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Lattice Boltzmann Methods - Pennsylvania State University1. Initialize macroscopic properties and...
Transcript of Lattice Boltzmann Methods - Pennsylvania State University1. Initialize macroscopic properties and...
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Lattice Boltzmann Methods DavidChasBoltonBrandonSchwartz
SrisharanShreedharan
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Introduction
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Introduction
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• Fictitiousassemblageofmolecules• Usesparticleprobabilitydistributionfunctioninsteadofsimulatingeverymolecule’spositionandvelocity• Particlescanonlymovefromnodetonodewithinalatticeorbetweenlattices,basedonprescribedboundaryconditions.• Incompressibleflowisassumedandparticles‘stream’&‘collide’
Introduction
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Introduction
LBM&FEM:• Lattice<-->Mesh• Boltzmannequation<-->Navier-Stokesequation• Weightingparameter<-->InterpolationfunctionLBM&DEM• Mesoscopicparametersareusedtoestimatemacroscaleproperties(density,velocity,internalenergy)
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Introduction
LBMvsconventionalCFD:- Uses1storderadvectionPDEinsteadof2ndorderconvectionPDE- DiscretizationisimplicitinBoltzmannequation- Solvedasa’stream’stepand‘collision’stepoveralllatticesandsimplekineticboundariesapplied
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Introduction
LBMadvantages:• Supportsmassiveparallelcomputingsincelocallattice-levelstepscanbesolvedindependentlyandsimultaneously• Noneedof‘interface’elementsformulti-component/multi-phasefluidflows• Multi-scalestudiesoverwiderangeofparticlesizespossible
LBMdrawbacks:• Needsmorememory/storagethanNavier-Stokessolvers• CannotstablyhandlecompressibleflowsorMachnumbershigherthan0.3• RequiresexternalpackagesforTHMcoupling
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Historical perspective
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HistoricalPerspective• LBMformulatedin1988byMcNamaraandZanetti
– 1859:Maxwell’sdistributionfunction– 1868:Boltzmanntransportequation– 1954:Bhatnagar,Gross,andKrook(BGK)collisionoperator– 1956:FEMbyTurner– 1973,76:Hardy,Pomeau,anddePazzis(HPP)model/LatticeGasAutomata(LGA)
– 1980:Finitevolumemethod(FVM)atImperialCollege
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MaxwellDistributionFunction• Measurestheprobability
thatacertainpercentageofapopulationofmoleculeswillbetravelingatacertainspeed
• Heaviermoleculestravelslower(onaverage)
• Theareaundereachdistributionis1
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BoltzmannTransportEquation/BGKCollisionOperator
• Ifnocollisions
• Sameequation,withcollisions
• Ifnoexternalforce
• BGKCollisionOperator
• LBMEquation:
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DiscretizedLBMEquation
• Turns1storderPDEintoalgebraicexpression
• AddresseschallengespreviousCFM’sdidnot
• Verystraightforwardtouse
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General principles & equations
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Lattice Arrangements
• DescriptionofthelatticeanddegreeofproblemisrepresentedviaDnQm
• m=speed,#ofthelinkagesofanode,numberofvelocitydirections
• N=dimensionoftheproblem
• Particlesarerestrictedtomovevialinkagesandareallowedtointeractatnodes
• Particlesmovealongthelinkagesatthelatticespeed;normallyassumethatinagiventimesteptheparticlesmovefromonecellnodetothenext.
Exampleofa1dproblemSource:A.AMohamad
Exampleofa2dproblemSource:http://www.cims.nyu.edu/~billbao/report930.pdf
D2Q9
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• D1Q3isdescribedwiththreevelocitiesc0,c1,c2andf0,f1f2.c0=0forcenterparticle
• Totalnumberofparticlesnotallowedtoexceed3
• Particle'sarefreetomovetotheleftorright
• Eachparticleisassignedaparticularweight,whichisafunctionofhowclosethatparticleistothecentralnodeandthevelocities.
• ForD1Q3theweightingfactors,𝛚I,are4/6,1/6,1/6forf0,f1,f2
• Speedofsound,Cs,is1/(3^.5)
• Thesumofallweightsmustequal1.
Lattice Arrangements
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Boundary Conditions
Bounce-Back:• Modelssolidstationaryormoving
boundaryconditions.• Whenaparticleencountersthe
boundaryitwillsimplybounceback.• Boundarycanbeplacedbetweenthe
nodesorgoingthroughthecenterofthenodes.
• Unknowndistributionsaftercollisionaref2,f5andf6.
• Focusingonbottomlayerweseethatf2=f4,f6=f8,f5=f7.
Igor2013
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Curved Boundary Conditions
Meietal.2000
Representthecurvedsurfacethroughasetofstairsteps.Requirestheboundarytoplacedbetweenthenodes.
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From Lattice Gas Automata to LBM
• ForLGAparticlesrestrictedtomovewithinalattice• Werepresenttheparticlesinspaceandtimevia• X=position,t=timeandi=directionoftheparticlevelocity• Ni=1=>particleispresentatsitexandtimetviceversaifNi=0• Candescribehowtheparticlesevolveinspaceandtimevia:• • ei=localparticlevelocities,𝜴i=collisionoperatori=collisionoperator• Collisionsarelocal
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Example of LGA
Attimet-1particleisoccupiedatsite1and4Attimet,particlescollideAttimet+1,particlesmoveoffindirectionsofe2ande5.(governedbyscatteringrules)
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Derivation of Lattice Boltzmann Equation from LGA
• RatherthandescribingparticlesviaBooleanalgebrawecanrepresentthemthroughadistributionfunction• Fk=average(nk)• Distributionfunction,f(x,e,t);wherex=position,e=velocity,t=time• Ifweapplysomeforce,f,ontheparticlestheirpositionsandvelocitieswillchangefromxx+edt;ee+F/Mdt
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Collison vs no Collison • Ifnocollisionsbetweenparticlestakeplace,thenthedistributionofparticlesshouldbethesamebeforeandafterforcewasappliedi.e
• • Withcollisionstherewillbeadifferencebetweeninitialdistributionandfinaldistribution:
• Dividethroughbydxdedt
• Where𝜴(f)isthecollisionoperator
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Lattice Boltzmann Equation final form • Rateofchangeofourdistributionfunctionisequaltothecollisionoperator
• Expandedform:
• Dividethroughbydt:
• Note,e=dx/dt;de/dt=F/m• IfweassumeF=0,i.enoexternalforcesthen:
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Collision Operator continued
• Ifparticlesinoursystemcollide,thenitmusttakesometimefortheparticlestoreachanequilibriumstate.
• Thetimetakentoreachtheequilibriumstateisafunctionofthetypeofcollisionandarelaxationtime
• DuetothecomplexityoftheCollisionOperatortheBoltzmannequationcanbedifficulttosolve.
• WecansolveforthecollisionoperatorbasedonBhatnagar,GrossandKrooksolution
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More on the collision operator
Thecollisionoperator𝜴(f)isreplacedwiththeBGKoperator:𝜏=istherelaxationratetowardsequilibriumandisrelatedtoviscosityby:shouldbeintherangeof.5-2fkEQ=equilibriumdistributionfunctionfkEQisanexpansionoftheMaxwellDistributionFunctionassumingalowMachnumber:M=u/cs<<1Whereu=macroscopicvelocityofthefluid,cs=speedofsound,𝛒=macroscopicfluidvelocity
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Equilibrium Distribution Function, fkEQ
• Note,TaylorsExpansionfore^-x=1-x+x2/2-x3/3!• UsingTaylorsExpansionwecanrewritetheequilibriumdistributionfunctionasfollows:
• k=numberofvelocities,𝛚k=weightingfactors
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Going from continuous form to discretized
Recall,thatthecollisionoperator,𝜴(f),istherateofchangeofparticledistributionfunction.Expandingtheparticledistributionfunctionoutintoitscounterparts,weobtaintheequationtotheright:Again,dividingthroughbydt,andassumingnoexternalforcesyieldsthefollowing:
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Continuous to discrete • Recall,thatthecollisionoperatorissimply:𝜴(f)=-1/𝜏(f-feq)
• -1/𝜏(f-feq)=∂f/∂t+∂f/∂x*c
• Nowmultiplythroughbydt
• -dt/𝜏(f-feq)=(∂f/∂t+∂f/∂x*c)dt (1)
• Note,Taylorseriesexpansion:f(x+∆x,t+∆t)=f(x,t)+∆f+c*(∆f/∆x)∆t
• SubstitutethesecondtermintheTaylorSerieswithEq1
• f(x+∆x,t+∆t)=f(x,t)-∆t/𝜏(f-feq)=DiscretizedversionofLBM
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Connecting microscopic quantities to macroscopic
quantities
• Basicidea:Torelatemicroscopicphenomenatomacroscopicbehavior• Wecanrepresentthedensityofafluidviathefollowingeq:
• Canrepresentthefluidvelocityviathefollowingeq:
• KineticEnergy:
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Hand calculation
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HandCalculation• ImaginealongtubeofgaswithaninitialtemperatureofT=0.
• Attimesgreaterthan0,theleftboundaryofthetubehasatemperatureT=1.
• Modelthechangeingastemperaturethroughoutthetubeastimeincreases– Assumethetubeisnon-conductivesuchthatallheattransferoccursthroughthegas
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ProblemDescription• Canbemodeledat1-Dproblem:
• Wewillmodelwith3elements:
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Workflow1. Initializemacroscopicpropertiesand
distributionfunctions1. Tw=1,allothers0.2. Makeaneducatedguessfordistributionfunction(for
diffusionequation,itdoesn’treallymatter)1. Forinitialfi’s,setfi’sinelement1towi’sandfi’sin
elements2and3toci’s.
2. Calculateequilibriumdistributionfunctions
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3. CalculateCollisions:
1. UsingtheBGKApproximationfortheCollisionOperator
4. CalculateStreaming:
AfterInitialization…
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UpdateMacroscopicProperties
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UpdateMacroscopicProperties
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Sameproblemfor100units
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FormofthesolutionwithincreasingT
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Summary
Collision• fi*
Stream• fiatnewlocation
Movetonexttimestep• t+dt
UpdateT,fieq
Initialize• T,fi,fieq
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Numerical example using OpenLB
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Example problem – 2D flow around cylinder
• Steadyflowaroundacylinderinachannel• Poiseuilleflowprofileatinlet• Dirichletboundaryofp=0atoutlet• Elasticbounce-backalongwalls• Reynoldsnumber=20and100forlaminarandturbulentflowsrespectively• D2Q9system
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0=Donothing1=Fluid2=noslip/bouncebackboundary
3=velocityboundary4=constant(zeroinourcase)boundary5=curvedboundary(cylinder)
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Re=20 (laminar flow)
t = 0 s t = 10 s
t = 5 s t = 15 s
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Re=100 (turbulence with Karman vortex street)
t = 0 s t = 10 s
t = 5 s t = 15 s
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Example applications
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Rayleigh-Benard flow
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Flow of particulates through nasal cavity
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Flow through lungs - parallel processing
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Turbulent flow in volcanoes
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Our favorite – flow in porous media