Lattice_Boltzmann_Chen

P1:ARK/aryP2:MBL/vksQC:MBL/bsaT1:MBLNovember24,199715:23AnnualReviewsAR049-12Annu.Rev.FluidMech.1998.30:329–64Copyrightc1998byAnnualReviewsInc.AllrightsreservedLATTICEBOLTZMANNMETHODFORFLUIDFLOWSShiyiChen1,2andGaryD.Doolen21IBMResearchDivision,T.J.WatsonResearchCenter,P.O.Box218,YorktownHeights,NY10598;2TheoreticalDivisionandCenterforNonlinearStudies,LosAlamosNationalLaboratory,LosAlamos,NM87545;e-mail:[e-mail]:latticeBoltzmannmethod,mesoscopicapproach,fluidflowsimulationABSTRACTWepresentanoverviewofthelatticeBoltzmannmethod(LBM),aparallelandefficientalgorithmforsimulatingsingle-phaseandmultiphasefluidflowsandforincorporatingadditionalphysicalcomplexities.TheLBMisespeciallyusefulformodelingcomplicatedboundaryconditionsandmultiphaseinterfaces.Recentextensionsofthismethodaredescribed,includingsimulationsoffluidturbulence,suspensionflows,andreactiondiffusionsystems.INTRODUCTIONInrecentyears,thelatticeBoltzmannmethod(LBM)hasdevelopedintoanalternativeandpromisingnumericalschemeforsimulatingfluidflowsandmodelingphysicsinfluids.Theschemeisparticularlysuccessfulinfluidflowapplicationsinvolvinginterfacialdynamicsandcomplexboundaries.Unlikeconventionalnumericalschemesbasedondiscretizationsofmacroscopiccon-tinuumequations,thelatticeBoltzmannmethodisbasedonmicroscopicmod-elsandmesoscopickineticequations.ThefundamentalideaoftheLBMistoconstructsimplifiedkineticmodelsthatincorporatetheessentialphysicsofmicroscopicormesoscopicprocessessothatthemacroscopicaveragedprop-ertiesobeythedesiredmacroscopicequations.Thebasicpremiseforusingthesesimplifiedkinetic-typemethodsformacroscopicfluidflowsisthatthemacroscopicdynamicsofafluidistheresultofthecollectivebehaviorofmanymicroscopicparticlesinthesystemandthatthemacroscopicdynamicsisnotsensitivetotheunderlyingdetailsinmicroscopicphysics(Kadanoff1986).Bydevelopingasimplifiedversionofthekineticequation,oneavoidssolving3290066-4189/98/0115-0329$08.00P1:ARK/aryP2:MBL/vksQC:MBL/bsaT1:MBLNovember24,199715:23AnnualReviewsAR049-12330CHEN&DOOLENcomplicatedkineticequationssuchasthefullBoltzmannequation,andoneavoidsfollowingeachparticleasinmoleculardynamicssimulations.EventhoughtheLBMisbasedonaparticlepicture,itsprincipalfocusistheaveragedmacroscopicbehavior.Thekineticequationprovidesmanyoftheadvantagesofmoleculardynamics,includingclearphysicalpictures,easyimplementationofboundaryconditions,andfullyparallelalgorithms.Becauseoftheavailabilityofveryfastandmassivelyparallelmachines,thereisacurrenttrendtousecodesthatcanexploittheintrinsicfeaturesofparallelism.TheLBMfulfillstheserequirementsinastraightforwardmanner.ThekineticnatureoftheLBMintroducesthreeimportantfeaturesthatdis-tinguishitfromothernumericalmethods.First,theconvectionoperator(orstreamingprocess)oftheLBMinphasespace(orvelocityspace)islinear.Thisfeatureisborrowedfromkinetictheoryandcontrastswiththenonlinearconvectiontermsinotherapproachesthatuseamacroscopicrepresentation.Simpleconvectioncombinedwitharelaxationprocess(orcollisionoperator)allowstherecoveryofthenonlinearmacroscopicadvectionthroughmulti-scaleexpansions.Second,theincompressibleNavier-Stokes(NS)equationscanbeobtainedinthenearlyincompressiblelimitoftheLBM.ThepressureoftheLBMiscalculatedusinganequationofstate.Incontrast,inthedirectnu-mericalsimulationoftheincompressibleNSequations,thepressuresatisfiesaPoissonequationwithvelocitystrainsactingassources.Solvingthisequationforthepressureoftenproducesnumericaldifficultiesrequiringspecialtreat-ment,suchasiterationorrelaxation.Third,theLBMutilizesaminimalsetofvelocitiesinphasespace.InthetraditionalkinetictheorywiththeMaxwell-Boltzmannequilibriumdistribution,thephasespaceisacompletefunctionalspace.Theaveragingprocessinvolvesinformationfromthewholevelocityphasespace.BecauseonlyoneortwospeedsandafewmovingdirectionsareusedinLBM,thetransformationrelatingthemicroscopicdistributionfunc-tionandmacroscopicquantitiesisgreatlysimplifiedandconsistsofsimplearithmeticcalculations.TheLBMoriginatedfromlatticegas(LG)automata,adiscreteparticleki-neticsutilizingadiscretelatticeanddiscretetime.TheLBMcanalsobeviewedasaspecialfinitedifferenceschemeforthekineticequationofthediscrete-velocitydistributionfunction.Theideaofusingthesimplifiedkineticequationwithasingle-particlespeedtosimulatefluidflowswasemployedbyBroadwell(Broadwell1964)forstudyingshockstructures.Infact,onecanviewtheBroadwellmodelasasimpleone-dimensionallatticeBoltzmannequation.Multispeeddiscreteparticlevelocitiesmodelshavealsobeenusedforstudyingshock-wavestructures(Inamuro&Sturtevant1990).Inallthesemodels,al-thoughtheparticlevelocityinthedistributionfunctionwasdiscretized,spaceandtimewerecontinuous.Thefulldiscreteparticlevelocitymodel,whereP1:ARK/aryP2:MBL/vksQC:MBL/bsaT1:MBLNovember24,199715:23AnnualReviewsAR049-12LATTICEBOLTZMANNMETHOD331spaceandtimearealsodiscretizedonasquarelattice,wasproposedbyHardyetal(1976)forstudyingtransportpropertiesoffluids.Intheseminalworkofthelatticegasautomatonmethodfortwo-dimensionalhydrodynamics,Frischetal(1986)recognizedtheimportanceofthesymmetryofthelatticeforthere-cov