.2.ComparisonofestimatedexperimentalvelocitywiththerespectivevelocityobtainedfromthespiralmodelshowstheirratherclosedvaluesseealsoFigure4.3.23

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Table4.1ResultsoftheNumericalSolutionsofSpiralEquations. Times Radiimm r0t r00t vst v0st vt t xt yt radian 0:00 0 200 0 0 0 200 0 0 0 1:570 0:13 20 265 357 115 783 289 3:44 6 19 1:162 0:19 40 382 1280 177 571 422 4:59 36 19 1:131 0:24 60 509 2287 241 432 564 4:84 59 4 1:129 0:28 80 536 2504 255 404 594 4:95 77 19 1:127 0:32 100 601 3009 287 339 665 5:20 88 46 1:125 0:35 120 663 3516 318 276 736 5:41 92 78 1:123 0:38 140 718 3950 345 222 797 5:59 90 109 1:122 0:40 160 763 4313 368 177 848 5:73 84 136 1:121 0:43 180 779 4605 391 133 871 5:85 74 163 1:119 0:46 200 781 5063 423 237 888 5:99 57 196 1:070 Table4.2ExperimentalRadialVelocity. Radiiatt Radiiatt+t RadialVelocity 160 167:7 770 159 165:6 760 163 170:7 770 161 168:8 780 160 167:8 780 158 165:6 760 162 169:5 750 160 167:6 760 163 170:7 770 4.2.2EstimationofAzimuthalVelocityComponentTheazimuthalvelocitycomponentisdeterminedbyva;theor=r0 tansee[19].Thisformulausestwovalueswhichareill-possedtodetermineexperimentally.Thevelocityr0twasestimatedaboveandtheinclinationanglewillbeestimatedbelow.Thus,usingthosetwoestimateanazimuthalvelocitycomponentcanbefound.Forinstance,iftheradialvelocityfortheradii160and167:7istakenfromTable4.2andthevalueoftheinclinationanglefromtheTable4.4thentheazimuthalvelocityva;theorr0 tan770 tan1:12530mm=s.Alsothelmthicknesshcanbecalculatedusingthefollowingformulah=q 2r0 R2;whereiskinematicviscosity.24

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Figure4.3Model-BasedSpiralPointsRedontheExperimentalVideoData.4.3WaveFrontDetectionThefollowingalgorithmsweredeveloped:1detectionofpointsonthewavesalongthesameradius,2wavelengthestimation,and3waveinclinationestimation.Theblock-schemeisgivenbelowinFigure4.4. Figure4.4Block-SchemeofWaveFrontDetection.Inordertodetectthepointsonthewavesthefollowingsequenceofoperationsisperformed:theimagescaling:thepixelswithintensityI>220areincreasedtoI+10andwithI220decreasedtoI20;thelocalhistogramequalization;theaverageintensitiesdenitionforfourradiiatanincrement1 57radian;thefteenrstmaximumsndingontheaveragingradiusliedintheenhancementwindow.TheimagewithfourdetectedlinesaandresultsoftheintensityaveragingoflinesbareshowninFigure4.3.Theresultingimageisshownin25

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a bFigure4.5aEnhancedImagewithSelectedLines.bIntensityDistributionoftheAverageResult.Figure4.6. Figure4.6DetectedPointsofWaves.4.3.1EstimationofInclinationAnglesandWavelengthsFigure2.2showsthedeterminationofaninclinationangle.Theproblemofestimationofisalsoill-posed[25].So,theso-calledquasi-optimalmethod[25]tominimizeanerrorofestimateundertheknownerrorofinitialdataareused.Inordertousethetheradiiasthefunctionofazimuthalangleinthevideoimages,whichallowseveryonetousearbitrarystepalongtheangle,therstandsecondderivativesofradiusoveranglehavetobecalculated.Usingformulae.13andTable4.1,therstandsecondderivativesarecalculatedforradiiintherange820cmseeTable4.3.Thequasi-optimalstepd=4 RR M21 2is1 25radian,where Risarelativeerrorofcameracalibration0:003andM2=maxd2r d2isfound26

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Table4.3CalculatedResultsofdr dandd2r d2. Times Radiimm 0 dr d d2r d2 0:00 0 0 0 0 0:13 20 5:7 46 63 0:19 40 4:5 86 186 0:24 60 4:0 126 354 0:28 80 3:2 168 514 0:32 100 2:9 209 710 0:35 120 2:7 250 922 0:38 140 2:5 291 1163 0:40 160 2:3 331 1433 0:43 180 2:1 358 1690 0:46 200 2:14 369 1695 fromtheTable4.3.Anerrorproducedbyimageprocessingwasestimatedas2 M1p M2R R17%,whereM1=maxr0370mm rad.Thevideodataofuidowoverrotatingdiskof520rpmandtheowrateof0.8lpmwereusedtocalculatethewaveinclinations.Thesequenceoftenframeswereprocessedtondtheaveragedinclinationanglesforradiiintherange8-20cm.ThecalculatedaveragedinclinationanglesareshownintheTable4.4.InTable4.4CalculatedAveragedWaveInclinations. Thecalculatedinclinationanglescentimeter R 8 10 12 14 16 18 20 1.32 1.19 1.12 1.00 0.99 0.97 0.93 ordertocalculatetheradialwavelengthl[37]thedierencebetweentwoneighboringwaves,inradialdirection,wascalculated:l=r2r1.Notethatvaluesr1andr2havetobeusedintheprocessoftheaveraging.4.3.2CorrespondencetotheMathematicalModelThesystem.6maybeappliedtodescribeaxis-symmetricorspiralwaveregimesiftwoparameters,2and"=aresmall;theseconditionsallowonetousetheboundarylayerapproximation.Thoseconditionswereexaminedfortheobservedspiralwavesdescribedabove.Thecoordinatesofpointsonthefrontofcurvesintheradialdirectionweretakenataconstantincrementof=6.TheestimatedradiiarepresentedinTable4.5.TheparametersrelevanttoTable4.5areshowninTable4.6,obtainedbyDr.Sisoev.Itcanbeseenfromthe27

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Table4.5RadiioftheFirstandSecondWavesR1andR2,Respectively. radian cm cm =6 7.994 8.304 =3 9.899 9.928 17=36 12.003 12.229 21=36 13.997 14.197 24=36 16.00 16.191 26=36 17.893 18.063 Table4.6InputDataandModelCoecients. R1, R2, 2103 "2 210 Re cm cm 7.994 8.304 7.284 0.6086 69.64 9.899 9.928 0.8604 0.5225 138.4 12.003 12.229 0.8290 0.5211 140.0 13.997 14.197 0.1765 0.4666 230.2 16.000 16.191 0.1543 0.4622 240.4 17.893 18.063 0.1002 0.4481 276.2 Table4.6thattheprincipalconditionsaboutsmallnessof2and"=arefullledforthespiralwavesaswellasforaxis-symmetricwaves.Thus,theboundarylayerapproximationextendedbythetermsaccountingfordependenceontheazimuthalanglemaybeformulated.Toestimateinclinationangleofnon-axis-symmetricwavestheeigenvalues!werecalcu-latedbyDr.SisoevfromEquations.11fordierentvaluesofradiusunderexperimentalconditionswith=520rpmandtherateofinitialuidowQcequalto0:8lpm.ExamplesofcalculationsaregiveninFigure4.7.Itisseenthatnon-axis-symmetricperturbationsaremoreunstablethanaxis-symmetricones.Thenforafewvaluesofradius,wavenumberandinclinationparameternwerefoundinthecaseofmaximumofamplicationfactors.4.3.3ComparisonThevevideoswiththesequenceoftenframesfortheowrateof0.8lpmandtherotationofdiskof520rpmwereusedfordeterminationofthewavelengthsseeTable4.7andTable4.8.AverageexperimentallyandtheoreticallypredictedresultsforwavelengthsareshowninFigure4.8.28

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Figure4.7aAmplicationFactors.bAxis-symmetricn=0andNon-axis-symmetricn=100Perturbationsat~r=10cm.Table4.7CalculatedAveragedWavelengths. Videos Thecalculatedwavelengthscentimeter n=radii 8 10 12 14 16 18 20 1 .32 .30 .24 .21 .19 .18 .16 2 .33 .31 .25 .23 .21 .19 .17 3 .29 .27 .20 .18 .18 .16 .14 4 .33 .31 .22 .20 .22 .18 .16 5 .28 .26 .18 .17 .16 .14 .12 ItcanbeseeninFigure4.8thatthereisgoodcorrespondencebetweentheoreticalpredic-tionofwavelengthsandexperimentaldata.Theaveragerelativeerroris1%.Inaddition,aspredicted,thewavelengthdecreasesasradiusincreases.Thevideodataofuidowoverrotatingdiskof520rpmandtheowrateof0.8lpmareusedtocalculatethechangesofwaveinclinations.Thesequenceoftenframeswereprocessedtondtheaveragechangeofinclinationanglesforradiiintherange8-20cm.ThechangesofinclinationanglesareshownintheTable4.9.Theexperimentalinclinationalanglechangesagreeratherwellwiththeresultsofthepredictedinclinationalanglechanges.Theaveragerelativeerroris2%.Inaccordancewithexperimentalobservationsandtheoreticalprediction,theinclinationangledecreasesasradiusgrows.29

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Table4.8CalculatedAveragedWavelengthOverFiveVideosandTheoreticallyCalculatedWavelength. Wave length 8 10 12 14 16 18 20 Predicted :24 :20 :18 :17 :15 :14 :13 Calculated :31 :29 :22 :20 :19 :17 :15 Figure4.8DependenceofWavelengthsOverRadii.Asterisksandx-sCorrespondtoExperi-mentalandTheoretical,Respectively.Table4.9InclinationAngleChangesfortheVariousRadii. Wave inclination 8 10 12 14 16 18 20 changes Predicted 0 :08 :06 :04 :02 :01 :02 Calculated 0 :09 :06 :04 :03 :02 :01 30

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CHAPTER5CONCLUSIONS5.1SummaryThisthesispresentsnovelvideo-basedalgorithmsfordetectionandtrackingofspiralwavesinaspinningdiskreactor.Ineachframe,pointsonthetopofmultiplewavesaredetectedandaspiralmodelttedtothepoints.Basedonthesecomputationthewavelengthsareestimated.Inaddition,theinclinationanglesbetweenspiralsandtherespectivecirclesandtheradiibetweenthecenterofadiskandpointsthatlieonthefrontofcurvesinthedirectionofspinningdiskarecalculatedusingtheso-calledquasi-optimalmethod,whichminimizeserrorofdierentiationestimateunderknownerrorofinitialdata.Toestimatevelocityofuidow,we,also,usethequasi-optimalmethod.Resultscomputedfromvideodataarecomparedwithnumberspredictedbythetheoreticalmodel.Theobtainedresultsareingoodaccordancewithlinearstabilityanalysisofthetheoreticalmodels.Inparticular,theaveragecomputedwavelengthiswithin1%ofthepredictedvaluesandaaveragerelativeerrorofinclinationanglechangecomputationiswithin2%.Thegoalofthisresearchtodevelopimage-baseduidowobservationalgorithmscapableofcomputinguidowparametersandsuitableforindustrialinspectionapplications.5.2FutureResearchSomeissuesthatneedtobeaddressedinthefutureinvestigationare:automateddetectionofpointsalongthesurfaceofwavesoftheuidow;experimentswithvariousphysicalparameters;analysisofthoseexperimentsbasedonnonlinearsolutionsoftheevolutionsystem;furthercomparingtheresultsoftheexperimentswiththerespectivemodels;theapplicabilityofthetheoryofcomputervisionfortheevolutionsystems.31

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REFERENCES[1]K.Aggarwal,Q.Cai,W.Liao,andB.Sabata.Nonrigidmotionanalysis-articulatedandelasticmotionreview.ComputerVisionandImageUnderstanding,70:142{156,1998.[2]A.AouneandC.Ramshaw.Processintensication:heatandmasstransfercharacteristicsofliquidlmsonrotatingdiscs.InternationalJournalHeatMassTransfer,42:2543{2556,1999.[3]A.Bunov,E.Demekhin,andV.Shkadov.Onthenonuniquenessofnonlinearwavesolutionsinaviscouslayer.ApplMathMech,48:691{696,1984.[4]A.ButuzovandI.Puhovoi.Onregimesofliquidlmowsoverarotatingsurface.JournalofEngineeringPhysics,31:217{224,1976.[5]H.ChangandE.Demekhin.ComplexWaveDynamicsonThinFilms.Elsevier,2002.[6]A.Charwat,R.Kelly,andC.Gazley.Theowandstabilityofthinliquidlmsonarotatingdisk.FluidMechanics,53:227{255,1972.[7]T.ClarkeandJ.Fryer.Thedevelopmentofcameracalibrationmethodsandmodels.PhotogrammetricRecord,16:51{66,1998.[8]T.Corpetti,E.Memin,andP.Perez.Denseestimationofuidows.IEEETransactionsonPatternAnalysisandMachineIntelligence,24:365{380,March2002.[9]C.DoeringandJ.Gibbon.AppliedAnalysisoftheNavier-StokesEquation,page217.CambridgeUniversityPress,1995.[10]L.Dorfman.Flowandheattransferinaviscousliquidlayeronaspinningdisc.JournalofEngineeringPhysics,12:309{316,1967.[11]H.EspigandR.Hoyle.Wavesinathinliquidlayeronarotatingdisk.FluidMechanics,22:671{677,1965.[12]D.ForsythandJ.Ponce.ComputerVision.AModermApproach.PrenticeHall,2003.[13]D.Goldgof,H.Lee,andT.Huang.Motionanalysisofnonrigidsurfaces.InIEEEConferenceonComputerVisionandPatternRecognition,volume8,pages899{904,1988.[14]D.GottliebandS.Orszag.NumericalAnalysisofSpectralMethods:TheoryandAppli-cations.SIAM,1977.[15]J.HeikkilandO.Silven.Afour-stepcameracalibrationprocedurewithimplicitimagecorrection.InProceedingsofCVPR'97,IEEE,pages1106{1112,1997.32

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Tracking fluid flow in a spinning disk reactor
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2006.
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ABSTRACT: The flow of a liquid film over a rapidly rotating horizontal disk has many applications inmedical, industrial, and engineering fields. A specific example is the heat and mass transfer processes between expanded liquid and surrounded dense gas. Diferent wave regimes of a liquid film depend on a flow conditions such as the properties of a liquid, its initial speed,parameters of environment, etc. Therefore, experimental investigation of the film flow over a spinning disk is needed to both validate theoretical predictions and establish methods for fluid flow monitoring.This thesis presents novel video-based algorithms for detection and tracking wave structural data of the liquid film flowing over a spinning disk reactor. The algorithms are based on the spiral model of wave and the quasi-optimal method for estimation of a wave velocity as ill-posed problem. Their performance is compared with results predicted by the fluid dynamics based on the Navier-Stokes equations in the case of thin film.Using experimental video data, the developed models and algorithms allow investigators to estimate the characteristics of wave regimes such as wavelengths, inclination angles, and the radial and azimuthal velocity components of the fluid. The accuracy of estimated characteristics was analyzed. It was shown that average distance between consecutive two waves,their spiral shapes, and the radial velocities of waves confirm the theoretical results and predictions. In particular, computed wavelength is within 1% and a change of the inclination angles is within 2% of the predicted values.
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Thesis (M.A.)--University of South Florida, 2006.
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653
Inclination angle.
Mathematical modeling.
Spiral wave.
Velocity components.
Wavelength.
690
Dissertations, Academic
z USF
x Computer Science
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.1529