The precise calculation of drag forces in the technical application of automatic balancing of rotating machinery provides important information about efficiency and stability. With in- creasing geometric complexity of the design, this poses a challenge that can be solved with computer-aided fluid dynamic approaches. In rolling element bearings, the influence of drag induced by the lubricant is predominantly considered in the context of efficiency loss estima- tions,whereasthemovementoftherollingelementsismainlyconstrainedbythecontactwith thebearingringsand,ifpresent,thecage.Automaticballbalancers,whichareinstalledinro- tatingmachinerytoreduceunbalanceexcitation,areindesignverysimilartofullylubricated ball bearings missing the cage, the inner ring and the majority of the balls. Inherent to the functional principle, the balancing efficiency and stability are significantly influenced by the choice of lubricant and resulting drag forces. Therefore, the estimation of the drag coefficient based on the geometry and lubrication of automatic ball balancers plays an important role in theengineeringprocess. WithafocusontheStokesflowregime,thedragcoefficientforasinglesphereinanannu- lar flow domain is determined numerically with finite volume discretization and the SIMPLE steady state solution scheme. Based on a parameter study utilizing the presented solution ap- proach, a simple empirical relation between the design of the automatic ball balancer and re- sultingdragcoefficientsisderived.Asaresult,adragforceformulationbasedonthebalancer geometry and the lubrication fluid properties is presented, which helps to supplement a large number of published kinetic models regarding the analysis of automatic ball balancer stabil- ityandtransientbehaviour,givingabetterunderstandingoftheinfluencesofdesigndecisions regardinggeometryandlubricant. KEYWORDS ballbearing;drag;Stokesflow;automaticbalancing 1. Introduction The calculation of drag forces resulting from flows around spheres and cylinders are widely used benchmarks to evaluate computational methods in the area of fluid mechanics due to the simplicity of the geometry, see for example Johnson and Patel (1999), Jones and Clarke (2008),andBayraktaretal.(2012).Whendefiningalargecomputationaldomaincomparedto the dimensions of the obstacle, the effect of domain boundaries on the solution is attenuated in order to represent the targeted free flow condition. In a further study by Martinez et al. (2009)thesingleplanecontactofasphereandtheresultinginfluenceoftheBlasiusboundary layer was examined and in the context of process engineering advances in numerical studies CONTACTL.Spannan.Email:lars.spannan@ovgu.deORCiD:0000-0003-3237-4405 1 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. environing fluidraceway contactwidthheightorbit- radius axis of rotationball diameter Figure1. Principledesignofautomaticballbalancers.Whentheouterringofaballbearingisinsertedtofunctionasaraceway withreducedfriction,aracewaygrooveispresentwhichisnotshowninthissketch. onparticleflowsinpipesarepresentedbyKrishnanandKaman(2010)andChenetal.(2019) forexample. In the study at hand, however, the drag of a sphere confined in an annular domain is ex- aminedinthecontextofautomaticbalancingofrotatingmachinery.Theconceptofautomatic balancingdescribesthearrangementofbalancingmasses,usuallyballs,inrelationtotheaxis ofrotationofarotorinordertocompensateitsunbalance.Dependingontheapplication,the unbalance can vary during operation, for example laundry in washing machines, samples in laboratorycentrifugesorabrasioninducedimbalanceinanglegrinders. Since thefrictionbetweentheballsandtherotorcavity in whichthey are orbiting around therotoraxishasnegativeinfluenceontheaccuracyoftheunbalancecompensation,theouter ringsofballbearingsareoftenusedinthedesigntodecreasefrictionaspresentedbyLindell (1996) for the application in angle grinders. Therefore, the basic design of the automatic bal- ancercanbedescribedasaradialballbearingmissingtheinnerring,thecageandthemajority oftheballs,seefigure1.Inadditiontotheballs,thecavityisfilledwithaviscousfluid,which hasinfluenceonthedynamicsandnoiseemissionoftheautomaticbalancingunit.Asaresult, themovementoftheballsintheannuluscavityissignificantlyaffectedbydragforces,which aretobedeterminedbynumericalanalysisinthepresentedstudy. Since the first publication by Thearle and Schenectady (1932) on the topic of automatic ball balancers, several research advances have been published on the analysis of the dynamic behaviour. For example, Sharp (1975) presents equations of motion, with which the stability ofthesystemsbalancedstatecanbeanalysed.MoredetailedanalysesfollowedbyNesterenko (1984), considering anisotropic supports of the rotor, Van De Wouw et al. (2005), discussing theinfluenceofdryfrictionontheresidualunbalance,Ishidaetal.(2012),presentingmethods to improve the balancing efficiency and Wright and Peng (2019), examining the influence of varyingangularvelocities. Commonly,theviscousforcesinducedbythefluidonthebalancingballsinsidethecavity aremodelledinlinearrelationtotheorbitvelocityoftheballsrelativetothecavitywalls.Even thoughthisenablesqualitativestudiesontheinfluenceofincreasedviscosityoftheusedfluid, nomethodsareproposedtoquantitativelydetermineasuitablelineardragparameterbasedon thebalancerdesignandthefluidparameters.Inthestudyathand,aCFDapproachispresented delivering the missing modelling aspect in order to describe the viscous forces based on the fluidviscosityandthegeometricpropertiesoftheannulusandthebalancingballs. The determination of drag on spheres considering contact with domain boundaries is ex- aminedinseveralresearchpublications.Goldmanetal.(1967),forexample,studiedtheslow motionofasphereincontactwithaplanewallusingasymptoticsolutionsoftheStokesequa- tions. Jan and Chen (1997) conducted experiments with a sphere rolling down an inclined 2 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. plane and derived an empirical relation between the drag coefficient cdand the correspond- ing Reynolds number Re. Miyamura et al. (1981) derived wall correction factors for spheres incontactwithcylindersandparallelplatesexperimentally.Numericalinvestigationsondrag in rolling element bearings were carried out by Marchesse et al. (2014) focusing on the tan- demeffectofmultiplespheresinarow.However,theauthorsarenotawareofanypublications presentingquantitativemodelstoexaminetheviscousdragthescopeofautomaticbalancers. The simulations presented in the following sections yield the drag coefficient of a single sphereincontactwiththegeometryofanouterradialballbearingringconfinedinanannular domain.TheflowconditionsarelimitedtolowReynoldsnumbers(Re <10)andtherefore,as willbepresentedintheresults,leadtoasteadyflowregimeinwhichtheproductofReynolds number and drag coefficient equals a constant, reinforcing the assumption of laminar flow. WiththetransitiontounsteadybehaviourforspheresincontactwithaflatsurfacenearRe = 250, see Martinez et al. (2009), sufficient distance to unsteady behaviour is assumed for the flowinthecurvedannulus,consideringtherestrictiontoRe <10. Section2describestheconfigurationofthesimulationscarriedoutwiththeOpenFOAM1 software. The resulting drag coefficients for different Reynolds numbers as well as a subse- quentparameterstudy,whichanalysestheinfluenceoftheballorbitradius,theballdiameter andthecavitydimensionsarepresentedinsection3.Withthechosenparameterrange,thede- signspaceofcommerciallyavailableregularradialballbearingsiscovered.Section4specifies howtheidentificationofthereciprocalrelationshipbetweenthedragcoefficientandReynolds numberinlaminarflowregimeshelpswiththeparametrisationofavarietyofkineticmodels ofautomaticballbalancerspublishedintheliterature.Aconcludingdiscussionandoutlookis giveninsection5. 2. Methods SincetheStokesflowregimeatlowReynoldsnumbersisconsidered,asymmetricflowregard- ing the axial geometric symmetry plane is assumed, see figure 2. The inlet and outlet of the curveddomainaredefinedinasuitabledistancetothespherespanninganorbitalangleof60° so that the assumption of a periodic flow condition can be implemented while reducing the computationaleffort.Anincreaseoftheorbitalangleto90°inthecalculationofthereference balancer design, section 3, showed no significant influence on the result, substantiating the proposedapproach. Sincethespherehasasinglecontactpointwiththeracewayoftheouterring,thegeometry of the domain is altered by introducing a cylindrical column connecting the sphere and the raceway,seefigure2.Thecolumndiameterissetto16%relativetothespherediameter.While the impact on the results is believed to be marginal, this modification improves the meshing and solution process. The marginal influence is attributed to the fact that the height of the cylinder is relatively small, even with a diameter of 16% of the ball diameter, due to the very similarradiioftheballandgroove.Onthebasisofseveralmodellingvariants,16%turnedout tobethesmallestminimumratioatwhichastablenumericalsolutioncouldbeobtained. Inordertoreducethecomputationaleffort,thecaseofasphereorbitinginsidetheannular domainwithanorbitalangularvelocityof ,anorbitradiusof Rorbitandaballradiusof rball, seefigure3,isviewedregardingacoordinatesystemoriginatinginthecenterofthespherewith theX-axis pointing outwardly in radial direction and the Y-axis pointing in axial direction. Duetothisorientation,thevectorsofangularvelocityareorientedinthenegative Ydirection to be consistent with the inlet and outlet velocity directions. As a result, an invariant mesh 1OpenFOAM7Version20190708 3 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. Figure2. Thecomputationaldomain.Wallsarenotshownbutindicatedbytheiroutlines.Thetransparentsemi-circleat Z= 0, Y= 0isresultingfromtheconnectingcolumnavoidingasingularcontactpoint. Ω RorbitRgrooverballrgroove W1H Y Xω W22fluid domainwall (lid) wall (outer ring of ball bearing with groove) wall (bottom)wallannulus symm. axis Figure3. Sliceofthedomaininthe XY-plane( Z= 0). can be utilized and under the assumption of ideal rolling of the sphere, i.e. no slip occurs in the contact point, the following boundary conditions are defined on the domain boundaries depictedinfigure2withimportantgeometricentitiespresentedinfigure3: symmetryplane:SymmetryboundaryconditionjustifiedbythelowReynoldsnumbers resultinginlaminarflow. inlet:Normalflowintothedomain.Thevelocitymagnitude vrefofeachcellfaceisdeter- minedbytheproductoftheorbitalangularvelocity andthedistanceoftheindividual elementfacetothesymmetryaxisoftheannulus. outlet:Zeropressureoutlet. walls:Constantwallvelocitydefinedbytheproductof andthedistanceofthecellface tothesymmetryaxisoftheannulus. spheresurface:Fixedvelocityboundarycondition.Thevelocityisdefinedbytheproduct oftheballangularvelocity !aboutthe Y-axisgoingthroughthespherecenterandthe distancetoit. !resultsfromtheidealrollingcondition (Rorbit+rball) =! rball. The OpenFOAM simpleFoam application uses the semi-implicit method for pressure– 4 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. Table1. Numberofcellsfordifferentmeshrefinements. Mlvl No.ofcells 1 142000 2 1136000 3 3834000 4 9088000 5 17750000 6 30672000 linkedequations(SIMPLE)forasteady-statesolutionofthecontinuityequation r ~ u= 0 (1) andthemomentumequation r (~ u ~ u) r =rpk+~Su; (2) for incompressible flow, where ~ u,pk,,~Sudescribe the velocity vector, the kinematic pres- sure, the stress tensor and the momentum source, respectively (Ferziger et al., 2020). Herein, theOpenFOAMincompressiblesolverstransformthestaticpressure pstokinematicpressure pk, i.e.pk=ps/, with being the fluid density. The equations for velocity and kinematic pressure are solved iteratively. As a result, for each mesh element three degrees of freedom for the velocity and one degree of freedom for the pressure are determined. The sought drag coefficient cdis then calculated from the sum of forces in the tangential direction ( Z) on the ballsurfacepatcheswithfaceareavectors ~ s,thefacialreferencearea Arefandthefreestream velocity vrefattheinlet,i.e. cd= 2X (i)pk;i~ si0 0 1T(Arefv2 ref)1: (3) Due to the exploitation of symmetry, the reference area is defined as Aref=r2 ball/2. In the conducted calculations, the iterations are stopped when the residuals fall below 2 106. A structuredhexaedralmeshisgeneratedwiththeOpenFOAMblockMeshutilityanddifferent mesh refinement levels are realized, see table 1. The mesh refinement level (Mlvl) scales the numberofcelldivisionsperblockinthecomputationaldomain.Themeshisdefinedforhalf ofthecomputationaldomain,seefigure4,andmirroredonthe XY-plane.Sincethepressure fieldonthespheresurfaceisevaluatedinordertogainthedragcoefficient,asmallelementsize isdefinedintheadjacentcells.Thesphereissurroundedbyaradialpaddingofmeshregions which are discretised gradually with the smallest element height at the sphere surface. The remainingcoarsermeshregionsarealsodiscretisedgraduallytowardsthesphere,resultingin improvedresultswithoutanincreaseindegreesoffreedom. 5 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. symmetry planewall ball surfacemesh mirror plane symmetry planewall mesh mirror plane ball surface Figure4. Defined mesh for the coarsest discretization setting (Mlvl = 1) prior to the mirroring operation on the XY-plane. Viewfromtheoutside(top)andfrominsidethedomain(bottom). Table2. Automaticballbalancerreferencegeometry. Property Value Rorbit 77mm rball/Rorbit0:08 rgroove /rball1:2 H/rball4:0 W1/rball1:554 W2/rball0:690 6 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. Flowvelocitymagnitude 0 maxinlet outlet Figure 5. Exemplary normalized result of the flow field in the symmetry plane. Arrows indicate the flow velocity direction in theplaneandthevelocitymagnitudeisindicatedbythecolourofthestreamlines. 3. Results The domain geometry for the study is determined by the properties given in figure 3 and ta- ble2representinganautomaticballbalancerprototypeavailabletotheauthors.Followingthe solution method described in section 2 the velocity and pressure fields are obtained and the drag coefficient according to equation 3 is returned by OpenFOAM. An exemplary normal- ized result of the flow field in the symmetry plane is shown in figure 5. The mesh refinement studyinfigure6showsthattheresultingdragcoefficientforaReynoldsnumberRe = 1does not change significantly for refinement levels above four. The relative deviation with respect to the result of level six is less than 0.4%, which is considered sufficient for the calculation of drag forces presented in section 4. Therefore, all subsequent simulations are performed with refinementlevelfour. 1 2 3 4 5 6360370380390400 Meshrefinementlevel(Mlvl)cd Figure6. MeshindependencestudyforthereferenceconfigurationofthebalancergeometryandaReynoldsnumberofone. FordifferentReynoldsnumbersintherangeofRe = 0:01: : :100thedragcoefficientsare determinedandareshowninfigure7alongwithdatafromtheliteratureforthefreeflowcon- ditionpublishedbyMorrison(2013),theflatsurfacecontactpublishedbyJanandChen(1997) and experimental results gained for the balancer prototype in the prevalent double logarith- micdiagram.InthepreviousworkofSpannanetal.(2017)theutilizedexperimentalmethod, basedontheequilibriumevaluationbetweendragforcesandgravitationalforces,ispresented. Inordertoderivethedragcoefficient cdfordifferentReynoldsnumbersthebalancingdevice prototype is driven on a horizontally aligned axis and the balance of forces is examined. For 7 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. the considered range of low Reynolds numbers, measurements with silicone oil of kinematic viscosities 1=50mm2s1,2=298mm2s1and3=715mm2s1areusedasreference fortheconductedCFDsimulations.Foradetaileddescriptionoftheexperimentalprocedures thereaderiskindlyadvisedtothegivenreference. 102101100101102103100101102103104105 Recdfreeflow:Morrison(2013) flatsurfacecontact:Jan(1997) presentCFDstudy measurementsacc.toSpannan(2017): 1=50mm2s1 2=298mm2s1 3=715mm2s1 Figure 7. Drag coefficient of flow around a sphere depending on the Reynolds number. Measurement results are gained from theexperimentalstudypublishedinSpannanetal.(2017). 10210110010110224100200322387500600 RecdRefreeflow:Morrison(2013) flatsurfacecontact:Jan(1997) presentCFDstudy measurementsacc.toSpannan(2017): 1=50mm2s1 2=298mm2s1 3=715mm2s1 Figure8. ProductofdragcoefficientandReynoldsnumberatlowReynoldsnumbers.Measurementresultsaregainedfromthe experimentalstudypublishedinSpannanetal.(2017). Figure7showsanincreaseddragcoefficientfortheconsideredrangeofReynoldsnumbers comparedtotheresultsoftheflatsurfacecontact.Thisresultisexpectedduetotheadditional confinementoftheflowinfluencedbythedomaingeometry.SinceonlytheStokesflowregime isconsidered,wheretheproductofdragcoefficientandReynoldsnumberresultinaconstant, theresultsaretransformedtofigure8showingtheproduct cdReforReynoldsnumbersupto Re= 100.Thesimulationresultspresentaconstantvalueof cdRe= 387andagreewellwith theexperimentaldatagainedforReynoldsnumbersRe = 0:1: : :0:7andtheoilwithakine- matic viscosity 3=715mm2s1. The deviations of the experimental results acquired with theoilofakinematicviscosity 2=298mm2s1arebelievedtobecausedbyfrictioninduced temperaturefluctuationsduringoperationandassociatedviscosityvariations.Inaddition,the agreementbetweenthenumericalandtheexperimentalresultsforReynoldsnumbersaround Re= 100is given so that it can be assumed that the solution approach is also applicable for thetransitionareaaftertheStokesflowregime.Sincetheoverallagreementisacceptable,the presentedsimulationmodelisconsideredsufficienttogainusefulresultsforasubsequentpa- 8 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. rameterstudy. 3.1. Parameterstudy Althoughthepresentednumericalmodelcansupplementthedesignprocessofautomaticball balancersbydeterminingdragcoefficientsforagivengeometryatlowReynoldsnumbers,the possibilityofreducingthecalculationtimeandeffortbyderivinganempiricalrelationbased onaparameterstudyisinvestigatedinthefollowing.Asdescribedpreviously,theutilizationof radialballbearingcomponentsinautomaticballbalancersisadvantageousduetolowfriction. With commonly available radial ball bearings in mind, the variance of the drag coefficient for orbit radii in the range Rorbit=5mm : : :300mm and relative ball sizes rball/Rorbit= 0:05: : :0:3 are considered in the parameter study. The annulus height compared to the ball radius varies only slightly between different bearing sizes and is therefore held constant with H= 4rball.Similarly,alsothegroovedepthratio W2/rball,seefigure3,isbelievedtovaryonly slightly and is held constant. The distance between the ball center and the inner boundary of thefluiddomainmayvaryfordifferentdesignsofautomaticballbalancerunits.Withregard to compact designs, low values for the clearance W1/rballto the inner annulus boundary are to be expected. By increasing the distance, it was checked at which point the influence of the innerboundarybecomesnegligible. BasedontheevaluationofthedimensionlessunitsReynoldsnumberanddragcoefficient, itisapparentthatthechangeintheorbitradius Rorbithasnoinfluenceontheresult,ifthere- mainingproportionquantitiesintable2remainconstant.Thecheckcanbecarriedoutquickly by using the converged velocity field of one solution and scale it inversely to the orbit radius magnification.Fromthisinitialfieldsolutiontheconvergenceisachievedpromptly. Increasingtherelativeballradius rball/RorbitresultsinanincreaseoftheproductofReynolds numberanddragcoefficient,seefigure9.Thecalculationsareconductedwithincreasingrel- ativeballsizewhileusingthesamenumberofcells. 0:1 0:2 0:3200300400500600 rball/RorbitcdRe sim.results d(rball/Rorbit) +e Figure 9. Variation of drag resistance depending on relative ball size. The parameters of the shown linear approximation are d= 813,e= 318. The study of increasing distance between the inner boundary of the annular domain and the sphere, figure 10, shows a decay in drag, which can be approximated by an exponential relationwithanasymptoticvalueforlargeradialclearances.Clearances W1/rball>2:5result 9 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. invalues cdRe<322,whichwasidentifiedbyJanandChen(1997)forthesingleflatsurface contact. The decreased drag for designs with high clearances to the inner wall is attributed to the groove in the outer ring. However, since minimal overall ABB dimensions are aspired, W1/rballis usially less than two. While increasing the domain volume, the number of cell divisionsinradialdirectionareincreasedaccordingly.Furthermore,themapFieldsapplication of OpenFOAM is utilized to copy a previous solution to the coincident part of the domain. Startingwiththesolutionfor W1/rball= 5andconsecutivelydecreasingthedomainvolume, thecollectivecomputationaleffortisreduced. 2 4300350400450 W1/rballcdResim.results a(W1/rball)b+c Figure10. Variationofdragresistancedependingondistancetotheinnerradialboundary.Theparametersoftheshownexpo- nentialapproximationare a= 248,b=1:375,c= 245. Basedonthepresentedsimulationresults,neglectinginterdependenceofthetwovariables, therelation cdRe=aW1 rballb +c+drball Rorbit0:08 ; (4) witha= 248,b=1:375,c= 245andd= 813,isproposedinordertoapproximatethere- sultingdragintheStokesregimeofasingleballinanautomaticballbalancer.Thecoefficients a,bandcare fitted with trust-region adjusted non-linear least-squares and the coefficients d andewithlinearleast-squaresmethods,respectively.Sinceequation(4)emergesfromthepa- rameterstudybasedonthereferencegeometrygivenintable2thebestagreementisexpected nearW1/rball= 1:554andrball/Rorbit= 0:08. 4. Application One of the most researched kinetic models for the application of automatic ball balancers is that of a planar oscillator, figure 11, with x,y,'rotor,'ballbeing the two translatoric degrees of freedom, the rotation of the rotor and the orbital position of the ball, respectively. In the derivation of the equations of motion the viscous damping forces due to the fluid inside the cavityiscommonlymodeledinalinearmanner,e.g.adragmoment MDrag= ( _'rotor_'ball); (5) proportional to the difference of the rotor angular velocity _'rotorand the ball orbit velocity _'ball,isintroduced.SeeforexampleSharp(1975)andSperlingetal.(2004). 10 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. 2rball Rorφrotor y Figure11. Exampleofaplanaroscillatorwithanautomaticballbalancer. In several works the importance of the viscous forces is recognised and simulations are conducted in order to identify values for the coefficient , which lead to optimal dynamic behaviour of the rotor system. See for example Ryzhik et al. (2003) for the analysis of rotor run ups and Bykov and Kovachev (2018) for the identification of stable operating speeds of automaticbalancingdevices.However,concretecorrelationsbetweentheintroducedconstant viscouscoefficient, inequation(5),andthedesignofthebalancingunitaswellasfluiddensity and viscosity are not discussed. The results of the parameter study described above, yielding a relation between the annulus geometry, the ball size, the fluid properties and the constant cdRe,canbeusedtoremedythissituationasspecifiedinthefollowing. Consideringthefamiliarequationfordragforces,thediscusseddragmomentcanbewritten intheform MDrag=FDragRorbit =1 2 Arefcdv2 refRorbitsign(vref) =cdRe 4  rballR2 orbit_'ref; (6) with,,Aref,vrefbeingthefluiddensity,thefluidkinematicviscosity,thecross-sectionalarea oftheballandthereferenceflowvelocity,respectively.TheReynoldsnumberisintroducedas Re=2rballvref : (7) Under the assumption that vref= Rorbitand _'ref_'rotor_'ball, meaning that the fluid is rotatingwiththerotorangularvelocity,thecoefficient canbeidentifiedasaconstant =cdRe 4  rballR2 orbit: (8) Forrunupprocessesofmachineryequippedwithautomaticballbalancingdevicestheas- sumptionoflowReynoldsnumbersdoesnotholdgenerallyandvelocitydifferencesbetween therotorandthefluidoccur.Butwhensteadystateoperationsofautomaticballbalancersare 11 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. considered, the stated assumptions are met. As a result, stability analyses, which incorporate equation (5), see for example Bykov and Kovachev (2018), can be conducted solely based on geometric properties of the balancer, fluid properties and the presented identification of the productofthedragcoefficientandReynoldsnumber,seeequation(4). 5. Discussion In the presented study a finite volume approximation solution for the laminar flow around a sphere in an annular domain is presented. The domain geometry is derived from an auto- maticballbalancerprototypeandaparameterstudyisconductedcoveringthedimensionsof commonradialballbearings.Thedragsimulationresults,intermsoftheproductofReynolds numberanddragcoefficientforareferencegeometry,showtheanticipatedconstantnumerical valueforReynoldsnumbersbelowten.Theincreaseindragcomparedtothecaseofaspherein contactwithaflatwalldiscussedintheliteratureshowstheinfluenceofthesidewallsandthe curvature of the domain. It is shown that the knowledge of the product of Reynolds number and drag coefficient is beneficial for the parametrisation of kinetic models for automatic ball balancers.However,duetothedependenceoftheresultsonthefluidviscositythecalculation ofdragforcesinautomaticballbalancermodelsissubjecttouncertaintiesindeterminingthe viscosityofthefluidandpossibletemperaturesensitivity. Incontrasttoapplicationsofautomaticballbalancersutilizinglargernumbersofbalancing ballsinordertoincreasethemaximumcompensableimbalanceoftherotor,thepresentedflow simulations consider only one ball neglecting possible tandem effects. Two balls in tandem arrangement on the orbit will reduce the drag forces on the downstream body (Marchesse et al.,2014)makingitworthconsideringappropriatecorrectionfactorstoequation(4). During run-up of rotors equipped with automatic ball balancers the angular acceleration maybesufficienttoleavetheStokesflowregime.Asaresult,theproductofReynoldsnumber anddragcoefficientdeviatesfromaconstantvalue.Withregardtocomputationalflowsimu- lations, the assumptions of symmetry and laminar flow will not be met, leading to increased efforttogainresultsforlargerReynoldsnumbers. Concludingfromtheselimitations,futureadvancesinnumericalsolutionofthefluidflow inautomaticballbalancersshouldwithdrawtheassumptionofsteadysymmetricflowtoallow for the consideration of turbulent flow at higher Reynolds numbers with multiple spheres in tandem orientation inside the annular cavity. Consequently, more advanced numerical solu- tionmethodsmustbeused. Inadditiontotheviscousforces,thefrictionforcesandpossiblegeometricaldeviationsof the raceway are influencing the dynamical behaviour of the automatic ball balancing system, seeChaoetal.(2005),Greenetal.(2006)andBykovandKovachev(2019)forexample.There- fore,afullycoupledfluidstructureinteractionofautomaticbalancers,i.e.inmultibodysimula- tionmodels,shouldonlybepursuedifthemodelingqualityoffrictionandimprecisegeometry is also enhanced. Compared to the increased effort resulting from a fully coupled simulation the presented method based on the widely utilized gas-particle flow approach (Henderson, 1976)representsagoodcompromiseintheviewoftheauthors. Acknowledgements ThesupportofMr.S.Engel,UniversityofMagdeburg(ISUT),inconductingthefluidviscosity measurementsandMr.S.Koch(IFME)inexperimentalrealisationisgreatlyappreciated. 12 PreprintPreprint: ThisarticlehasbeenacceptedforpublicationinEngineeringApplicationsof ComputationalFluidMechanics,publishedbyTaylor&Francis. Nomenclature a,b,c,d,ecoefficientsresultingfromleast-squaresfittingmethod Aref ballfacialreferencearea,(m2) cd dragcoefficient FDrag dragforce,(N) H annulusheight,(m) MDrag dragmoment,(Nm) Mlvl meshrefinementlevel pk kinematicpressure,(m2s2) ps staticpressure,(Nm2) Rorbit orbitradius,(m) rball ballradius,(m) rgroove ballbearinggrooveradius,(m) Re Reynoldsnumber ~ s faceareavector ~Su momentumsource,(Nm3) ~ u flowvelocityvector,(ms1) vref freestreamreferencevelocity,(ms1) W1 radialdistancebetweenballcenterandinnerannuluswall,(m) W2 radialdistancebetweenballcenterandouterannuluswall,(m) X,Y,Zballcenteredcoordinatesystem x,y translatoricdegreesoffreedomofautomaticbalancermodel,(m) Greek lineardragmomentcoefficient,(Nmsrad1)  kinematicviscosity,(Nm2)  fluiddensity,(kgm3)  stresstensor,(Nm2) 'rotor rotorrotationangle,(rad) 'ball ballorbitalposition,(rad) _'ref referenceangularvelocity,(rads1) orbitalangularvelocityofthespheresrelativetothecavity,(rads1) ! ballangularvelocityaboutthe Y-axis,(rads1) Notes TheOpenFOAMcasefilesthatsupportthefindingsofthisstudyaswellastheresultspresented inthefiguresofthispublicationareopenlyavailableinZenodoathttps://doi.org/10.5281/zen- odo.3741868.