fluctuation

Largeeddysimulationofvortexsheddingandpressure

fluctuationinaerostaticbearings

JinchengZhua,HanChena,b,XuedongChena,n

aStateKeyLaboratoryofDigitalManufacturingEquipmentandTechnology,HuazhongUniversityofScienceandTechnology,

Wuhan430074,China

bDepartmentofMechanics,HuazhongUniversityofScienceandTechnology,Wuhan430074,China

articleinfo

Articlehistory:

Received21May2012

Accepted5March2013

Availableonline28April2013

Keywords:

Aerostaticbearing

Largeeddysimulation

Vortexshedding

Pressurefluctuation

Vibration

abstract

Inaerostaticbearings,highspeedairflowmayinducesmallvibration,whichhasbeen

harmfultotheimprovementofmovingandpositioningaccuracyofaerostatically

paper,thetransientflowfield

intheaerostaticbearingisnumericallyinvestigatedusingthelargeeddysimulation

entstructuresarestudiedandvortexsheddingphenomenonisdiscovered

putationalresultsdemonstratethatvortexsheddingcaus

onshipbetweenpressurefluctuation

andbearingvibrationistablishedbadonoursimulationresultsandexperimentally

measuredvibrationstrength.

&htsrerved.

uction

Aerostaticbearingshahemeritof

near-zerofrictionandlowheatgeneration,applicationsofaerostaticbearingsmakeitpossibleforsupporteddevicesto

r,withtheincreasingdemandofpositioningaccuracy,theinherentsmall

vibrationontheorderofnanometers(Kawaietal.,2005)verelydamagesstabilityandprecisionofthebearing,especially

rstandandeventuallysuppressthisharmfulvibration,traditionaldesign

andanalysismethodsformassflowrateandloadcarryingcapacitydonotsufficeanymore,andlotsofrearcheffortshave

beenmadeontheairflowfieldinaerostaticbearings.

Recently,therelationshipbetweenthehighspeedairflowandthesmallvibrationinaerostaticbearingshasbeen

tal.(2005)studiedthenano-vibrationinultra-precisionmachinetoolsandattributed

dHe(2006)foundvortexflowstructuresinthebearingrecess

bycomputationalfluiddynamics(CFD)simulationofthesteadyairflowfield,andsuggestedthattheairvorticesare

etal.(2006)alsoobrvedthisairvortexflowbyCFD

simulationandreachedasimilarconclusion,andaccordinglypropodanewrestrictordesigntoweakenthevibration.

Zhangetal.(2007)analyzedthehighReynoldsnumber(Re)flowinthebearingclearance,andreducedthevibrationof

entwork,Yoshimuraetal.(2012)attributednano-vibrationofaerostatic

bearingswithsurfacerestrictiontoghthe

flow-inducednatureofbearingvibrationhasgenerallybeenrecognized,thepreviousworksonlyassumedasteadyflow

ContentslistsavailableatSciVerScienceDirect

journalhomepage:/locate/jfs

JournalofFluidsandStructures

0889-9746/$-efrontmatter&htsrerved.

/10.1016/structs.2013.03.012

./fax:+862787557325.

E-mailaddress:chenxd@().

JournalofFluidsandStructures40(2013)42–51

fieldoraveragedtheflowfieldinaReynoldsAveragedNavier–Stokes(RANS)hisflow-inducedvibrationis

apparentlyatimedependentprocess,timedependentisnecessarytoinvestigatethetransientairflowfieldinthebearing

clearanceinordertofurtherunderstandthisharmfulsmallvibration.

Tonumericallyanalyzethedetailedflowcharacteristicsinaerostaticbearings,thefullNavier–Stokequationsfor

hehighspeedairflowinthebearinggapneartheorificeoutletisturbulent,

RANSsimulationisusuallyemployed,andnumericalresultsdemonstrateadequateaccuracyinpredictingmeanflow

characteristics(ChenandHe,2006;LiandDing,2007).Pressuredepressions(Eleshaky,2009;Yoshimotoetal.,2007)and

vortexflowstructures(Chenetal.,2011)r,

RANSsimulationadoptsastatisticy,direct

numericalsimulation(DNS)canresolvethewholespectrumofturbulentscalesasnoturbulentmodelisassumed,butits

eeddysimulation(LES),largescaleturbulenteddiesaresolveddirectlyand

,coherentturbulentstructurescanbeobtainedwith

acceptablecomputationalcostinLES,whichhasbeenvalidatedinvariousapplications(Chengetal.,2012;Lametal.,2010;

Tucker,2011).

Inapreviousstudy(Chenetal.,2011),steadyRANSsimulationmethodwasudtostudyairflowfieldsinvarious

aerostaticbearingswithdifferentparametersandrecessshapes,andtherelationshipbetweenvortexstrengthandvibration

r,notransientflowcharacteristicsintheaerostaticbearingcouldberesolved.

Inthispaper,theulationresultsreveal

ionofthebearingisalsomeasuredexperimentally,

anditisdemonstratedthatvibrationstrengthofthebearingincreaswithincreasingpressurefluctuationinducedby

vortexsheddinginthebearingrecess.

calmodeling

InLES,largeeddiesofturbulencearedirectlyresolvedandeddieswithscalessmallerthangridspacingaremodeled.

ThegoverningequationmployedinLESarethetime-dependentFavreFilteredNavier–Stokequations,including

continuityandmomentumequations:

∂ρ

∂t

þ

∂

∂xi

ðρ

~u

iÞ¼0;ð1Þ

∂

∂t

ðρ

~u

iÞþ

∂

∂xj

ðρ

~u

i

~u

jÞ¼−

∂p

∂xi

þ

∂

~

sij

∂xj

−

∂

∂xj

ð~τijÞ;ð2Þ

theFavrefilteristhedensity-weightedfilter,wheredensityandpressurearespatialfiltered(denotedby“

–”)whilevelocity

isdensity-weighted(

e

üρÃ=ρ,ndenotesageneralvariable).InEq.(2),s

ij

istheviscousstresstensorandτ

ij

isthesubgrid-

scale(SGS)stress,whicharedefinedas

~sij¼μ

∂

~u

i

∂xj

þ

∂

~u

j

∂xi

−

2

3

δij

∂

~u

k

∂xk