FlowTurbulenceCombust(2014)92:673–698689
Fromtheexperimentalpointofview,severalstudiesinvestigatingtheseparatedshear-layerinthenearwakeofacircularcylinderareavailableintheliterature.Ontheonehand,thepresentfindingsareconsistentwiththehot-wireanemometrybyBloor[2],whoestablishedthattheratiofsl/fvsyieldedtoRe0.5dependence.Further,thehot-wireandflowvisualizationsbyKourtaetal.[21]revealedadistributionforfsl/fvsidenticaltothatofBloor[2].MihailovicandCorke[28]confirmedaRe0.5variation.Ontheotherhand,investigationsobtainingmeasurementsbyNorberg[30],PrasadandWilliamson[35],Brede[3]andRajagopalanandAntonia[37]demonstratedaRe0.67relation.Recently,Brunetal.[6],whohaveperformedLESandlaser-Dopplervelocimetrymeasurementsfortheflowaroundasquarecylinder,reportedthattheratiofsl/fvsvariedaspowerlawwithavalueofn=0.7.Thisfitsfullywiththeexistingresultsforaroundcylindercase,n=0.67[35,37].RajagopalanandAntonia[37]foundthatthereappearstobenouniversalReynoldsnumberdependenceoffsl/fvs.
Fromthenumericalsimulationspointofview,onlyfewLES/DNSstudiesareavailableintheliterature.Jordan[18]haveinvestigatedthephysicsoftheshear-layerseparationbyLESforRe=8×103andpredictedfsl/fvs=9.4.Dongetal.[8]haveperformedacombinedPIV/DNSforReequalto3.9×103/4×103and104andreportedfsl/fvs=7.33andfsl/fvs=11.83,respectively.Rai[36]haveusedahigh-orderaccurateupwind-biasedfinite-differencemethodtoinvestigatetheinstabilitiesoftheseparatedshearlayersinthewakeofacircularcylinderatRe=3.9×103andM=0.1.ThecomputedvaluesbyRai[36]werefsl/fvs=5.71andfsl/fvs=5.14,dependingonthedownstreamprobe’slocation.
Finally,Fig.9summariesallthediscusseddata.TheresultsofthenumericalsimulationsseemstobeyieldedtotheRe0.5relation,whiletwopowerlaws(Re0.5andRe0.67)couldbeidentifiedfortheexperiments.
10
2Bloor(1964)Kourta(1987)Mihailovic/Corke(1997)Norberg(1987)Prasad/Williamson(1987)Rajagopalan/Antonia(2005)Maekawa/Mizuno(1967)0.67fsl/fvs≈ReDNSRai(2010)LESJordan(2002)f/f
slvs
1DNSDong(2006)PresentLES0.5fsl/fvs≈Re10
10
310
4Re
10
5Fig.9Variationoffsl/fvswiththeReynoldsnumber
690FlowTurbulenceCombust(2014)92:673–698
Meanwhile,analyzingthesimulatedwaveletsignaturesinFig.8b,onecanfindanotherinterestingfeature.Thesignaturesoftheshear-layervorticesplottedinFig.8binaformofthelocalizedpeaksintime,hadalsothefinitebroadbandfrequencydistribution,whichresultedfromvortexpairingandbreakdowninawakeshearlayersimilartoamixinglayer[8,37].Thisobservationsuggeststhatthenearwakezonedevelopsviaaconvectiveinstabilitymechanism[37].RajagopalanandAntonia[37],aswellasDongetal.[8],estimatedtheupperandlowerlimitsoffslas2fslandfsl/2,respectively,basedondetailedFourieranalysis.Atthesametime,RajagopalanandAntonia[37]mentionedthatthesubharmonicfrequencieswerenotalwaysequaltohalfoftheKHinstabilityfrequency.Theestimationsoffsl/2suffersfromuncertainties,similartothoseinestimatingfslfromspectra,whicharisefromtherandomflappingoftheshearlayerandtherandommovementofthelocationatwhichtransitionbegins.Thepresentresultsestimatedthesesubharmonicfrequenciestofsl/2=11.5and2fsl=22.6.Surprisingly,ifweconsidertheuppervariationlimit2fslasthefundamentalKHfrequency,thanfsl/fvsdependencywillbecharacterizedbytheRe0.67relation.4.2.5InfluenceofSGSmodel
Theinfluenceoftheturbulenceviscosityintroducedduetosub-gridscalemodelingcouldbeassessedbycomparingtheresultsfromtheNSGS-IandTKE-Isimulations.BothrunsintroducedthesameStrouhalnumber(St=0.2),whichisslightlylowerthantheexperimentalone.TheexperimentalevidencetellsthattheStrouhalnumberforRe=2×104shouldbearound0.19.Whilethemeandragcoefficientwasapproximatelythesame,thedynamicTKEmodelledtoalowervaluefortheRMSliftcoefficientandtoalargervaluefortherecirculationzonelengthcomparedtothevalues,predictedwithoutanSGSmodel.??Lr??andCl??aretypicallyrelated.ThehighlevelofCl??islikelyrelatedtothesmallrecirculationbubblebehindthecylinderthroughthehighlevelofturbulenceintheleadingstagnationregion.4.2.6Someremarksonthecomparisonwithexperimentaldata
AccordingtoBreuer[4,5]thefollowingshouldbementionedwhencomparingandassessingthepresentresultswithexperimentaldata.ItiswellknownthattheinvestigatedflowdependsnotonlyontheReynoldsnumber,butonavarietyofinfluencefactorsaswell,suchastheaspectratioofthecylinder,theblockageratio,theroughnessofthecylinder,thefree-streamturbulenceandtheMachnumber,etc.Thesefactorstypicallyleadtohighlyscatteredexperimentaldata,especiallyfor??????
??Cd??,ClandCp,b,whichcanbeobservedclearlyfromTable2.Thepredictionoftherecirculationzonelengthiscriticalforsuchsimulations,however,therewerenoavailableexperimentalmeasurementsof??Lr??forthisparticularReynoldsnumberatall.Ofcourse,thereweresomediscrepanciesbetweenexperimentaldataandthepresentresults.However,oneshouldkeepinmindthoseassumptions,whichwereusedinthepresentsimulations:thelimitedspan-wiselengthofthecomputationaldomain(duetoresourcelimitations)andthelaminarinletboundaryconditions(duetothegridlimitations).Thelastassumptionisnotfeasibleatall.Thespan-wiselengthisthemostimportantand,asdiscussedinSection2,itshouldbeinvestigatedinthefuturework.
FlowTurbulenceCombust(2014)92:673–698Fig.10EffectoftheReynoldsnumberonthespuriousoscillations.25contoursof
p
normalizeddilatationfield??tforRe=3.9×103(a)andRe=2×104(b)andM=0.2obtainedonthe300×300×64grid
691
4.3Compressibleflowresults
4.3.1Reynoldsnumberinfluenceonspuriousoscillations
p
obtainedfortheReynoldsFigure10displaysnormalizedpressuredilationfields??t34
numbersRe=3.9×10andRe=2×10usingtheTKEmodel.TheconventionalCDS-2schemeforconvectivetermapproximationswasusedforbothcases.Forcon-sistency,Fig.11presentsthedistributionoftheroot-meansquarepressurecoefficientforRe=3.9×103.Itcanbeobservedthatthecalculatedlevelofthepressurefluctuationsattheforwardstagnationpointθ=0?isnotzeroandinconsistentwiththeexperimentaldatainspiteoftheinletlaminarboundaryconditions.
4.3.2Influenceofconvectiveschemesonspuriousoscillations
ThecomparisonofthesolutionsobtainedbytheconventionalCDS-2andthefilteredCDS-2schemesaswellastheGammaandLUSTschemesispresentedinFig.12.Onecanclearlysee,thatallexaminedcases(exceptLUST)producesspuriouswavesinthevicinityofthecylinder.ThefilteredCDS-2schemedecreasedoscillationsslightly.TheGammaschemedampedoscillationssignificantly,howevernotentirely,andsomeoscillationswerestillpresentinthedilatationfield.OnlytheLUSTschemeeliminatedoscillationscompletely.Sinceallschemesintroducedsomeamountofdissipation,itisinterestingtoaccesstheirdissipativeproperties.Comparisonoftheone-dimensionalpowerenergyspectrainthewakeofacircularcylinderobtainedbytheCDS-2,GammaandLUSTschemesispresentedinFig.13.TheLUSTscheme
0.08Norberg (2003)
0.060.040.02020406080θ
100120140160180LESFig.11Root-meansquarepressurecoefficientdistributiononthecylindersurface(θ=0isthestagnationpoint)fortheflowoveracircularcylinderatRe=3.9×103andM=0.2(TKErunfrom[24])
692FlowTurbulenceCombust(2014)92:673–698
Fig.12Effectofconvective-termsapproximationonthespuriousoscillationsfortheflowoveracircularcylinderatRe=2×104andM=0.2:CDS-2(a),filteredCDS-2(b),Gamma(c)andLUST
p
(d)schemes.Contoursofnormalizeddilatationfield(??t)from?1:0.08:1obtainedonthe440×440×64grid
andtheGammaschemeprovidedthemostdissipativepowerenergyspectrumcomparedtothepureCDS-2scheme,whichindicatesalimitedapplicationforLES.4.3.3Influenceoflinearalgebrasolvers
TheimpactofthelinearalgebrasolversonthenumericalwavesispresentedinFig.14.TheconventionalCDS-2schemewasusedforbothcases.Itisobservedthatthebothcasessufferedfromartificialwaves.Fromthepresentsimulationsresults,itseemsthatthechoiceoflinearalgebrasolverdoesnothaveamajorimpactontheexistenceoftheartificialoscillationsinthesolution.
102100Evv/U2D10–210–410–610–1–5/3 lawCDS-2LUSTMesh cut–off100f/fvs
101Fig.13Effectofconvective-termsapproximationonone-dimensionalspectraofthetransversevelocityinthewakeofacircularcylinderatRe=2×104andM=0.2atx/D=3
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