Efficient Numerical Solution of Steady Free-Surfac

CentrumvoorWiskundeenInformaticaEfficientnumericalsolutionofsteadyfree-surfaceNavier-StokesflowE.H.vanBrummelen,H.C.Raven,B.KorenModelling,AnalysisandSimulation(MAS)MAS-R0103March31,2001ReportMAS-R0103ISSN1386-3703CWIP.O.Box940791090GBAmsterdamTheNetherlandsCWIistheNationalResearchInstituteforMathematicsandComputerScience.CWIispartoftheStichtingMathematischCentrum(SMC),theDutchfoundationforpromotionofmathematicsandcomputerscienceandtheirapplications.SMCissponsoredbytheNetherlandsOrganizationforScientificResearch(NWO).CWIisamemberofERCIM,theEuropeanResearchConsortiumforInformaticsandMathematics.Copyright©StichtingMathematischCentrumP.O.Box94079,1090GBAmsterdam(NL)Kruislaan413,1098SJAmsterdam(NL)Telephone[phone]Telefax+31205924199EfficientNumericalSolutionofSteadyFree-SurfaceNavier-StokesFlowE.H.vanBrummelenandB.KorenCWI,P.O.Box94079,1090GBAmsterdam,TheNetherlandsH.C.RavenMaritimeResearchInstituteNetherlands,P.O.Box28,6700AAWageningen,TheNetherlandsABSTRACTNumericalsolutionofflowsthatarepartiallyboundedbyafreelymovingboundaryisofgreatimportanceinpracticalapplicationssuchasshiphydrodynamics.Theusualmethodforsolvingsteadyviscousfree-surfaceflowsubjecttogravitationisalternatingtimeintegrationofthekinematiccondition,andtheNavier-Stokesequationswiththedynamicconditionsimposed,untilsteadystateisreached.ThispapershowsthatatsubcriticalFroudenumbersthistimeintegrationapproachisnecessarilyinefficientandproposesanefficientiterativemethodforsolvingthesteadyfree-surfaceflowproblem.Thenewmethodreliesonadifferentbutequivalentformulationofthefree-surfaceflowproblem,involvingaso-calledquasifree-surfacecondition.Theconvergencebehaviorofthenewmethodisshowntobeasymptoticallymeshwidthindependent.Numericalresultsarepresentedfor2Dflowoveranobstacleinachannel.Theresultsconfirmthemeshwidthindependenceoftheconvergencebehaviorandcomparisonofthenumericalresultswithmeasurementsshowsgoodagreement.2000MathematicsSubjectClassification:primary:65N12.secondary:35R35,76D05,76D27,76D33.KeywordsandPhrases:numericalsolutionmethods,free-surfaceflows,incompressibleNavier-Stokesequa-tions.Note:ThisworkwasperformedunderaresearchcontractwiththeMaritimeResearchInstituteNetherlandsandwascarriedoutunderCWI-projectMAS2.1”ComputationalFluidDynamics”.1.IntroductionThenumericalsolutionofflowsthatarepartiallyboundedbyafreelymovingboundaryisofgreatimportanceinshiphydrodynamics[1,6,7,12],hydraulics,andmanyotherpracticalapplications,suchascoatingtechnology[15,16].Inshiphydrodynamics,animportantareaofapplicationisthepredictionofthewavepatternthatisgeneratedbytheshipatforwardspeedinstillwater.Thiswavegenerationisresponsibleforasubstantialpartoftheship’sresistanceand,therefore,itshouldbeminimizedbyaproperhullformdesign.Computationalmethodsplayanimportantroleinthisdesignprocess.Mostcomputationaltoolsthatarecurrentlyinuseforsolvinggravitysubjectedfree-surfaceflowsaroundasurface-piercingbodyrelyonapotentialflowapproximation.Presentdevelopmentsprimarilyconcernthesolutionofthefree-surfaceNavier-Stokes(orRANS)flowproblem.Fortime-dependentfree-surfaceflows,generallythereisnoessentialdifferenceinthetreatmentofthefreesurfacebetweennumericalmethodsforpotentialfloworNavier-Stokesflow.Typically,thesolutionoftheflowequationsandtheadaptationofthefreeboundaryareseparated.Eachtimestepbeginswithcomputingtheflowfieldwiththedynamicconditionsimposedatthefreesurface.Next,thefreesurfaceisadjustedthroughthekinematiccondition,employingthenewlycomputedvelocityfield.Forsteadyfree-surfaceflows,however,suchaconformityofapproachesforviscousandinviscidflowcannotbeobserved.Forinstanceinshiphydrodynamics,whereasdedicatedtechniqueshavebeende-velopedforsolvingthefree-surfacepotentialflowproblems(see,e.g.,[14]),methodsforNavier-Stokes2flowusuallycontinuetheaforementionedtransientprocessuntilasteadystateisreached(see,e.g.,[1,7]).However,thistimeintegrationmethodisoftencomputationallyinefficient.Inparticular,atsubcriticalFroudenumberstheconvergencetosteadystateisretardedbyslowlyattenuatingtran-sientsurface-gravity-waves.Moreover,theseparatetreatmentoftheflowequationsandthekinematicconditionyieldsarestrictionontheallowabletime-step.Duetothespecifictransientbehavioroffree-surfaceflowsandthetime-steprestriction,atsubcriticalFroudenumberstheperformanceofthetimeintegrationmethoddeterioratesrapidlywithdecreasingmeshwidth.Inpracticalcomputations,tensofthousandsoftimestepsareoftenrequired,renderingthetimeintegrationapproachprohibitivelyexpensiveinactualdesignprocesses.Alternativesolutionmethodsforsteadyfree-surfaceNavier-Stokesflowexist,buttheyhavenotbeenwidelyappliedinthefieldofshiphydrodynamics.Inthefieldofcoatingtechnologysucces-siveapproximationtechniquesareoftenemployed,inparticular,kinematiciterationanddynamiciteration[16].Kinematiciterationimposesthedynamicconditionsatthefreesurfaceandusesthekinematicconditiontodisplacetheboundary.Dynamiciterationimposesthekinematicandthetangentialdynamicconditionsatthefreesurfaceandusesthenormaldynamicconditiontoadjusttheboundaryposition.However,theconvergencebehaviorofbothiterationschemesdependssensi