Lattice Boltzmann Method for the Elder Problem

[e-mail]~[e-mail]~sukopmLatticeBoltzmannMethodfortheElderProblemCMWR2004DepartmentofEarthSciencesMotivationCMWR2004•Groundwatermodelingofhighlynon-linearcoupledequationsthatgovernsolutetransport/density-drivenflowproblemsarechallenging.•LatticeBoltzmannmodels(LBMs)cansimulatemanyfluiddynamicsproblemsthattraditionallyrequiredmorecomplexapproaches.•WeuseamulticomponentLBMtoaddresstheclassicsolute-inducedbuoyancyElderproblem,whichhasbeenwidelyadoptedasa“benchmark”amongdensity-dependentflowmodelers.•ThisworkrepresentsfirststepsinthedevelopmentofLBMsforapplicationtogroundwaterproblems(e.g.salt-waterintrusionincoastalregions).Thorne&SukopLEHC=1C=0ElderProblemCMWR2004Elder,J.W.(1967)J.FluidMech.27(3)609-623Voss,C.I.,W.R.Souza(1987)Wat.Resour.Res.23,1851-1866Thorne&SukopE/H=4L/H=2Temperature-inducedbuoyancySolute-inducedbuoyancyHeaterSaltSourceLBMBasicsILatticeUnit,luCMWR2004012345678D2Q9e1e2e3e4e5e6e7e8f1f2f3f4f5f6f7f8f0Histogramviewofthedistributionfunction,f.Thorne&Sukopf1f5125aff2f3f434678f6f7f8Notes:•Basedonstatisticalmechanicsandkinetictheory.•Soluteandbuoyancywillbecoveredinlaterslides.StreamingStepCMWR2004Thorne&Sukop(f*←f)CollisionStepCMWR2004f*Thorne&SukopCollisionStepCMWR2004feqThorne&SukopCollisionStepCMWR2004t-1(feq-f*)Thorne&SukopCollisionStepCMWR2004f=f*+t-1(feq-f*)Thorne&SukopLBMBasicsIICMWR2004Thisfeqsimulatesacompressiblefluid.(Itcanbemodifiedtosimulateincompressible.)(BGK=Bhatnagar,Gross,andKrook.)Wolf-Gladrow,D.A.,Lattice-GasCellularAutomataandLatticeBoltzmannModels:AnIntroduction,Springer,Berlin,2000,308pp.Succi,S.,TheLatticeBoltzmannEquationforFluidDynamicsandBeyond,ClarendonPress,Oxford,2001,288pp.Initializer,uandfeq.Initializef←feq.Streamf*←f.Computerandufromf*.Computefequsingrandu.Collidef=f*+t-1(feq–f*).12345678Thorne&SukopLBMBasicsIIICMWR2004fluidsolidTimet:Initialconfiguration.Latticenodeadjacenttowallboundarybeforestreamingstep.Bounce-backno-slipwallboundaryThorne&SukopLBMBasicsIIICMWR2004fluidsolidBounce-backno-slipwallboundaryTimet:Afterstreaming.Directionaldensitiesstreamintothesolid(fortemporarystorage).Thorne&SukopLBMBasicsIIICMWR2004fluidsolidBounce-backno-slipwallboundaryTimet:Bounceback.Temporarilystoreddensitiesreflectbacktowardthefluidlatticenode.Thorne&SukopLBMBasicsIIICMWR2004fluidsolidBounce-backno-slipwallboundaryTimet+Dt:Afterstreaming.Densitiesstreambackintothefluidpointinginthedirectionexactlyoppositetheirinitialdirection.Thorne&SukopBodyForcesCMWR2004Bodyforcessuchasgravityareimplementedasanadditiontothevelocityvectorwithffromthelasttimestep.Martys,N.S.andH.Chen(1996)Phys.Rev.E53,743Thorne&Sukop•Soluteissimulatedbyaseconddistributionfscalledthe“solutecomponent”ors-component.•Thiscorrespondstothefluiddistributionfunctionexceptwithasimplerequilibriumdistribution•Concentration://Analogoustofluiddensity.•Diffusioncoefficient://Analogoustoviscosity.•Modifiedbodyforce:SoluteTransportCMWR2004Drivesbuoyantflow.Yoshino,M.andT.Inamuro(2003)Int.J.Numer.Meth.Fluids43,183.Thorne&SukopPorousMediaCMWR2004Thorne&Sukopns(x)•Specifysoliddensityvalueateachlatticenode.//ns=1-f•Simulateporousmediumby“probabilisticbounce-back”basedonsoliddensity.Requiresprohibitivelylargelatticestotranscendporescale!Dardis,O.andJ.McCloskey(1998)Phys.Rev.E57,4834-4837Dardis,O.andJ.McCloskey(1998)Geophys.Res.Let.25,1471-1474Notes•Probabilisticbounce-back.•Fractured/heterogeneousporousmedia.•Dispersion/diffusionunderPM?//FutureworkPorousMediaCMWR2004Considerthetraditionalcollisionstepasasecondintermediatestepafterstreaming,denotedbyf**Thentheporousmediumstephastheformwherea´istheindexofthedirectionoppositeea.Permeability:ns=0,free-fluid.ns=1,no-flow.a´axx+eaDtDardis,O.andJ.McCloskey(1998)Phys.Rev.E57,4834-4837Dardis,O.andJ.McCloskey(1998)Geophys.Res.Let.25,1471-1474Thorne&SukopLEHC=1C=0ElderProblemCMWR2004Elder,J.W.(1967)J.FluidMech.27(3),609-623Thorne&Sukop//ControllingparameterLEHC=1C=0ElderProblemCMWR2004Thorne&SukopElder,J.W.(1967)J.FluidMech.27(3),609-623//ControllingparameterResultsCMWR2004Notes•Nofullyacceptedresults(computerorlab).•Maybenouniquesolution.Elder,J.W.(1967)J.FluidMech.27(1),29-48Elder,J.W.(1967)J.FluidMech.27(3),609-623Woods,J.A.,etal.(2003)Wat.Resour.Res.39,1158-1169Thorne&SukopThorne&Sukop(2004)Elder(1967)20%60%20%60%60%20%60%20%60%20%Year1Year2Year10Year4Year15Year20CMWR2004ResultsFrolkovič,P.,H.DeSchepper(2001)Adv.Wat.Res.24,63-72Thorne&SukopThorne&Sukop(2004)Frolkovič&DeSchepper(2001)20%40%60%80%20%40%60%80%20%40%60%80%20%40%60%80%20%40%60%80%20%40%60%80%80%80%Year15Year4Year1Year2Year10Year20Results(year15)CMWR2004Thorne&SukopThorne&Sukop(2004)Elder(1967)Thorne&Sukop(2004)Frolkovič&DeSchepper(2001)Year15Year1520%40%60%80%80%80%20%60%Conclusions•LBMsappeartobeahighlyviabletoolformodelingnon-linearcoupleddensity-dependentflowproblems.•OurresultscomparefavorablywithElder'sclassicresultsandwithmodernstate-of-the-artresults,suchasthosefoundinFrolkovičandDeSchepper(2001),thoughnodefinitivesolutionexists.•