J. FluidMech. (2009), vol. 620, pp. 89119.c2009CambridgeUniversityPressdoi:10.1017/S0022112008004904 PrintedintheUnitedKingdom89Steady separated ow past a circular cylinderat low Reynolds numbersSUBHANKARSEN1, SANJ AYMI TTAL2ANDGAUTAMBI SWAS11DepartmentofMechanicalEngineering, IndianInstituteofTechnologyKanpur, Kanpur208016, India2DepartmentofAerospaceEngineering, IndianInstituteofTechnologyKanpur, Kanpur208016, India(Received26March2008andinrevisedform5November2008)Thesteadytwo-dimensional laminarowaroundastationarycircularcylinderhasbeeninvestigatedviaastabilizednite-elementmethod. TheReynoldsnumberReisbased on the cylinder diameter and free-stream speed. The results have been presentedfor6 6Re 640andtheblockagesbetween0.000125and0.80. TheblockageBistheratioofthecylinderdiametertothedomainwidth.ThereislargescatterinthevalueofRes,reportedintheliterature,markingtheonsetoftheowseparation.Fromthepresent studytheResis foundtobe6.29, approximatelyfor B =0.005. Theeectof the blockage onthe characteristic owparameters is foundtobe insignicantfor B 60.01. The bubble length, separationangle andResexhibit non-monotonicvariationwiththe blockage. It is for the rst time that suchabehaviour for theseparationangleandResisbeingreported. Twotypesofboundaryconditionsatthelateral walls havebeenstudied: theslipwall andtowingtank. Ingeneral for highblockage, theresultsfromtheslipboundaryconditionareclosertotheonesfortheunboundedow. Inthatsense, theuseoftheslipboundaryconditionasopposedtothetowingtankboundaryconditiononthelateral walls is advocated. Thebubblelength, separationangle, base suction, total drag, pressure drag, viscous dragandmaximumvorticityonthe cylinder surface for the steadyoware foundtovaryasRe, Re0.5, Re1, Re0.5, Re0.64, Re0.60andRe0.5, respectively. Theextrapolatedresults for the steady ow, for higher Re, are found to match quite well with the otherresultsfromtheliterature.1. IntroductionThe incompressible ow past a stationary cylinder is a classical blu body probleminuidmechanics. Itsenrichedphysicsandreal-lifeapplicationshaveattractedtheattention of the engineers and scientists for over a century, leading to many theoreticalandexperimentalinvestigations. Despiteitssimplegeometry, theowpastacircularcylinder is consideredtobe abaseline case of more complexows (Zdravkovich1997).Tothis day, thereis noagreement ontheexact valueof thelaminar separationReynoldsnumber, Res, forthesteadyunboundedow. ThedependenceofRes, forconnedow, ontheboundaryconditionsandblockagehasreceivedlittleattentionin the past. The blockage,B, due to the cylinder is dened as the ratio of the cylinderEmailaddressforcorrespondence:smittal@iitk.ac.in90 S. Sen, S. Mittal andG. BiswasEE = Forward stagnation pointF = Rear stagnation point or base pointG = Wake stagnation pointF GPsQP, Q = Separation pointsUD L = 0 = 0 = 0x2x1Figure 1. Schematicrepresentationofthecylinderandseparationbubblecross-section.diameterDtothewidthHoftheexperimentalapparatusorcomputationaldomain.Inthepresentworkweconductasystematicnumerical investigationtoaddressthedependence of Resand numerical treatment of the boundary conditions on the lateralwalls. Recentnumericalinvestigationsinvolvingtheblockageeectontheoweldof acircular cylinder at lowRe include Kumar &Mittal (2006a,b) andPrasanthet al. (2006). Kumar &Mittal (2006a,b) investigatedtheeect of theblockageonthevortexsheddingfrequencyandthecritical ReynoldsnumberRecleadingtothevortexshedding. Prasanthetal.(2006)studiedtheblockage eectontheow eldofafreelyvibratingcylinderatlowRe.Beforeproceedingtothecurrentnumerical work, wedescribethestructureoftheseparationbubble or standingvortexpair andpresent anoutline of the previousexperimental, numerical, analytical andsemi-analytical investigationsinvolvingthisowbydierentresearchers. Aschematicofthesteadyseparationbubbleisshowningure1. Theowisfromlefttoright. ThekeygeometricalparametersofinterestthatcharacterizetheseparationbubblearethebubblelengthLandseparationangles(expressedindegrees). Theseparationanglesismeasuredinacounterclockwisedirectionfromtherear stagnationpoint or basepoint F. Thebubblelength(alsocallededdylength) is denedas the distance measuredalongthe wake centrelinebetweenthebasepoint, F, andthewakestagnationpoint G. Thetwosymmetric,counter-rotatingrecirculationzones remainstablyattachedtothe cylinder till theinitiationofthevortexshedding.Nisi &Porter (1923) conducted a pioneering investigation to determine theseparationReynoldsnumberandfoundRes =3.2. Theirexperimental investigationswerebasedonsmokevisualization. Usingoil astheworkinguid, Homann(1936)studied the ow around the cylinders and provided ow visualization picturesbetween Re =1.95 and Re =140.5. He estimated Res =6 for B =0.067. Taneda(1956) conductedexperiments inatowingtank. Fromtheplot of theeddylengthwithRe, hedeterminedRes =5byextrapolation. Groveet al. (1964) andAcrivoset al. (1965, 1968) conducted a series of oil tunnel experiments for the steadySteadyseparatedowpastacircularcylinder 91separatedow. For the constant blockage, these experiments consistently showedlinear variation of the bubble length with Re even in the presence of a splitterplate. Ashorter bubblewiththewall proximitywas alsonoted. Nishioka&Sato(1974) conducted wind tunnel experiments and determined the detailed structureof the wake velocity distribution for Re in the range of 1080. Coutanceau &Bouard (1977) employed owvisualization to explore the closed wake structureanditsevolutionwithRe for5 0.70), onlythecentral, leftandrightblocksconstitutethemeshes;theupperandlowerblocksaredropped.6. Validation of the method and convergence of the results6.1. ComparisonwithearlierresultsThe nite-element formulation and its implementation has been validated bycomparingthe pressure distributiononthe surface of the cylinder at Re =15, 30,36and40andthetotal dragcoecient Cdat Re =40at verylowaswell aslargeblockages with those reported by the earlier studies. The pressure coecient CpSteadyseparatedowpastacircularcylinder 971.51.00.500.51.01.52.00 30 60 90 120 150 1801.51.00.500.51.01.52.00 30 60 90 120 150 1801.51.00.500.51.01.52.00 30 60 90 120 150 180 0 30 60 90 120 150 180CpPresent, B = 0.015Hamielec & Raal (1969), B = 0.015Present, B = 0.015Hamielec & Raal (1969), B = 0.015Present, B = 0.05Homann (1936), B = 0.051.51.00.500.51.01.52.0 (deg.) (deg.)Present, B = 0.076Thom (1933), B = 0.076Present, B = 0.05Grove et al. (1964), B = 0.05(a) (b)Cp(c) (d)UURe = 15 Re = 30Re = 40Re = 36UUFigure 4. Steadyowpast acircularcylinder: comparisonof predictedCpwithdatafromliterature. (a) Cp plotatRe =15;(b) Cp plotatRe =30;(c) Cp plotatRe =36;(d)CpplotatRe =40.computedwiththeslipboundaryconditionis showningure4. As expected, theforwardstagnationpoint is the maximumpressure point, andthe correspondingpressure coecient Cp0is greater thanunity inall cases. For constant blockage,Cp0decreaseswithincreasingReandapproachesunity, whilethepointofminimumpressuremovesupstream,andtheminimumpressureincreases.Thiscanbeobservedfromgure4(a,b)forB =0.015.Figure 4(a) compares the calculated Cpat Re =15 and B =0.015 with thoseobtained by Hamielec & Raal (1969) at the same Reynolds number and blockage. Thecomparisonrevealsexcellentagreement. Figure4(b)demonstratesthatthepredictedsurfacepressuredistributionat Re =30is veryclosetotheresults of Hamielec&Raal (1969) forB =0.015 and Homann (1936) forB =0.05. Close agreement betweenthe present results with those of Thom (1933) at Re =36 is seen from gure 4(c). ThepredictedvaluesofCpatRe =40arecomparedwiththeexperimentaldataofGroveet al. (1964) asshowningure4(d). Asatisfactorymatchbetweenthepresent andearlierresultsisobserved.At Re =40 and B =0.000125, the present calculations result in Cpb =0.4782.Following an articial stabilization of the wake by a splitter plate, Grove et al.(1964)obtainedCpb =0.45forthesteadyowcondition. Theyconcludedthatthisvalueremainsconstantfor25 6Re 6177.Fornberg(1980)estimatedCpb =0.46forRe =40. The recent study by Posdziech & Grundmann (2007) reported Cpb =0.4736at Re =40 and B =0.000125. The values ofCdobtained from the present simulationsat B =0.000125arecomparedwiththeexperimental andnumerical datafromtheliterature at Re =40intable 2. For this value of lowblockage, the Cdpredictedusing,both,theslipandtowingtankboundaryconditionsisthesame.Thepredicted98 S. Sen, S. Mittal andG. Biswas11010010000.01 0.10 1.00CdBPresentSahin & Owens (2004)1.41.61.80.01 0.10Figure 5. SteadyowatRe =40pastacircularcylinderwithno-slipsidewallsandparabolicinow:variationofCdwithblockage. ThenumericalresultsfromSahin&Owens(2004)arealsoshown. TheCdaxisintheinsetisinthelinearscale.Studies B CdKawaguti(1953) 1.618Tritton(1959) 1.58Thoman&Szewczyk(1969) 1.572Takami&Keller(1969) 0.059 1.5359Son&Hanratty(1969) 1.51Dennis&Chang(1970) 1.522Fornberg(1980) 1.4980Henderson(1995) 0.0178 1.54Posdziech&Grundmann(2007) 0.000125 1.4942Present 0.000125 1.5093Table2. SteadyowatRe =40pastacircularcylinder:Cdfromthepresentstudyandtheearliereorts.Cdliesintherangeof thevaluesreportedbytheearliereorts. It isinverygoodagreement withthemore recent valuesreportedby Fornberg (1980)andPosdziech &Grundmann(2007).For large blockages (0.10 6B 60.90), Cdobtainedfromthe present simulationsusingtheno-slipsidewallsandthosereportedbySahin&Owens(2004)atRe =40are comparedingure 5. Excellent agreement is observedfor the entire range ofblockage. As is shown in the inset, the present results predict a non-monotonicvariationofCdbetweenB =0.005andB =0.10.6.2. StreamwiseextentofthedomainandmeshconvergenceTo study the inuence of the location of the upstream and downstream boundaries onthecharacteristicsoftheow, computationsarecarriedoutwithtwomeshesL1andL2withdierentstreamwiseextents. TheblockageforboththemeshesisB =0.01,andthespatial resolutionforthetwocasesiskept identical.Thedetailsof thetwoSteadyseparatedowpastacircularcylinder 99Mesh LuLdNodes ElementsL1 50D 100D 93 488 92 800L2 100D 200D 121 780 120 976Table 3. Steady owpast a circular cylinder: details of the various meshes employedtostudytheeect of thestreamwiselocationof theinowandoutowboundaries. Theotherparametersforthemeshesarehr1= 0.001, Nt= 368andB= 0.01.L sCdCp0CpbMesh Re= 7 40 7 40 7 40 7 40 7 40 ResL1 0.0926 4.4995 14.62 53.63 3.3523 1.5121 1.6901 1.1626 0.7843 0.4758 6.28L2 0.0909 4.4935 14.49 53.61 3.3428 1.5093 1.6835 1.1591 0.7832 0.4758 6.29Table4. Steadyowpastacircularcylinder:eectofthestreamwiselocationoftheoutowandinowboundaries onthecharacteristics of theow. Theresults correspondtotheslipboundaryconditionandB= 0.01.Mesh Nodes Elements hr1NtM1 47 498 47 000 0.01 236M2 93 488 92 800 0.001 368M3 184 416 183 424 0.001 564Table5. Steadyowpastacircularcylinder:descriptionofthevariousmeshesemployedforthegridconvergencestudy. TheotherparametersforthemeshesareLu=50D, Ld=100DandB= 0.01.meshes are given in table 3. The streamwise location of the upstream and downstreamboundariesofmeshL2istwicethatofthoseofL1. TheresultsforvariousRewiththetwomeshesaregivenintable4. ThedierenceintheLandRespredictedbythetwomeshes are1.87 %and0.16 %, respectively. Theresults fromthetwosetsof computations for the other parameters are verysimilar. This demonstrates theadequacyof thestreamwiseextent of meshL1. Therefore, thecomputationsinthispaperarecarriedoutforLu =50DandLd =100D.Tocheckforthegridindependence, severalmesheswithincreasingrenementareutilizedtocompute the Re =7and40ows for B =0.01blockage, usingthe slipboundarycondition. These meshes have astreamwise extent equal tothat of L1.Startingfromacoarsemesh, thesubsequentnemeshesareconstructedbyuniformglobal renement. Out of the many meshes studied, table 5 lists the various parametersformeshesM1,M2andM3.ThespatialresolutionincreasesfromM1toM3sothatthe number of nodes almost doubles in each successive mesh. Mesh M2 is the same asmeshL1describedinthepreviousstudy. Theresultsofthemeshconvergencestudyfor M1throughM3arepresentedintable6. It is foundthat theowparametersforM2andM3arevirtuallyidentical.Basedonthegridindependencetestshownintable6, meshM2with93 488nodesand92 800elementsischosenasthereferencemeshcorrespondingtoB =0.01.The computations involving dierent blockages, B 60.20, employ meshes that dieronthenumberof elementsonlyintheupperandlowerblocks(see , 5). Thetotalnumbers of nodes andelements inthecentral, left andright blocks haveconstant100 S. Sen, S. Mittal andG. BiswasL sCdCp0CpbMesh Re= 7 40 7 40 7 40 7 40 7 40 ResM1 0.0924 4.4999 13.37 53.12 3.3511 1.5100 1.6922 1.1642 0.7841 0.4748 6.32M2 0.0926 4.4995 14.62 53.63 3.3523 1.5121 1.6901 1.1626 0.7843 0.4758 6.28M3 0.0929 4.5025 14.64 53.64 3.3519 1.5120 1.6892 1.1623 0.7847 0.4758 6.28Table6. SteadyowpastacircularcylinderatB= 0.01, usingtheslipboundarycondition:owparametersforthethreemeshesforgridconvergencestudy.valuesof 68 768and68 160, respectively. Thecentral blockisasquareof edge4Dandcontains29 808nodesand29 440elements.For0.30 6B 60.80thecentralblockis reducedtoasquare of edge length1.25D. It contains 24 656nodes and24 288elements, whilethesumsofnodesandelementsinthecentral, leftandrightblocksequal 91 302 and 90 512, respectively. For B =0.90 the edge length of the central blockfurtherreducesto1.11Dandconsistsof 23 184nodesand22 816elements. All themesheshavebeendesignedtomaintainthesamemeshdensityforall theblockagesasfaraspossible.7. ResultsThe keynon-dimensional parameters governingthe steadylaminar owaroundastationarycircular cylinder are the Reynolds number andblockage. The resultsarepresentedfor thecomputations for 6 6Re 640andawiderangeof blockage(0.000125 6B 60.80)withboththeslipandtowingtankboundaryconditions. TheRe =40 and0.005 6B 60.90 results for the bubble parameters using the no-slipsidewallsarealsoshown. ItisfoundthattheresultsforB =0.005aredevoidoftheblockageeectandrepresentsolutionsfortheunboundedow.The element level matrixandvector entries have beencomputedbyemployingthe 2points2points GaussLegendre quadrature formula. All the computationsare carriedout onacomputer withAMDOpteronprocessor withfour CPUs of2.4GHz clockspeed. Double-precisionarithmetic is usedfor all the calculations.The Re =40 owcomputed on a mesh with 93 488 nodes and 92 800 elements(B =0.01) requires 582 MBof RAMand, approximately, 4 hours of CPUtimewhenadirectsolverisutilizedtosolvethealgebraicequationsystemresultingfromthe nite element discretization. Incontrast, withthe matrix-free implementationof the generalizedminimal residual (GMRES) methodof Saad&Schultz (1986),the memoryrequirement reduces toamere 34MB, andthe solutiontime is 1.24hours. ThedimensionoftheKrylovsubspaceis30, andadiagonalpreconditionerisemployedtoacceleratetheconvergencerateofGMRES.7.1. DevelopmentofthesteadyseparationbubblewithReIn order to demonstrate the growth of the steady separation bubble with Re, contoursofthestreamwisecomponentofthevelocity(for6.3 6Re 66.5)andstreamfunction(for6.3 6Re 640)inthewakearepresentedingure6. Theresultsintheseguresare for B =0.01andthe slipboundarycondition. The calculations showthat theseparationbubble rst appears at Re =6.28. At Re =6.3, the bubble lengthis ofthe order of the element thickness onthe cylinder surface inthe radial direction.As expected, thecontours of aresymmetricabout the x1-axis. Theenlargementof theseparationbubblewithincreasingRe isclearlyseenfromtheseimages. TheSteadyseparatedowpastacircularcylinder 101(a)(c)(b)Re = 6.3 Re = 6.4 Re = 6.5Re = 8 Re = 15Re = 40 Re = 30Re = 6.3 Re = 6.4 Re = 6.5Figure 6. Steady owpast a circular cylinder at B =0.01 and using the slip boundarycondition:(a)Re =6.36.5, u1contours,(b)Re =6.36.5, contoursand(c)Re =8 40, contours, showingthedevelopmentoftheseparationbubblewithRe.(a)(b) (c)Re = 40, slip BC Re = 40, towing tank BCRe = 40, noslip BCFigure 7. SteadyowatRe =40pastacircularcylinderatB =0.80, usingthe(a)no-slip,(b)slipand(c)towingtankboundaryconditions:contoursoffortheseparationbubble.separationbubbleforlargeblockage(B =0.80)isshowningure7fortheRe =40owwithvariousboundaryconditions. Theverysignicant eect of theboundaryconditionsonthelengthoftheeddiesforhighblockageisevidentfromthisgure.7.2. VariationofLandswithReFigure 8 shows the variation of the bubble length with ReforB =0.04, 0.11 and 0.20.The results have been computed with the towing tank boundary condition. The valueofLisestimatedbylocatingthepointalongthewakecentrelineatwhichu1goestozero. Forconstant blockage, it isobservedthat thebubbleelongatesapproximately102 S. Sen, S. Mittal andG. Biswas0123455 10 15 20 25 30 35 40LRePresent, B = 0.04Present, B = 0.11Present, B = 0.20TA, B = 0.03GR, B = 0.10GR, B = 0.20AC, B = 0.025AC, B = 0.05T-K, B = 0.059C-B, B = 0.024C-B, B = 0.07Figure 8. Steadyowpastacircularcylinder, usingthetowingtankboundaryconditionforRe =6.540:comparisonofthebubblelength, L, withtheexperimental andnumerical datafromtheliterature. TheabbreviationsusedareTA, Taneda(1956);GR, Groveetal. (1964);AC, Acrivosetal. (1968);T-K, Takami&Keller(1969);C-B, Coutanceau&Bouard(1977).linearly with Re. This is in agreement with the earlier observations. The best linear t,usingtheleastsquaresapproximation,isalsoshowninthegure.ThebestlineartsforB =0.000125, 0.005, 0.01and0.04overlap, andforclarityonlytheB =0.04caseis shown. The linearity ofL with Reis utilized in this work to determine Resfor eachvalue of blockage (see , 7.4).Except atRe =40, the predicted eddy length atB =0.04exhibits satisfactory agreement with those from Takami & Keller (1969) atB =0.059.The datafromTaneda(1956) for B =0.03, Acrivos et al. (1968) for B =0.05andCoutanceau&Bouard(1977) for B =0.024and0.07liewithinour dataspectrumfortherangeof B =0.040.20. ForB =0.025, Acrivoset al. (1968)predictedlargerbubble comparedtothe present predictions. The eddylengths fromthe studybyGroveetal. (1964)forB =0.10and0.20arequitedierentfromtheonespredictedbyeveryoneelse. Itisfoundthatboththeslipandtowingtankboundaryconditionsfor0.000125 6B 60.80leadtolinearvariationoftheeddylengthwithRe.The slope of the LReproles obtained by using the slip and towing tank boundaryconditionsarelistedintable7. It isseenthat boththeboundaryconditionsresultinnearlythesameslopefor0.000125 6B 60.04. Theeectoftheblockagebecomesimportant for B >0.04. Thedierenceintheslopeobtainedbyusingtheslipandtowing tankboundary conditions increases withincreasing blockage. Also, a keyobservation is that using the slip boundary condition the slope decreases up toB =0.30 and then increases. In contrast, the towing tank boundary condition predictsmonotonicdecreaseoftheslopewithincreasingB.The least squares t for the LRe relationshipprovides the followingempiricalequationfor the predictionof the bubble lengthconcerning the unboundedow(B =0.005):L = 0.847 + 0.1336Re forRes 20arealsoveryclosetothoseof Coutanceau&Bouard(1977) attheblockageof 0.07. Afewdiscrepancieswiththedatapointsof Thom(1933) forB =0.10, Homann (1936) for B =0.10 and Coutanceau & Bouard (1977) for B =0.12areobserved.Wuet al. (2004)proposedthefollowingempirical equationforthedeterminationofs:s= 78.5 155.2Re0.5for10 6Re 6200. (7.3)Figure 10 shows the variation ofswith Re0.5for 10 6Re 640 and 0.005 6B 60.20forthepresentdata. Itisobservedthatforbothtypesofboundaryconditions, theseprolesarelinearforallblockages. Further, theyarevirtuallyidenticalforB 60.11.Theempirical equationforsfortheunboundedowobtainedbytheleastsquarescurvetiss= 77.66 152.65Re0.5for10 6 Re6 40. (7.4)104 S. Sen, S. Mittal andG. Biswas15253545555 15 25 35 45s (deg.)RePresent, B = 0.04Present, B = 0.11Present, B = 0.20Thom (1933), B = 0.10Homann (1936), B = 0.10Takami & Keller (1969), B = 0.059Coutanceau & Bouard (1977), B = 0.07Coutanceau & Bouard (1977), B = 0.12 Wu et al. (2004)Figure 9. Steadyowpastacircularcylinder, usingthetowingtankboundaryconditionforRe =6.540: comparisonof theseparationangle, s, withtheexperimental andnumericaldatafromtheliterature.253545550.14 0.18 0.22 0.26 0.30 0.34s (deg.)Re0.5253545550.14 0.18 0.22 0.26 0.30 0.34Re0.5B = 0.0050.010.040.070.110.150.20(a) (b)Figure 10. Steadyowpastacircularcylinder:variationofswithRe0.5for10 6Re 640and0.005 6B 60.20forthe(a)slipand(b)towingtankboundaryconditions.The associated r.m.s. error for this t is 0.04. This equation is in very good agreementwith(7.3)proposedbyWuetal. (2004).The LRe andsRe curves for the unboundedow(B =0), obtainedvia theleast squares curve t, are showningure 11. The data points for the curve taredeterminedviaRichardsons extrapolationof order two. Thecomputedresultsfor B =0.005and0.01areutilizedfor this purpose. Theempirical equations thusobtainedaregivenbyL = 0.848 + 0.1336Re forRes< Re6 40, (7.5)s= 77.66 152.70Re0.5for10 6 Re6 40 (7.6)Steadyseparatedowpastacircularcylinder 1050123455 10 15 20 25 30 35 40LReExtrapolatedEquation (7.5)ExtrapolatedEquation (7.6)2535455510 15 20 25 30 35 40Re(a) (b)s (deg.)Figure 11. Steadyowpastacircularcylinder:(a)LReand(b)sRecurvesfortheunboundedow(B =0)obtainedbyttingtheextrapolateddata.01.02.03.04.05.05 10 15 20 25 30 35 40LReB = 0.0050.010.110.200.300.400.500.600.700.8001020304050605 10 15 2025303540Re(a) (b)s (deg.)Figure 12. Steadyowpastacircularcylinder, usingthetowingtankboundaryconditionfor0.005 6B 60.80:variationofthe(a)bubblelengthand(b)separationanglewithRe.These equations are verysimilar to(7.1) and(7.4) obtainedviacurve t throughthecomputedresults. ThevariationofLandswithRe, fortherangeofblockagestudiedinthiseort, isshowningure12.7.3. EectoftheblockageandboundaryconditionsonLandsThe variation of L and swith the blockage has been investigated for 0.005 6B 60.80forboththeslipandtowingtankboundaryconditions. Theresultsarepresentedingure13forRe =7,20,30and40.ForB 60.01, Landsarefreefromtheblockageeect. BetweenB =0.01andB =0.15, anon-monotonicvariationof LandswithBisobserved. Werefertothisbehaviourastherstnon-monotonicvariation. Thevalues ofL andsincrease with the blockage for small values ofBand then decrease.The blockage atwhichL andsachieve the maximumvalues isa function of Reandtheboundarycondition. Werefertothisvalueasthecritical blockage. Forinstanceat Re =7 the maximum is achieved at B =0.07 and 0.11 with the towing tank and slipboundaryconditions, respectively. As Re increases, thecritical blockagedecreases.Fornberg(1991)investigatedthesteadywakeforowpastacylinderforRe 6800.Eventhoughit was not explicitlypointedout, careful observationof gure 11inhis paper shows non-monotonicvariationof thebubblelengthwiththeblockage.For0.01 6B 60.04, theslipandtowingtankboundaryconditionspredict identicalincreaseofLandsforallReconsidered. ForallblockagesbeyondB =0.04, Landsobtainedwiththeslipboundaryconditionarehigher thanthoseobtainedwiththetowingtankboundarycondition. Ingeneral, forlargeBcomparedtothetowingtankboundarycondition, slipboundaryconditionresults invalues that areclosertotheones for theunboundedow(for B =0.005). For B >0.15, thetowingtank106 S. Sen, S. Mittal andG. Biswas0.060.120.180.240 0.2 0.4 0.6 0.8LPresent, slip BCPresent, towing tank BCTakami & Keller (1969)7111519230 0.2 0.4 0.6 0.80.51.52.53.54.536404448525644485256601.02.03.04.05.06.0B B1.02.03.0LLLPresent, slip BCPresent, towing tank BCTakami & Keller (1969)Coutanceau & Bouard (1977)20304050Re = 7 Re = 7Re = 30 Re = 30Re = 40 Re = 40Re = 20Re = 200 0.2 0.4 0.6 0.80 0.2 0.4 0.6 0.80 0.2 0.4 0.6 0.80 0.2 0.4 0.6 0.80 0.2 0.4 0.6 0.80 0.2 0.4 0.6 0.8s (deg.)s (deg.)s (deg.)s (deg.)Figure 13. Steady ow past a circular cylinder for 7 6Re 640 and 0.005 6B 60.80: variationof L and swith B. Also shown are the results from the earlier studies. BC: boundary condition.boundaryconditionpredicts monotonicdecreaseof Landsfor all Re. However,theslipboundaryconditionleadstoamuchmorecomplexvariationthat dependsonRe. For Re >30andB >0.30, Landsare foundtoincrease withBleadingtoasecondregime of non-monotonicity. At verylowRe, sayRe =7, the secondregime of non-monotonicity occurs at relatively lowblockage. Also, a regime ofthirdnon-monotonicityisobservedforlargeblockageinthevariationofeddylengthwithB.Steadyseparatedowpastacircularcylinder 107123450 0.2 0.4 0.6 0.8 1.02030405060LBLss (deg.)Figure 14. Re =40steadyowpastacircularcylinderfor0.005 6B 60.90usingtheno-slipsidewalls:variationofLandswithB.15101520250 0.3 0.6 0.9 1.2 1.5 1.8x2u1u1Slip BC, B = 0.04Towing tank BC, B = 0.04Slip BC, B = 0.20Towing tank BC, B = 0.20Slip BC, B = 0.30Towing tank BC, B = 0.300.51.52.53.50 2 4 6 80.4, S0.4, T0.5, S0.5, T0.6, S0.6, T0.7, S0.8, S0.7, T0.8, T(a) (b)B = 0.3, TB = 0.3, SFigure 15. Steadyowat Re =20past acircular cylinder: variationof the x1componentof velocity with x2at x1 =0 for the slip and towing tank boundary conditions for (a)0.04 6B 60.30and(b)0.30 6B 60.80.Thesurfaceofthecylinderisshowninsolidline.In(b),SandTdenotetheslipandtowingtankboundaryconditions, respectively.BC:boundarycondition.At Re =20 and 30, a satisfactory match is found with the results of Takami & Keller(1969)forB 0.20,the presence of the lateral walls leads toanoverall accelerationof the ow. The108 S. Sen, S. Mittal andG. BiswasB = 0.04B = 0.20B = 0.50(a) (b)B = 0.80B = 0.50B = 0.20B = 0.04B = 0.80Figure 16. SteadyowatRe =20pastacircularcylinder:vorticityeldsobtainedwiththe(a)slipand(b)towingtankboundaryconditionsfor0.04 6B 60.80.Thebrokenlinesdenotenegativewhilethesolidlinesrepresentpositivevaluesofthevorticity. ForB =0.50and0.80,thehorizontallinesnearthecylinderrepresentthesidewalls.specication of the free-streamspeed at the lateral walls for owin the towingtankboundaryconditionleadstoanadditional owacceleration. Forlowblockage(B =0.04), theslipandtowingtankboundaryconditionsleadtovirtuallythesamevelocityproles. However, for B>0.20thetowingtankboundaryconditionleadstosignicantlylargeraccelerationasseeningure15. Thisresultsindelayedowseparationforthetowingtankboundaryconditionandthereforeashorterbubble.To further investigate the eect of the blockage and boundary conditions, we studythevorticityeldsoftheRe =20owforcertainvaluesofB. Thevorticityeldsfortheslipandtowingtankboundaryconditionsareshowningure16, whilegure17showsthevorticitydistributiononthecylindersurface. Forlowblockage(B =0.04)the vorticity elds obtained with both the boundary conditions are virtually identical.However, for B> 0.20 the vorticity is generated at the sidewalls due to the impositionof the free-streamspeed for the towing tank boundary condition (gure 16b).Thestrengthofthewall-generatedvorticityincreaseswithincreasingblockage. Thisinterfereswiththeadvectionofthevorticitygeneratedonthesurfaceofthecylinderand leads to the delay of owseparation. This is reected by the downstreammovement of the zero-vorticity point as seen from the inset of gure 17(b). In contrast,the slip boundary condition results in very similar vorticity elds for B =0.04 and 0.20(gure16a). ForB>0.50, theshorteningof thestreamwiseextent of thevorticitycontoursisassociatedwiththeupstreammovementofthezero-vorticitypoint. TheSteadyseparatedowpastacircularcylinder 109100204060800 30 60 90 120 150 180 210B = 0.040.200.300.400.500.600.700.8010035701051300 30 60 90 120 150 180 2100.200.2130 135 1400.200.2135 145 155 160(a) (b)UUFigure 17. Steady owat Re =20 past a circular cylinder: vorticity distributionon the cylindersurfacefor0.04 6B 60.80, usingthe(a)slipand(b)towingtankboundaryconditions.90961021080 0.20 0.40 0.60 0.8090100110120 (a) (b)0 0.20 0.40 0.60 0.80B BSlip BCTowing tank BCUURe = 40Re = 20 (deg.)Figure 18. Steadyowpastacircularcylinderfor0.005 6B 60.80:variationwithBofthelocationat whichCp/ =0.05isachievedfor(a)Re =20and(b) Re =40. BC: boundarycondition.reversalofthemovementofthezero-vorticitypointforB> 0.50isclearlyseenfromtheinsetofgure17(a). Thelocationofthezero-vorticitypointisameasureoftheseparationangle(see , 7.2).The adverse pressure gradient plays a major role in the owseparation. Toinvestigate the possible cause of the non-monotonic variation of Land swiththeblockage, westudytheadversepressuregradientonthesurfaceofthecylinder.Figure 18 shows, for various blockages, the location on the cylinder surface atwhichapressuregradientcorrespondingtoCp/ =0.05occur. TworepresentativeReynolds numbers are chosen: Re =20 and40. For bothRe it is seenthat thelocations corresponding to Cp/ =0.05 shownon-monotonic variation with B.This, therefore, leadstonon-monotonicvariationofLandswiththeblockage.7.4. ResfortheconnedandunboundedowIt was shownin , 7.2(gure 8) that the eddylengthvaries linearlywithRe. Weutilize thisvariationtoestimate Resforeach blockage. Figure 19shows the variationof ReswithH/D( =1/B) fortheslipaswell astowingtankboundaryconditions.Thevariationof ReswithH/Dfor thetwoboundaryconditions is quitedierentfor H/D0.04. While boththe boundaryconditions result inanon-monotonicvariationofReswithH/D, thetowingtankboundaryconditionpredictshigher values of Res. This can be clearly seen in gure 19. The present results with thetowing tank boundary condition predict a decrease in Reswith H/Dfor H/D 25(B 60.04). TheblockageeectbecomesnegligibleforB 60.01.Figure19canbeutilizedtoestimateResfortheunboundedow. ItisfoundtobeRes6.29.RichardsonsextrapolationofordertwoandthevaluesofResatB =0.005and0.01are utilized to predict Resfor the unbounded ow (B =0). This method also results inRes6.29. Fromtable1itisseenthatthisvalueisclosetothepredictionofDennis&Chang(1970). Anotherobservationthat canbemadefromgure19isthat forhighblockage(H/D40andcomparedwiththeempirical equationsandnumericaldatafromtheliterature.Figure25comparesthebubblelengthLextrapolatedfrom(7.1)withtheformulaproposedbySobey(2000), givenby(7.2), andthenumericaldatafromtheliteraturefor 20 6Re 6400. It is found that the relation proposed by Sobey (2000) underpredictsthebubblelengthfor theRe rangeconsidered. For Re 6100, thepredictedresultsmatchwellwiththeresultsofDennis&Chang(1970). UptoRe =300, Lpredictedbythe proposedequationdisplays satisfactoryagreement withthose predictedbySteadyseparatedowpastacircularcylinder 1150.20.61.01.41.82.22.60 100 200 300 400 500 600CdRePresent, equation (7.8)Son & Hanratty (1969)Dennis & Chang (1970)Fornberg (1980)Smith (1981), equation (7.7)Ghia et al. (1986)Fornberg (1991)Gajjar & Azzam (2004), B = 0.02Figure 26. Steady owpast a circular cylinder: comparisonof Cdextrapolatedfromtheproposed equation (7.8) and numerical data from the literature for 20 6Re 6600. Also plottedistheformulaproposedbySmith(1981).Fornberg(1980, 1991) andGajjar &Azzam(2004). At Re =400, Lpredictedby(7.1) is 9.62 %smaller thanthat predictedbyFornberg(1991) and7.07 %shortercomparedtothepredictionofGajjar&Azzam(2004).Ingure26, thevariationof CdwithRepredictedby(7.8)iscomparedwiththeformulaproposedbySmith(1981) andthenumerical datafromtheliteratureforthesteadyowfor 20 6Re 6600. ForRe 6100, thepredictedresultsdisplaycloseagreement withthoseof Son&Hanratty(1969) andDennis&Chang(1970). ForRe 6400, Cdpredictedbytheproposedequation(7.8)comparesfavorablywiththedatafromFornberg(1980, 1991), Ghiaetal. (1986)andGajjar&Azzam(2004). Ingeneral, thepredictionswiththepresentequationarebetterthan(7.7)proposedbySmith(1981).The pressure and viscous drag coecients extrapolated from the proposed equations(7.9) and (7.10), respectively, are presented in gure 27 and also compared with variousnumerical resultsfor6 6Re 6100. ItisobservedthatwithintheRerange, CdpandCdvareof thesame order of magnitudewithCdpexceedingCdv. Thecomparisonreveals a satisfactory agreement between the predicted results and the results ofDennis&Chang(1970), Henderson(1995)andPosdziech&Grundmann(2007).The maximumvorticityonthe cylinder surface extrapolatedfromthe proposedrelationgivenby(7.12) for 7 6Re 6400is comparedwithvarious numerical datafromthe literature as showningure 28. For Re 6100, maxpredictedby(7.12)matchwell withtheresultsofDennis&Chang(1970). ForRe 6200, thepredictedmaxcompares favorablywiththe results of Fornberg(1980, 1991) andGajjar &Azzam(2004).8. ConclusionsAstabilizednite-element methodhas beenemployedtoinvestigate the steady,laminarowaroundastationarycircularcylinder. Twosetsofboundaryconditions,namelytheslipandtowingtank, havebeenstudied. Resultshavebeenpresentedfor116 S. Sen, S. Mittal andG. Biswas1.02.03.04.00 10 20 30 40 50 60 70 80 90 100Cdp, CdvRePresent, Cdp, equation (7.9)Present, Cdv, equation (7.10)Dennis & Chang (1970), CdpDennis & Chang (1970),CdvHenderson (1995),Cdp, B = 0.0178Henderson (1995),Cdv, B = 0.0178Posdziech & Grundmann (2007),Cdp, B = 0.000125Posdziech & Grundmann (2007),Cdv, B = 0.000125Figure 27. Steady ow past a circular cylinder: comparison of Cdpand Cdvextrapolated fromtheproposedequations(7.9) and(7.10), respectively, andnumerical datafromtheliteraturefor6 6Re 6100.26101418220 100 200 300 400RePresent, equation (7.12)Dennis & Chang (1970)Fornberg (1980)Fornberg (1991)Gajjar & Azzam (2004), B = 0.02 maxFigure 28. Steadyowpastacircularcylinder:comparisonofmaxextrapolatedfromtheproposedequation(7.12)andnumericaldatafromtheliteraturefor7 6Re 6400.6 6Re 640and0.000125 6B 60.80. TheresultsforB =0.005closelyrepresenttheunboundedow.Resfrom the earlier studies varies between 3.2 and 7. The critical Reynolds numberfor the onset of the ow separation has been found to be Res6.29 for the unboundedow. The separation initiates from the base point, i.e.s =0, at Re =Res. The bubblelengthisfoundtovaryapproximatelylinearlywithRe. Irrespectiveoftheboundaryconditionsandblockage, thesteadyowdisplayssymmetryaboutthex1-axis.For the rst time, it is foundthat sandResdisplaynon-monotonic variationwiththeblockage. Althoughitwasnotexplicitlystated, non-monotonicvariationofLwithBcanbeobservedfromgure11inthepaper byFornberg(1991). Usingthetowingtankboundarycondition, Landsincreasewiththeblockageforsmallvalues of B. Beyonda critical blockage, a decrease inboththe parameters withincreasingBisobserved. Incontrast, thevariationofLandswithBisassociatedSteadyseparatedowpastacircularcylinder 117withmultipleregimesofnon-monotonicitywhentheslipboundaryconditionisused.Thenon-monotonicvariationof LandswithBarisesduetothenon-monotonicmovementoftheseparationpointwiththeblockage. ForB 60.01, Landsarefreefrom the blockage eect. For high blockage, compared to the slip boundary condition,the towing tank condition leads to signicantly larger acceleration of the ow near thecylinder.Also,thevorticitygeneratedatthesidewallswiththetowingtankboundarycondition interferes with the advection of the vorticity generated on the surface of thecylinder. Theoutcomeisadelayedowseparationand, therefore, ashorterbubblewith the towing tank boundary condition. For large blockage, compared to the towingtankboundary condition, the resultspredicted by the slip boundary condition forthebubbledimensionsareclosertotheonesfortheunboundedow. ForB 60.04, boththeboundaryconditionsyieldverycomparableresults.The values of Resfromthe present computations, for each blockage, exhibitclose agreement withthe value obtainedviathe criterionproposedbySrinivasan(2006). Also, excellent agreement is foundwiththepredictions of Chen(2000) for0.10 6B 60.70, using the towing tank boundary condition. For B 60.01, Resisinsensitive to the blockage. In general, Resobtained with the slip boundary conditionare closer to the ones for the unbounded ow than those obtained with the towing tankboundary condition. Therefore, the use of the slip boundary condition is recommendedforcomputationofowsifoneisforcedtousehighblockage(B >0.17)topredicttheunboundedowcharacteristics.The variation of various owparameters with Re has been studied for theunbounded ow, and empirical relations via curve t have been proposed. The bubblelength varies as L=0.847+0.1336 Re for Res