Numerical solution of steady free-surface Navier-S

CentrumvoorWiskundeenInformaticaNumericalsolutionofsteadyfree-surfaceNavier-StokesflowE.H.vanBrummelenModelling,AnalysisandSimulation(MAS)MAS-R0018June30,2000ReportMAS-R0018ISSN1386-3703CWIP.O.Box940791090GBAmsterdamTheNetherlandsCWIistheNationalResearchInstituteforMathematicsandComputerScience.CWIispartoftheStichtingMathematischCentrum(SMC),theDutchfoundationforpromotionofmathematicsandcomputerscienceandtheirapplications.SMCissponsoredbytheNetherlandsOrganizationforScientificResearch(NWO).CWIisamemberofERCIM,theEuropeanResearchConsortiumforInformaticsandMathematics.Copyright©StichtingMathematischCentrumP.O.Box94079,1090GBAmsterdam(NL)Kruislaan413,1098SJAmsterdam(NL)Telephone[phone]Telefax+31205924199NumericalSolutionofSteadyFree-SurfaceNavier-StokesFlowE.H.vanBrummelenCWIP.O.Box94079,1090GBAmsterdam,TheNetherlandsABSTRACTNumericalsolutionofowsthatarepartiallyboundedbyafreelymovingboundaryisofgreatpracticalim-portance,e.g.,inshiphydrodynamics.Theusualtimeintegrationapproachforsolvingsteadyviscousfreesurfaceowproblemshasseveraldrawbacks.Instead,weproposeanecientiterativemethod,whichreliesonadierentbutequivalentformulationofthefreesurfaceowproblem,involvingaso-calledquasifree-surfacecondition.Itisshownthatthemethodconvergesifthesolutionissucientlysmoothintheneighborhoodofthefreesurface.Detailsareprovidedfortheimplementationofthemethodinparnax.Furthermore,wepresentamethodforanalyzingpropertiesofdiscretizationschemesforthefree-surfaceowequations.Detailednumericalresultsarepresentedforowoveranobstacleinachannel.Theresultsagreewellwithmeasurementsaswellaswiththepredictionsoftheanalysis,andconrmthatsteadyfree-surfaceNavier-Stokesowproblemscanindeedbesolvedecientlywiththenewmethod.2000MathematicsSubjectClassication:35B20,35R35,65R20,76D05,76D33.KeywordsandPhrases:free-surfaceows,incompressibleNavier-Stokesequations,numericalsolutionmeth-ods,discretedispersionrelations.Note:ThisworkwasperformedunderaresearchcontractwiththeMaritimeResearchInstituteNetherlandsandwascarriedoutunderCWI-projectMAS2.1\ComputationalFluidDynamics.1.IntroductionThenumericalsolutionofowsthatarepartiallyboundedbyafreelymovingboundaryisofgreatpracticalimportance.Thenumericaltechniquesavailabletosolvesuchfree-surfaceows,canbecate-gorizedintosurfacetrackingmethods,themostprominentbeingthemarkerandcellmethod[13]andthevolumeofuidmethod[15],interfacecapturingmethods,e.g.,[17,21],andsurfacettingmeth-ods[9].Itisgenerallyacknowledgedthatifthefree-boundaryissmooth,inparticularifthesurfacecanberepresentedbyaso-calledheightfunction,surfacettingmethodsareunsurpassedinaccuracy.Sincethefree-surfacesoccurringinmanypracticalapplications,forinstance,shiphydrodynamics,aresmooth,surfacettingmethodshavereceivedmuchattention.Iftime-dependentsurfacettingmethodsareconsidered,generallythereisnoessentialdierenceinthetreatmentofthefree-surfaceinpotentialoworNavier-Stokesow.Then,independentoftheowmodel,thesolutionoftheowequationsandthegeometryofthefree-boundaryareusuallyseparated.Theowequationsareintegratedoverasmalltimeinterval,withthedynamicconditionsimposedatthefree-surface.Subsequently,thepositionofthefreesurfaceisdeterminedthroughthekinematiccondition,employingthenewlycomputedvelocityeld.Forsurfacettingmethodsforsteadyfree-surfaceows,suchacommonapproachforviscidandinviscidowsdoesnotexist.Whereasdedicatedtechniqueshavebeendevelopedforsteadypotentialow[5,6,24],methodsforNavier-Stokesowsimplycontinuetheaforementionedtransientprocessuntilasteadystateisreached.In[23]severaldrawbacksofthisprocessarediscussed,suchasslowconvergencetosteadystate.Inparticular,fromtheresultsin[19]oneinfersthatatsubcriticalFroudenumbers,dispersioncausesasymptotictemporalbehavioroftheamplitudeoftransientwavesinRdofO(t(d1)=2).Hence,iftheobjectiveistoreducetheamplitudeoftransientwavestotheorder2ofspatialdiscretizationerrors,theeciencyofthetimeintegrationapproachdeterioratesrapidlywithdecreasingmesh-width.Inpracticalcomputations,thousandsoftimestepsareusuallyrequired,renderingthetransientapproachprohibitivelyexpensiveinactualdesignprocesses.Foreciency,themethodsdevelopedforsteadyfree-surfacepotentialowexploitthefactthatduringthesolutionprocessneitherthekinematicnorthedynamicconditionneedstobesatised.Insteadofimposingthedynamicconditiononthesub-problems(i.e.,theowproblemscorrespondingtoagivenfree-surfaceposition)andusingthekinematicconditiontodetermineanewapproximationtothefree-surfacelocation,anycombinationofboundaryconditionscanbeimposedonthesub-problemsandanyoperatorthatlocatesthefreesurfacecanbeemployed,providedthatthesub-problemsarewell-posed,theresultingiterativeprocessconvergesandtheconvergedsolutionsatisesboththedynamicandthesteadykinematicconditions.Thispermitstheconstructionofiterativealgorithmsthatforeachsub-problemevaluationprovideamoreaccurateapproximationtothesteadyfree-surfacepositionthanwouldbeobtainediftheusualtime-dependentapproachwerefollowed.Eortscanthenbedirectedtosolvingthesub-problemseciently.Toreducethecomputationalexpenseofsolvi