Lattice Boltzmann Models for Axisymmetric Flows.pdf

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1、LatticeBoltzmannModelsforAxisymmetricFlowsTimPhillipsCardi®UniversityCollaborator:TimReis(UniversityofOxford)Cardi®LBMWorkshop3-4September2009MotivationIStandardLBMforincompressible°owsisbasedontheCartesiancoordinatesystemIManyimportant°owproblemspossessaxialsy

2、mmetrye.g.°owpastasphereIComputationaldemandfor3DLBMsisconsiderablygreaterthanfor2DLBMsIReductionindimensionalityallowsforgreaterspatialre¯nementRationaleITheideaistoaddsuitablesourcetermstotheLBGKevolutionequationtorecovertheaxisymmetricNavier-Stokesequationsi

3、nthemacroscopiclimit(Hallidayetal.(2001)).IPremnathandAbraham(2005)usedasimilarideaformultiphase°owstoaccountforaxisymmetriccontributionsoftheorderparameterofthe°uidphasesandinertial,viscousandsurfacetensionforcescf.continuumsurfaceforcemodelofBrackbilletal.(19

4、92).HistoricalOverviewIHallidayetal.(2001)proposedanaxisymmetricD2Q9modelforaxisymmetric°ows.Sometermsaremissinginthemodel-theNavier-Stokesequationsarenotrecoveredinthemacroscopiclimit.IPremnathandAbraham(2005)adoptasimilarapproachformultiphase°ows.IGuoetal.(20

5、02)presentageneralapproachforincorporatingforcingtermsintotheLBMframework-discretelatticee®ectsareconsidered.IReisandPhillips(2007)followthisapproachtoderiveanaxisymmetricLBM,whichwasvalidatedonsomebenchmarkproblems.IHuangetal.(2009)generalisetheseapproachestoi

6、ncludeproblemsforwhichthereisabodyterminthemomentumequationAxisymmetricFlowConsiderthe°owofanincompressible,isotropic°uidthroughathree-dimensionalpipe.Leter,eµandezbethestandardorthonormalunitvectorsde¯ningacylindricalcoordinatesystem:xyyxer=;;0;eµ=;¡;0;ez=

7、(0;0;1);(1)rrrrpwherer=x2+y2,x=rcosµandy=rsinµ.IfthesolutiontotheNavier-Stokesequationisoftheformu=ur(r;z)er+uz(r;z)ez;(2)thatisthevelocity¯elddoesnotdependonµ,thenthe°owissaidtobeaxisymmetric(withoutswirl).2DNavier-StokesEquationsThecontinuityequationincylindr

8、icalcoordinatesis@urur@uz++=0;(3)@rr@zandthecomponentsofthemomentumequationare:@ur@ur@ur1@P+ur+uz=¡(4)@t@r@z½@r22@ur1@urur@ur+º+¡+;@r2r@rr2@z2@uz@uz@uz1@P+ur+uz=¡(5)@t@r@z