The lattice Boltzmann method: Fundamentals and acoustics

233 Pages·2014·1.32 MB·English

Erlend Magnus Viggen
The lattice Boltzmann method:
Fundamentals and acoustics
Thesis for the degree of Philosophiae Doctor
Trondheim, February 2014
Norwegian University of Science and Technology
Faculty of Information Technology, Mathematics and
Electrical Engineering
Department of Electronics and Telecommunications
NTNU
Norwegian University of Science and Technology
Thesis for the degree of Philosophiae Doctor
Faculty of Information Technology, Mathematics and Electrical Engineering
Department of Electronics and Telecommunications
© Erlend Magnus Viggen
ISBN 978-82-326-0036-6 (printed ver.)
ISBN 978-82-326-0037-3 (electronic ver.)
ISSN 1503-8181
Doctoral theses at NTNU, 2014:55
Printed by NTNU-trykk
Abstract
The lattice Boltzmann method has been widely used as a solver for
incompressibleflow,thoughitisnotrestrictedtothisapplication. More
generally, it can be used as a compressible Navier-Stokes solver, albeit
with a restriction that the Mach number is low. While that restriction
mayseemstrict,itdoesnothindertheapplicationofthemethodtothe
simulation of sound waves, for which the Mach numbers are generally
verylow. Evensoundwaveswithstrongnonlineareffectscanbecaptured
well. Despitethis,themethodhasnotbeenaswidelyusedforproblems
whereacousticphenomenaareinvolvedasithasbeenforincompressible
problems.
The research presented this thesis goes into three different aspects
of lattice Boltzmann acoustics. Firstly, linearisation analyses are used
to derive and compare the sound propagation properties of the lattice
Boltzmann equation and comparable fluid models for both free and
forced waves. The propagation properties of the fully discrete lattice
Boltzmannequationareshowntoconvergeatsecondordertowardsthose
ofthediscrete-velocityBoltzmannequation,whichitselfpredictsthesame
lowest-orderabsorptionbutdifferentdispersiontotheotherfluidmodels.
Secondly, it is shown how multipole sound sources can be created
mesoscopicallybyaddingaparticlesourcetermtotheBoltzmannequa-
tion. ThismethodisstraightforwardlyextendedtothelatticeBoltzmann
method by discretisation. The results of lattice Boltzmann simulations
ofmonopole,dipole,andquadrupolepointsourcesareshowntoagree
very well with the combined predictions of this multipole method and
thelinearisationanalysis. Theexceptiontothisagreementistheimme-
diatevicinityofthepointsource,wherethesingularityintheanalytical
solutioncannotbereproducednumerically.
Thirdly, an extended lattice Boltzmann model is described. This
modelalterstheequilibriumdistributiontoreproducevariableequations
of state while remaining simple to implement and efficient to run. To
compensateforanunphysicalbulkviscosity,theextendedmodelcontains
a bulk viscosity correction term. It is shown that all equilibrium distri-
butions that allow variable equations of state must be identical for the
one-dimensionalD1Q3velocityset. UsingsuchaD1Q3velocitysetandan
isentropicequationofstate,bothmechanismsofnonlinearacousticsare
capturedsuccessfullyinasimulation,improvingonpreviousisothermal
simulationswhereonlyonemechanismcouldbecaptured. Inaddition,
the effect of molecular relaxation on sound propagation is simulated
usingamodelequationofstate. Thoughtheparticularimplementation
usedisnotcompletelystable,theresultsagreewellwiththeory.
iii
Preface
ThereareafewthingsthatIwouldliketotellyouaboutthisthesisbefore
youdivein.
Thisthesisissplitintotwoparts. PartIcoverstheunderlyingtheory:
Fluid mechanics, acoustics, the kinetic theory of gases, and finally the
latticeBoltzmannmethoditself. PartIIbuildsdirectlyonthisbackground,
andcoverstheresearchthatwasdoneinthecourseofmyph.d.project.
Throughout this thesis you will occasionally see small notes in the
margin. Whenevernewandimportanttermsareintroduced,thesemargin
notesgiveashortdefinition. Occasionally,thesenotesmayberepeated
forvariousreasons: Readersmayhaveskippedpastapreviousdefinition
inanearlierchapter,adifferentformulationmightmakemoresensein
lightofthesurroundingtext,orImaysimplyhaveconsideredaconcept
criticalenoughtorepeat.
AtconferencesIhavesurprisinglyoftenmetotherstudentswhotell
methattheyhaveusedmyMaster’sthesistolearnthelatticeBoltzmann
method. Thishasbeentremendouslyinspiring,andhasledmetotakeex-
tra care to make Part I of the thesis thorough (though hopefully not
off-puttingly thorough) and readable. As introductions to the lattice
Boltzmannmethodthatareeasilyreadable,thorough,andfreelyavailable
arethinontheground,oneofmygoalshasbeentomakePartIjustsuch
anintroduction. IhopethatIhavesucceededinthisgoal,thoughthisis
ofcourseuptoyoutodecide.
WhilemyMaster’sthesiswillstillofferaquickerandsimplerintro-
ductiontothelatticeBoltzmannmethod,PartIofthisthesisputsmore
emphasis on the physical background of the method, which is in my
opinionessentialtotrulyunderstandit.
ThisthesisissubmittedtotheNorwegianUniversityofScienceandTechno-
logy(NTNU)inpartialfulfilmentoftherequirementsforthedegreeof Philo-
sophiaeDoctor(ph.d.). Theph.d.projectranforfouryears,withoneyearspent
onteachingduties. TheworkwascarriedoutattheAcousticsResearchCenter
at the Departmentof Electronics and Telecommunications, with Professor Ulf
Kristiansenassupervisor.
Ihopeyoufinditinteresting.
iv
Acknowledgements
Throughout these four years as a ph.d. student, there have been many
peoplewhohaveaidedmeinvariousways. ForthisIamdeeplygrateful.
IwouldhavehadamuchhardertimeearlyonifJorisVerschaevehad
notspentmanyanhourhelpingmetounderstandthelatticeBoltzmann
method. Similarly,mylaterworkwouldhavebeenmuchmoredifficultif
not for the occasional comment from Paul Dellar. His deep insight has
beeninvaluabletomeandmywork.
TherearemanyotherresearcherswithwhomIhavehadinteresting
andusefuldiscussionsonawidevarietyofscientifictopics. Iwouldespe-
ciallyliketothankTimmKrüger,JonasLätt,TimReis,TorYtrehus,David
Packwood, Manuel Hasert, Martin Schlaffer, and Guillaume Dutilleux.
Also,IwouldliketothankalltheotherfriendlypeoplewhosecompanyI
haveenjoyedatconferences. Thankstothem,thenon-scientificaspectsof
theseconferenceshaveneverbeendull.
I am grateful for the company and friendship of some of my fellow
travellersthroughtheph.d.programatNTNU.Whiletheyaretoomany
for me to start listing names, I would especially like to thank Anders
Løvstad for being excellent company in the office throughout much of
ourtimeasph.d.students,thoughIdonotmisstheroarofhiscomputer.
Finally,IamverygratefultoUlfKristiansenforbeingasavailableand
affableanadvisorasanyph.d.studentcouldhopetohave.
Toallofyou: Thankyou.
ErlendMagnusViggen
Trondheim,September2013
v
Contents
I Background 1
1 Introduction 2
1.1 Microscopic,mesoscopic,andmacroscopicscales . . . . . 3
1.1.1 Connection . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Asimplemicroscopicmodel: TheFHPlatticegas . 6
1.2 Thisthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Thesisstructure . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Relatedpublishedarticles . . . . . . . . . . . . . . . 12
1.2.4 Mathematicalnotationandlistofsymbols . . . . . 14
2 Fundamentaltheory 18
2.1 Indexnotation . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Fluidmechanics . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 TheEulermodel . . . . . . . . . . . . . . . . . . . . 23
2.2.2 TheNavier-Stokes-Fouriermodel . . . . . . . . . . 24
2.3 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Idealwaveequation . . . . . . . . . . . . . . . . . . 26
2.3.2 Viscousandthermoviscouswaveequation . . . . . 28
2.3.3 Molecularrelaxationprocesses . . . . . . . . . . . . 32
2.3.4 Acousticmultipolesandaeroacoustics . . . . . . . 37
2.3.5 Nonlinearacoustics . . . . . . . . . . . . . . . . . . 43
3 Thekinetictheoryofgases 46
3.1 Thedistributionfunctionanditsmoments . . . . . . . . . 47
3.2 Pressureandheat . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.1 TheMaxwell-Boltzmanndistribution . . . . . . . . 53
3.3.2 Peculiarvelocitymomentsatequilibrium . . . . . . 55
3.4 TheBoltzmannequation . . . . . . . . . . . . . . . . . . . . 56
3.5 Thecollisionoperator . . . . . . . . . . . . . . . . . . . . . 57
3.6 Macroscopicconservationequations . . . . . . . . . . . . . 59
3.6.1 Massconservation . . . . . . . . . . . . . . . . . . . 60
3.6.2 Momentumconservation . . . . . . . . . . . . . . . 60
vi
Contents vii
3.6.3 Energyconservation . . . . . . . . . . . . . . . . . . 61
3.7 Equilibrium: TheEulermodel. . . . . . . . . . . . . . . . . 62
3.8 TheChapman-Enskogexpansion . . . . . . . . . . . . . . . 63
3.8.1 Findingthedistributionfunctionperturbation . . . 65
3.8.2 Findingthemomentperturbations. . . . . . . . . . 68
3.8.3 TheNavier-Stokes-Fouriermodel . . . . . . . . . . 69
3.8.4 Higher-orderBoltzmannequationapproximations 70
H
3.9 Boltzmann’s -theorem . . . . . . . . . . . . . . . . . . . . 72
4 ThelatticeBoltzmannmethod 74
4.1 Thediscrete-velocityBoltzmannequation . . . . . . . . . . 75
4.1.1 Momentsandconstraints . . . . . . . . . . . . . . . 76
4.1.2 Moment-basedChapman-Enskogexpansion . . . . 79
4.1.3 Velocitysets . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.4 Digression: LinearisedDVBE . . . . . . . . . . . . . 87
4.2 ThelatticeBoltzmannequation . . . . . . . . . . . . . . . . 87
4.2.1 Firstorderdiscretisation . . . . . . . . . . . . . . . . 88
4.2.2 Secondorderdiscretisation . . . . . . . . . . . . . . 89
4.2.3 Summary: ThelatticeBoltzmannmethod . . . . . . 91
4.2.4 LatticeBoltzmannunits . . . . . . . . . . . . . . . . 93
4.3 Alternativecollisionoperators . . . . . . . . . . . . . . . . 96
4.3.1 Multiplerelaxationtime . . . . . . . . . . . . . . . . 97
4.3.2 Regularised . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.3 Entropic . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4 Simpleboundaryconditions. . . . . . . . . . . . . . . . . . 102
II Research 107
5 Acousticlinearisationanalysis 108
5.1 IsothermalNavier-Stokes-Fouriermodel . . . . . . . . . . 111
5.1.1 Absorptionanddispersion . . . . . . . . . . . . . . 112
5.1.2 Magnituderatiosandphasedifferences . . . . . . . 113
5.2 Discrete-velocityBoltzmannequation . . . . . . . . . . . . 115
5.2.1 Linearisationprocess . . . . . . . . . . . . . . . . . . 116
5.2.2 Propertiesofforcedandfreewaves . . . . . . . . . 119
5.2.3 Comparisonwithrelaxationprocesses . . . . . . . 121
5.2.4 Comparisontoothermodels . . . . . . . . . . . . . 122
5.2.5 Anisotropyintwodimensions . . . . . . . . . . . . 128
5.3 LatticeBoltzmannequation . . . . . . . . . . . . . . . . . . 137
5.3.1 Linearisationprocess . . . . . . . . . . . . . . . . . . 140
5.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.3.3 Example: Exactwaveinitialisation . . . . . . . . . . 146
5.4 Summaryanddiscussion . . . . . . . . . . . . . . . . . . . 149
viii Contents
6 Mesoscopicacousticsources 151
6.1 SourcetermsfortheBoltzmannequation . . . . . . . . . . 153
6.1.1 Macroscopicconservationequations . . . . . . . . . 154
6.1.2 Linearwaveequation . . . . . . . . . . . . . . . . . 155
6.2 SourcetermsforthelatticeBoltzmannequation . . . . . . 156
6.2.1 Firstorderdiscretisation . . . . . . . . . . . . . . . . 157
6.2.2 Secondorderdiscretisation . . . . . . . . . . . . . . 158
6.2.3 Multipolebasis . . . . . . . . . . . . . . . . . . . . . 159
6.3 Numericalexperiments . . . . . . . . . . . . . . . . . . . . 163
6.3.1 Planewaves . . . . . . . . . . . . . . . . . . . . . . . 163
6.3.2 Multipolesintwodimensions . . . . . . . . . . . . 169
6.4 Summaryanddiscussion . . . . . . . . . . . . . . . . . . . 177
7 Variableequationofstate 179
7.1 Theextendedmodel . . . . . . . . . . . . . . . . . . . . . . 181
7.1.1 Momentsandconstraints . . . . . . . . . . . . . . . 182
7.1.2 Macroscopicequations . . . . . . . . . . . . . . . . . 183
7.1.3 Bulkviscositycorrection. . . . . . . . . . . . . . . . 185
7.1.4 Generalequilibriumrequirements . . . . . . . . . . 189
7.1.5 Linearisationanalysis . . . . . . . . . . . . . . . . . 190
7.2 Isentropicequationofstateandnonlinearacoustics . . . . 194
7.2.1 TheisentropiclatticeBoltzmannmodel . . . . . . . 196
7.2.2 D2Q9stability: Comparisontoanothermodel . . . 197
7.2.3 Physicalnonlinearacousticscase . . . . . . . . . . . 198
7.2.4 Nonlinearacousticssimulation . . . . . . . . . . . . 200
7.3 Molecularrelaxation . . . . . . . . . . . . . . . . . . . . . . 203
7.3.1 Verificationbysimulation . . . . . . . . . . . . . . . 204
7.4 Summaryanddiscussion . . . . . . . . . . . . . . . . . . . 205
8 Discussionandconclusion 209
Bibliography 213
Part I
Background
1
1 Introduction
Many scientific articles on the lattice Boltzmann method begin with a
fairlydenseparagraphonthemethodanditscapabilities,whichtypically
goessomethinglikethis:
ThelatticeBoltzmann(LB)methodisarecentadvanceincom-
putationalfluiddynamics(CFD).WhiletraditionalCFDmeth-
ods directly discretise and solve the macroscopic equations
of fluid mechanics, the LB method solves a discrete kinetic
equationwhichreproducesthefluidmechanicsequationsin
themacroscopiclimit. Itisstraightforwardtoimplementand
paralleliseefficiently,whilebeingversatileenoughtosimulate
multiphase flows, multicomponent flows, flows of complex
fluids, flows in complex geometries such as porous media,
thermalflows,andturbulentflows.
A paragraph this succinct can of course not give a full picture of the
method. However,itdoesmanagetopaintmuchofthispictureinbroad
strokes. Letusnowpaintsomeofthefinerstrokesbyexpandingonthe
threesentencesofthisparagraph.
Asthefirstsentencestates,theLBmethodhasnotbeenaroundforas
longasmostotherCFDmethods. Historically,itgrewoutofthefieldof
Cellularautomaton cellularautomata,andspecificallylatticegases,whichwewilllookatbriefly
Adiscretemodelwithvery in section 1.1.2. The first lattice gas was described in 1973 [1], though
simplerulesthatcantypically
resultinverycomplex it was not until 1986 that a lattice gas that could be used to correctly
behaviour simulatefluidflowwasproposed[2]. Anarticlewaspublishedsoonafter
Latticegas
in1988onamodificationtolatticegasesinordertoavoidsomeoftheir
Acellularautomatonfor
simulatinggases,basedon problemswhensimulatingfluidflow[3]. Thisarticlecanbeconsidered
particlesmovingaroundona thefirstarticleonthelatticeBoltzmannmethod.
lattice,theircollisions
conservingmassand The second sentence of the paragraph implies that the LB method
momentum
solvestheequationsoffluidmechanicsindirectlybysolvingsomething
else, somethingsimpler. Whilethismayseemtoogoodtobetrue, there
is indeed a good physical reason why it works. The lattice Boltzmann
methodisadiscretisationoftheBoltzmannequation,anequationwhich
describesgasesatamoredetailedlevelthantheequationsoffluidmech-
anics,whilestillhavingasimplerform. Ifwesmoothawaythesedetails
intherightway,weendupwiththeequationsfamiliarfromfluidmech-
2