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Applications in Engineering Science
journalhomepage:www.elsevier.com/locate/apples
Why homogenization should be the averaging method of choice in hydrodynamic lubrication
Michael Rom
a,∗, Florian König
b, Siegfried Müller
a, Georg Jacobs
baInstitute for Geometry and Applied Mathematics, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany
bInstitute for Machine Elements and Systems Engineering, RWTH Aachen University, Schinkelstraße 10, 52062 Aachen, Germany
a r t i c le i n f o
Keywords:
Homogenization
Patir and Cheng’s average flow model Reynolds equation
Hydrodynamic lubrication Textured bearing Numerical investigation
a b s t r a ct
Forthecomputationofthehydrodynamicpressuredistributioninfluidfilmlubricationproblems,averaging techniquesarefrequentlyusedforapplicationswithroughortexturedsurfacestoreducethecomputationalcost.
Thepresentworkshowstheexcellentsuitabilityofhomogenizationforthispurpose.Forcomparison,themost frequentlyappliedaveragingmethodisused:theaverageflowmodelbyPatirandCheng.Thetwomodelsare brieflysummarizedtoshowthesimilarityoftheformulations,resultinginthesameimplementationeffort.By meansoftwotexturedapplications,thesuperiorityofthehomogenizationmethodisdemonstratedanderrors inherentintheaverageflowmodelarequantified.Homogenizationprovidesaveragedpressuredistributionswith asignificantlyhigheraccuracyandallowsforefficientlyrestoringlocalinformationtoresolvepressurepeaksby computingahigher-ordersolution.ThelatterisanaccurateapproximationofthesolutionoftheoriginalReynolds equationandcanbeobtainedwithnegligibleadditionalcomputationalcost.Furtherimportantadvantagesofthe homogenizationmethodsuchasitsextensibilitytosimultaneouslyaccountforroughnessandtexturesarepointed out.
1. Introduction
Forthecomputationofthehydrodynamicpressuredistributionin fluidfilm lubricationapplications, theReynolds equation (Reynolds, 1886)wasderivedfromtheNavier-Stokesequationsbyneglectingsev- eraltermsandaveragingoverthelubricatedgap.Thisreductionofthe dimensionfrom3Dto2DandthelinearityoftheReynoldsequationsig- nificantlyreducethecomputationalcost.However,adequatelycaptur- ingthesurfaceroughnessoftheapplicationrequiresahighly-resolved computationalmeshsuchthatevensolvingthe2Dlinearproblemcan betootime-consuming.Forthisreason,PatirandChengempiricallyde- velopedtheaverageflowmodel(PatirandCheng,1978;1979) which splitstheproblemintoonemacroscaleproblemwithoutroughnessand severalmicroscaleproblemswithroughnessandallowsforthecompu- tationofanaveragedpressuredistribution.Sinceamicroscaleproblem onlyrepresentsasmallpartofthewholeapplication,itcanbesolved efficientlydespitetheroughness.
ItquicklybecameclearthatPatirandCheng’saverageflowmodel onlyprovidesaccurateaveragedpressuredistributionsincaseswhere thesurfaceroughnessisisotropicorwherethemaindirectionsofthe roughnessanisotropyareparalleltothecoordinateaxes(orthotropic).
ThiswasforinstanceshownbyElrod(1979),whoconductedamultiple-
∗Correspondingauthor.
E-mail addresses: rom@igpm.rwth-aachen.de (M. Rom), florian.koenig@imse.rwth-aachen.de (F. König), mueller@igpm.rwth-aachen.de (S. Müller), georg.jacobs@imse.rwth-aachen.de(G.Jacobs).
scaleanalysisfortwo-dimensionalroughness,orbyTripp(1983),who extendedtheaverageflowmodelbyderivingthecorrecttensorialform ofthemodelusingastochasticapproach.Lo(1992)proposedtotrans- formperiodicanisotropicsurfacesintoisotropiconesbyusingamap- pingfunctionsuchthattheaverageflowmodelisapplicable.Worksby TealeandLebeck(1980)andLundeandTønder(1997)discussedsev- eralboundaryconditionsforthemicroscaleproblemstoimprovethe resultsoftheaverageflowmodel.
After Patir and Cheng’s average flow model, further averaging techniques with respect to the Reynolds equation were presented.
Phan-Thien (1982) used the mathematical concept of homogeniza- tion(Allaire,1992;PavliotisandStuart,2008)todevelophisso-called two-spacemethod.Mitsuyaetal.(1989)introducedanapproachbased ontheaveragingofthefilmthickness.Pratetal.(2002)appliedthe volumeaveragingmethod(Whitaker,1999)andobtainedamodelthat takesintoaccountcontactareasnotinvolvinganyfluidandcombines timeandspatialaveraging.
ApartfromPatirandCheng’saverageflowmodel,thehomogeniza- tiontechniquereceivedthemostattentionbecauseofitsaccurateav- eragingresults forsurfaceroughnessof arbitraryanisotropy. Several derivationsfortheReynoldsequationexistintheliterature,forinstance byBayadaandChambat(1988),BayadaandFaure(1989),Jai(1995),
https://doi.org/10.1016/j.apples.2021.100055
Received18December2020;Receivedinrevisedform1April2021;Accepted7May2021 Availableonline19May2021
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Buscagliaetal.(2002),AlmqvistandDasht(2006)andAlmqvistetal.
(2007,2012).Thesederivationsdifferintheintendeduseofthehomog- enizedReynoldsequation,e.g.,forstationaryortransientproblems.For applicationswithisotropicororthotropicsurfaceroughness,thehomog- enizationmethodandtheaverageflowmodelprovidetheexactsame averagedpressuredistributions.Furthermore,homogenizationleadsto thesameaveragedpressuresasthevolumeaveragingapproachbyPrat etal.(2002)ifnocontactoccursandifonesurfaceissmoothandmov- ing,whereastheotheroneisroughandstationary.
Acomprehensiveliteraturereviewonaveragingtechniquesforthe Reynoldsequationisbeyondthescopeofthiswork,buttheoverview givenaboveshows thattheshortcomingsoftheaverage flowmodel arewellknownandthatnumerousimprovementsoralternativesexist.
However,forthestudyoflubricationapplicationswithroughsurfaces, theoriginalmodelbyPatirandChengisstillthemostcommonlyused method(Fatuetal.,2012;Gropperetal.,2016).Thisalsoappliesto thestudyoftexturedapplications,whichareofincreasingimportance duetotheirpotentialtoimprovehydrodynamicperformance(Gropper etal.,2016).
Thepopularityof theaverageflowmodelhasledtothedevelop- mentofmanyextensions,e.g.,byHarpandSalant(2001)toaccountfor cavitation,byWuandZheng(1989)orWilsonandMarsault(1998)to incorporatesurfacecontactorbyMengetal.(2009)tocombineelas- ticdeformation,inter-asperitycavitationandthermaleffects.Sincethe methodofhomogenizationiscertainlynotascommoninthefieldoftri- bologyastheaverageflowmodel,onlyfewextensions,e.g.,byBayada etal.(2005)forincorporatingcavitation,orimplementationsasinthe mixedlubricationmodelbySahlinetal.(2010a,b)areavailableinthe literature.
Comparisonsofaveragingmethodsmainlyconcentrateontheeval- uationofthesolutionsof themicroscaleproblems,i.e.,theresulting pressureandshearflowfactors.Suchcomparisonswerepresentedby Sahlinetal.(2010a), Almqvistetal.(2011b)andFatuetal.(2012). Macroscaleresults,i.e.,averagedpressuredistributions,obtainedfrom thehomogenizationmethodandfromthefilmthicknessaveragingap- proachbyMitsuyaetal.(1989)werecomparedbyJaiandBou-Saïd (2002).
Mostoftheaforementionedworksdiscussingthederivationofthe homogenizedReynoldsequationorthecomparisonofhomogenization withotheraveragingtechniquesmissonemajoradvantageoftheho- mogenizationmethodoratleastdonotquantifyitsbenefit:byasim- pleupscalingoftheaveragedpressuresolution,whichhasnegligible computationalcost,local informationarerestoredsuchthatpressure peaksareresolvedwithahighaccuracy.Thisallowsforamoreaccu- ratepredictionofperformanceparameterssuchasmaximumpressure orfrictionforcethanpossiblewithotheraveragingtechniques.Theup- scalingwasstudiedbyBayadaandFaure(1989)forajournalbearing withdifferentgroovesetups.Theirinvestigationmainlyconcentrateson theevolutionofrelativeerrorsintheload-carryingcapacity,obtained fromthehomogenizedortheupscaledpressuresolution,forincreas- ingroughnessfrequency.Sincetheload-carryingcapacityiscomputed byintegratingtheaveraged(homogenized)ortheoscillating(upscaled) pressure,theresultingvalues usuallydeviateonly slightly.However, theauthorsstateintheirconclusionthat“theanalysisofsomeother bearingcharacteristicswhicharerelatedwiththefloworthegradient ofthepressuremustnotbecomputedfromthegradientof𝑝0”,where 𝑝0denotestheaveragedpressuresolution.Thegradientofthepressure solutionisforinstanceinvolvedinthecomputationofthefrictionforce.
AnotherworkbyBayadaandJai(1996)demonstratestheupscalingby showingpressuredistributionsforultra-thingasfilmbearingsobtained withandwithoutupscaling.
Themainobjectiveofthisworkistodemonstratetheadvantageof thepressureupscalingofthehomogenizationmethod.Wealsoaimat emphasizinghoweasilytheupscalingisobtained.Thehomogenization methodiscomparedwiththeaverageflowmodelbyPatirandCheng sincethelatter isthemostwidely usedaveragingtechnique. Onthe
onehand,atheoreticalcomparisonisintendedtounderlinethesim- ilarityofthetwomethodsregardingthegoverningequations,bound- aryconditionsandcomputationalprocedure,whichleadstothesame implementationeffort.Eventhoughtheformulasarewellknown,this overviewwill helppointoutthesimilarityof thetwomethodsand, inparticular,howtoimplementthehomogenizationmethodwithup- scaling.Ontheotherhand,acomparisonofnumericalresultsismeant toverifythesignificanceof theerrorsintroducedbyusing theaver- age flowmodelinstead ofthehomogenization method.These errors do notonly originatefrom themissingupscalingor fromtheuse of anisotropicroughnessortexturesbutalsofromthepressureaveraging.
Hence,eveninthecaseofisotropic/orthotropicroughnessortextures,a considerableerroriscausedbyapplyingtheaverageflowmodel,which isavoidablewiththehomogenizationmethodatvirtuallynoadditional cost.
Tokeepthetheoreticalcomparisonandthenumericalinvestigation simpleandcomprehensible,wefocusonstationaryhydrodynamiclubri- cationwithiso-viscousandincompressiblelubricantsanddonottake intoaccountcavitation.Thisallowsfortheunadulteratedidentification oftheerrorsinherentintheaverageflowmodel.
Inaddition,wediscussfurtheradvantagesofthehomogenization method overtheaverage flowmodel. These comprisetheavoidance of modelinguncertaintiesoftheaverageflow modelpresentforsur- faceswithdeepdimples(Grützmacheretal.,2018;Königetal.,2020a;
Tomaniketal., 2020) andtheextensibilityoftheconcept byreiter- atedhomogenization(AllaireandBriane,1996),whichallowsforthe efficientinvestigationofsurfacesthatareroughandtextured.Thepro- cedureofreiteratedhomogenizationfortheReynoldsequationwasde- rivedbyAlmqvistetal.(2008)andMartin(2008),thelatterincorpo- ratingacavitationmodel.
Forthecomparisonofnumericalresults,weinvestigatetwotextured applications,namelyajournalbearingandaplane-inclinedsliderbear- ing.Inbothcases,twodifferentsetupsregardingthetexturesarestud- ied:(𝑖)withsymmetrictextures(isotropic/orthotropiccase)and(𝑖𝑖)with nonsymmetrictextures(anisotropiccase).Thesolutionsoftheaverage flowmodelandthehomogenizationmethodarecomparedwithcorre- spondingreferencesolutionsobtainedbysolvingtheoriginalReynolds equationwithhighly-resolvedcomputationalmeshes.Inaddition,the performanceparametersmaximumpressure,load-carryingcapacityand frictionforcearecomputedfromthesolutionsforaquantitativeerror investigation.Tothebestoftheauthors’knowledge,suchaquantitative comparisonisnotavailableintheliteraturesofar.
InSect.2,thetwoaveragingtechniquesareintroducedintermsof theirgoverningequationsandboundaryconditions.Thisisdonewith particularfocusonthegeneralcomputationalprocedureandtohigh- lightthesimilaritiesanddifferencesbetweenthetwomethods.Thead- vantagesofthehomogenizationmethodovertheaverageflowmodel aresummarizedinSect.3.Theapplicationofthemethodstogetherwith aqualitativeandquantitativecomparisonisdemonstratedforthetest casesmentioned aboveinSect.4.Finally,thearticle isconcludedin Sect.5.
2. Averageflowmodelandhomogenizationmethod
Inthiswork,we focusonhydrodynamic lubricationproblemsfor which thelubricatedgap isboundedby onestationarytexturedsur- faceandonemovingsmoothsurface.TheReynoldslubricationequa- tion(Reynolds,1886)describingsuchaproblemisbrieflyintroducedin Sect.2.1.Itisthestartingpointforthetwoaveragingtechniquesstudied inthefollowing.Thegeneralcomputationalprocedureforbothaverag- ingmethodsisdemonstratedinSect.2.2.Afterintroducingthedifferent filmthicknessrepresentationsusedbythetwomethodsinSect.2.3and somenotationfortheparameterdependencyregardingthemacro-and microscalecoordinatesinSect.2.4,wesummarizetheempiricallyde- rived averageflow modelbyPatir andCheng (1978,1979) andthe mathematically derived homogenizedReynolds equation, see for in-
Fig. 1. Illustration of the macro- and mi- croscaledomainsandthecomputationalpro- cedurefortheaverageflowmodelandtheho- mogenizationmethod.
stance(AlmqvistandDasht,2006;Almqvistetal.,2007;2012;Bayada andChambat,1988;BayadaandFaure,1989;Buscagliaetal., 2002;
Jai,1995),inSects.2.5and2.6,respectively.Thesimilaritiesanddif- ferencesbetweenthetwomethodsareemphasizedinSect.2.7. 2.1. Startingpoint:Reynoldsequation
Thetwo-dimensionalReynoldsequation(Reynolds,1886)iswidely used for fluid film lubrication problems. Its derivation from the three-dimensionalNavier-Stokesequationscan be foundforinstance inKhonsariandBooser(2017).Duetounderlyingassumptionssuchas negligibleinertiaforces,thevalidityoftheReynoldsequationislimited.
ItdependsontheReynoldsnumberoftheinvestigatedproblemandin caseofatexturedsurfaceontheaspectratioofthetextures(Dobrica andFillon,2009).Inthiswork,weonlyconsiderproblemsforwhich theReynoldsequationisvalid.
Asmentionedabove,westudyhydrodynamiclubricationproblems withonestationarytexturedsurfaceandonemovingsmoothsurface.
Neglectingcavitation,thestationaryincompressibleReynoldsequation foraniso-viscousandincompressiblelubricantreads
∇⋅(
ℎ(𝒙)3∇𝑝(𝒙))
=6𝜇𝒖𝑎⋅∇ℎ(𝒙) (1)
andissolvedforthehydrodynamicpressure𝑝(𝒙).Thelocalfilmthick- nessℎ(𝒙)isgiven,e.g.,byafunction.Thedynamicviscosity𝜇andthe velocityvector𝒖𝑎=(𝑢𝑎,1,𝑢𝑎,2)𝑇 ofthemovingsurfaceareconstantand alsogiven.
2.2. Generalcomputationalprocedurefortheaveragingtechniques
Theobjective of theaveraging methodsis to avoid theneed for highly-resolvedcomputationalmeshesand,hence,toreducethecom- putationtimewhilepreservingahighsolutionaccuracy.Thisisdone bydividingtheprobleminto onemacroscaleproblemfor whichthe roughnessorthetexturesdonothavetoberesolvedandseveralsmall microscaleproblemswhichcanbesolvedefficientlyandprovidecoef- ficientsenteringthemacroscaleproblem.
ThecoefficientscomputedbytheaverageflowmodelbyPatirand Chengarecalledpressureandshearflowfactors.Thesearedetermined by solvinga reference problem on the microscale being representa- tivefortheroughness/texturesoftheproblemandoccurringperiodi- cally.Theflowfactorsdescribethedeviationoftheflowthroughthe rough/textured gap from the flow through a gap with smooth sur- faces.Incontrasttothis,thehomogenizationmethodis basedonan
asymptoticexpansionofthepressurewhichispluggedintotheoriginal Reynoldsequation(1).Scaleseparationthenleadstosimilarreference problemscalledcellproblems.Thedifferencesinthemethodsstemfrom theirderivations:theaverageflowmodelwasempiricallyderivedby investigatingtheflowbehavior,whereashomogenizationisavalidated mathematicalconcept.
Figure 1 exemplarily illustrates how an averaged pressure 𝑝avg
can becomputed withbothmethods. Thetwo-dimensionalCartesian macroscaledomainΩwithcoordinates𝒙=(𝑥1,𝑥2)𝑇 isoflength𝑙and width𝑤.Itcontains𝑛𝑡=8squaretextures(𝑛𝑡,1=4in𝑥1-directionand 𝑛𝑡,2=2in𝑥2-direction)oflength𝑙𝑡andwidth𝑤𝑡=𝑙𝑡.Thedashedlines indicate the distribution into eight elements of the same size with length 𝑙𝑡𝑒andwidth𝑤𝑡𝑒=𝑙𝑡𝑒,which wecalltexture elements.These areidentifiedbyindices(𝑖,𝑗)asgiveninFig.1.Themicroscaleproblem correspondstoonetextureelementandisrepresentedbythedomain withcoordinates
𝒚= (𝑦1
𝑦2
)
∶=
⎛⎜
⎜⎜
⎝ 𝑥1−𝑖𝑙𝑡𝑒
𝜀 𝑥2−𝑗𝑤𝑡𝑒
𝜀
⎞⎟
⎟⎟
⎠
. (2)
Thescale𝜀oftheroughness/texturesisdefinedby𝜀∶=𝑙𝑡𝑒.Hence,the microscaledomainisscaledsuchthatitisoflengthoneandwidth 𝑤𝑡𝑒∕𝜀.Forsquaretextureelementswith𝑤𝑡𝑒=𝑙𝑡𝑒,thedomainisthe unitsquaresince𝑤𝑡𝑒∕𝜀=1.
Fortheaverageflowmodel,thevelocitymustbealignedtothe𝑥1- direction,i.e.,thecomponent𝑢𝑎,2 in𝑥2-directionhastobezerosuch that𝒖𝑎=(𝑢𝑎,1,0)𝑇.Thereisnosuchrestrictionforthehomogenization method.Thevelocityofthestationarysurfaceisgivenby𝒖𝑏=𝟎.
The only quantity which enters the microscale problems with macroscaleinformationisthelocalnominalfilmthicknessℎnom(𝒙)(av- erageflowmodel)orthelocalgapheightℎ0(𝒙)(homogenization),see Sect.2.3forthedefinitions.Onthemicroscale,ℎnomorℎ0isconstant, whichresultsinaconstantgapheightforthecorrespondingmicroscale problem.Forthebest-possibleaccuracyofthemacroscalesolution,one microscaleproblemhastobesolvedforeachvalueofℎnom(𝒙)orℎ0(𝒙) whichoccursinthediscretizedmacroscaledomain.Alternatively,the microscale problemsareonlysolvedforaselectionoftheheightsin whichcasethemissingvaluescanbeinterpolatedfromthecomputed ones.Onthemicroscale,thecoefficients(see𝜙𝑠(𝒙),etc.inFig.1)are computed bysolvingthemicroscale problemwithvarying boundary conditionsandsubsequentlyaveragingtheparticularsolutionoverthe reference/celldomain.Thesecoefficientsenterthemacroscaleproblem
Fig.2. Localfilmthicknessℎasdefinedintheaverageflowmodel(a)andthe homogenizationmethod(b).
whichisfinallysolvedwithzeroboundaryconditions,i.e.,𝑝avg=0on allfourboundaries.
2.3. Filmthicknessrepresentations
Theaverageflowmodelandthehomogenizationmethodusediffer- entrepresentationsofthefilmthickness.Thedifferencesaredepicted inFig.2foronetextureelementonthemicroscale.Notethat,forthe purposeofillustration,thefigurecontainsthe𝑧-coordinatewhichdoes notexistintheactual2D-domain,cf.Fig.1.Intheaverageflowmodel, seeFig.2(a),thenominalfilmthicknessℎnomisdefinedasthedistance betweenthetwocenterlinesoftheroughness/textures.Inoursetting, thelowersurfaceissmoothsuchthatthesurfaceitselfdefinestheend- pointofℎnom.Thelocaldistancefromtheroughness/texturecenterline isdenotedby𝛿.Incontrast,inthecaseofthehomogenizationmethod, thefilmthicknessiscomposedoftheactuallocalgapheightℎ0andthe localroughness/textureheightℎ1,seeFig.2(b).Insummary,thefilm thicknessesaredefinedby
ℎ(𝒙,𝒚)=ℎnom(𝒙)+𝛿(𝒚), (averageflowmodel) (3)
ℎ(𝒙,𝒚)=ℎ0(𝒙)+ℎ1(𝒚). (homogenization) (4) Inbothcases,thelocalfilmthicknessℎ(𝒙,𝒚)isdividedintoamacroscale (𝒙)andamicroscale(𝒚)part.Notethatthenominalfilmthicknessℎnom
changesiftheroughness/textureheightchangesand,hence,formally alsodependson themicroscale(𝒚).However,aslongasno asperity contactsoccur,wehave
ℎnom(𝒙,𝒚)=ℎ0(𝒙)+const.≡ℎnom(𝒙). (5) 2.4. Notationforparameterdependency
Sections 2.5 and 2.6, respectively, summarize the average flow modelandthehomogenizationmethodin theorderof theircompu- tationalsteps.Theformulasofthetwomethodsaresimilarandgivenin awaysuchthattheyareeasilycomparable.Thedependencies(𝒙,𝒚)and (𝒚;𝒙)canbeexplainedbymeansofthefilmthicknessℎ:basically,the filmthicknessisinfluencedbyboththemacroscaleandthemicroscale, i.e.,ℎ(𝒙,𝒚).However,asdescribedabove,ℎnom(averageflowmodel)or ℎ0(homogenization)isaconstantvalueinthemicroscaleproblemsuch thatℎisvariedonlyby𝛿(averageflowmodel)orℎ1(homogenization), seealso(3)and(4).Consequently,thenotationℎ(𝒚;𝒙)indicatesthatℎ isafunctionin𝒚dependingonaparticularfixedparameter𝒙. 2.5. AverageflowmodelbyPatirandCheng
1. Foreachlocalnominalfilmthicknessℎnom(𝒙): (a) Solvethethreereferenceproblems(microscale)
∇𝒚⋅(
ℎ(𝒚;𝒙)3∇𝒚𝜓0(𝒚;𝒙))
=6𝜇 (𝑢𝑎,1
0 )
⋅∇𝒚ℎ(𝒚;𝒙), (6)
∇𝒚⋅(
ℎ(𝒚;𝒙)3∇𝒚𝜓1(𝒚;𝒙))
=0, (7)
∇𝒚⋅(
ℎ(𝒚;𝒙)3∇𝒚𝜓2(𝒚;𝒙))
=0 (8)
withDirichletandNeumannboundaryconditions
In(6),theright-handsideisgiveninitsgeneralformulation.In ourcasewithnoasperitycontacts,thegradient∇𝒚ℎ(𝒚;𝒙)canbe replacedby∇𝒚𝛿(𝒚).
(b) Computethethreeflowfactors 𝜙𝑠(𝒙)=− 1
6𝜎𝜇𝑢𝑎,1∫ℎ(𝒚;𝒙)3𝜕𝜓0(𝒚;𝒙)
𝜕𝑦1
𝑑𝒚, (shearflowfactor) (9)
𝜙𝑝,1(𝒙)= 1
ℎnom(𝒙)3∫ℎ(𝒚;𝒙)3𝜕𝜓1(𝒚;𝒙)
𝜕𝑦1 𝑑𝒚,
(pressureflowfactor1) (10)
𝜙𝑝,2(𝒙)= 1
ℎnom(𝒙)3∫ℎ(𝒚;𝒙)3𝜕𝜓2(𝒚;𝒙)
𝜕𝑦2 𝑑𝒚
(pressureflowfactor2) (11)
fromthereferenceproblemsolutions𝜓0,𝜓1,𝜓2,where𝜎 isthe standarddeviationoftheroughness.
2. SolvethemodifiedReynoldsequation(macroscale)
∇𝒙⋅ ((
𝜙𝑝,1(𝒙) 0 0 𝜙𝑝,2(𝒙)
)
ℎnom(𝒙)3∇𝒙̄𝑝(𝒙) )
=6𝜎𝜇𝑢𝑎,1∇𝒙⋅ (𝜙𝑠(𝒙)
0 )
+6𝜇 (𝑢𝑎,1
0 )
⋅∇𝒙̄ℎ(𝒙) (12)
with ̄ℎ(𝒙)=∫ℎ(𝒚;𝒙)𝑑𝒚 and boundary conditions as shown in Fig.1with𝑝avg∶= ̄𝑝.Notethat̄ℎ(𝒙)=ℎnom(𝒙)inthecaseofnoas- peritycontacts.
2.6. Homogenization
1. Foreachlocalgapheightℎ0(𝒙):
(a) Solvethethreecellproblems(microscale)
∇𝒚⋅(
ℎ(𝒚;𝒙)3∇𝒚𝜒0(𝒚;𝒙))
=6𝜇 (𝑢𝑎,1
𝑢𝑎,2
)
⋅∇𝒚ℎ1(𝒚), (13)
∇𝒚⋅(
ℎ(𝒚;𝒙)3∇𝒚𝜒1(𝒚;𝒙))
=−∇𝒚⋅( ℎ(𝒚;𝒙)3𝒆1
), (14)
∇𝒚⋅(
ℎ(𝒚;𝒙)3∇𝒚𝜒2(𝒚;𝒙))
=−∇𝒚⋅( ℎ(𝒚;𝒙)3𝒆2
), (15)
where𝒆1=(1,0)𝑇 and𝒆2=(0,1)𝑇,withperiodicboundarycon- ditions
(b)Computethesixcoefficients 𝑏1(𝒙)=
∫ℎ(𝒚;𝒙)3𝜕𝜒0(𝒚;𝒙)
𝜕𝑦1
𝑑𝒚, 𝑏2(𝒙)=
∫ℎ(𝒚;𝒙)3𝜕𝜒0(𝒚;𝒙)
𝜕𝑦2
𝑑𝒚, (16) 𝑎11(𝒙)=
∫ℎ(𝒚;𝒙)3
(𝜕𝜒1(𝒚;𝒙)
𝜕𝑦1
+1 )
𝑑𝒚, 𝑎12(𝒙)=
∫ℎ(𝒚;𝒙)3𝜕𝜒2(𝒚;𝒙)
𝜕𝑦1 𝑑𝒚, (17)
𝑎21(𝒙)=
∫ℎ(𝒚;𝒙)3𝜕𝜒1(𝒚;𝒙)
𝜕𝑦2
𝑑𝒚, 𝑎22(𝒙)=
∫ℎ(𝒚;𝒙)3
(𝜕𝜒2(𝒚;𝒙)
𝜕𝑦2
+1 )
𝑑𝒚 (18)
fromthecellproblemsolutions𝜒0,𝜒1,𝜒2.
2. SolvethehomogenizedReynoldsequation(macroscale)
∇𝒙⋅ ((
𝑎11(𝒙) 𝑎12(𝒙) 𝑎21(𝒙) 𝑎22(𝒙) )
∇𝒙𝑝0(𝒙) )
= −∇𝒙⋅ (𝑏1(𝒙)
𝑏2(𝒙) )
+6𝜇 (𝑢𝑎,1
𝑢𝑎,2
)
⋅∇𝒙ℎ0(𝒙) (19) withboundaryconditionsasshowninFig.1with𝑝avg∶=𝑝0. 3. Computetheupscaledpressure
̃𝑝(𝒙,𝒚)=𝑝0(𝒙)+𝜀𝑝1(𝒙,𝒚) (20)
=𝑝0(𝒙)+𝜀 (
𝜒0(𝒚;𝒙)+𝜕𝑝0(𝒙)
𝜕𝑥1 𝜒1(𝒚;𝒙)+𝜕𝑝0(𝒙)
𝜕𝑥2 𝜒2(𝒚;𝒙) )
(20)
inapost-processingbysimplycombiningthecellproblemsolutions andthegradientoftheaveragedpressuresolution.Asdescribedin Sect.2.2,𝜀istheroughness/texturescalewhichinoursettingisthe lengthofonetextureelement,i.e.,𝜀=𝑙𝑡𝑒.Theupscaling(20)orig- inates from the asymptotic expansion 𝑝(𝒙,𝒚)=𝑝0(𝒙)+𝜀𝑝1(𝒙,𝒚)+ 𝜀2𝑝2(𝒙,𝒚)+…forthepressure,whichisinsertedintotheReynolds equation(1)toderivethehomogenizedReynoldsequation.Sincethe termsstartingfrom𝜀2𝑝2(𝒙,𝒚)aresmallcomparedwiththeleading terms,(20)providesaveryaccurateapproximationofthepressure𝑝 computedfromtheReynoldsequation(1)whileallowingforsignifi- cantlycoarsercomputationalmeshes.ThisisshowninSects.4.2and 4.3.
2.7. Similaritiesanddifferencesbetweenthemethods
Table1summarizesthedifferencesintheformulasoftheaverage flowmodelandthehomogenizationmethod.Theaveragedpressureso- lutions ̄𝑝and𝑝0 canonlybe equaliftheroughness/textureelements aresymmetric(Almqvistetal., 2011b) andiftheflowisattachedto
Table1
Differencesbetweentheaverageflowmodelandthehomogenizationmethod.
average flow model homogenization local film thickness ℎ = ℎ nom+ 𝛿 ℎ = ℎ 0+ ℎ 1
surface velocity 𝒖 𝑎= ( 𝑢 𝑎,1, 0) 𝑇 𝒖 𝑎= ( 𝑢 𝑎,1, 𝑢 𝑎,2) 𝑇 reference problem solutions 𝜓 0, 𝜓 1, 𝜓 2 𝜒0, 𝜒1, 𝜒2
coefficients 𝜙𝑝,1, 𝜙𝑝,2, 𝜙𝑠(flow factors) 𝑎 11, 𝑎 12, 𝑎 21, 𝑎 22, 𝑏 1, 𝑏 2
averaged pressure ̄𝑝 𝑝 0
upscaled pressure - ̃𝑝 = 𝑝 0+ 𝜀 𝑝 1
the𝑥1-coordinate.Then,thecoefficients𝑎12,𝑎21and𝑏2andtheveloc- itycomponent𝑢𝑎,2arezero.Aroughness/textureelementissymmetric ifmirroringtheelementatthecoordinateaxesresultsinthesameel- ement.With𝑎12=𝑎21=𝑏2=𝑢𝑎,2=0,thehomogenizedReynoldsequa- tion(19)reducesto
∇𝒙⋅ ((
𝑎11(𝒙) 0 0 𝑎22(𝒙)
)
∇𝒙𝑝0(𝒙) )
= −∇𝒙⋅ (𝑏1(𝒙)
0 )
+6𝜇 (𝑢𝑎,1
0 )
⋅∇𝒙ℎ0(𝒙). (21) AcomparisonofthemodifiedReynoldsequation(12)of theaverage flowmodelwith(21)revealsthatfor̄𝑝(𝒙)=𝑝0(𝒙)thefollowingrelations musthold:
𝜙𝑝,1(𝒙)ℎnom(𝒙)3=𝑎11(𝒙), 𝜙𝑝,2(𝒙)ℎnom(𝒙)3=𝑎22(𝒙), 6𝜎𝜇𝑢𝑎,1𝜙𝑠(𝒙)=−𝑏1(𝒙),
̄ℎ(𝒙)=ℎnom(𝒙)=ℎ0(𝒙)+const. (22) 3. AdvantagesofthehomogenizedReynoldsequation
Regarding the applicability and accuracy, the homogenization method has several considerable advantages over the average flow modelbyPatirandCheng.Thesearesummarizedinthefollowinglist.
• Homogenizationisamathematicalconcept, whichis validatedby two-scaleconvergence(Allaire,1992; PavliotisandStuart,2008), whereastheaverageflowmodelwasempiricallyderivedbyinvesti- gatingpressureandshearflowforreferenceelementswithsymmet- ricroughness.Hence,incontrasttothemodifiedReynoldsequation (12)oftheaverageflowmodel,thehomogenizedReynoldsequa- tion(19) alwaysprovides anaccurateaveragingof theoriginal Reynoldsequation(1).
• Theboundaryconditionsforthecellproblemson themicroscale (periodic)arealwayscorrect(Almqvistetal.,2011b),whiletheyare notforthereferenceproblemsoftheaverageflowmodel(Dirichlet andNeumann).Inparticular,theperiodicboundaryconditionsallow fornonsymmetricroughness/textures.Then,thecoefficients𝑎12, 𝑎21and𝑏2,see(16),(17)and(18),arenonzero.
• Theupscaling(20)providesahigher-ordersolutionandapprox- imatesthesolutionoftheoriginalReynolds equation(1)well.It allowsforresolvinglocalpressurepeakswhichisimportantfor thedetectionofcriticalstatessuchaslubricationbreakdownorfail- ure(ZhuandWang,2011)orforthedeterminationofaccurateval- uesofthefrictionforce.ThelatterisdemonstratedinSect.4.3.The computationofthehigher-ordersolution ̃𝑝onlyinvolvestheeval- uationof knownquantities,namelythegradient oftheaveraged pressuresolution𝑝0 andthecellproblemsolutions𝜒0,𝜒1and𝜒2. Hence, ̃𝑝canbeevaluatedinapost-processingstepwithnegligible computationalcost.
• Byapplyingtheconceptofreiteratedhomogenization,seeforin- stance(AllaireandBriane,1996),furtherscalescanbetakeninto account.Hence,therangeofapplicationsregardingthestudyoflu- bricationproblemswithanaveragingmethodisextendedfromsys- temswithroughortexturedsurfacestoacombinationofboth.This isdonebydividingtheproblemathandintomacroscale,mesoscale
(textures)andmicroscale(roughness)problems.Aderivationforthe Reynoldsequation(1)withoutandwithacavitationmodelcanbe foundinAlmqvistetal.(2008)andMartin(2008),respectively.In additiontothefirstlocalscaleforthetextureswith𝒚=𝒙∕𝜀,asec- ondlocalscaleisusedfortheroughnesswith𝒛=𝒙∕𝜀2.Notethat forsimplicitytheserelationsaregivenfor(𝑖,𝑗)=(0,0)here,cf.(2). Applyingtheasymptoticexpansionthenleadstocellproblemsfor eachofthetwolocalscales.
Insummary,reiteratedhomogenizationallowsfortheefficientin- vestigationofapplications withtextured surfaceswhile alsocon- sideringthe naturalsurfaceroughness.Up tonow,usually semi- deterministicsimulationsareconductedforthispurpose,seeforin- stance(Maetal., 2017;Profitoetal.,2017),forwhichoftenthe averageflowmodelisusedforthesurfaceroughness,whilethetex- turesarediscretizedbyhighly-resolvedcomputationalmeshes.The computationtimeforsuchsimulationscanbesignificantlyreduced byusingtheconceptofreiteratedhomogenizationinstead.
• AsdescribedinTomaniketal.(2020),anotherproblemoccurswhen applyingtheaverageflowmodeltosurfaceswithdeepdimplessuch assuperficialporesinathermalspraycoatingforplainbearings.The deepdimplesresultinadistortionofthenominalfilmthickness.The nominalfilmthickness,whichistheprimaryreferenceplaneforall calculationswiththeaverageflowmodel,affectsthehydrodynamic filmthickness,thehydrodynamicpressure,theshearrateandthehy- drodynamicfriction.Furthermore,thedistortionofthenominalfilm thicknessleadstoawrongestimationoftheasperitycontactpressure inthecaseofmixed-frictionsimulations,forinstanceinjournalbear- ingswithlowslidingspeed(Grützmacheretal.,2018;Königetal., 2019;2020a)orinstart-stopoperation(Königetal.,2020b).Similar observationsweremadeformicro-coinedandmulti-scaletextured surfaces,viz.acombinationofmicro-coininganddirectlaserinter- ferencepatterning(Königetal., 2020a).Incontrast,byapplying theconceptofhomogenization,neitherforthecellproblemnorfor thehomogenizedproblemthenominalfilmthicknessℎnomorthe standarddeviationof theroughness𝜎 arerequired.Toconclude, thehomogenizationtechniqueavoidsthemodelinguncertainties andthelimitedinterpretabilityoftexturedsurfaceswiththeaverage flowmodel.
Despitetheseadvantagesofthehomogenizationmethodandthepo- tentiallackofaccuracyoftheaverageflowmodel,thelatteriswidely usedintribologicalstudies,mostlyinitsoriginalformulationbyPatir andCheng(1978,1979),eventhoughnumerousmodificationsforits improvementwereproposed(Fatuetal.,2012;Gropperetal.,2016).
4. Applicationofthemethodsandcomparisonofresults
Forthecomparisonofthetwoaveragingmethods,wenumerically studytwoapplications,namelyajournalbearingandaplane-inclined sliderbearing.ThesesetupsaresummarizedinSect.4.1.Afterabrief meshconvergencestudyinSect.4.2,thenumericalresultsarepresented inSect.4.3.Sincebothaveragingmethodsarederivedfromtheoriginal Reynoldsequation(1),thesolutionof(1)isusedasreferencesolution formeasuringtheaccuracyoftheaverageflowmodelandthehomoge- nizationmethod.
4.1. Numericalsetups
ThesimulationparametersarelistedinTab.2.Testcase1isbasedon thejournalbearingapplicationdescribedinGrützmacheretal.(2018), whereastestcase2forasliderbearingissetuptoverifythegeneral- ityofthefindings.Forbothcases,weinvestigatedifferenttextures:(a) symmetricduetosquare/rectangularsurfacearea(isotropic/orthotropic case)and(b)nonsymmetricduetoadiagonallayout(anisotropiccase).
Theparticulartexture elementsareillustratedin Fig.3.Thetexture heightℎ𝑡isconstantforbothtestcasessuchthateitherℎ1(𝒚)=ℎ𝑡or
Table2
Parameters for the simulations of the journal bearing andtheplane-inclinedsliderbearing.TheReynoldsnum- ber𝑅𝑒iscomputedwithℎ∗=𝑐fortestcase1andℎ∗= 0.5(ℎ0,min+ℎ0,max)fortestcase2.
test case 1 2
bearing type journal slider
bearing parameters
diameter 𝑑[mm] 80∕ 𝜋 -
(unfolded) length 𝑙[mm] 80 6
width 𝑤 [mm] 20 2
number of textures 𝑛 𝑡,1[-] 160 60 number of textures 𝑛 𝑡,2[-] 40 50 total number of textures 𝑛 𝑡[-] 6 , 400 3 , 000 lubricant parameters
density 𝜌[kg/m 3] 820 820
dynamic viscosity 𝜇[Pa s] 0.014 0.014 operating conditions
radial clearance 𝑐[ μm] 17.5 - minimum gap height ℎ 0,min[ μm] 3.5 5 maximum gap height ℎ 0,max[ μm] 31.5 10 relative eccentricity 𝑒 rel[-] 0.8 - convergence ratio 𝑘 [-] - 1 surface velocity 𝑢 𝑎,1[m/s] 0.2 4 Reynolds number 𝑅𝑒 = 𝜌𝑢 𝑎,1ℎ ∗∕ 𝜇[-] 0.205 1.757
ℎ1(𝒚)=0inthemicroscaledomain.Theratio𝐴𝑡∕𝐴𝑡𝑒ofthetexture areatotheareaofthewholetextureelementissimilarforallfourtex- tures.AsdiscussedinSects.2and3,wecanonlyexpectaccurateresults oftheaverageflowmodelincaseofthesymmetrictextures,whereasthe homogenizationmethodshouldprovideaccurateresultsforalltextures.
Thegapheightsforthejournalbearing(testcase1)andtheslider bearing(testcase2)areprescribedby
ℎ0(𝒙)=𝑐 (
1+𝑒relcos (2𝑥1
𝑑 ))
(23)
and
ℎ0(𝒙)=ℎ0,min−ℎ0,max
𝑙 𝑥1+ℎ0,max, (24)
respectively.
All numericalsimulations areperformedon quadrilateral meshes with finiteelement solvers we implemented usingthe C++ library deal.II(Bangerthetal.,2007).Detailsontheweakformulationsforthe cellproblems(13)-(15)andthehomogenizedproblem(19)aswellas informationontheirimplementationcanbefoundinRomandMüller (2018).
4.2. Meshconvergence
Forareliablecomparisonof thenumericalresults,weperformed a thorough meshconvergence study for both test cases for each of thetexturesetups.Thiscomprisesthemeshesforsolvingtheoriginal Reynoldsequation(1),whichservesasreferencesolution,themeshes forthereference/cellproblemsonthemicroscaleandthemeshesforthe macroscaleproblems.Exemplarily,wepresenttheresultsofthestudy fortestcase1withsquaretexturesinthissection.
Figure 4 (a) shows thesolution 𝑝 of the Reynolds equation (1), Fig.4(b)thesolution𝜒0ofthecellproblem(13)forℎ0=31.5μmand Fig.4(c)theupscaledsolutioñ𝑝,see(20),ofthehomogenizedproblem (19),eachintheareaofthemaximumoftheparticularsolutionandfor anincreasingnumberofmeshcells.IncaseoftheReynoldsequation(a) andthecellproblem(b),themaximumincreaseswithincreasingmesh resolution, butthedistancesbetween thecurves decrease.Themesh resolutionof10,240× 2,560fortheReynoldsproblemcorrespondstoa resolutionof64× 64pertextureelement.Foragoodcomparabilitywith thesolutionofthehomogenizedproblem,wechoose64× 64alsoasfinal resolutionforthecellproblems.Thisisdoneanalogouslyforthethree
Fig.3. Textureelementsforthetwotestcases.
Fig.4. Meshconvergencefortestcase1with squaretextures.
othersetups,i.e.,fortestcase1diagonalandtestcase2rectangularand diagonal.AsFig.4(c)shows,thehomogenizedproblemislessmesh- dependentthantheReynoldsandthecellproblem,andaresolutionof 1,280× 320isclearlysufficient.RelatedtotheReynoldsproblem,thisis areductioninthenumberofmeshcellsof98.4%.Themeshconvergence behavioroftheaverageflowmodelisanalogoustothatofthehomog- enizationmethod.Notethatthediagonaltexturesrequireafinermesh toadequatelyresolvetheedgesofthetextures.Togiveanexample,the numberofmeshcellsweusefortheReynoldsproblemfortestcase1 withdiagonaltexturesis16,000× 4,000,whichcorrespondsto100× 100 pertextureelement.
4.3. Numericalresults
Inthefollowing,wecomparethesolutionsoftheaverageflowmodel andthehomogenizationmethodwiththesolutionoftheReynoldsequa- tion.Foraquantitativeinvestigationoftheaccuracyoftheaveraging methods,weconsiderthemaximumpressure𝑝maxandtheperformance parametersload-carryingcapacity𝑊 andfrictionforce𝐹. Thelatter twoaredefinedby
𝑊 ∶=
∫Ω
[𝑝]+𝑑𝒙, (25)
whereonlypositivevalues[𝑝]+=max(0,𝑝)ofthepressurearetakeninto account,and
𝐹∶=
∫Ω𝜏 𝑑𝒙=
∫Ω
(ℎ 2
𝜕𝑝
𝜕𝑥1
+𝜇𝑢𝑎,1
ℎ )
𝑑𝒙, (26)
respectively,seeforinstance(WenandHuang,2018).Here,𝑝represents thesolutionoftheReynoldsequation.Itcanbereplacedbȳ𝑝(average flowmodel),𝑝0(homogenization)or̃𝑝(homogenizationwithupscaling).
4.3.1. Testcase1(journalbearing)
Figure5(a)showsapartofthedifferentpressuresolutionsforthe journalbearing oftestcase 1withsymmetric (square)textures.The curveisextractedattheposition𝑥2=10.25mm.Asexpected,theav- eragedpressures ̄𝑝and𝑝0 oftheaverageflowmodelandthehomog- enizationmethod,respectively,areidenticalandprovideanaccurate averagingoftheReynoldssolution𝑝.Theupscaledpressurẽ𝑝oftheho- mogenizationmethodagreeswellwiththeReynoldssolution𝑝.Inthe caseofthenonsymmetric(diagonal)texturesinFig.5(b),thesolution̄𝑝 oftheaverageflowmodelshowsalargedeviationfromtheReynolds solution𝑝,whilethehomogenizationmethodagainleadstoaccurate results.
These findings aresubstantiated by aquantitative comparison of 𝑝max, 𝑊 and 𝐹. The values are listedin Tab. 3. Inthe symmetric case (a), 𝑝max and𝑊 arealmost thesamefor all foursolutions. In contrast,𝑝max=0.4139MPaand𝑊 =73.27Ncomputedbytheaverage flowmodelforthenonsymmetrictextures(b)deviatefromthecorre- spondingReynoldsvaluesby9.8%and7.1%,respectively.Thefriction forces𝐹 whicharecomputedfromtheaveragedsolutions ̄𝑝and𝑝0are alwaysinaccuratefortexturedapplications,regardlessofthesymme- try.Thisisduetonotresolvingthepressurepeakswhichleadstoan errorinthepressuregradiententeringthecomputationof𝐹,see(26). ThedeviationsfromtheReynoldsvaluesliebetween8.4%(symmetric case,𝐹=0.4068N)and10.9%(nonsymmetriccase,averageflowmodel,
Fig.5.Pressuresolutionsat𝑥2=10.25mmfor testcase1.
Table3
Maximumpressure𝑝max[MPa],load-carryingcapacity𝑊[N]andfrictionforce𝐹[N]computedfrom thedifferentpressuresolutionsfortestcase1.
(a) symmetric textures (square) (b) nonsymmetric textures (diagonal) Reynolds avg. flow homog. upscaled Reynolds avg. flow homog. upscaled 𝑝 max 0.5990 0.5987 0.5987 0.5991 0.4591 0.4139 0.4586 0.4604 𝑊 102.51 102.45 102.45 102.48 78.85 73.27 79.20 79.23 𝐹 0.4443 0.4068 0.4068 0.4443 0.4231 0.3771 0.3804 0.4222
Fig.6. Pressuresolutionsfortestcase1withnonsymmetric(diagonal)textures.
𝐹=0.3771N).Notethateveninthenonsymmetriccase,thevalueofthe averageflowmodel(𝐹=0.3771N)isalmostequaltothevalueofthe homogenizationmethod(𝐹 =0.3804N).Thisindicatesthattheslopesof theaveragedpressuresolutionsin𝑥1-directionand,hence,thepressure gradientsaresimilar.
Figure6visualizesthereasonfortheinaccurateaveragingoftheav- erageflowmodelincaseofthenonsymmetrictextures.Thesetextures leadtoanasymmetryofthepressuresolution,seeFig.6(a).Thisbe- haviorcannotbecapturedbytheaverageflowmodel,seeFig.6(b).In
contrast,thehomogenizationmethodisabletoreproducetheasymme- try,seeFig.6(c).Thisisduetotheadditionalcoefficients𝑎12,𝑎21and 𝑏2,cf.(12)and(19),whicharenonzerointhiscase.
4.3.2. Testcase2(plane-inclinedsliderbearing)
ThepressureplotsinFig.7fortestcase2,extractedat𝑥2=1.02mm, revealthattheinfluenceofthetexturesonthepressureislargerthanin testcase1.Apartfromthat,thebehaviorissimilar.Thehomogenization methodprovidesanaccurateaveragingandaverygoodupscalingfor boththesymmetric(rectangular)andthenonsymmetric(diagonal)tex- tures.Incontrast,theapplicationoftheaverageflowmodelleadstoa strongdeviationfromtheReynoldssolutioninthenonsymmetriccase.
Duetothelargerinfluenceofthetextures,themaximumpressures 𝑝max resultingfromtheaveragingtechniquesaretoolow,cf.Tab.4, by4.3%(symmetriccase),4.6%(nonsymmetriccase,homogenization withoutupscaling)andeven17.2%(nonsymmetriccase,averageflow model).Similarly,thereisonlyonelargedeviationregardingtheload- carryingcapacity 𝑊: inthenonsymmetric case,thevaluecomputed fromtheaverageflowmodel(𝑊 =1.4208N)deviatesfromtheReynolds value(𝑊 =1.6178N)by12.2%.Thedeviationsofthefrictionforces𝐹 resultingfromtheaveragingtechniquesfromtheReynoldsvaluesare slightlylargerthanfortestcase1andliebetween10.8%(nonsymmet- riccase,homogenizationwithoutupscaling,𝐹=0.07615N)and12.9% (symmetriccase,𝐹=0.07489N).
Asfortestcase1inFig.6,apressureasymmetryisalsovisibleinthe contourplotsfortestcase2inFig.8.Whilethereishardlyanydifference visiblebetweentheReynoldssolution(a)andtheupscaledsolution(c) ofthehomogenizationmethod,theasymmetryagaincannotbecaptured bytheaverageflowmodel(b).
Finally, Fig. 9 demonstrates how the upscaled pressure solution (20)is composedoftheaveragedmacroscalesolution𝑝0 andthemi- croscale solution 𝜀𝑝1. The pressure distribution in Fig. 8 (c) is the sumofthepressuresdepictedinFig.9(a)and(b),i.e., ̃𝑝=𝑝0+𝜀𝑝1. The texture-inducedasymmetry is visiblein both theaveragedsolu- tion 𝑝0 and themicroscale solution 𝜀𝑝1. The absolute values of 𝜀𝑝1
reachuptoapproximately14%oftheaveragedpressurevalues𝑝0with (𝜀𝑝1)max=0.047MPaand𝑝0,max=0.332MPa.
Fig.7.Pressuresolutionsat𝑥2=1.02mmfor testcase2.
Table4
Maximumpressure𝑝max[MPa],load-carryingcapacity𝑊[N]andfrictionforce𝐹[N]computedfromthe differentpressuresolutionsfortestcase2.
(a) symmetric textures (rectangular) (b) nonsymmetric textures (diagonal) Reynolds avg. flow homog. upscaled Reynolds avg. flow homog. upscaled 𝑝 max 0.3550 0.3399 0.3399 0.3551 0.3483 0.2885 0.3322 0.3501 𝑊 1.6378 1.6359 1.6359 1.6383 1.6178 1.4208 1.6300 1.6317 𝐹 0.08594 0.07489 0.07489 0.08595 0.08534 0.07606 0.07615 0.08531
Fig.8. Pressuresolutionsfortestcase2withnonsymmetric(diagonal)textures.
5. Conclusion
This work demonstrates the advantages of the homogenization methodovertheaverageflowmodelbyPatirandCheng.Thequali-
Fig.9. Componentsoftheupscaledpressurẽ𝑝from(20)fortestcase2with nonsymmetric(diagonal)textures.
tativeandquantitativecomparisonsofnumericalresultsforajournal bearingandaplane-inclinedsliderbearingclearlyshowthathomoge- nizationshouldbethemethodofchoice.
Theaverageflowmodelandthehomogenizationmethodarevery similarintermsoftheirgoverningequationsandcomputationalproce- dure.Theymainlydifferintheboundaryconditionsofthemicroscale
