THE LATTICE BOLTZMANN EQUATION:THEORY AND APPLICATIONS

R. BENZIa, S. SUCCIb and M. VERGASSOLACaDipartimentodi Fisica, Università “Tor Vergata”, Via E. Carnevale,1-00173Rome,italy

hiBM EuropeanCenterfor Scientificand EngineeringComputing,Via Giorgione 159, 1-00147Rome,

ItalyCCNRS Observatoirede Nice, B.P. 229, F-06304Nice Cedex, France

NORTH-HOLLAND

PHYSICSREPORTS(Review Section of PhysicsLetters)222, No. 3(1992)145—197.North-Holland PHYSICS REPORTS

The lattke Boltzmann.equation:theory andapplications

R. Benzia, S. Succi” and M. Vergassolac~Diparcimentodi Fisica, Università“Tor Vergata”, Via E. Carnevale,1-00173Rome,ItalybJBM EuropeanCenterfor Scientificand EngineeringComputing,Via Giorgione 159, 1-00147Rome,ItalyCCNRS ObservatoiredeNice, B.P. 229, F-06304Nice Cedex,France

ReceivedMay 1992; editor: I. Procaccia

Contents:

1. Introduction 147 4.5. The limit of zeroviscosity 1722. Lattice gasdynamicsand the Boltzmannapproximation 148 4.6. Numerical efficiency 173

2.1. Lattice symmetriesandcollision rules 148 5. Applications 1772.2. The equilibrium distribution 150 5.1. Two-dimensionalhomogeneousturbulence 1772.3. Hydrodynamicequations 152 5.2. Flows in complex geometries 1792.4. The Boltzmann approximation 153 6. Generalizationsof themethod 183

3. Somerigorous results 155 6.1. Multiphase flows 1833.1. Introduction 155 6.2. Magnetohydrodynamics 1843.2. A model with stochasticstirring 156 6.3. Subgndscalemodelling 1863.3. Spuriousinvariants 161 6.4. Boltzmannequationin non-uniformlattices 190

4. The lattice Boltzmann equation 162 7. Conclusions 1944.1. From the booleanmicrodynamics to the Boltzmann AppendixA 194

equation 162 AppendixB 1954.2. Lattice Boltzmann equationwith enhancedcollisions 164 References 1964.3. Macrodynamicequations 1664.4. Hydrodynamic behaviourof the lattice Boltzmann

equation 169

Abstract:The basicelementsof the theory of the lattice Boltzmann equation,a speciallattice gas kinetic model for hydrodynamics,are reviewed.

Applicationsarealsopresentedtogetherwith somegeneralizationswhich allowone to extendtherangeof applicability of themethodto a numberof fluid dynamicsrelatedproblems.

ElsevierSciencePublishersB.V.

1. Introduction

Overthe last decades,therehasbeenagrowing consensuson the fact that grandchallengescanbeencounteredin classicalareasof macroscopicphysics,evenif thebasicequationshavebeenknownfor along time. Fluid mechanicsis onesucharea.The main reasonis thathydrodynamicflows can exhibit anextremelyrich variety of complexphenomena;so rich that, in many respects,fluid flows can still beconsidereda paradigmof complexityin naturalsystems.Sucha complexitycan beassociatedto a set ofnon-linearpartial differential equations,the Navier—Stokesequations,which are, ultimately, nothingbut a statementof massandmomentumconservation.The Navier—Stokesequationsareuniversalin thesensethat their structureis fairly independentof the details of the underlyingmicroscopicdynamics,which only affect the numericalvaluesof the transportcoefficients.This universalityof the equationsplays an invaluable role in trying to devise stylized microdynamicalmodelswhich, while giving up asmuch irrelevantdetails as possible,still retain the basicingredientsof the physicsof fluids. Lattice gasmodelsbelongpreciselyto sucha classof models.The commonaim of latticegasmodelsconsistsin thedefinition of a fictitious simplified microworld whichallows oneto recover,in the macroscopiclimit, theequationsof fluid dynamics.By “macroscopiclimit”, one meansthat the equationsdescribingtheevolutionof the largescalesdiffer from Navier—Stokesequationsby spuriousterms,becomevanishing-ly small as the ratio betweenthe meanfree pathand the scale length is decreased.

Hystorically, the first step in the direction was performedby Broadwell [9, 30], with the discretevelocity modelswherethe velocity is discrete,but spaceandtime arestill continuous.Subsequently,amasterequationwith acontinuoustime, andleadingto soundwaves,was introducedby KadanoffandSwift [46]. In 1973, Hardy, de Pazzisand Pomeau[32,33] defineda fully discretemodel (HPP)on asquarelattice allowing theobservationof typical featuresof fluid flows, suchas soundwaves.However,due to alack of isotropy of the lattice, the HPPmodelreproducesneitherthe non-linearterm nor thedissipative term of the Navier—Stokes equations.This difficulty was first removed for the two-dimensionalcaseby Frisch, Hasslacherand Pomeau[27]and thenfor the three-dimensionalcasebyd’Humières,Lallemandand Frisch [42]. Once the correctsymmetriesof the lattice areobtained,onehasto define the evolution rules. Here, two possibilitiescan be chosen:lattice gas automataandthelattice Boltzmannequation.

In lattice gas automata,the variablesare the booleanpopulations,indicating the presenceor theabsenceof particlesin the link of the lattice- The evolutionis essentiallydefinedby a setof collisionrules, possessingsuitable conservationlaws. The passageto the macroscopicworld is performedthrough an averageover spaceand!or time. Since their evolution only involves logical operations,lattice gas automatahavediscloseda markedlyinnovativesimulationstrategy,wherebyfloating-pointcomputingis replacedby booleancomputing.Moreover,dueto the locality of the updatingrules, theyare ideal candidatesfor massively parallel processing.Lattice gas automatahavebeen usedfor thesimulationof Navier—Stokesequationsandotherrelatedproblems,such as two-dimensionalmagneto-hydrodynamics,three-dimensionalcomplexflows, multispeciesand multiphasesystems,and Burgers’model (seerefs. [5,28] and referencestherein).

Besidestheir relevanceto hydrodynamics,latticegas automataaresuitableto analyzefundamentalissuesin statistical mechanics,such as the long-time-tail problem [23,25]. In particular, one can

147

148 R. Benziet a!., The lattice Boltzmannequation:theory and applications

considertheBoltzmannapproximationof thedynamics,by making thesameassumptionleadingto theBoltzmann equationin continuouskinetic theory: particlesenteringa collision haveno correlations.The latterassumptioncanbe rigorously justified in thecaseof a lattice gasautomatonwith stochasticstirring [19].With respectto the dynamicswithout stirring (which is the one used in practice),bycomparingthemeasuredvalueof the viscositywith the theoreticalpredictions,McNamaraandZanetti[53] remarked that hydrodynamic behaviour is recovered even with very small lattices. In theBoltzmannequation,the evolution is defineddirectly in termsof the ensembleaveragedpopulationswhich, being real variables,do not lend themselvesto the possibility of booleancomputing. On theother hand, the advantageis the elimination of statisticalnoise usually affecting lattice gasautomatasimulations,while retainingthesameidealamenabilityto parallelcomputing.Actually, in ref. [37],it isshown that onecan further simplify the model by using the linearizedform of the collision operator.Finally in ref. [39],startingfrom this linearizedform, it is proved that one can define a Boltzmannequation,calledlattice Boltzmannequatioi~with enhancedcollisions, independentlyof any underlyingbooleanmicrodynamics.The collisions aredefinedby a matrix, whosestructureis essentiallydictatedby symmetryarguments.The resulting numerical free parametersgovern the value of the transportcoefficients,which can be easilytuned. It can be shownthat the macrodynamicsof the model,undersuitable conditions, converges to Navier—Stokes equations [26,69]. The schemehas proven itsnumericalefficiency in thesimulationof two-dimensionalhydrodynamics[2]. Moreover,it lends itselfto some interestinggeneralizations,such as magnetohydrodynamics[661,two-phaseflows [31], theanticipatedvorticity schemefor the subgridscalesparameterizationof two-dimensionalturbulence[3]and Burgers’model.

2. Lattice gasdynamics andthe Boitzmannapproximation

2.1. Lattice symmetriesand collision rules

Let us beginwith thedescriptionof themodel (FHP) introducedby Frisch,HasslacherandPomeau[27]for two-dimensionalfluid dynamics.Considera triangularlattice with unit lattice spacing,asshownin fig. 1. In eachnode thereare six links (denotedby c~{i = 1,. .. , 6)) to its nearestneighbours.Particlesof unit massandvelocity movealongtheselinks in sucha way that they residein thenodesatintegertimes. An exclusionprinciple governstheoccupationofthe links: no morethanone particlecanoccupy at a given time a given link. The state of a node is then a six-bit boolean variablen = {n1, i = 1,. . . , 6), wherethebit n1 indicatesthepresenceor theabsenceof a particlein the ith link.The evolutionof the systemfrom time t to time t + 1 (the natural time of the lattice is takenasunity)canbe decomposedin two phases:propagationand collision.

The propagation is very simple. Each particle is displaced in the direction of its velocity:n~(r)—~n1(r + c1), wherer denotesthe sites of the lattice.

The collisions representthemostdelicatepart of theevolution.In order to reproduceNavier—Stokesequationson the macroscopicscales,it is essentialto impose mass and momentumconservation,characteristicof a real fluid (in one-speedmodels, which are the only onesconsideredhere, energyconservationis obviously coincidingwith massconservation).At thesametime, it is necessaryto avoidthe presenceof spuriousinvariants. Consider, for example,binary head-oncollisions (see fig. 2). If(i, + 3) denotesthecoupleof links occupied,massandmomentumconservationrestrict the possiblefinal statesto (i + 1, i + 4) and (i — 1, i + 2). Onecan decideeitherto choosealwaysonepossibility or

R.Benziet a!., The lattice Boltzmannequation: theoryand applications 149

Fig. 1. Elementarycells of theFHPlattice. Fig. 2. A binary collision in the FHPlattice.

to makea randomor pseudo-randomchoice, e.g. dependenton theparity of time. Despitemassandmomentumconservation,a model with only binary collisions hasamacroscopicdynamicscompletelydifferent from the onedescribedby Navier—Stokesequations.This is due to thepresenceof a spuriousinvariant: the differenceof particlenumbersin any pairof oppositedirections.Thesimplestremedytothis problemis the introductionof triple collisions and/orrestparticles,as discussedin ref. [28].

Let us now considerthe three-dimensionalproblem. In this case,the difficulty is that no Bravaislattice, havingenoughsymmetriesto reproducehydrodynamics,exists.The way out of this problemisto move to a higher-dimensionallattice. A suitable four-dimensionallattice, the face-centeredhy-percubic (FCHC), was in fact found in ref. [42].The nodes(x1, x2, x3, x4) of the lattice satisfy thecondition: x1 + + x3 + x4 even,wherethex.s areintegernumbers.In eachnodethereare24 links toits nearestneighbours(indicated as c. {i = 1,... , 24)) of length V~.Propagationand collisions aredefinedin the samespirit as in FHP.

In the applicationsto three- (two-)dimensionalhydrodynamics,periodic boundaryconditions areimposedalongthex4 directionin a layerof thickness2 (andalongthex3 direction).Theresultinglatticeis shown in fig. 3. The thick black lines have a different weight in the sensethat two particles canpropagatealongthesedirections,accordingto the two valuesof the fourthcomponentof thevelocity

V4=OV4 =+—l

Fig. 3. Three-dimensionalprojectionof theelementarycell of theFCHClattice.Thesolidlinescarry two particles,havingspeedv4= ±1alongthefourth dimension.

150 R. Benzieta!., Thelattice Bo!tzmannequation: theory andapplications

(c,)~= ±1. It canbe shown that the conservedfourth componentof momentumbehavesasa passivescalarnot influencingthe otherthree[28].For subtle effectsassociatedwith the presenceof spuriousinvariantsgeneratedby theperiodicity imposedon the fourth direction, seerefs. [6,7].

A generalclassof latticessuitablefor lattice gasmodels,includingFHP andFCHC, can bedefined.We do not list thedefinitionof this classand the resultingpropertiesdiscussedin ref. [28],which will bereferredto wheneverneeded.Oneof themain hypothesesto be recalledis semi-detailedbalance.Lets = {s~,i = 1,. . . , b } and s’ = {s~, i = 1,. . . ,b} denotethe stateof a node before and after thecollisions respectively,with b the numberof links per node,and let

A(s—+s’) (1)

denotethesite-independenttransitionprobability from s to s’. An obviousconditionofnormalizationis~ A(s—s.s’) = 1,Vs. The semi-detailedbalanceconditionreads as

~.A(s—~.s’)=1, Vs’. (2)

This equalityimpliesthat thesituationwhereall thestateshavethesameprobability is stationarywithrespectto collisions. Recently,a new classof lattice gasautomata,where(2) is not satisfied[21],hasbeen introduced.As it will becomeapparentin the following, (2) is essentialfor the resultswe aregoing to summarizeand, in particular,for the argumentsdevelopedto prove the convergenceof themicrodynamicsto Navier—Stokesequations.Nevertheless,the newmodelsdefinedin ref. [21]showagood empirical agreementwith hydrodynamics.The eliminationof thesemi-detailedbalanceconditionleadsto the interestingconsequencethat the viscosity is no longerforced to be positive [34],but canassumealso negativevalues[35].

2.2. Theequilibrium distribution

As in statistical mechanics,it is convenientto take a probabilistic point of view. The classicalprocedureis: (i) definition of thephasespace,(ii) introductionwithin this setof a statisticalensembleof initial conditionswith a given probability distribution, (iii) evolution of the probability distributionfrom the Liouville equation, (iv) calculation of the averagevalues using the evolved probabilitydistribution. Specifically, let £~?be afinite lattice. The phasespaceF is definedby all thepossiblestatess(.) = {s1(r)}, where i = {1,. . . , b) and r denotesthe nodesof the lattice. An initial probabilitydistributionP(O, s(.)) is assignedin F.The evolutionof theprobabilitydistributionP is governedby theprobabilistic equivalentof the Liouville equationin statisticalmechanics

P(t+1, Efs())= ~ P(t,s’(.)) LI A(s’(r)-*s(r)). (3)rESt

Here, A(s—* s’) has been defined in (1) and &° is the streaming operator whose action is.~:n1(r)i—*n1(r+ c1). The averagevalues of an observablef(s(~))can be calculatedas (f(t)~=

~s(~) f(s(-))P(t,s(.)). In particular, onecan considerthe quantities

N1(t,r)= (n1(t,r)) , p=~N~, J(t,r)anpv(t,r)=~N,.c~, (4)

which correspondto the averagepopulation, thedensity and the masscurrent.

R. Benziet a!., The lattice Boltzmannequation:theory and applications 151

It is remarkablethat thegeneralclassof modelsmentionedin section~2.1 admitsthe equivalentofthe Maxwell statesin statisticalmechanicsand that a H-theoremcanbe proved. Indeed,as shownbyHénonin appendixF of ref. [28], from thesemi-detailedbalancecondition it follows that the localentropyfunction

S(r) — ~ P(t, s(r)) log P(t, s(r)) (5)s(r)

can be defined. The function neverdecreasesundertheeffect of collisions. The entropy is constantifandonly if theprobability distributionis a Boltzmànndistribution, i.e. if P is completelyfactorizedoverthe links of the site,

b N~’(1—N~)~’. (6)

Here, N1 is the averagepopulationdefinedin (4).

By definition, theequilibrium distributionpeq is a time-independentsolutionof eq. (3). The spatialhomogeneityand the H-theoremsuggestto look at p~’in a fully factorizedform, i.e. peq(5(.)) =

Hr P(s(r)), with P(s(r)) given by (6). Indeed,it can be provedthat suchasolutionexistsand that thecorresponding~ hasthe Fermi—Diracexpression

N~=[1+exp(h+q.c~)]1. (7)

The positive sign in front of the exponentialreflectsthe fermionic natureof theparticles due to theexclusionprinciple. The quantitiesh andq aretheLagrangemultipliers associatedwith theconservedquantities.In principle,they canbe obtainedfrom thedefinitionof thedensity andthemasscurrent.Inthecaseof small Machnumbers(I v 4 c,, wherec~is thespeedof sound),onecan exDlicitlv computehand q perturbativelyin v. Thanksto the mirror symmetryof the lattice, ~ is invariant under thetransformationv‘-~ — v and c

1 ‘—~ — c,. The scalarfunction h containsonly evenpowers of v andthevector function q only odd powersof v, i.e. h = h0 + h2(v . v) +... and q = q1v +.... Let us nowsubstitutethis last expressioninto (7). For thegeneralclassof modelsdefinedin ref. [28]the followingrelationshold:

C. 0, ~ Cj~Cjp= (bc2ID)öas . (8)

Here,b is thenumberof links per site, c = c. andD is the dimensionof the lattice. Using (8), onefinally obtains

= p/b + (PD/bC2)VaCja+ pG(p)Qj~pv~vp+ 0(v3), (9)

where

G(p)= (D212bc4)(b— 2p)/(b P)’ Qiap = CiaCip — (c2/D)ôap . (10)

Let us remark that in the case of the FCHC lattice, for the calculation of three dimensionalcomponents,D hasto be takenequalto 4.

152 R. Benziet al, The lattice Boltzmannequation: theory andapplications

2.3. Hydrodynamicequations

In this section, the macrodynamic equations and the assumptions necessaryto recoverthe Navier—Stokes equationswill be discussed.For this purpose,it is convenientto introduce a multiscaleformalismwhere spaceand time derivativesare expressedin termsof multiscalevariables,

= ear, , a1 = + + . (11)

Here, e is a small parameter which may be identified with a local Knudsen number: e I VFIF, where 1is theparticle meanfreepath andF a generic macroscopic field. The physical meaningof (11)is thatvarious phenomena(e.g., inertialpropagation,dissipation)takeplace atdifferent time scales.

Since our aim is to describe transport phenomenatriggered by weak departuresfrom localequilibrium, it is convenientto decomposethe averagepopulationN1 as

N, = N~(p,v) + N~’(Op, Ov), (12)

whereN~ is the local equilibrium population,while N~ is the non-equilibriumcontribution,which issupposedto be 0(s). In particular, one is interestedin the first term of the expansionlinearlydependenton thegradientsof theequilibriumfields, p andJ. By using thesymmetrypropertiesof thegeneralclassof latticesdefinedin ref. [281,it is possibleto showthat,at the first orderin e, thegradientlip does not contributeto N~ and the only contribution by öa comesfrom the product with thesymmetrictensorQjap~Thus ~ne~ = ifr(p)Q.~aJ+ O(s2), where ~1i(p)is still to be determined.

Massand momentumconservationcanbe expressedin the following form:

~N~(x+c1,t+1)—~N,(x,t)=0, (13)

~N,(x+c~,t+1)c~—~N,(x,t)e1=0. (14)

In the limit of small Knudsennumbers,onecan expressthedifferencesby a seriesof time andspacederivatives.By using theexpansion(11) and equatingtermsof thesameorder in sand ~2, one finallyobtainsthe partial differential equations

a,p+li.J=o, (15)

ätJa+dp(pG(p)Tapy~vyv&~ a7(pv8)]_—0, (16)

where

Tap.y~= ~ ctacjpQj~a. (17)

Note that in (16) we haveusedtheexpression(9) for the equilibrium population. It follows thathereafterwe shall restrict to the small Mach numberregion.

Equation(15) is the continuityequation.As usual, in the limit of small Mach numbersthe density

R. Benzietal., The lattice Boltzmannequation:theoryandapplications 153

can be considered constant except for the fluctuations in thepressureterm.Undertheseconditions,eq.(15) reduces to the divergence free condition (li. v) = 0.

Let us now consider eq. (16). In order to recover Euler and Navier—Stokes equations, the isotropyofthe fourth-order tensor Tapy& is crucial. From the symmetrypropertiesof the classof latticesdefinedinref. [28],it follows that the most general form of an isotropic fourth-ordertensoris

bc4 / 2Tap.,,

8 = D(D + 2) ~ + ~aö~P7— D ~ap

6y8). (18)

The tensorhasindeedthis expressionin thecaseof FHP andFCHC lattices.In the caseof thesquarelattice, Tapy

8 has not the form (18) and this explainswhy the lattice is not suitable to simulatehydrodynamicphenomena.Substituting (18) in (16), one obtains:

atJa + äpPap = apSap , (19)

where

= p(c2ID)(1 — g(p)v2Ic2)e5~p+ pg(p)v~vp

(20)Sap = ~[äaJp + — (2/D)(t3yJy)t5ap].

Onethus obtainsthe Navier—Stokesequations,apartfrom the presenceof a factor in the advective

term

g(p)= [D/(D + 2)](b — 2p)/(b — p). (21)

This difficulty canbe overcomebecausein the limit of small Machnumbersthe factorg(p) reducesto a

constant.This factorcan be easilyrescaledout by the transformationt—*tg(p), v—tv/g(p). (22)

The viscosity ii is equal to

p = —[bc4/D(D + 2)]~fr(p) — c212(D+ 2). (23)

The first contribution, ~ p), is positive and depends on the collision details through the function i/i,

while the second, v~, is negative and is essentially due to the discretenessof the lattice [34].

2.4. The Boltzmannapproximation

Let n.(x, t) be the boolean variable denoting the number of particles in x at time t for the state i (for

rest particles,c1 = 0). The evolution equationfor n. is

n~(x + c1, t + 1) — n1(x, t) = 41(n) , (24)

154 R.Benziet a!., The lattice Boltzmannequation: theoryand applications

where4(n) is the contribution of collisions involving productsofthen.s. Let usnow considerN1(x,t),obtained as the ensemble average of the boolean n.(x, t). When the evolution equation (24) isaveraged,a hierarchy of non-closedequationsis generated..With respectto the usual BBGKYhierarchyobtainedin thecontinuouscasewith two-bodycollisions [12],thefollowing differencesarise:thediscretecaseis simpler becausethereareonly b degreesof freedomperpoint but, at thesametime,the analysisof the collisions is complicatedby the fact that, in themost generalcase,b particlesareinvolved. Whateverorder one goesto, there are alwayshigher-ordercorrelationsappearingand thesystemremainsunsolvable(for more details on the effects of correlationsseeref. [67]).In order towrite an evolution equationfor the N,s, one hasto close this hierarchyby somekind of statisticalassumptionon the high-ordercorrelations.The simplestway to obtain a kinetic equation (a closedequationfor the N~s)is to usethe Boltzmannapproximationof molecularchaos:neglectcorrelationsbetweenthe particlesentering a collision. This, assumptionimplies that all the expectationvaluesfactorize.The resultingBoltzmannequationis

N,(x+ c~,t + 1) — N,(x, t) = 4(N) . (25)

Due to the factorization of the expectationvalues, the collision operator4(N) is obtained from thebooleancollision operator4(n) by simply replacing thebooleanpopulationswith the correspondingaveragepopulations.The expressionof 4(N) in termsof the transitionprobabilitiesA(s‘—~ s’) is

4(N) = ~ (s~— s1)A(s‘—~ s’) Ill N~’(1— N)~1~’~. (26)

There are two main consequencesof the Boltzmannapproximation.The first is the irreversibleapproachto equilibrium which immediatelyfollows from the H-theorem

discussedin section2.2.The secondis an explicit derivationof thevalueof the viscosity. Onehasto determinethe function

çli( p) usedin section2.3 in theexpressionN~ = ~ P)Qiapöa J,9. This canbe doneby expandingat the

first orderin s and v the finite differencesin the lhs of (25) andusing theexplicit Boltzmannform forthe collisionsin the rhs. By using theequationsfor the conservedquantitiesto expresstime derivativesin terms of spacederivativesone finally obtains

(DIbC2)Qiap — 4!1(P) ~ S4

11Q ja~= 0. (27)

Here, the matrix is the linearized collision matrix a4(N)!aN~,,evaluated at the zero velocityequilibrium stateN1 = p/b. The explicit expressionof thematrix in termsof the transitiQnprobabilitiesof theautomatonwas foundin ref. [34].

From (27), assumingthe isotropyof the fourth-ordertensors,onecanobtain a closedexpressionforthe contribution v~to the viscosity,

2 ~c “ian ~ia~= — D + 2 ~ijap Qtap~4jQjap

Once the viscosity is known, the ReynoldsnumberR associatedto a given flow can be easilyestimated. The Reynoldsnumberis by definition the ratio betweenthecharacteristicscale1~times the

R. Benziet a!., The lattice Boltzmannequation: theoryand applications 155

characteristicvelocity v0 of the flow and the kinematic viscosity. Taking careof the rescaling(22)necessaryto recoverNavier—Stokesequationsfrom (19), one obtainsR = l0v0g(p)/v(p). Introducingthe Mach numberM = v0/c,, one finally gets

R=M10R*(p), (29)

where R* (p) = c,g(p) Iv( p) containsall the informationon the lattice dynamics.In the limit of large Reynoldsnumbers,a huge rangeof scalesis excited by thenonlinearenergy

transfer. Let1d denote the dissipativelength, i.e. the length whereviscouseffectsbecomecomparable

to nonlineareffects. In a spectralsimulation all themodes at leastup to the inverseof 1d have to beresolved. In the case of lattice gas, one has to check that thewavelengthof all relevantscalesis muchlarger than the lattice spacing.Accordingto the 1941 Kolmogorov [48]and the Batchelor—Kraichnan[1,49] theories of three- and two-dimensional turbulence *), in the limit of large Reynoldsnumbers,thescale ‘d is relatedto the inertial scalel~as = CI

0Rm (m = 1/2 in D =2 and m=3/4 in D =3). By

using (29), one obtains

ld = C(MR*)~Ri’2 in2D, ld = C(MR*)~R~~4in 3D. (30)

In both cases, in the limit of R—~ ~, the condition of scale separationbetween1d and the lattice spacing

is satisfied. The validity of lattice gasmodelsfor high-Reynoldsnumbersimulationsis then ensured.

3. Some rigorous results

3.1. Introduction

In this section,we review somerigorous results. The readernot interestedcan skip this section,without compromising further reading of the paper.

A discretekinetic equationof the form

N,(x+c1,t+ 1)—N,(x,t)=4(N), (31)

can be rigorously investigatedfrom variouspointsof view. First of all, onecanexaminethevalidity of(31) in ‘describingensembleaveragesof theunderlying lattice gasdynamics.The correspondingissue incontinuousspaceis the classicalproblemof thevalidity of kineticequations.On theotherhand,moreclosely to the spirit of this report, one could take (31) as the starting equation and study itsconvergence,on suitablespace-timescales,to the incompressibleNavier—Stokeslimit. Thecorrespond-ing problemin thecaseof a real fluid hasa long traditionin the literaturedating backto thepapersbyHilbert [40]andEnskog [22].

Recently,De Masi et al. [19]haveinvestigatedboth problemsin thecaseof theHardy, de Pazzisand Pomeau (HPP) squarelattice [32,33]. Concerningthe first issue, a new featureappearsin thedynamics of the model with respect to the original formulation: thestochasticstirring. This operation

*) We omit any discussionon the possible correctionsdue to intermittency(29,561 becausewe are just interestedin estimatingordersofmagnitude.

156 R. Benziet al., The lattice Boltzmannequation: theory andapplications

consists of random exchanges between neighbouringcells, with the constraintof momentumconserva-tion. The stochastic stirring has been designedto prevent the build-up of correlations due tointeractions,which could destroythe validity of the Boltzinann approximation.The needto enhancestochasticity in order to rigorously prove the limit to theBoltzmannequationis not limited to theHPPmodel, but it seems a generalrequirementfor all latticegasautomata.The stirringprocess,while usefulfrom a theoretical point of view, has the practicaldrawbackof increasingthe kinematic viscosity. Byintroducing the stochasticstirring operation,it is possible. to prove that the kinetic equation(31), in thecontinuousversion, correctly captures the long-time and large-scalebehaviourof the automaton.Concerning the second problem, the authors of ref. [19]prove that, undersuitablelimits, thekineticequations converge initially, for suitablespace-timescales,to Euler equations,while for longer timesone can observe incompressibleNavier—Stokesequations.The aim of section3.2 is to summarizealltheseresults,giving in particulara sketchof the Hilbert—Chapman—Enskogexpansionusedto derivehydrodynamicequations.In section3.3, we discuss the issue of spurious invariants,arising from thediscretization of a partial differential equation.

3.2. A modelwith stochasticstirring

Let us now discuss in some detail the main resultsobtainedin ref. [19]for the HPPmodel. Particlesmove on a square lattice L x L with unit vectorsc~(i = 1,. . . , 4). There are four copies of the latticelabelled with cr E (1,. . . , 4). The couple (x, o) denotes a point x of integercoordinates(x, y) on thelattice cr. The standardexclusionprinciple of LGA is extendedto the newcoordinatea-: at mostoneparticlecanlive on each“point” (x, c1, a-) in thephasespace.The streamingandcollision updatingareperformedas usual (cf. section 2.1). The novel feature is the presenceof the stirring updating. Itconsistsof the following two-step process.At each point x, one chooseswith equalprobability anumber p(x) in the set {1, 2,3,4}. The index a- is then changed to a-’ = a- —p(x) {modulo 4). Next,eachparticlelocatedin (x, a-, c1) moves into (x + c~, a-’, c,). In order to get a betterunderstandingofthe dynamicsinducedby this process,let us considerthe casewhere thereis just oneparticle in thewhole space. The stirring processmakestheparticlejump from one planeto theother. If one looks atthe x position, the resulting motion is of theBrownian type. Note that thestirring updatingdescribedhere is the two-dimensionalversion of the processintroducedby Boghosianand Levermore[4].Theideaof parallellatticeshasalso beenusedin ref. [35].An interestingphenomenonis the presenceofstrong correlations betweenthe variouscopiesasthe viscosityapproacheszero.

The full evolution of theautomatonis a Markov chain: at eachtime step and for each site (x, a-) thecollision updatingis applied with probability P(L), while with probability 1 — P(L) nothing happens.Next, regardlesswhetherthe collisionupdatinghasbeen applied or not, the stirring andthestreamingupdatingare applied. The probability P(L) is chosen such that P(L) = i/vT. In the limit L —~ ~, theprobability P(L ) —~0, i.e. the stirring updating is acting much more often than collisions. Weshall show(cf. theorem1 below) that this is enoughto ensurethe validity of theBoltzmannapproximationin asuitable window of space-time.

Let us now consider an ensemble of initial conditions and define the averagenumberN,(x, a-, t) ofparticlesin (x, ci) in the direction c, at time t. If correlationswere absent,the evolutionof the meanpopulationswould be governedby the lattice Boltzmannequation

N1(x, a-, t + 1) = ~ N,(x — c1 — ç, a-’, t) + (1I\.[L)4(N(x, a-, t)) , (32)

R. Benziet a!., The lattice Boltzmannequation:theory and applications 157

where 4(N) is the collision operator of the HPPdynamics (cf. section 2.1),

4(N) = N~+1N1~3(1— N~)(1— ~+2) — N~2(1— N1~1)(1— ~ (33)

Sincethemolecularchaosassumptionis not exactly true,eq. (32) will notdescribeexactly theevolutionof the averagepopulations for all times and for all scales.Onecan howevershow that the kineticequationscorrectlycapturethemacrodynamicbehaviourof thestochasticautomaton.In particular,it ispossibleto show that the systemis governed,in different time regimes,by variouspartial differentialequations:for t � L ‘~,with f3 < 1/2, a diffusive equationdue to stirring; for t — L h/2, the space-timecontinuous Boltzmannequationand finally, for longertimes, theHPP hydrodynamicequatiotis(whichare neither the Euler nor the Navier—Stokesequations,becauseof the lack of isotropyof the lattice).Note that 4(N) appears in (32) with a multiplication factor P(L) = it’lL which is the expectednumberof times the collision updatingis appliedin themodel. In the limit L —~ ~, this multiplicationfactor implies that the negativecontribution to theviscosity due to the lattice disappears.

Let us now give the precise statements of the theorems. In the following sectionE will denote theexpectation with respect to the Markov chain describing the time evolutionof the automaton.

Theorem1. Let us choosem points in the phasespace y~,= (x, c1, a-)1 and denoteby n(y1, t) thepopulation in the point y~at time t. For suitably smoothinitial conditions,

lim {sup{7} E([T n(y1, \/L t) — LI N,(x/VL, t)) } = 0, (34)

where N~(x, t). satisfiesthe equation

ö,JV~(x,t) + (c1 . O)N,(x,t) = 4(N). (35)

By “suitably smooth initial conditions”, one meansessentiallythat the probability densityfunctionofthe ensembleat time t = 0 doesnot dependon a- (this fact explainswhy in thenotationof theN~stheindex a- is dropped) and that the initial velocity and density profile varies on theaverageon a distanceof the orderof VL. In theproof of the theorem,the role of the stirring processis essential.

Theorem2. Let N~(h(x),a(x)) = 1 /{1 + exp[h(x) + a(x). C1]) be the solution of the equation4(N~) = 0 and let A(L) = 1/log~(vT),with 0<~<1. Then, the analogueof theoremI holds with‘vT substitutedby V7JIA(L). Moreover,N,(xA(L) /\/L, t) fulfills the condition

lim N,(xA(L)/V7J, t) — N~(h,(x), a,(x))~ = 0, (36)

whereh,(x) and a(x) satisfy theHPP Eulerequations(seebelow).

Theorem 3. Let N,A(x, t) anN,(A(L)~x, A(L)2t), whereA(L) is thesameasin theorem2. For suitably

smooth initial conditions (N~— A2), thereexist a(x), C and r0 suchthat for t~

sup1~N~(x,t) — N~(h0, A(L)a(x))I ~ CA(L)~. (37)

158 R. Benziet a!., The lattice Boltzmannequation: theoryand applications

Moreover, the fields

p0 anA(L)...~o~ N~(h0,A(L)a(x)), p0van A~).O~ N~(h0,A(L)a(x))c1tA(L) (38)

solve the incompressibleHPP Navier—Stokesequations.

A brief discussionof thephysicalmeaningof the threetheoremsis nowin order. The first theoremimplies that, on a spatial scale of the order V~L,correspondingto time scale V’L, the lattice gasdynamicsis describedby theBoltzmannequation (35). Thentheorem2 statesthat, for time scalesofordervT log~vT,with 0<~< 1, and acorrespondingspatial scaleof thesameorder, the lattice gasdynamicsis describedby theEulerequation.Note that for theorems1 and2 both spaceandtime arerescaledin the sameway. The differencebetweenthe two theoremslies in the different valueof theKnudsennumber(the ratio betweenthemeanfree pathin theautomatonand the typicaldistanceDLover which macroscopicfields vary). Accordingto theorem1, the typical distanceDL = ~VL,while inthesecondtheoremDL = VIIA(L). Thus theKnudsennumberis orderone for theorem1 andorderA(L) 1/log~\/L4 1 for theorem2. Note that the Eulerequationsareinvariant underthe scaling

xE—~yx,t~-+y’”t, v~—~y”v. (39)

Theorem2 makesuseof this propertywith h = 0. The scalinglaw invariance(39) doesnot hold for theNavier—Stokesequationswhich satisfythediffusive scaling,i.e. the scaling(39) with h = —1. By settingy = ~s/Llog~\/L,one recoversthe scalingassumptionusedin theorem3. The diffusive scaling is alsoconsistentwith theChapman—Enskogexpansionof theBoltzmannequation:at thesecondorder in theexpansionone recoversthe Navier—Stokesequation.

From a numericalpoint of view, theorem3 implies that if wesolve eq. (31) on a grid of dimensionL,this is equivalentto solving the incompressibleNavier—Stokesequationson a grid of dimensionA(L )L,i.e. the smallesthydrodynamicscaleis of order 1I(A(L)L). For L E (100,1000),A(L)E (0.66,0.53).

This is not far from what hasbeenobservedin numericalsimulationsof the lattice Boltzmannequationwithout the stirring process.

Let us nowsketchtheproofof theorem3 (for details,seeref. [19]).Let usremarkthat thepresenceof thestirring hasno relevancein the proof.

The startingpoint is eq. (35). As before,one can define

N~(x,t) = N1(A(L)’x, A(L)2t), N~(h, a) 1/(1 + exp(h+ a~ce)]. (40)

For simplicity, we will skip the argumentL of A. Inserting (40) into (35), one obtains

+ A’(c1 . O)N~= A

24(NA). (41)

Becausewe expectthe systemto be in theneighbourhoodof the equilibrium state,it is convenientto

write the populationsasN~(x,t) = N7~(h,Aa) + A2g

21(x,t) + A3g

31(x, t) + A3f~

1(x,t). (42)

The theoremis proved onceone is able to determineuniquely g21 andg3~,the functions h(x, t) and

R. Benzietat., Thelattice Boitzmannequation: theoryand applications 159

a(x, t) andfinally showthat f~1(x,t) — 0(A) for A—~ 0. This resultis accomplishedby inserting (42) into(41) and thenimposingthat eachorder in A, not containingf~,,vanishes.

To follow this idea,it is necessaryto developboth N~(h, Aa) and4(N~) in powerseriesof A. Theresult is

dNeq d2Neq

N~(h,Aa)=N~+A-~--+~A2dA~+A3RAI, (43)

where thefunctionsarecalculatedat A = 0 and RAE includesthe remainderof the expansion.From eq.

(40), it follows (hereafterthex and t dependenceof h and a arerelaxedfor thesakeof conciseness):

N~”(x,t) = 11(1 + eh) (44)

(dN~/dA)’(x,t) = —e”(a c1)/(l + e”)

2, (45)

d2N~IdA2= (e21’ — e”)(a . c~)2I(l+ e~)3. (46)

Onecan now define

p(x, t)=4/(l+e’), (47)

p(x, t)v(x, t) = [2 eh/(1+ e”)2]a (48)

whenceit follows

dN~ 1 ~ f2~P’\ 2

=2pvc,, dA

2 =2pt,,-~----—-1(vc1). (49)

The same power expansionin A has to be performed for ~(N~). One can use the fact that

~(N~(h, Aa)) = 0 for any A. Let us then define the linear operator LA by the identity

L5N, = (d/d6)4(N~(h, Aa) + 6N~.)I00, (50)

dependingon A throughN~(h, Aa). The expressionfor thecollision operatoris then

= L~(A2g

21)+ L~(A3g

31)+ LA(A3f~t)+

= A2L0g2~+ A

3L0g3,,+ A

3L1g21+ A

3LA(f~~)+ RAe, (51)

whereL0 = LA~AQ,L1 = dLAIdAIA,o andRAE is at least0(A

3). TheoperatorL0 canbe computedfrom

the abovedefinition of 4(N1),L0N1 = A(--N, + N~~5— N~~2+ N1~3), (52)

where A is a negativequantity. It turns out that L0 hasthreezero eigenvalues,with eigenvectors

160 R. Benziet at., The lattice Boltzmannequation:theoryand applications

‘P~= (1, 1, 1,1), ‘P~= (1,0, —1,0), ‘P2 = (0, 1,0, —1) and a negativeeigenvaluewith eigenvector‘P3 =(1, —1, 1, —1). Inserting(43) and(51) into (41) andequatingthe termshavingthesameorderin A andnot containingf ~, one has:

(c1.li)p =0, ~a,p+ ~(c1.O)(pv.c1) L0g21

~a1(pv.c~) + (c1 .a)[g21 + p(~-=_P)(v. c1)2] = L0g31 + L1g2~, (53)

+ (c, O)f~= -~ LAf~,+ R,,~1 + SAl.

Here, SAL is at least 0(1). The final step consists in proving that eq. (53) correspondsto theincompressibleNavier—Stokesequations.First of all, it is easy to show that the first equationcorrespondsto Op = 0.

Letus now considerthesecondequation.The right-handside containsthe operatorL0. In ordertosolve theequationfor g21, the left-handside shouldbe orthogonalto ~ ‘P~and ‘P2. The orthogonalityrequirementleadsto the following equation

a1p--O.(pv)=0. (54)

This equation,together with Op = 0, are the continuity equationsfor an incompressiblefluid. By

inverting the operatorL0 on the complementof thenull space,one immediatelygets:

g2~= L~[~p(c1. O)(v . c1)] + ~(x~ t)’Pkl. (55)

This equationshould be understoodin the following way. The only componentof (c, . O)(v c,) whichentersin the determinationof g21 is proportionalto ‘P3,1’ i.e. g21 = a(x, t)’P31 + ~ f3~~t)’Pk~.Thefunctiona is computedby applyingtheoperatorL0 to both sidesof theequation,multiplying by~‘P3,1 andsummingover i. Because‘P3.1 is an eigenvectorof L0 with a negativeeigenvalue(let us call it a), weobtain

a(x, t) = —(p/4a)(a~v~— a~v~). (56)

Let us consider the third equationof (53). In order to solve the equation for g31 (note that theoperatorL0 is acting onceagainon g31) the unknown functions

13k haveto be determined.First, weremarkthat c~= (‘P11, ‘P2,1). Next, it is easyto showthat L

1’P0, = 0 andthat L1’P11 and L1’P21 arebothproportionalto ‘P3,1. It follows that the orthogonalityof the third equationof (53) with respectto ‘P~and ‘P2.i hasto be required.This requirementis matchedmultiplying by c1 andsummingover i. Using(56), one finally obtainsthe incompressible’HPP Navier—Stokesequations,

pö,v~+ ~ ~ = —app + va~pvp. (57)

Here, v is afunction of p anda and thepressurep is determinedby theincompressibilityequation.Theorthogonalitywith respectto ~ providesthe expressionof the functionsf

3k~

R. Benziet at., The lattice Boltzcnannequation:theoryand applications 161

Once the functionsg21 andg3,areobtained,it is necessaryto controlthe remainderf~1.This last stepis performed as in the classicalBoltzmann equation, where one can calculate the order of theremainder.The detailsof sucha computationare reportedin ref. [19].

3.3. Spuriousinvariants

As discussedin the previous section, under suitably smooth initial conditions, the continuousBoltzmann equation (35) convergesto the incompressibleNavier—Stokesequations.The proof’ oftheorem3 is generalanddoesnot dependon thepresenceof the stirring stochasticprocess,which isonly neededto provethe validity of the Boltzmannapproximation.Whenthe continuousBoltzmannequationis consideredas a model in its own, thehydrodynamicbehaviourof the macroscopicscalesisthenensured.Equation(35) is a particulardiscretization,in space,time andmomentum,oftheclassicalBoltzmannequation.Thekinetic equation(31), which is theonepracticallyused,canbe consideredasa particularfinite-differencesapproximationof thecontinuousversion(35).This discretizationmayleadto difficulties similar to those encounteredin other numericalschemes.In particular,as we shall seebelow, one shouldtakecarein identifying the invariantsof the discretizedmodel.

For simplicity, let us considera threestatesystemin onedimension,with c. (1, —1, 0) andlabelthe lattice site x by the integerm. This model hasalso beenstudiedin detail in ref. [57].Considerthequantities

QE(t) = ~ [N1(2m, t) — N2(2m,t)] , Q0(t)= ~ [N1(2m+ 1, t) — N2(2m+ 1, t)] , (58)

which correspondto the momentumevaluated at even (QE) and odd (Q0) sites of the lattice

respectively.From thedefinition of the model, it follows

QE(t + 1) = ~ N1(2m— 1, t) + 41(2m—1)— N2(2m+ 1, t) — 42(2m + 1) = Q0(t), (59)

where in the latter equality we have used the conservationof mass and momentum. Similarly,Q0(t + 1) = QE(t) so that we havea new invariantof motion: (—l)’[QE(t) — Q0(t)]. This newinvariantis an artifact of the discretizedversion of the lattice Boltzmannequation.In fact, in the continuousversion, eq. (31) becomes

+ (c~. 8)N = 4(N). (60)

Using thedefinitionJ(x, t) = ~ !V~(x, t)c,, we find thatthe invariant is proportionalto J au i3x dx whichis trivially equalto zero.

It hasbeenshown [8,43, 44, 71, 72] that invariantssimilar to the one previouslydiscussedcan befound in both two- and three-dimensionallattice gas automata. In the case of the automata,anon-linearcoupling betweenthe densitiescorrespondingto spuriousinvariantsand thehydrodynamicdensitiescanproducemajordrawbacksin thecontinuouslimit. Sucha couplingis in fact ableto changethe equationsof motion in the hydrodynamic limit. Insteadof the incompressibleNavier—Stokesequations,one finds some other partial differential equations.For the FHP lattice, the spuriousdensitiesalso introduceanisotropiceffects in the evolutionequationfor the hydrodynamicvariables.Howeverthe spuriousfields do not self-generate:if they areinitially zero,they will stayzerofor latertimes [14].

162 R.Benziet a!., The lattice Boltzmannequation: theory and applications

From the abovediscussionit follows that thehydrodynamiclimit for thediscretizedmodel is not asstraightforwardasit might seemat a first glance.In particularthechoiceofthe initial conditionsand/orboundaryconditionscan producemacroscopicunwantedeffects.In themodel discussedin section3.2,suchproblemsdo not arise,becauseof thestochasticstirring processaddedin thedynamics.Actually,for most applications,the stirring processrequirestoo much computationaleffort to be chosenas apractical tool to handle spurious invariants. On the other hand, a suitable choice of the initialconditions, i.e. a choice which avoids spurious invariants, is not readily implemented,becausetheexplicit form of the invariantsdependson the lattice symmetriesandthe dynamics.Onecould naivelyreachthe conclusionthat (31) and the lattice gasautomataare not interestingas a numericaltool tointegrateNavier—Stokesequations,becausethey requiretoo muchcarein their application. Such aconclusionwould howeverbe too pessimistic.Indeed,thespuriousinvariantsare generallyvery smallquantities.This statementfollows from the fact that thespuriousinvariantsareroughly proportionaltothegradientsof thehydrodynamicfields. We haveseenthat only for suitably smoothinitial conditionsthehydrodynamiclimit canbe identifiedwith the incompressibleNavier—Stokesequations.In the limitof small Knudsennumbers(which is the regionwherelattice gasautomataareused),suchquantitiesarethenof higher ordercomparedto the hydrodynamicfields themselves.Moreover, this implies thatthe effects of spurious invariants are of the same kind as those introduced by the other non-hydrodynamicfields (ghostfields) presentin themodels.We discussthesevariablesin thenext section,wherewe showthat their dynamicstoo is stronglyaffectedby anisotropiceffects.So long asthe initialconditionsaresuitably smoothand the free parametersof themodel arechosenin suchaway that thescalesof theorderof the lattice site arenot excited,we havenot to worry aboutunwantedmacroscopiceffectsdue to spuriousinvariantsand/orghostfields. A counterexampleis providedby thestochastical-ly perturbedLBE,

N,(x+ c1, t + 1) — N1(x, t) = 4(N)+ m(x~t) , (61)

where ~1(x,t) is an externalstochasticnoise.Here,theeffectsof spuriousinvariantscanbe relevantifthenoisehasin theaveragea non-zeroprojectionon thespuriousfields. This is thecase,for instance,ifE[r~,(x, t)r1~(x + y, t + T)] = S11K( I vI’ r) and thecorrelationfunctionK is a rapidly decayingfunctionofboth T and I yI~In this case,theanisotropicdynamicsof both spuriousinvariantsandghost fields arelikely to producemacroscopiceffectsin thehydrodynamiclimit.

4. The lattice Boltzmann equation

4.1. From the booleanmicrodynamicsto the Boltzmannequation

Becausethe dynamics of lattice gas automata(LGA) is based on booleanvariables, all thesimulationsperformedby this techniqueare affectedby thepractical drawbackof statisticalnoise. Inorder to get reasonablyresolved macroscopicfields, it is necessaryto averageover a possiblecombinationof large regions of the lattice, long times and a wide rangeof initial conditions. Thispenalty is particularly felt wheneverquantities involving derivatives,such as vorticity, need to beexamined.On the otherhand,considerableadvantagescan begainedby building dedicatedmachines,suchastheMIT cellularautomaton[54]or theRAP machineattheEcoleNormaleSupérieurein Paris[17],taking advantageof the booleancharacterof the variables.As usual, no numerical schemeis

R. Benziet a!., The lattice Boltzmannequation: theoryand applications 163

perfectandthechoiceof the techniquestronglydependson theparticularphenomenonto be simulatedand on the availablemachine.

We will now show how the problem of noise can be overcomeby switching from the booleandynamics to the Boltzmann equation. In section 2.4, we have shown that by the assumptionofmolecularchaosit is possibleto write the following kinetic equation,

N1(x + c1, t + 1) — N,(x,t) = 4(N). (62)

Here, 4(N) is obtainedfrom thebooleancollision termby simply substitutingthebooleanpopulationn, with the ensembleaveragedpopulationN,.

Theproblemof noiseis obviously absentin eq. (62), becauseN, is a realvariableandno averageatall is neededto get themacroscopicfields. For this reason,McNamaraandZanetti[53]proposedto useeq. (62) directly for hydrodynamicsimulations,with the4 arisingfrom theknownbooleanmodels.Inparticular,they studiedthe model definedby theFHP-III rules [41],by simulating thedecayof shearand soundwavesof definite wavelength.The comparisonbetweenthe experimentalvaluesand theChapman—Enskogmultiscalepredictionsshowsthat thehydrodynamicvalueis accurateto betterthan5%, evenfor a lattice assmall as4. Also the behaviourof soundwavesis definitely in agreementwiththe expectations.

Actually, as observedby Higuera and Jimenezin ref. [37],it is possibleto further simplify themodel. The reasonis that macrodynamicequationsin lattice gas models(and then Navier—Stokesequations)formally arisein thedouble limit of smallKnudsennumbersandsmall Mach numbers.It isthenconvenientto considertheexpansionof the collision termin the rhsof (62) correspondingto theseconditions. Let us write N~asin (12),

= N~(p,v) + N~~(Op,liv), (63)

and further decompose~eq as

= N~°~+ N~1~+ N~2~+ 0(v3), (64)

wheretheupperindex refers to theorder in v. The explicit expressionof thevarioustermsis given in

(9). The correspondingexpansionof the collision operatoris

4(N) 4(N°~)+ ~ N~’~+ ~J(N~2~+ N~~)+ ~ aN~,aNkN~’~N~’~ (65)

where all the derivativesare calculatedat the stateof zero velocity N~= d = p/b. We now usethe

propertythat, for any equilibrium distribution, the following equality holds:

4(N~)= 0. (66)

Specializingeq. (66) to the caseof uniform equilibria (zero velocity), we have4(N~°~)= 0. For a

genericequilibrium distribution, eqs. (65) and (66) yield

8~ãf N”~N”~—0 67~ aN1 ~ 2aNJaNk j k —

164 R. Benziet a!., The lattice Boltzmannequation:theory and applications

Plugging (67) into (65), we finally obtain

4(N)— K.J(NJ— Np), (68)

where

K11an~jj- (69)

= — ~ (s1 — s~)A(s—*s’)d~’(1— d)~P’(s1 — s), (70)

andp = E,s~.Remarkthat, dueto thesymmetriesof the collision matrix K,~, the rhs of eq. (68) canbewritten as K1J(N~,— N~

2~),which is more convenientfor computationalpurposes.Despiteits apparentlinearity, theexpression(68) accountsfor second-ordertermsin the expansion

of the collision operator.The passagefrom the completecollision operatorto the form (68) is clearlyadvantageousfrom the pointof view of simplicity andstoragerequirementsof the numericalscheme.On the other hand, as will becomeclear in the following, the reduced expression(68) is exposedtonumericalinstabilities.

4.2. Lattice Boltzmannequationwith enhancedcollisions

The starting point in the definition of the lattice Boltzmann equation with enhancedcollisions is thelinearizedkinetic equationobtainedin the last section,

N~(x+c1,t+l)—N1(x,t)=.11(N1—N~),i=1,...,b, (71)

governingtheevolutionof themeanparticlepopulationN, in theb possiblestatesper site.Here,C, arethe velocities, sf41 is the collision matrix (the reasonsfor the changeof notationwith respectto theprecedingsectionwill becomeclear in a while), whose elementa~1determinesthe scatteringratebetweendirections i and j, and N~ is the equilibrium population. In the definition of the latticeBoltzmann equationwith enhancedcollisions a different point of view is taken: the choice of thequantitiesin (71) is no moreforced by an underlyingbooleanmicrodynamics,but it is ratherdictatedby the macroscopicequationsto be reproduced.This possibility was usedin ref. [39]in the caseofhydrodynamicsand it is reviewedhere. Furtherspecializationsfor different hydrodynamicalequationsaregiven in sections4.5, 6.2 and6.3.

The argumentsof theprevioussectionsuggestthe following definitions:(i) The equilibrium population*)hasthe sameexpressionas (9),

= ~ (i + ~ vacia + Qiap1~at~p), Q,~= CiaCip — (C2ID)i5ap . (72)

(ii) The generalelementa,1 only dependson the anglebetweendirectionsc, and c1.

(iii) Collisions conservemassandmomentum,i.e.

*) The constantin thenonlineartermof theequilibriumpopulationcouldactuallybearbitrarily chosen.By aconvenientchoice,onecould then

eliminatethespuriousfactorg(p)which arisesin thehydrodynanucequationsof themodel. Thiscouldbeuseful in multi-phaseflows. As notedinref. 145] this freedomcan alsobeusedto removethe quadraticcorrection in the pressureterm(by the addition of rest particles)or to uselessregularlattices,a rectangularonefor instance147].

R. Benziet at., The lattice Bolrzmannequation:theory andapplications 165

~ j=1,...,b. (73)

Notice that the matrix elementsa,1 are now numerical parameterswhich can be changedat will.Apart from this freedom,the matrix .cfIq defined by (ii) and (iii), hasthe samesymmetriesas thecollision matrix K,1 obtainedin theprevioussectionfrom the booleanmicrodynamics.In the followingwe will thenskip the label “with enhancedcollisions” wheneverthis doesnot lead to confusion.

The possible anglesbetweenthe various directions are 0, ‘rr/3, 2ir/3, ~rfor the FHP lattice and0, ir/3, ‘rr/2, 2’ir/3, ir for theFCHC lattice. Accordingto (ii), thenumberof possiblydifferent elementsa,1 of thecollision matrix is four for the6-particleFHP and five for the24-particleFCHC. The inclusionof a rest particle implies the introduction of two new independentparameters,to account for theinfluenceof the restingparticle on itself andon theotherdirections.In the following, for simplicity, weshall considerthe casewithout restparticles.

The condition (iii) imposessomeadditionalconstraintson thenumberof independentelementsof

~ Denotingby a0 thematrix elementsa,1 suchthat c, C1 = c2 cos0, it is easyto verify that eqs.(73)

are equivalentto

a0+2a60+2a120+a180=0, a0+a60—a120—a180=0, (74)

for the FHP model. In the FCHC case,the correspondingequationsare

a0 + 8a60+ 6a90+ 8a120+ a180 = 0, a0 + 4a60 — 4a120— a180 = 0. ‘ (75)

Thanksto the propertythat the matrix .t~ is symmetricand cyclic, the independentcoefficientsa0canbe expressed[70]in termsof the non-zeroeigenvaluesof the matrix. For the FHP, the non-zeroeigenvaluesare

A=6(a0+a60), cr=—6(a0+2a60), (76)

with multiplicities 2 and 1 respectively.For the FCHC, the samecalculationsleadto

A = a0 — 2a90+ a180, a- = ~(a~— a180), r = ~(a0+ 6a90 + a180), (77)

with muhiplicities9, 8 and2. In both casestheeigenvectorsaremutuallyorthogonalanddo not dependon the matrix coefficients. In particular, the eigenvectorsassociatedto the eigenvalueA are theD(D + 1)12 — 1 linearly independentelementsof the setof vectors

Let usnow consider’the rangeof thepossiblevaluesof thecoefficientsa0. In thehomogeneouscase,an obvious condition for the stability of the schemeis that the non-zeroeigenvaluesof the collisionmatrix benegativeandlargerthan—2. In thenon-homogeneouscase,the first checkonehasto makeisthe linear stabilityof thescheme.In thelinear approximation,oneneglectsin (71) thenon-lineartermin theequilibrium population.By Fourier transformingthe resultingequation,oneobtains

{[exp(i(k. c, — w)r) — 1]ô,1 — N1 an G.1(w, k)N1 = 0. (78)

For the sake of clarity, the time step r is explicitly reintroduced.As usual, the normalmodesare

166 R. Benziet a!., The lattice Boltzmannequation:theoryand applications

determinedby imposingthat the determinantof the matrix G,1 is zero. By letting w = (1 + iy andrequiring‘y <0, we concludethat theschemeis linearly stableunderthecondition—2 < ~< 0, ~beingany eigenvalueof

Thepreviousconditionon theeigenvaluesis also sufficient to ensurethat collisions act in thesenseof restoringequilibrium. Let us in fact considerthe quantity

~(x, t) = — ~ (N~~)2, (79)

and considerits evolutionunderthe actionof collisions. By using eq. (71), one has:

N~(x,t + 1) = N~(x,t) ~ne~(~ t + 1) = N’~(x t) + if41N1 (x, t) . (80)

Inserting(80) into (79), one easily finds that the contributionsto thevariation of ~‘(x, t) are of theform — ~~.2 + 2~)P, where P is a positive (or zero) quantity and ~is the genericeigenvalueof thecollision matrix. Providedthat —2<~<0, the quantity ~(x, t) is then always growing during thecollision phase.The only casewhen ~ doesnot grow is at the equilibrium state,where it attains itsmaximumvalue. Weremarkthat the sign of variationof ~ during thestreamingphaseis not fixed. Themain differencewith respectto a true H-theoremis that the global quantity H(t) = E~~4x,t) is notconstantduring thestreamingphase.Therelation of this resultwith theobservednumericalinstabilitiesof the schemewill be discussedlater.

4.3. Macrodynamicequations

In order to study the macrodynamicsof the lattice Boltzmannequation,multiscale expansions,alreadydiscussedin section2, canbe used.We recall that theexpansionparametere is the inverseofthe wavelengthof the typical spatialvariation of the fields expressedin thenaturalunitsof the lattice.Assumingthat s is small enough,onecan keeponly first-ordertermsin themultiscaleexpansion.Theresultingdifferential equationsare

a~N1+ (c, .li)N = sf41(N1 — Np). (81)

Correctionsto theseequationsarisingfrom higher-ordertermsof theexpansionwill be discussedin thenextsection.*)

As suggestedby Frisch in ref. [26],it is nowconvenientto projecteqs.(81) onto theeigenvectorsofthe collision matrix. The advantageis that a natural distinction betweenhydrodynamic and non-hydrodynamicfields arises.A clear analysisof the hydrodynamicbehaviourof the model is thenallowed.

In thefollowing two subsectionswe will perform theprojectionfirst for the FHP and then for theFCHC lattice. In thenext sectionthehydrodynamiclimit of theequationsis discussed.Thecalculationswere performedin ref. [261for the FHP lattice and in ref. [69]for the FCHC lattice.

*) A different pointof view couldbe taken. Equation(81) is thebasic equationandthe discreteequation(71) is just a particularnumerical

schemeof solution.

R. Benziet a!., Thelattice Boltzmannequation:theoryand applications 167

4.3.1. The FHPcaseThanksto the symmetriesimposedon the collision matrix, all the eigenvaluesand five of the six

eigenvectors(1, c~,Cj)~and the two independentcomponents0f Qiap) of the collision matrix arealreadyknown. Onecan showthat theothereigenvectoris (—1)’. The genericpopulationN~canthenbe decomposedas

= ~p + ~pciav + ~QIaP~ap+ ~ (82)

Equations (81) can now be projectedonto the orthogonal basis of the eigenvectors.Multiplying

successivelyby 1, C,a~Qiap’ (—1)’ and summingover i, one obtains

3tP+äaJa=0~ atJa+aa(p12)+apSap=0,(83)

+ ITIap + 1?a~y~9yP~= A(S’ap — S~),a~+4RapyäySap=

Here

Rap..,, ~ (~)‘Qiap’~iy, S~= pg(p)(v~vp— ~V2ôap),

(84)

g(p)=(3—p)I(6—p), Hap=aaJp+apJa—(a.yJy)oap

Remarkthat the non-isotropy introduced by the rotational discretizationof the lattice affects theexpressionof thenon-hydrodynamicfields, i.e. theghost field ~ andthehigher-ordercontributionstothestresstensor.

4.3.2. The FCHC caseDue to the dimensionof thematrix, thecalculationof thespectralpropertiesof theFCHC collision

matrix is muchlengthierthanin FHP. A significant advantagecan be achievedby defininga reducedcollision matrix. For simplicity, let us considerthe two-dimensionalcase.When the four-dimensionallattice is projectedonto the x—y plane,somevectorsproject onto thesametwo-dimensionalvector[forinstance, the four vectors (1,0,1,0), (1,0,—1,0), (1,0,0,1), (1,0,0,—1) all project onto thetwo-dimensionalvector (1,0)]. Only the sum of the four populationsis relevantfor the calculationoftwo-dimensionalfields. Without lossof generality,it canbe assumedthatall thepopulationsprojectingonto the sametwo-dimensionalvectorhavethe samevalue. We are thenleft with only nine distinctpopulationsN~,ranging from zero to one. It must however be rememberedthat eachpopulationcontributesto the calculationof the fields with a weight

— [1 for diagonaldirections, 85— I. 4 otherwise.

For instance,the density p is calculatedas

p4(N1+N3+N5+N7+N9)+N2+N4+N6+N8, (86)

where N1 propagatesalong the positive x-axis, N9 is the rest particle and the remaining seven

168 R. Benziet al., The lattice Boltzmannequation: theoryand applications

populationsare numberedcounterclockwise.It is now natural to introducea 9 X 9 reducedcollisionmatrix ~‘, expressingthe effect of collisions on theaforementionednine populations.The valueof theelementsof thematrix ~ asa functionof theoriginal a0 is reportedin appendixA. The reductionin thedimensionof thematrix ‘~ is very convenientalso from a computationalpoint of view.

The matrix ~ is no longersymmetricandcyclic and its eigenvectorsarenotorthogonal.The problemcan howeverbe solved by introducingthe following weightedscalarproduct,

A.Ban>..p1A~B,. (87)

By standardcalculations,one canverify that ~ hasthe samesetof eigenvalues(0, A, a- and r) astheoriginal matrix .~1,with multiplicities 3, 3, 2 and 1. The eigenvectorsconstitutean orthogonalbasisaccording to the scalarproduct defined by (87). It follows that the generic populationN~can beexpressedas

N1(x,t) = ~ A~(x, t), (88)

whereA~arethe eigenvectorsand~ (n) arethe correspondingfields.It is readily checkedthat the subspaceof the conservedvectors(zero eigenvalue)is spannedby

A~’~=(1,1,1,1,1,1,1,1,1)an11,A~2~=(1,1,0,—1,—1,—1,0,1,0)=c~~,

(89)A~3~=(0, 1, 1, 1,0, —1,—i, —1,0)=c,~

The identification of ~~ with the densityp and of ~ (2), ~ (3) with the x—y projectionof the currentI = pv is immediate.After somealgebra,one finds also

A~9~= (1, —2, 1, —2, 1, —2, 1, —2, —2) an r,(90)

= r.c,~, = r,c,ywith eigenvaluesr, a- and a- respectively.Formula (90) suggeststhe identificationof the fields ~7) and

(8) asthex—yprojectionof adensitycurrentire, whosedensityfield is 4~(9), hereafterdenotedby ~. We

shallrefer to i andp~asghostfields, for reasonswhich will becomeclearin the following. Finally, A~4~and A~5~coincide with Q,~and Q~and the vector A~6~has the following expression:A~6~=

Q,,,., + ~Q1~.Hereafter,we find it more convenientto usethe tensorQiap insteadof ~ A~

5~and~ In this case,weloosetheorthogonalityof the basisbut, asweshall seebelow, thedefinitionof thestresstensorSap becomesstraightforward:Sap = E

1 Pi1”4Qiap

Backedup by the list of the eigenvectors,one can project the equations(81), asin the FHP. Theresult is the following setof non-linearpartial differential equations:

ä~p+ aaja =0, a~1.i.+8afla = ~ , atJa + aa(p/2)+ apSap 0,

(91)+ 0a(/~L/2)+ apTap= t7~1a, a~S

0~+ ~ + Gap= A(SaP— S~~)

R.Benziet a!., The lattice Boltzmannequation: theoryand applications 169

Here

g(p) = ~(12 — p)/(24.—p), S~= pg(p)(vavp— ~v26ap), Tap = ~ PITINiQIaP

1 1 (92)

Hap = aafp+ apJa — ~13y1yôap, ~ an PiTiCiaCipl~iyCj8 , G~,p= Rap..,~ay

We remark the formal similarity of the lhs of the equationsfor the ghost fields and for thehydrodynamicfields, which will beuseful in thefollowing. In particular,it is possibleto switchfrom oneblock to the other by changingthe weights of the variousdirections:p, ~ p1r1. The different weightsimply thenon-isotropyof the ghost stresstensor:

~ ~ T~~=—2S~~, (93)

whereTr denotesthe trace of a matrix.Using eqs. (91), it is also possibleto prove the uniquenessof the symmetriesused to define the

collision matrix in LBE. Let us start with the most general9 x 9 matrix .s~’,and let us imposethefollowing requirements:(i) theconservationof massandmomentum,(ii) ~ hasthesamesymmetriesofthe lattice, (iii) in themacroscopiclimit theLBE reproducesNavier—Stokesbehaviour.It is possibletoverify that, undertheaboveconstraints,the resultingmatrix hasthesamestructureas ~. We arguethatthis uniquenessis not peculiarto the FCLIC, but can be extendedto otherlattices.

By similarbut lengthier,calculationsonecantreatalsothe three-dimensionalFCHC. The dimensionof the reducedcollision matrix is 18 x 18. The expressionof the eigenvectorsis given in appendixB.The resulting macrodynamicequationshave essentially the same structure as in 2D. The resultsobtainedfor the lattercasecanbetransposedto 3D with only minor modifications.In the following weshall then restrictourselvesto the two-dimensionalequations.

4.4. Hydrodynamicbehaviourof the lattice Boltzmannequation

Let us now discussthe macrodynamicequationsobtainedin the last section,focusing on thepossibility of recoveringthe equationsof hydrodynamics.

In order to obtain the incompressibleNavier—Stokesequationsfrom the setsof equations(83) or(91), it is sufficient to satisfy the following threerequirements.The first is thepossibility of neglectingghost fields with respectto hydrodynamic fields. The secondis the usual incompressibilityconditionp const.The third is the validity of the adiabaticapproximation,i.e. the time derivativetermin theequationof thestresstensorSapbe muchsmallerthantheothertermsin thesameequation.Let usnowsubstitutetheexpressionof thestresstensorcorrespondingto theseassumptionsin theequationfor thecurrent. One can easily check that Navier—Stokesequationsare recovered,apartfrom the usualconstantfactor g(p), which can be easilyrescaledout.

The first condition is strictly relatedto the issue of higher-ordertermsof the multiscaleexpansionusedin the last section.Equations(83) and (91) suggestthat the various fields havegenerallythefollowing order:

S—v2+ev, ,~— ev2+s~v,p.--.O(ei). (94)

170 R. Benzieta!., The lattice Boltzmannequation:theoryand applications

Becausethe correctionsto (81) are at leastof secondorder, the rates ~IS and j.~IS tendto zero ase—~ 0. Figures4 and 5 providea visual confirmationthat theactivity of ghost fields is mainly confinedtosmall scales.When theadiabaticexpressionof thestresstensoris substitutedinto theequationfor thecurrent, a viscous term0(e2) is generated.It is thennecessaryto includethe second-ordercontribu-tions to this equation,coming from the expansionof the streamingoperator. In section 2, thesecond-orderexpansionof the streamingoperatorwasperformedin order to obtain the macroscopicequationsfor a latticegasmodel.Thesecalculationscanbetransposedwith no modificationto ourcase.Onecan thus verify that the final result is the inclusion of thepropagationviscositycontribution.

Concerningthe incompressibilityissue,it is well known [50]that this approximationrequiresthat thevelocitiesof thefluid be muchsmaller thanthe speedof soundand hasalreadybeendiscussedin thegeneralcontext of lattice gasmodels.

The physicalmeaningof theadiabaticapproximationis that the typical hydrodynamictime scaleismuchlonger thanthe time scaleof relaxationto the local thermodynamicequilibrium. Non-equilibriumfields are then adiabaticallyslavedto the equilibrium fields.

To obtainaquantitativeconstrainton theeigenvaluesof thecollision matrix, let usobservethat, thespeedof sound being of the order one, the inverse time scale of propagationassociatedto aperturbationhavinga wavenumberk is of the orderk. In the continuousversionof theequationsforthestresstensor,the relaxationtime is equalto 1/IA. It is thenreasonablethat, in orderto verify theadiabaticapproximation,the inequality

IAl~k (95)

mustbesatisfied.Condition (95)is in agreementwith theexperimentalobservationthathydrodynamicbehaviouris alreadyattainedfor the modeshaving a wavelengthof few lattice spacings.

With respectto the dynamicsof the other non-equilibriumfields, one should requirethat theircharacteristictime scalesaresmaller or atleastcomparableto that of the stresstensor. Denotingby ~the genericnon-zeroeigenvalueof thecollision matrix, the condition I ~l I Al naturally arises.In thediscretecase,the maximumspeedof relaxationis attainedfor ~= —1. The useof this value for theeigenvaluesof the ghost fields was suggestedin ref. [39].On the other hand, we expect that the

~ ~ c

~

Fig. 4. Typical snapshotof thehydrodynamicvorticity field (from ref. Fig. 5. Typical snapshotof theghostvorticity field (from ref. 169]) at169]) for a 1282 simulation with Mach numberM = 0.2. the same time and with the same parametersas in fig. 4. The

amplitudesarescaleddown by six ordersof magnitudewith respecttothe fields in fig. 4.

R. Benziet a!., The lattice Boltzmannequation:theoryand applications 171

behaviourof thesystemmay changein theneighbourhoodof A = —2, wheretheviscosityis very small.It is in fact well known in statisticalmechanicsthat, whenone of thekinetic coefficientstendsto zero,macroscopicequationsmust be modified, by including higher ordertermsof the multiscaleexpansion[12].

A simple examplewherethepreviousobservationsarerelevantcanbe provided.Let us considertherelaxation of a macroscopicperturbationas a function of A in the two-dimensionalFCHC model.Non-linearterms,which do not contributeto dissipation,are discarded.The dynamicsis well repre-sentedby the following set of equations

a~J~=0, atJa +aa( p12)+apSap+ ~apapJa=0,(96)

aiSap+ ~(aaJp+ öpJa)= A(Sap— S~).

Notice thatwehaveincludedthecontributioncorrespondingto the propagationviscosity(otherwisethevalueA = —2 would not be critical). Thanksto theisotropy, thereis no lackof generalityin consideringthe casek~= 0 and the substitutionk~‘—~ k is sufficient to obtain the eigenvaluesin full generality.Bylooking for thesolutionin termsof a FourierseriesE k exp(ikx— xkt), the following dispersionrelationis easily found,

(97)

In the limit k ~ I Al, the squareroot can be developedin termsof k/I Al. At the leading order, the

expressionof the eigenvaluesis

X~A+0(k2), ~~‘—~~k2(1+2IA)+O(k4). (98)

Onealso finds that, at theleadingorder,theevolutionof the fields, asa functionofthe initial values,is

J~(t) = J~(0) exp(—x~’t)+ ~ — exp(—x~t)],(99)

S~~(t)= S~~(0)exp(—x~t) + (ik~I3A)J~(0)exp(—x~’t)

The resultsarethe sameasthoseonewould get by using theadiabaticapproximation.In theoppositecase(IA I sk), the relaxationtime is very long, thenon-equilibriumfield S~,is not slavedto J~,andthebehaviouris not hydrodynamic.This confirmsthe validity of (95).

Let us now considerthe limit whereA is very closeto —2. Within thescalesverifying the inequality(95), thereis a rangeof modessuchthat the term0(k4) in theexpressionof x~hasthesameorderofmagnitudeasthe first term. Thesescalesarein fact no longerdescribedby eqs. (96) andhigher-ordertermsin e mustbe included. In thegeneralcasethesecontributionsarenot isotropic (at leastfor FHPandFCHC lattices) and the model is no longer representativeof thebehaviourof a real fluid in thisrange of scales.

In summary,wehaveshownthat thelattice Boltzmannequationis a hyperbolicapproximationof theNavier—Stokesequations,i.e. thestresstensoris slavedto its hydrodynamicexpressionwithin a delaytime of theorder 1/I Al. When theviscosity is not too small, the lattice Boltzmannequationconvergesto Navier—Stokesequationsbetterandbetteras the viscosity is decreased.When the eigenvalueA is

172 R.Benzieta!., The lattice Bo!tzmannequation: theoryand applications

very close to —2, non-isotropic and non-universalhigher-orderterms of the multiscale expansionbecomeof relevance.This issue will be discussedin more details in thefollowing section.

4.5. The limit of zero viscosity

The observationsmade in the previous section concerning the limit of zero viscosity can bequantitativelyclarified by the analysisof a simplified one-dimensionalmodel. The advantageis that asolutionof themodel in its fully discreteform canbe found,with no needof any multiscaleexpansion.

The model is definedon a one-dimensionallattice with two statesper node.Particlescan moverightwardsandleftwardswith a unit speed.The equationscorrespondingto (71) are

N,(x + c,, t+ 1) — N,(x, t) = .s141(N3— Np). (100)

Becauseour aim is to obtaina diffusive behaviouron themacroscopicscales,weimposethat collisionsconservethemass.Consequently,thecollision matrix only dependson thecoefficienta, governingthescatteringratebetweenthenghtwardsandleftwardsdirections.In particular,thematrix elementsarea11 = a22 = —a and a12 = a21 = a. The eigenvectorsare (1, 1) with zeroeigenvalueand c, = (1, —1)

with eigenvalue—2a. Thecorrespondingfields are ~ = 1N1 and J = N,c,. The equilibrium popula-tion neededto obtainadiffusive behaviour*)is simply N~= 4 /2. Thesameprojectionprocedureusedpreviouslycan be obviously repeatedin this case.The equationscorrespondingto (96) are

~a~a~=0,a1J+a~4=_2aJanAJ, (101)

whereJ = E~N,c,. In themacroscopiclimit and in theadiabatichypothesis,thebehaviourof themodel

is purely diffusive and thediffusion coefficient D( A) isD(A)=—(1IA+~). (102)

Remarkthat D( A) dependson theparameterA in thesameway as theviscosity in hydrodynamics.Inparticular,in the limit A—~ —2, the kinetic coefficient vanishes.

Let us now considerthediscretesolution.The modelis linear and thesolutionis mostconvenientlysoughtin termsof planewaves,

exp[ikm — Xkfl], (103)

where m andn areintegers.It is easyto obtainthedispersionrelation.Oneof theeigenvaluesdescribesa pure relaxation,while the otheris associatedto diffusive behaviour.The frequencyof the diffusivemodecan be recastin the following form

Xk = [D(A)/Q(A, 0)]k2. (104)

HereD( A) is themacroscopicvalueof thediffusion coefficient, 0 is thewavenumbernormalizedto thelattice spacingand

*) Remarkthat by usingN~”= 4il2 + c4.214at the leadingorderswe obtain the Burgersequation.

R. Benziet a!., The lattice Bo!tzinannequation:theory and applications 173

Q(A,0)—~02[(2+A)IA]log’{fl(2+A)cosO+\fA2—(2+A)2sin20]} (105)

is a functionwhich embodiestheeffect of the lattice discreteness[52](theparameterappearingin ref.[52] is equalto —A /2). As onecancheckby directinspectionQ(A, 0) = 1, in agreementwith (102). It isnow of interestto examinethebehaviourof this function.A seriesof curvesfor different valuesof A inthe interval (0, —2) are shownin fig. 6. From this figure, we seethat:

(i) The regionof convergenceto themacroscopicvalueextendsas I Al grows,providedthat A is nottoo closeto —2. For valuesof A far enoughfrom —2 and 0, the correctionsto the hydrodynamicvalueQ = 1 are very small for the scalesof interest.

(ii) For small valuesof A, the correctionstend to overdampthe modes. In the oppositelimit therelaxationrateof the modesis decreased.

(iii) In theneighbourhoodof —2, for a fixed k, thecorrectionstendto increasewhen I Al growsandthe transition to non-dissipativebehaviourbecomesmoreabrupt.

Thebehaviourof thesystemis definitelyin agreementwith thephenomenologicalpredictionsmadein the previoussection.Whenthe kinetic coefficient approacheszero (A—p —2), thereis obviously noproblemswith isotropy as in the multidimensionalcaseand the higher-ordertermsonly reflect in thefact that the model function Q(A, 0) displaysa sharperdependenceon 0.

We argue that (iii) has a certain generality.Also in hydrodynamicsthe value —2 is indeedthefrontier betweenthe stableand the unstableregions. When transferredto the context of hydro-dynamics,theseconsiderationssuggestthat, by pushing the viscosity too low, undampedscalesareexcitedand the schemeis prone to numerical instabilities,as observedin numericalexperiments.

4.6. Numericalefficiency

Although lattice gasmodelsbearanintrinsic interestasfictitious ab initio modelsoffluid flows, it isclear that much of the attentionthey havebeen capturingin the last few years is to be attributedtotheir potentialasa new computationaltool for numericalfluid dynamics.The purposeof this sectionis

I ________________________theta

Fig. 6. The function Q( A, 0) expressingthe effect of lattice discretenesson thevalue of the diffusion coefficient versus9 for variousAs. Frombottomto top: A = —1.02; A = —1.2; A = —1.4 and A = —1.6.

174 R. Benzieta!., The lattice Boltzmannequation: theoryand applications

therefore to outline the main conceptswhich allow a semi-quantitativeassessmentof the numericalefficiency of the lattice Boltzmannequation.

We begin with some definitions. Let L be the integral scaleof the flow, 1d the dissipativescale;1!mfp = v/c, the particlemeanfree path,a the latticespacing.Accordingto thediscussionof section2.4,the ReynoldsnumberR achievablewith N lattice sites (L = Na) is given by:

R=MR*N, (106)

whereR* = c,alv — a/lmfp andM = v/c, is thesoundMachnumber.The dissipativelength 1d is relatedto the integral length by the relation Id LIRtm (m= 1/2 in d = 2 and m = 3/4 in d = 3), whenceweobtain

R= (LIld)Vm = (aIld)~mNhJm . (107)

From theseexpressionswe seethat the Reynoldsnumberis basically controlledby theparticle meanfreepath(R* = in latticeunits). As shownin section4.4, theviscosity ii = lmfpCs is relatedto theleadingeigenvalueof the collision matrix, A, by

v—[c2/(D+2)](1IA+ ~), (108)

which shows that, by letting A come close to —2, one could hope for the possibility of reachingarbitrarily high valuesof the Reynoldsnumber,evenworking in a finite lattice.

It is useful to identify two main working-regimesfor LBE: (a) adiabatic regime: 1mfp ‘~ ‘d’ (b)

numericallyresolved:a ~ ‘d~ Obviously, the aim is to work in theadiabaticregime(in orderfor LBE tobe a true picture of the Navier—Stokesequations)and having enoughresolution to guaranteethevalidity of themultiscaleexpansion.Hencetheright orderingis ‘mfp ~ a ~ ‘d• For the sakeof clarity, letus specialize to the two-dimensionalcase.The numerically resolvedregime is attainedfor a givenReynoldsnumberwheneverthe numberof nodesexceedsa minimum thresholdNNR, given by

ld/a=N/R�1, N�NNR=R~2. (109)

On theotherhand, the adiabaticregime is characterizedby thecondition

1 Ia 1 10* ~1I2iafp mfp = III~ =~__>~ (110Id Id/a N/RV2 NR* — ‘

which identifies the adiabaticthresholdNAD as

NAD = RII2IR*. (111)

From this equationwe seethat for R* > 1 (the usuallattice gasregime),oncenumericalresolutionhasbeenensured,adiabaticitywill ensueautomatically,while the reverseis not true.Differently restated,this meansthat, atvariancewith theearliestlattice gasschemes(R* 1), thelimits of LBE arefixed bycomputationalresolutionratherthan by the discretenessof velocity space.

It is now instructive to representthe equation(106) in the R—N plane. In this plane, (106) is

R. Benziet a!., The lattice Boltznsannequation: theoryand applications 175

representedby a seriesof straight lineswhoseslope is given by 1 /MR*. Thus by increasingR~, for afixed R, one linearly reducesthe numberof computationalnodes.However, in view of eq. (111),representedin fig. 7 by thesolid line, weseethat this reductionis really effectiveonly aslong asN doesnot fall below the thresholdNNR. Otherwise1d becomessmallerthanthe lattice spacingand the wholemultiscaleexpansionbreaksdown. This clarifies the reasonwhy the limit v—*0 with A —* —2 is purelyformal.

Let us now examinemore closelythe numericalpropertiesof LBE.First of all, noticethat, asusualfor numericalschemesworking at a unit Courantnumberyr/a (r was

introducedin (78)), the LBE schemeis free of numericaldispersionand diffusion.The secondissue concernsthe numericalstability of the scheme.The linear stability discussedin

section4.2 applies only to quasi-uniformflows, with small departuresfrom zero flow speeduniformequilibria, N~(v= 0) = d. The dynamicsof the fluctuationsaroundnon zero-speedequilibria is moreand more exposedto the effectsof non-linearmode—modecoupling betweenthevarious populations

Ni. As is well known,non-linearmode—modecoupling can give rise to high-frequencymodes,whosecorrect representationin the discretegrid is mandatoryto preservethepositivity of the distributionfunction. Violation of this condition may result in catastrophicinstabilities,becausetermswhich arephysically stabilizing changesign and destabilize the system. This is easily seen by expliciting thedefinition of the flow current v = E 1s1

1c1 in the expressionof the non-linearterm in the equilibriumpopulation.This yields

+ (c1 O)N, = ~ — N~)— P4Jk(N1 — Nr)(Nk — Np), (112)

where

P4jk = P(J(P)~9(4hQhapt~jaCkp, (113)

which clearly revealsthe non-linearityhidden in the expressionof the equilibrium population. At aqualitative stage,the scenariofor the developmentof non-linear instabilities can be devised byidentifying the rhsof eq. (112) with a restoringforcewhich dragsthe systemto equilibrium. When vincreases,counterfiowpopulations (v~ c1 <0) get depletedso that a smalleramplitudeof fluctuations

a

-i-Nod 91.-tO)E1E+02 .‘U.N~d(R.1) ...

. . .....

I::: ~ ITT$ub~rid1E+OO 1~+Ot 1E+02 1E+03

Reynolds number

Fig. 7. Thresholdsof numberof grid points neededto attainadiabaticity(Nad) andnumerically resolvedregimes,asa function of the Reynoldsnumber. In the region labeled “Subgrid” theflow is no longernumerically resolved. The dotted lines, whoseslope is 1/R*, indicatethe linearrelation betweenthe numberof grid points and thecorrespondingReynoldsnumber.This figure refersto Mach number0.1.

176 R. Benziet a!., The lattice Boltzmannequation: theory and applications

aroundthecorrespondingequilibriumwill suffice to makethe restoringtermnegative,turning it into arepellingone andthus triggering the instability.

Specifically,it is a commonfinding of theauthorsthat whenevertheviscositybecomestoosmall, i.e.so small that scalescomparableto the lattice pitch areexcited, numericalinstabilities developin theflow. Theseinstabilities may proceedvia two mechanisms:either a conspicuousflow speedsetsuplocally (local violationof the low Machnumberlimit) orstrong gradientsaregenerated(local violationof theKnudsenlimit). In bothcases,a carefulmonitoringhasevidencedthat they areinvariably relatedto theviolation of the constrainton thedistributionfunction: 0 � Ni � 1.

These instabilities could be curedby using the full non-linearBoltzmann collision term, but thiswould considerablytax thecomputationalefficiencyof themethod.This operatorwould indeedrequireof theorderof

2b flops insteadof b2. Intermediatesolutions,i.e. higher-orderexpansionofthecollision

operatormight prove effectivein solving this problem.Having discussedthestabilityof LBE, we cannow turn to its numericalefficiency.The dynamicsof

LBE is driven by thestreamingoperatorand the collision operator.The streamingoperatoris local invelocity spaceandnon-localin configurationspace,while theoppositeis truefor thecollisionoperator.One can arrangethe distribution function in a 4D array F(I, LX, LY, LZ) with I representingthevelocity index andLX, LY andLZ thespatialindices.Thestreamingstepinvolvesb address-shiftingsofthe spatialindices, while thecollision steprequiresof theorder of b2 floating point operations(flop)betweendifferent Is for thesamegrid point. Assumingfor a while that thecost of datatransferfrom/tomemory can be neglected,it appearsthat the computationalkernel of LBE consists of 0(b2)flops Ispatial site. Remarkthat, for theFCHC lattice,b does not coincide with the number of links ofthe four-dimensionallattice. Thanksto the degeneraciesdiscussedin section4.2, in two dimensions9fields are needed,while in threedimensionsthis numberincreasesto 18. Actually, it is worth notingthat thestoragerequirementsof LBE aresomehowincreasedby thepresenceof theghostfields. In 2Dwe haveto solve for 9 fields, but only 6 (density,velocity andstresstensor)would be neededfor theaim of a hydrodynamicanalysis.In threedimensionsthesituationis 18 fields against10. Onecould tryto avoid this waste by storing only the hydrodynamic fields and converting them to/from thecorrespondingb populationfields just before/afterthe “move” cycle. This choice obviously corre-spondsto thesuppressionof the ghost field dynamics.

The scheme being explicit in time and local in space,oneconcludesthat LBE is an idealschemeforvectorandparallelcomputing,in preciselythesameway asbooleanlatticegasesare.When theschemeis implementedin a specificcomputerarchitecture,somecautionis needed,however,in orderto fullyexploit the potential of computational concurrency offered by the scheme. Of particular importance, tothis concern,is theminimization of the costs associatedwith datatransferfrom/to storageto/from theprocessingunits.

In the caseof IBM 3090vector-multiprocessor,themachineto which mostof theauthors’experiencerefersto, data-transferminimization is achievedby accessingdataat “stride one”, i.e. alongcontiguousmemorylocations(stride2 is equallygood).This allows a full bandwidthtransferfrom centralmemoryto a high-speedbuffer (cache)from which operandsaredeliveredto theprocessingunit at a rateof oneword per machinecycle. The collision stepis particularlywell behavedin this respect,in that it requiresmany (— b2) flops per eachitem reference.The move-step,while muchlessdemandingin principle (noflops at all), is of coursemuch moreexposedto the negativeeffectsof awkwarddataaccess.This isparticularly true in threedimensions,wherestridesareof orderofN2. Thereforesomecareis neededinDO loops orderingand arrangementbefore the computationalcost becomesreally negligible, as itshouldbein view of themuchsmallercomputationalwork required[65].To date,on a singleprocessor

R. Benziet a!., Thelattice Bo!tzmannequation:theoryand applications 177

of the IBM 3090-VF,sustainedratesof 30—50Mflops arecurrentlyachieved,which correspondto about1 Msite/s in two dimensionsand 0.1 Msite/s in three.More recently, LBE has beentimed on theIBM RISC system6000superscalarworkstation,whereprocessingratesof theorderof0.3 Msite/ s havebeenmeasuredin two dimensions.

A detailed comparison for the case of a 1282 grid (which will be discussed in section 5.1) hasprovedLBE to be roughlyasfast asa pseudo-spectralcode.This comparisonis howeverof little importanceascompared to the real advantageof LBE, which is linear scalability for parallel processing.It isremarkablethat this linear scalabilitycanalreadybe observedon non-dedicatedmachines,suchas theIBM 3090, where speedup factorsof 5.5 havebeenachievedwhenpassingfrom themonoto the fullsix-processorconfiguration. The use of massively parallel SIMD (single instruction multiple data)architecturesshould allow the full exploitation of the capabilitiesof the scheme.

5. Applications

5.1. Two-dimensionalhomogeneousturbulence

The validity of LBE asa numerical schemefor fluid flows wasfirst shown for thecaseof moderateReynoldsnumber (R < 100) two-dimensionalflows (see figs. 8a and 8b) past a cylinder [38].Thesesimulationswere limited to a maximum value of R* = 7.57, for the LBE schemewas still in aone-to-onecorrespondencewith theunderlyingFCHC lattice gasautomaton.

The ability of LBE to reproducetwo-dimensionalturbulencewas investigatedin detail in ref. [2]with themodel in the“enhanced”version (cf. section4.2). As a first insightin thecapabilityof LBE,the schemewascomparedagainsta pseudo-spectralsimulationfor the caseof a homogeneousforcedturbulentflow. Threelevelsof resolutionswereadopted:low (64x 64), moderate(128 X 128)andhigh(512x 512). In fig. 9 weshowthe time seriesof theenergyE and the enstrophyLI for bothLBE andpseudo-spectral(PS)simulations,in thecaseof low numericalresolution.Fromthis figureweseethat,apartfrom somespiky fluctuationsin theearly stageof the evolution,thetwo methodsyield quite asimilar answerin termsof the statistical behaviourof the flow. The comparisonwas subsequentlypushedforth by inspectingthe energyspectrumE(k) at moderateresolution.This is shownin fig. 10,which clearly displaysthe excellentmatch betweenthe two methods(it is to be mentionedthat thisspectrumpertainsto the time-asymptoticregime). In particular,it is worth noting thatbeyonda given

SIREAKLII[S ~ II

0 ~ 21

Fig. 8. (a) Streaklinesrepresentationof atypicalvortex sheddinginstability computedwith the lattice Boltzmannequation(from ref. 1381). (b) Thevorticity field correspondingto thesamesituation asin (a) (from ref. [38]).

178 R.Beaziet a!., Thelattice Boltzmannequation: theoryand applications

Energy Spectrum 64x84 grid

~BE II 10—07

— ~~~___] 10—02

10—03-10—04- . .. ... LO

lE~O5 ~ — SpecS,I

~ ~

10—72Energy 10-13

____________________________________________________________ 10—16 ____________________________________________________________

50 100 150 200 250 0 10 20 30 40 50 60 70time Wavenumber

Fig. 9. Energyandenstrophytime seriesfor a 642 lattice Boltzmann Fig. 10. Energyspectrumfor a 642simulationwith thepseudospectralandspectralsimulation (from ref. [2]). (solid line) andlatticeBoltzmannmethod(dottedline) (from ref. [2]).

maximumwavenumbera flat spectrumsetsin. It is remarkablethat the landmarkwavenumberfor thislack of dissipationis basicallythesamein the two cases.The last step,andthemostsevereone,of thisanalysiswas focusedon the ability of LBE to reproducethe inertial statisticalproperties,where theenergyspectrumhasthe form of apower law E(k)= k~.The universality of the exponenta is stillunder debate.The numericalparameterswere chosenin sucha way asto yield a nominal Reynoldsnumberof about5000 (basedon the global size of the computationalbox). The energyspectrumisshownin fig. 11 from which no clearevidenceof a scalingregimecanbe deduced.This contrastswithpreviousfindings of k3 spectraobservedin pseudo-spectralsimulationswith thesamenumberof gridpoints [51]. A tentative explanationof this discrepancycan be tracedback to the use in the PSsimulationof superviscositytermsproportionalto 48, which reducedissipationin the high-frequencyregion of the spectrum,and inversedissipationterms41 which enhancedissipationat large scales.Obviously, noneof thesemechanismsis activein theLBE simulation,aswitnessedby the absenceofsmall-scaleeddiesandthepresenceof largeagglomeratesof vorticity resultingfrom the inverseenergycascade(see fig. 12). Hyper- and hypo-viscosityallow one to exploit the full potential of numericalresolutionoffered by the computationalgrid in a fairly easymannerand haveno counterpartin theLBE method.Nevertheless,limitations ofartificial viscositymethodsarenot well understoodat presentand it is thereforenot clearwhetherthe inability of theLBE methodsto incorporatethem is a trueshortcoming.

LB Energy Spectrum 512x512 grid

70—23-

20+00 10+01 10+02

Wavenumber

Fig. 11. Energyspectraat varioustimesfor a 5122 lattice Boltzmannsimulation (from ref. [2]).

R.Benz!eta!., The lattice Bokzmannequation:theoryand applications 179

~ / (

c:~~-~ I ~N~J’ :E=~-~

- .:..~ ~ ‘~zi C) )— -. ..-— 7 ~ ~~_7 .‘-.

-

~. ~ (~Fig. 12. Vorticity contoursfor a 5122 lattice Boltzmannsimulation (from ref. [2]).

5.2. Flows in complexgeometries

Turbulentflows arecertainly very complex. It is neverthelessnot necessaryto havehigh Reynoldsnumbersto get interestingphenomena.As a matterof fact, remarkablycomplexflows canbe obtainedeven at very low Reynoldsnumbers(of the order of one or less)providedthat thegeometryof themediumhostingthe fluid is sufficiently intriguing. Such is the case,for example,in porousmedia.TheLBE is particularlywell suitedfor this kind of simulations,mostly in view of its flexibility with respectto complex boundary conditions. This is easily understoodby recalling that LBE is based on aparticle-trackingtechniqueand thereforethe intricaciesassociatedwith complexboundaryconditionsareeasilymanagedin termsof simple reflectionsandbouncesat appropriatespatiallocationsflaggedassolid sites. In this respect,solid sites are just grid locations where a different (and much simpler!)collision operatoris to be applied.

To be more specific, let us illustrate how a solid wall no-slip boundarycondition is implemented.Assuminga six-stateFHP latticeand theoutwardnormalto thewallalignedwith thehorizontalaxis, atall solid siteswehaveN2 = N5 andN3 = N6 for theno-slip condition.For thefree-slipconditionN3 = N5

and N2= N6 (seefig. 13).

~

Fig. 13. Boundarycondition at a solid wall. The wall lies entirely on grid points, whereparticles arelocally reflected.

180 R. Benzieta!., The lattice Boltzmannequation: theoryand applications

An alternativeinterpretationof thenumericalschemehasbeensuggestedin refs.[58,59]. It consistsof consideringthe populationsin thepoint x as half the sum of the populationson a staggeredgrid,interleavedby half a lattice spacingwith respectto the fluid grid. Obviously,one losessomesimplicityof implementationsbut the schemeis now somehowcloser to a second-ordercenteredscheme.Concerningthe boundaryconditions,thesolid siteslie on the staggeredgrid. If we imagineto let theparticlestravel one-timestepto their targetsite, it is clearthat oppositespeedparticleswill meeton thesolid sites half-wayalongtheir flight. Both thestaggeredand theunstaggeredmethodpresentedpertainto the simple caseof a wall lying entirelyon a row of grid points. The situation becomessomewhatmorecomplicatedfor thegeneralcaseof a boundarygoing acrossthe lattice,not coincidingnorbeingparallelto anylattice row orcolumn.This situationhasbeenexaminedin detailfor the caseof theFHPlattice in ref. [18].Theseauthorshavefoundthat underspecialconditions,spuriouswall modes areexcitedwhich can propagatefar inside the bulk flow. Theyshow that the problemexhibits beautifulanalogieswith the Schrodingerequationin a quasi-periodictiling (Andersonlocalization). In thisanalogythe dangerousmodes contaminatingthe bulk flow correspondto extendedsolutions,whilethoseremainingconfinedcloseto the walls correspondto localized non-conductingsolutions.

A practicalway to escapethesesubtleeffectsis to approximatethesolid boundarywith a polygonalpath lying entirely on the lattice. It shouldbe noted that, as we haveformulatedthem above,bothmethodsrequirea knowledgeof theoutward normal ñ to the solid wall in order to select the inwardpropagatingpopulations(c~.ñ <0), which needto be reflectedat thesolid wall. This requiresstoringsomeextra informationto enabletheschemeto performtheseselectivereflections.Alternatively, onecouldsimply invert all thepopulationson a givensolid site, irrespectiveofthedirectionof theirspeed.This would be simpler to program and more economical, but it engendersthe potential risk ofgeneratingspuriousparticlespropagatinginward, from thesurfaceto the interiorof the solid region.With the first method,this risk canbe completelyruledout by simply initializing thepopulationslivingon solid sitesat zero. With thesecondmethod,thespuriousparticlescanbe madeharmlessby simplyexcludingthe interior of thesolid regionfrom themovestep. This might seeman obviouschoice,wereit not for thefactthaton a pipelinedmachineit mayprovemoreefficientto processsolid sitesaswell tonot hampervector processing.The point is that in orderto selectthe interior points within the solidregionwe needpreciselythesameinformationwhich is requiredto performtheselectivereflectionswewantedto get rid of. For this reason,it seemsthat if onewantsto keepthe staggeredgrid algorithm,one hasto live with the amountof extrainformationbroughtaboutby its direction sensitivity. In thecodes,this meansthat onehasto usea setof boundaryarrayswhosedimensionis equalto the numberof surface solid sites. These arrays are then usedas indirect addressesto selectively move thepopulationliving on the surfaceof thesolid, while leavingthe interior onesat rest.

5.2.1. Three-dimensionalflows in porousmediaThe first lattice gas simulationsof flows in porous media were performedby Rothman, using a

two-dimensionalFHP automaton[61].Subsequently,this work hasbeenextendedto three-dimensionalflows using the LBE [64]in its standardFCHC version,on a 32~cubic lattice. The porousmediumwasmodeledas arandomsequenceof elementaryblocks, four lattice units wide. Thedistributionwassuchthat no freeporewith a crosssectionnarrowerthan4 x 4 latticeunits could befound. This minimal sizeof the poreswas identified by running a seriesof preliminary testsaiming to ascertainunderwhichparameterconditions (R*, Mach numberand the pore size h) LBE reproducesa Poiseuilleflow in asquare-section3D channel.In fact, the poresizehasto fulfill two oppositerequirements.On oneside,h shouldbe as small as possiblein order to havea large “granularity”, L Ih, of the porousmedium

R. Benziet a!., The lattice Bo!tzmannequation: theoryand applications 181

(here, L is the global size of the computationaldomain). On theother hand,h cannotbe too small,becauseotherwisegradientsof theorderof h developwhich would impair theconvergenceof LBE totheNavier—Stokesequations.

Hereagain,thepossibility of raisingthevalueof R*, i.e. reducingtheparticlemeanfree path,turnsout to be very important in orderto improve convergenceto hydrodynamicsat small scales.This isdocumentedin figs. 14a and 14b in which we show the ratio QN/ QT betweenthe volumetric flow

FL(Y~WRATES(ve1~rO.O5,den~rO.38.Rt~?.5fl

120-

100- ~- - -. U— U- — — a——-—--

60-

— Esp./Theo.(X)

60 —44— Thea. Floe

—4-- Exp. Floe

0 100 200 300 400 500 600 700 600 000chmnnel ~4rea

(a)

FLOW PATES(vel=O.1.deno.05.R*=1.~1)

200~ J.~Exp./Rieo.(I)I

i ‘. l44lheo.!~l40~ ~. I100- — . — . — . ~. . — . — . —.

~e0.

0 00 40 50 80 110 140 170 800 ~0 320 300 380~h&ine1Area

(b)Fig. 14. (a) Ratio of numericalto theoreticalrate in a Poiseuilleflow asa functionof thechannelarea(from ref. [24]). (b) The sameas(a) withR* = 1.31, at a muchhighermean free path (from ref. [241).

182 R. Benz!eta!., The lattice Boltz,nannequation: theoryand applications

(Q Vh~)measuredin the numerical simulation (QN) and the one predictedby analytical theoryQT = 0.035h2G4~.Here, G is the total pressuregradient acrossthe channel,~ = pv is the dynamicviscosityof the fluid and V is the velocity. Darcy’slaw statesthat V= (kIp~G, wherek is knownasthepermeability. From fig. 14b, we see that with R* = 1.71 deviationsfrom fluid behaviourstart to beobservedalreadyfor h < 10. By increasingR* to 7.57, fluid dynamic behaviouris achievedfor muchnarrowerchannels(h 3) [24].

The main goal of theearlier investigation[64]wastwofold: (i) assesstheparameterrangein whichDarcy’s law is fulfilled, (ii) measurethepermeabilityof themedium.Theresultsarecollectedin fig. 15which showsthe numericalflux QN as a function of the pressuregradientG, for threedistinct valuesoftheporosity 1’ (void volume!total volume): 0.635, 0.5, 0.375. From this figure oneseesthatDarcy’slawis well fulfilled for G up to 1.4 x ~ which correspondsto an effective speedV= Q RPh2of a fewmilli-lattice units per step,i.e. Reynoldsnumberof orderone.The resultingpermeabilityk canalsobecomputedand comparedwith theoreticaland seminumericalestimates(cf. ref. [64]and referencestherein). The numerical values are k = 0.5,0.18,0.07, which agreewithin a factor two with theaforementionedtheoreticalestimates.The quantitativecomparisoncould not be pushedany furtherwithin this model mainly becauseof the absenceof exact results. To this purpose,a much wiserexperimentconsistsin designingthe randommediumas a randomsequenceof penetrablespheres,forwhich a numberof rigorousresultsareavailablein the literature (seeref. [11]andreferencestherein).This kind of experimenthasbeenperformedrecently. In order to avoid spuriouseffects of the kindmentionedin the previoussection,thespheresaremodeledasa superpositionof cubicblocksentirelyresiding on the lattice sites. In addition, to improve the granularity of the medium (poor in a 32~simulation),the lattice sizewasdoubledfor a numberof casesandevendoubledtwice, with anincreaseof a factor 8—64 in computerstorageand64—256 in computertime (see fig. 16). However,due to thelargeamountof centralstorageavailableon the IBM 3090/ 600 (256Mbyte) eventhelargestsimulationscould be accommodated“in core” (a 128~simulation requiresabout160Mbyte of storageand takesroughly tenhoursto completean about 1000 stepslong simulation).Fifteenindependentsimulationswereperformed,eachwith a different solid fraction a- = 1 — 1, rangingfrom a- = 0.02to a- = 0.978.The

l’.kr,erlcal Verification of DercU s Law

:: ~r::=Ogj

2.00

a

-

0O

Pressi.reGradient * 1E4

Fig. 15. Numericaldemonstrationof Darcy’s law in a 32~lattice Boltzmannsimulation for different porosities(from ref. [64]).

R. Benz!et al., The lattice Boltzmannequation:theoryand applications 183

102

101

lao~~ ~ (3)

101

(1)

~10-2 (2)

~ g=

iO~ (4)

00 02 04 06 08 10solid fraction

Fig. 16. Typical flow configurationin a 128~latticeBoltzmannsimula- Fig. 17. DimensionlesspermeabilitykIR2 asa function of the solid

tion in a porousmedium (from ref. [64]). fraction. Curve (1): Weissberger—Pragerbound; curve (2):Berryman—Miltonbound; curve(3): Brinkman estimate;curve (4):Kozeny—Carmanestimate;squares:numerical calculation (from ref.[11]).

numericalresultswerecomparedwith four distinct theoreticalestimates.Two are variational bounds(Weissberger—Prager)anda refinementdue to BerrymanandMilton (BM) and the remainingtwo areestimatesdue to Brinkman (B) and to Kozeny—Carman(KC). Mathematically,thesefour theoriesresult in the following expression:

ks ~ = —~R2~!log~, ~ = ~P3Is3, kB = k0[1 + ~a-(1— V8!a-—3)]. (114)

Here,R is the radius of thesphere,k0 = 1 /6irRp (Stokespermeabilityof an infinite diluted mediumatdensityp) ands = 41TR

2pi~I.The resultsareshownin fig. 17. First of all, onenoticesthat thenumericalresultsarealways below the theoreticalupperbounds,asthey should.The boundis tightestat low a-,wherethe WP theorypredictsk —* k~

2.At higher solid fractions,theboundsdepartsignificantly fromthe numerical results, which remain howeverwithin one order of magnitudeof the BM bound.Remarkablygood agreementwas obtainedwith the KC formula in the rangeof intermediatesolidfractions,up to a- <0.9. It is worthemphasizingthat, while small a-agreementwith kB is a mustfor thenumericalmethod,theagreementwith theKC formula is not expectedin general.This work permittedthe conclusionthat variationalboundsprovidean overestimateof thepermeabilitythat is usuallywithinan orderof magnitudeof the true value.

6. Generalizationsof the method

6.1. Multiphaseflows

Multiphaseflows have beenin the focus of lattice gasautomataresearchsince the earliest days[10,15, 16]. Multiphase flows canbesimulatedby addinga new“statenumber”,color, to the lattice gas

184 R. Benziet a!., The lattice Boltzmannequation:theory and applications

formulation.Thus eachsite of theautomatonis characterizedby two setsof booleanvariables,sayn1andf~,wheren~indicatesparticles(n1 = 1) or holes (n1 =0) andf indicatescolor (ft 0~GREEN,ft = 1 RED) An enlargedsetof collision rules is thendefinedm which, besidesthe usualconservation of massand momentum,color caneither be conservedor it canbe redisttibutedamongparticles[62]. In the former caseit behaveslike a passivescalar,while in the latter it simulateschemicalreactions.Collision rules and recoloring can also be tunedin such a way as to minimize transportcoefficients In addition, collision rules can be devlsedthat preservemterfacial boundanes,thusaccountingfor phase-separationphenomena[60].This is achievedby biasingthe rulesby meansof thegradientof thecolor flux F = E, c, f,, in sucha way asto encourageparticle motiontowardscolor-likeregionsof the flow. Owing to their dependenceon the gradient of the color flux, theserules areinherentlynon-localandthereforecomputationallyratherexpensive.Locality canhoweverbe restoredwith an ingenioustrick dueto Rem and Somers[60].It consistsin letting, not only particles,but alsoholes, carrycolor information The collislon rulesarethentunedin sucha way that color mforniationmoving along the direction c. is positively correlatedwith the bulk color in the direction — c1. Thisallows us to locate theparticleinterfaceon the basis of a purely local color currentF, with no needofinspectingthe neighborhoodof the interface.Severalinterestingtwo-phasephenomena,suchasbubblepropagationandinteraction,havebeensuccessfullysimulatedwith this technique.

Recently,theessentialingredientsof multiphaselattice gashavebeencombinedwith thebasicideasbehindLBE, thus leading to new interestingmultiphasevariantsof LBE [31].In theseschemes,thecollision rule is split into two separatesteps:in the first step,a non-isotropicperturbationis addedtothe particledistributionnearthe interfacewith the intent of depletingmassalonglinks parallel to theinterface,while addingmass to the perpendicularones.Subsequentlythe mass is recoloredin suchaway as to avoid diffusivity of onecolor into theother.This new schemehaspasseda numberof testsincluding surfacetensionmeasurementsandcapillary wave experiments.

6.2. Magnetohydrodynamics

Among the other generalizationsof the method, an interesting case is magnetohydrodynamics(MHD). The first automatonaimed to reproduceMHD wasproposedby Doolen et al. [55].Theyprovideda hybrid schemein which the MHD automatonis assistedby a finite-differencingstage,inorder to obtain the currentin termsof derivativesof the vector potential. The needof this hybridschemeis due to thenon-localnatureoftheLorentzforceI x B. Subsequently,ChenandMatteus[13]provideda new automatonwhich, by treatingthe magneticfield B on the samefooting as the fluidcurrentJ, doesnot requireany finite differencestage.However,for theFHP two-dimensionallattice,onehasto consider6 x 2 = 12 statesnil (i = 1~..., 6; j = 1,2), with theindex i referringto I andjto B.

In thecaseof LBE [66],thesituationis somehowdifferent.A hybrid scheme,which involvesfinitedifferencescan be easily inserted into the LBE, becausethe model itself can be regardedas afinite-differenceschemefor the Navier—Stokesequations.Let us considerthecaseof two-dimensionalMHD

(115)

a~(~i)+(v~ô)(z~i) =(B.O)(~b)+~7z~(L~fr). (116)

Here,B = ~x 84s is themagneticfield and v = £ x oq~is thevelocity, ~beingtheunit vectoralongthethird ignorablecoordinate.Note that eq. (116) is the equationfor the vorticity, rot(v) =

R. Benzieta!., The lattice Boltzmannequation:theoryand applications 185

Let us first prove that eq. (115), correspondingto the advectionof the scalarfield 4 by thetwo-dimensionalvelocity field v, can be easilyreproducedin the frameworkof LBE. As we know,thefour-dimensionalFCHC lattice can beprojecteddown to two and threedimensionswith acorrespond-ing increasingreductionof the numberof independentfields. In threedimensions,by imposingtheappropriatedegeneracies,one is left with 18 populationsandthe fourth componentof themomentumJ4is zero. If we removethesedegeneracies,then.14 ~ 0 and behavesasa passivescalaradvectedby thethree-dimensionalvelocity field v = I/p (cf. ref. [28]). Similarly, in two dimensionsone can imposeadditional degeneracies,leaving only 9 independentpopulationsand J4 = J3 = 0. Again, by removingthe degeneracyon the third axis J3 ~ 0 andthe velocity is definedby thevector(vi, v~,v5) wherethethreecomponentsof thevelocity only dependon x andy. Wewant to showthat v~behavesasa passivescalar. In the macroscopiclimit, the equationsdescribingthe evolution of J~are

+ a~s~+ = 0, a,S~+ = A(SXZ — pg(p)v1v~)

+ ~9~JZ= A(S~~— pg(p)v~v~),

where we haveomitted, for simplicity, the propagationviscosity. In the adiabaticlimit, one finds

a,J~+g(p)(v.O)J~=—(113A)iXJ5. (117)

If we nowidentify J~with 4’ and the diffusivity xwith —1/3A, we obtain eq. (115). If thecontributioncoming from the propagationviscosity is included, then x= —1 /3A —

In orderto obtain eq. (116), we observethat the magnetictension (B •ô) i~4’ hasexactly the samestructureastheadvectionterm (v . ô) z~i/i. Becausetheadvectionis obtainedfrom theequilibrium partof the stresstensorS~,it follows that, in orderto model themagnetictension,it is sufficient to give anew definition of the equilibrium populationas

N~’—+(N~)’= (p!b)[1 +2cjav~+ bG(p)Qja~(UaUp— BaBp)]. (118)

Here a = x, y and f3 = x, y. It turns out that the new expressionof the equilibrium stresstensoris

(s:~)’=pg(p)[v~v~~BaBp — ~(~2B2)t5,~]. (119)

From the usual techniques,it follows that in the adiabaticlimit the equationsfor the velocity fieldbecome

a,va +(v.ö) v,, ~(B.ô)Ba 77~Va, (120)

where a= x, y and the non-galileanfactor g(p) hasbeen reabsorbedin the definition of the timevariable.By using thedefinitionof streamfunction,we finally obtaineq. (116)with i~= x = —1/3A. Asbefore,inclusion of the propagationviscosityresults in ~j = x = —1 I3A —

Note that in theLBE schemethemagneticfield B is obtainedby finite differences,using the relationB = ~ x 04’ and4’ = J

3. It shouldbe notedthat theschemeproposedherecaneasilybeextendedto nonunit Prandtlnumbers~ For this purpose,one only needsto changethe collision matrix by imposingtwo distinct eigenvaluesA1 and A2 to the fields S~,Sfl,, S~,andS~,~ respectively.By doing so, onehasin fact ~ = (1/A2 + ~)I(1IA1+ ~).

186 R. Benzieta!., The lattice Boitzmannequation: theoryand applications

As anexampleofthe resultswhich can-beobtainedwith this scheme,a typicalvorticity field is shownin fig. 18, togetherwith the correspondingspectrum.The schemecould be very useful to simulatetwo-dimensionalmagnetohydrodynamicsin complexgeometries.

6.3. Subgridscalemodelling

As previouslydiscussed,the lattice Boltzmannequationcan be interpretedas a finite-differenceapproximationfor a set of partial differential equationsdescribingboth the hydrodynamicfields(density, velocity and stresstensor)and non-hydrodynamicfields, i.e. ghost fields. Ghost fields aresmall-scalefields, whoseintensity is proportionalto the local departurefrom thermodynamicequilib-rium. Their dynamicsis strongly anisotropic,asonecanreadily checkfrom thedefinition of theghoststresstensorTap.Oneusuallydefinestheparametersof LBE in sucha waythat theghostfields areassmall as possible. The physical meaningof this choice is rather simple. The dynamics of eacheigenvectorwith eigenvalue~of the linear collision matrix st~hasa characteristicrelaxationtime toequilibrium of order1 / I ~I(in the discretetime formulation 1/ 1og~1 + ~ . For ~ —~ (in thediscretecase~ —1) the relaxationof thecorrespondingeigenvectoris asfastaspossible.In theLBE thevalueof theparametersis tunableandone caneasilychoosethemin order to reducetheeffectsof the ghostfields dynamics. In LGA such a choice is not always possible, or at least is not always easilyimplementedwith booleancollision matrix. -

Letus now takea different look to theghost field dynamics.As we havealreadynoted,ghost fieldsare small scalevariables.They satisfy an equationsimilar to Navier—Stokesequation,although thestresstensoris not isotropic.Using the languageof computationalfluid dynamics,onecanalso think ofthe ghost fields as some kind of subgrid scale variables. However such variables, besidesbeinganisotropic, do not play any role in the dynamicsof hydrodynamic fields, at least in the LBEformulationgiven so far. If we reallywant to view ghost fields assomekind of subgridscalevariables,we needto give them a hydrodynamicmeaning,andalso get rid of all the anisotropiceffects.

The first and mostdifficult problemto be solved is the anisotropyin thestresstensorT0~of ghostfields. The eigenvectorsA7~andA8i associatedwith i~can bewritten asA91c,~andA91c,~respectively,

1E23~~~~

~ / ~/ //~\.\ ~E+D0 5 ir+o, 5 1E+02 5

(a) (b)WAVENUI~IOER

Fig. 18. (a) Typical vorticity mapfrom a 10242 magnetohydrodynamicssimulationwith the latticeBoltzmannscheme(from ref. [66]). (b) Typicaldecayspectrumfrom a 10242 magnetohydrodynamicssimulation. The initial conditionscorrespondto k’ energyspectrumwith randomphases.

R. Benz!et a!., The lattice Boltzmannequation:theoryand applications 187

whereA9~is theeigenvectorassociatedto ~. It is thennaturalto think of fla asthecurrentassociatedtotheghostdensityj.t. Onecanthenconsiderthe latticedefinedby thevectorsd.~= A7~andd,~= A8i~Intermsof d,, we can define theLBE

N,(x + d,, t+ 1) = N,(x, t) + ~ ~ — Nr), (121)

where,for the time being,theequilibriumpopulation is unchanged.Equation(121)canbe analyzedinthe sameway as the usual LBE, i.e. by projectingthe equation onto the eigenvectorsof sf1~,and byassumingas a first approximationthat

N1(x+d1,t+1)—N1(x,t)—~a~N,+(d~a)N,. (122)

By restrictingthe analysisto the fields ~1a’ we obtain:

+ ~(i — ~ + + ~ + 4a~S~~=(123)

+ a~(p— ~ + + ~ + 4a~S~~=

Hence,eq. (121) producesin theghostfield dynamicsa stresstensor~ (“d” standsfor dual) which isstill anisotropic.Howeverusing eqs. (91) it is easyto showthat:

Tap + ~ =2Sap~ (124)

Thus wecan setup anisotropic form of thestresstensorfor theghostfieldsby consideringthemodifiedLBE

~ (125)

where is the projectionoperatoron the eigenvectorsd.. The projectionoperatori~neededbecauseotherwisethe stresstensorof the hydrodynamicalmomentumbecomesanisotropic,as ~necan ensilycheck.Once againonecananalyze(125)by standardprojectionontotheeigenvectorsof s4,~.The tinulresult is

~ + 0 .J = atJa + äpSap+ aa(p12) = 0,

~~t’~ap+ ~(aaJp+ apJa — ~a~J~)+ Rap78= A(Sap— S~), (126)

t9

1’tla +t9pSap+äa(P/2)= 2°~7a’ a,/1 +0. tJ=1~/L.

Here,S~= pg(p)(u~vp— ~(U

2)Sap) and Rapy8 is thesameasin (92). Theisotropyof theequationfor

q is thenachieved.Note howeverthat the term Rapy&~correspondingto termsof higherorderin s, isnot isotropic.

188 R. Benz!et a!., The lattice Boltzmannequation:theory and applications

The ghost field i is divergencefree. Shouldthe divergenceof ig be non-zeroat time t .= 0, it wouldbeannihilatedin a very shorttime of theorder lIT. Now it is temptingto considerir~asthesubgridscalevariableof .1. In particular,let us define w q Ip which can be consideredasthe subgridprojectionofthevelocity. In this case,S~shoulddependbothon v and w. The mostnaturaldependenceof Sap on wis through the combinationu = v + w. This identifiesu asthe full, velocity field and w asthe subgridprojectionof u. We obtain

S~= pg(p)[(Va + Wa)(Up + w~)— ~(v,+ W)2f5ap]. (127)

Becausewe haveassumedthat w ~v, we canapproximateeq. (127) to the first order in w:

= pg(p)[v~v~+ VaWp + UpWa — ~(t~2 + 2v• w)ö~~], (128)

sothat S~now includesterms0(s)(this explainsthechangeof notations).By assumingp const.and

computing Sap up to the first order in e, we obtain:Sap = s:~ + (lI3A)(t3aJp + ‘~p~a)’ (129)

wherewe haveusedthe fact that (1/A)a7 ~ = o(s2)and 0• J = 0. Insertingeq. (129)into theequations

for I and i~we finally obtain

atfa +pa~(VaWp+VpWa +VaVp+P)=(l/3A)~Ja, (130)

~ +Pôp(UaWp+VpWa+VaVp+P) =(lI3A)1~Ja+O~?la, (131)

where the factorg(p) hasbeenrescaledout. Next, to first order in s, we obtain

W = U tVpt3pVa . (132)

Equations(130) and (132) representa subgrid scalemodel of the original two-dimensionalNavier—Stokes equations.This model has been derived by assumingthat the equilibrium distribution isdependentboth on the hydrodynamicalvariables and on the ghost velocity, i.e. the subgrid scalevelocity. In the languageof turbulent modelling we can say that the velocity field u has beendecomposedasv + w wherew is a fluctuatingquantity.Usuallyin turbulenttheoriesw= u — (U) whereu is the full velocity field and ~ is someaveragingprocedure,suchasspaceorensembleaverage.Inour approachv and w are different projectionsof u, i.e. they areprojectionson large andsmall scalerespectively.The identificationof w with the ghost variableenablesus to close the modellingfor thesubgridvariable.

The effect of w on v is twofold, correspondingto the terms: (i) Up iJpWa~advectionof small scalesfrom large scales;(ii) ~ a~Va~advectionof the large scalesfrom the small scales.The secondtermconservesenergy,while the first actsasan energydissipationterm. Enstrophyis alsodissipated.In fact,onecan easilyshowthat

~8Jv2dxdy= — !J[(v.a)v]2dxdy<0, (133)’

R. Benz!et al., The lattice Boltzmannequation: theory andapplications 189

~a, f [rot(v)]~ dx dy = — ! f [(v 0) rot(v)]2 dx dy <0 (134)

However, the rate of enstrophydissipation is much larger than the rate of energydissipation. Thisfollows from the fact that enstrophy rather than energy is transferredfrom large to small scales.Moreover, for coherentstructures,namelyfor flows where (v . 0)v = 0, the subgrid scale velocity isidentically zero,as it should be.

Surprisingly,eqs.(130) and(132) havebeenalreadyderivedby SadournyandBasdevant[63],whohave introduced the anticipatedvorticity scheme(AVS) for two-dimensionalturbulence.We shallbriefly reviewthephysicalassumptionof AVS in the following. Let us considerthevelocity field v asdecomposedin two parts,tY andv’, where tY is explicitly resolvedby thenumericalsimulationandv’ isthe subgridvelocity fluctuation.Oneusuallyhasv’ 4 t3 and w’ 4 ~, where w’ and~ are thevorticityfields correspondingto u’ andö respectively.Thetwo-dimensionalvorticity Navier—Stokesequationcanbe written as

13,(ti + w’) + (iii + v)a1(t3 + w’) = v ~ + w’). (135)

Assumingan enstrophycascadeon small scales,theenergyspectrumfol!ows a k3 law anddissipdtion

plays no role in the evolution of the large scales Under theseconditions,the following inequalitieshold: - -.

v231w’ 4 U~ä~O.)’, u;a1w4 ~i1a1w’. - (136)

Theinequalitiesfollow from the fact that enstrophygradientson small scalesaremuchlarger,becauseof theenstrophycascade,thanenstrophygradientson large scales.Hencethe evolutionequationforthe resolvedvorticity i is ‘ -

-(137)

where the -term ö~a1w’,responsiblefor the- enstrophytransferto small- scales,needsto be parame-

terized.To this purpose,AVS assumesthat: - - -a~’+ L[(ö. a)~iJ= —w’/O, (138)

where 0 is a suitablesubgridrelaxationtime usually expectedto be small and L is a positive definiteoperator.-Thentheapproximation - - -

- w’=—OL[(iJ.a)~i] - - (139)

is made,correspondingto the adiabaticlimit. Equation(137) finally becomes(the bar is omitted forsimplicity):

a~w+(v-O){w —OL[(v0)w]} ~0. (140)

Note that eq. (139)-with L =1 ando = 0 providesexactlythesameresults-as-eq. (126),written in term’sof vorticity. - - - - -

190 R. Benz!et al., The lattice Boltzmann equation: theory andapplications

Let us now discusshow theLBE schemefor AVS could be implementedin practice.The operationof projectiononto the ghost fields proceedsas follows. Given a set of b populationsN~,the ghostcomponentHgNj is:

[IgNj = ~ ~ (141)

with the samenotations asin section4.3.2. The updatingof theschemefrom time t to time t + 1 isperformedin two steps:

(1) Collisions are performedaccording to the usual rules with the new equilibrium populationobtainedusing the definition (127). ProvisionalpopulationsN,(x,t + 1) are generated.

(2) The final populationat time t+ 1 at point x is obtainedas

N,(x,t+1)=1S’~(x—c1,t)_HgfV~(X~dj,t+1). (142)

Note that the populationsat time level t and t + 1 needto be stored.It is interestingthat (139), which is assumedas an empirical ansatzin the derivationof AVS by

SadournyandBasdevant,canbesomehowdeducedin the frameworkof LBE. The assumptiononehasto makeis that theequilibrium dependson theghostvelocity. From a moregeneralpoint of view, wecanarguethat thesameresultscould be obtainedstartingfrom thecontinuous(bothin spaceandin themomentum)Boltzmannequation.

6.4. Boltzmannequation in non-uniform lattices

While it is clear that the bestuseof LBE is in theareaof grosslyirregulargeometries,it would bevery interesting to extend the range of applicability of the method to the case of body-fittedcalculations. Most flows of practical interest are in fact characterizedby the presenceof solidboundaries,wherebythespatialactivity of theflow tendsto concentrate(boundarylayers).Undertheseconditions,it is clear that a selectiveclusteringof thecomputationalnodesin therelevantregions,anda correspondingrarefactionin the irrelevantones,canresult in a dramaticreductionof thecomputa-tional work. As it standstoday, LBE is restrictedto regularuniform lattices.In this section,we willportray a numberof ideaswhich may help in filling the presentgap.

Let us beginwith non-uniformgrids. The first difficulty which ariseswhentrying to formulateLBE ina non-uniformgrid is theemergenceof inertialforces,relatedto themetricof the lattice. In auniformlattice the grid lines can be thoughtof as beinggeneratedby the motion of free particlesalongthecharacteristicsof theHamiltonianof a freeparticle.Similarly, a non-uniformmeshcanbe thoughtof asbeinggeneratedby trajectoriesin which, ateachnode,theparticleundergoesa kick, which makesitstrajectorydeviatefrom a straight line. As a result, the correspondingLBE must include a streamingterm also in the velocity space:

N,(x + c, c7 + a, t + 1) — N1(x, c1, t) = 4, , (143)

wherea, is theaccelerationimpartedto the ith speedatnodex. This reflectstheemergenceof inertialforces which may be difficult to dealwith. The difficult may howeverbe overcomeby relaxingtheconstraintthat themacroscopicquantitieshaveto bedefinedon thesamegrid wheretheLBE dynamics

R. Benziet a!., The lattice Boltzmann equation: theoryand applications 191

takes place. In other words, one can introduce a two-grid procedure,in which one grid, fine anduniform, is theusuallatticehostingtheLBE dynamics,while theother,coarseand non-uniform,servesasa controlgrid for the evaluationof the fluid quantities[36].In this way, thedifficulties mentionedabove are automatically solved, while the task of accumulatingthe nodeswhere they are neededistakenin chargeby the controlgrid. Technically,this procedureis tantamountto solving theLBE by afinite-volumeprocedure.

Let us start from thecontinuousversionof LBE,

a,N1+(c,.o)N,=4. (144)

The first stepconsistsin applyingthe Gausstheoremto eachcontrolvolume (cell) of the coarsegrid.

With referenceto fig. 19 (for simplicity we treat the two-dimensionalcase),we obtain

(145)

where

F~~=~-fN1dV,cbjp=3-~N,(ct.ñ)dS,i”1~~=.~—f4dV. (146)

Thecontourintegral is takenover the genericcontrolvolume centeredaroundthepoint F, wherethemacroscopicfield F,,, is defined.The expressions(146)define oneequationper eachunknownF,,,,so that a closedsystemof equationsresults. This systemis howeverempty as long as onedoes notspecifyhow to expressthe integrandof thesurfaceintegralasafunctionof thenodalvalues in thevariouscells.This callsfor aninterpolationprocedure(finite-differencing),whosechoiceis by no meansuniqueand controlsthe degreeof accuracyof the method. We can write the flux term as

çb1~= ~ g,,,iç, (147)

where g,~= (c, n,,)S~IV~is a metric factor measuringthe projection of the surfaceSe,. onto thestreamingdirection c, and the index U = 1,. . . , 4 correspondsto the four boundariesof the cell(denotedhereafteras“e”, “n”, “w” and“s”, for east,north,west,south). Accordingly,N1~representsthe valueSof N~at thosenodesof the fine lattice which lie within ~,, while their neighborsalongc, donot. Confiningour attention,for the time being,to thecaseof a singletime step,the time-differenced

north

wes/~I~st

south

Fig. 19. Typical macroscopiccell for a finite-volume lattice Boltzmanncalculation.

192 R. Benz!et a!., The lattice Boltzmannequation: theory andapplications

versionof eq. (145) now takesthe form

F~’— F7~+~r~ ~ = (2~T. (148)

It is clearthat theschemerepresentedby eq. (148)hasall theconnotationsof a MUSCL-[68] scheme,in which the solution at time t + ‘r is obtainedby repeatedapplicationsof three operators: areconstructionoperatorPh(•), an evolution operator~(.) and an averagingoperator.~f().Equation(148) representsthe combination of the operators~ and s~.What remainsto be specified is thereconstructionoperatorPh, i.e. theoperatormappinga sequenceof discretevaluesF,, into apiecewisepolynomialfunctionP~(x). Typical reconstructionoperatorsoneconsidersarezeroth-order(piecewiseconstant)and first-order (piecewise linear). Once a reconstructionoperatorhasbeen chosen,thefine-grainedunknownsN~ffareuniquely identified in termsof the coarse-grainedF,,,. As a result,theconcreteform of the finite-volume LBE is given by

F~1— F + r ~ ~ = (2T, (149)crl cr

where ~ is the matrix representationof the reconstructionoperator Ph and P~describestheneighborhoodof P involved in the interpolationprocedure.Note that in generalPh implies a certaindegreeOf non-locality,which increaseswith theorderof the interpolation.Nearest-neighborcommuni-cation (the one usedin the original fine-grainedformulation) can Only be preservedby using thelowest-orderinterpolator.

Let usnow computethe specific form of theaveragedcollision term12.,,. In the linearizedversionofLBE, the coarse~grainedcollision operatoris given by

Q,,,= ~- J~1~1(IsI~,_N~)dV, (150)

where N~= PG(P)Q1aPVaVP. In order to get this formula we have used the expressionof theequilibrium populationand thepropertiesof the collision matrix describedin section4. It is convenientto expressthevelocity field Va as the sum of a coarse-grainedaverageVa = (Va) v;

1 S ~ dv, plus afluctuationu~.Using Va = V~+ v~in eq. (150), we obtain

(li,, = s4,,[F, F,~(V)— (151)

Here, F’(V) = QjapVaVP while F~= Q jap (v~v~)representsthenon-equilibriumcontributioncomingfrom fluctuationswithin thecontrolvolume

At this stage,it is worth stressingthat thesefluctuationsareperfectlywell definedonceone specifiesthe type of reconstructionoperatorto be used.In somesense,thechoiceof thereconstructionoperatoris equivalentto a turbulence-closurewithin the cell ‘c,,. For instance,if we takethepiecewiseconstantreconstructionoperator,it is clear that v~= 0. When selecting the piecewise linear reconstructionoperator,the field v~will be proportionalto thegradientof Va acrossthecells.

The final form of the finite-volume LBE schemesreads as follows (suffix n refers to the timediscretization):

R. Benziet a!., The lattice Boltzmannequation: theoryand applications 193

F~1— F~+ r ~ g10R~~F= ~ A~(F,~— F~— ~ (152)

A few commentsare now in order. The first point concernstheconvergenceof theschemedefinedby (152) to the Navier—Stokesequations.For the free-streamingstep, the requirementis that in thelimit of a uniform LBE lattice the streamingoperatorreducesto the “bare” form of a laddermatrix5,,,,, whereP~= P — c~. It is not difficult to prove that this condition is fulfilled by choosingupwindspatialdifferencingschemes.For the collision step,convergenceto the Navier—Stokesequationsshouldbe ensuredby the fact that thenon-equilibriumtermsassociatedwith a cell size4 areof order1 //i~,i.e.higher-ordercorrections to the usual LBE collision term. A second point concernsthe range ofapplicability of the method.This leadsto examinationof the important issueof numericaldiffusion,which in turn fixes thehighestvaluesof the Reynoldsnumberachievablein the simulation.It can beproventhat the simplestversionof thescheme,i.e. theonebasedon a piecewiseconstantrepresenta-tion, is affectedby a large amountof numericaldiffusion. In lattice units, onefinds

(153)

This showsthat for largecellsnumericaldiffusion is of the orderof manylattice units, i.e. atleast twoordersof magnitudehigherthanthe oneattainablein the original LBE lattice. Preliminarywork showsthat the situation can be dramaticallyimprovedby moving to the next order in the reconstructionoperator,i.e. a piecewiselinear representation.In this case,aschemecanbe foundwhich allowsone tovirtually eliminatenumerical diffusion, thus ensuringa mesh-independentvalue of the viscosity. Theeffect of numericaldiffusion is clearlyshownin figs. 20a and20b, which depict the flow speedfor aPoiseuille flow in a channelof width 48 lattice units, for different valuesof thephysicalviscosity. Thesimulationwasperformedwith 32 points distributedas follows: 16 in the periphericalregions(4= 1)and 16 in the centralone (4 2). The dramaticimprovementbroughtaboutby the piecewiselinearschemeis well visible. This suggeststhat thepiecewiselinear versionof theschemecanhandlethesamerangeof Reynoldsnumbersaccessibleto LBE ata sensiblyminor cost in termsof computationalnodesand with the additionaladvantageof geometricalflexibility. Manifestly,theseadvantagesareboughtatthe expensesof a weak lossof locality of both thestreamingandcollision operator.In particular,theformer can result in a significant increaseof the computertime spentper eachnode update.Future

0.020 (a) - 0.020 (b) ,._~-. .. ..

0.00- l1O152O2~53~,54a45 °°~ llO152O253~~54~~5

Fig. 20. (a) Parabolicprofiles obtainedwith the piecewiseconstantmethodfor different valuesof the“physical” viscosity (NU). Thiscan be freelytunedaccordingto theformula v = —[c

21(D+ 2)](1 IA + fl, whereA is oneof theeigenvaluesof thecollisionmatrix. A reasonableagreementwiththeanalyticalresultsis obtainedonly in the limit NU ~ ~NUM (b) The sameas in (a) with thepiecewiselinear scheme.The physicalviscosity is nowabouttwo ordersof magnitudesmaller.

194 R. Benzietal., The lattice Boltzmannequation:theory andapplications

work will indicate the best compromisebetweenthe conflicting issuesof geometricalflexibility and

computationalefficiency.

7. Conclusions

Sincetheir relatively recentappearance,lattice gasmodelshaveknown considerableprogress.Yetmuchwork remainsto be done. For instance,it is widely acceptedthat in the caseof high Reynoldsnumbershomogeneousand isotropic turbulencespectralmethodsare to be preferred.However,thissituationis likely to changein the futurewith theadventof massivelyparallelcomputers.For today, itis certainthat for heterogeneousflows in grosslyirregular geometries,lattice gasmodelshavereachedthepointwherethey canprovidestate-of-theart simulations.In this respect,either latticegasautomataor the lattice Boltzmann equationcan be chosen.Lattice gasautomatahavethe peculiarmerit ofintrinsic numericalstability, stemmingfrom the useof logical “exact” operationsfree of anyround-offerror. On the other hand, the advantagesof the lattice Boltzmann equation are the elimination ofstatisticalnoiseanda muchgreaterflexibility in thechoiceof thesimulationparameters.Bothmethodsareideally placedto matchthecurrenttrendstowardsmassivelyparallelcomputing.From thephysicalpointof view, the lattice Boltzrnannequationhasprovided, insofar, a quite simple tool to understandsomepropertiesof non-equilibriumstatisticalmechanics.For instance,the analysispresentedin section4 could be useful to clarify the effect of non-hydrodynamicdegreesof freedomon the dynamicalevolutionof thehydrodynamicfields.This is justbut anexamplethat illustratesour generalfeelingthatfurther improvements in the numerical solution of the Navier—Stokesequations(via the latticeBoltzmannequation)canonly be achievedby a betterunderstandingof thebasicphysicalmechanisms,ratherthan by the developmentof numericalrecipes.

Acknowledgements

B. Alder, D. d’Humieres,M. Ernst, U. Frisch,F. Higuera,A. Noullez, R. Marra andE. Presutti,arekindly acknowledgedfor illuminatingdiscussions.R.B. and M.V. aregrateful to IBM ECSECwheremost of the computationalwork discussedin this paperhasbeencarried out. This work hasbeensupportedby contractCEE SCI-0212-C.

Appendix A

The 9 x 9 reducedcollision matrix of the two-dimensionalFCHC lattice taking into accountthedegeneraciesresulting from the projectionfrom the four-dimensionalspaceis presentedhere. Thepopulation N

1 propagatesalong the positive x-axis, the rest particle is associatedto N9, and theremainingsevenpopulationsare numberedcounterclockwise.

a11 = a33 = a55 = a77= a”~, (A.1)

a12= a18= a32 = a34 = a54 = a56= a76 = a78 = a60, (A.2)

a13= a17 = a31 = a35= a53 = a57 = a71 = a75 = ~ (A.3)

R. Benz!ft a!., The lattice Boltzmannequation: theoryand applications 195

a14 = a16= a36 = a38 = a52 = a58 = a72 = a74 = a120, (A.4)

a15 = a37 = a51 = a73 = ~ (A.5)

a19 = a39= a59 = a79 a91 = a93 = a95 = a9,7= ~ (A.6)

a21 = a23 = a43 = a45 = a65 = a6,7 = a81 = a87 = 4a60, (A.7)

a22 = a44 = a6,6 = a88 = a0, (A.8)

a24= a28 = a42 = a46 = a64 = a68 = a82= a86 = a92 = a94 = a96 = a9,8 = a~, (A.9)

a25 = a27= a41 = a47= a61 = a63 = a83 = a85 = 4a120, (A.1O)

a26 = a48= a62 = a84 = a180 , (A.11)

a29 = a49 = a69 = a89 = 4a~, (A.12)

a99= a0 + 2a90 + a180. (A.13)

Here, -

aW= a0 + 2a60 + a90, a~2~= a~+ 2a

90+ a120, (A.14)

a~3~= a

90 + 2a120 + a180 , a~4~= 2a

60+ 2a120 . (A.15)

Appendix B

In this appendixwelist theeigenvectorsof the reduced18 collision matrix of the three-dimensionalcase.The reductionfrom the original 24 matrix is due to the degeneraciesimposedon the fourthaxis.

Null eigenvalue:

= (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1), (B.1)

A~2~=(1,—1,0,O,O,0,1,1,--1,—1,1,1,—1,—1,o,o,o,o), (B.2)

A~3~=(O,0,1,—1,O,O,1,—1,1,—1,O,O,O,O,1,1,—1,—1), (B.3)

A~4~=(0,0,O,O,1,—1,O,O,O,O,1,—1,1,—1,1,—1,1,—1). (B.4)

EigenvalueA = a0 — 2a90+ a180:

A~5~= Q

1,,, , A~6~= Q.,,., , = Q,,~, (B.5)

A~8~= Q,,., , A~9~= Q,~, = Q,., . (B.6)

196 R. Benziet a!., The lattice Holtzm.ann equation:theory and applications

Eigenvalueo• = ~(a0— a180):

A,~11~= A~2~k~, A,Y2~= A~3~k~, A,~13~= ~ , (B.7)

A~’4~= A~2~k,A~15~= ~ A~’6~= A~4~k. (B.8)

Eigenvaluer = ~(a0+ 6a90+ a180):

A~17~= ~ A~8~= A~1~k. (B.9)

Here, -

k, = (1, 1, 1, 1, —2, —2, —2, —2, —2, —2, 1, 1, 1, 1, 1, 1, 1, 1) , (B.10)

= (1, 1, —1, —1,0,0,0,0,0,0, —1, —1, —1, —1, 1, 1, 1, 1) . (B.11)

References

[1] G.K. Batchelor, Phys.Fluids (Suppl.2)12 (1969) 233.[2] R. Benzi andS. Succi,J. Phys. A 23 (1990) Li.

[3] R. Benzi, S. Succiand M. Vergassola,Europhys.Lett. 13 (1990) 727.[41B.M. BoghosianandCD. Levermore,ComplexSystems1 (1987) 17. -[5] J.P. Boon, PhysicaD 47 (1991) 3; in: Lattice Gas Methods: Theory,Applications,arid Hardware,ed. 0. Doolèn (MIT Press,Cambridge,

1991).[6] R. Brito andM.H. Ernst, Lattice Gasesin Slab Geometries,Phys. Rev. A, to appear.[71R. Brito andM.H. Ernst, Ring Kinetic Theory for TaggedParticle Problemsin Lattice Gases,preprint, Universityof Florida (1991).[8] R. Brito, M.H. Ernstand T.R. Kirkpatrick, in: DiscreteModels of Fluid Dynamics,ed. AS. Alves (World Scientific,Singapore,1991)pp.

198—207.[9] i.E. Broadwell, Phys.Fluids 7 (1964) 1243.

[10] C. BurgesandS. Zaleski,Complex Systems1(1988) 31.[11] A. Cancelliere,C. Chang,E. Foti, D.H. RothmanandS. Succi, Phys.Fluids Lett. A 2 (1990) 2085.[12] C. Cercignani,The BoltzmannEquationandits Applications(Springer,Berlin, 1988).[13] H. Chenandw.H. Matthaeus,Phys. Rev.Lett. 58 (1987) 1845.[14] S. Chen,Z.S. She, L.C. HarrisonandG. Doolen,Phys. Rev. A 5(1989)2725.[15] P. Clavin, D. d’HumiIres,P. LallemandandY. Pomeau,C.R. Acad.Sci. Paris II 303 (1986) 1169.[16] P. Clavin, P. Lallemand,Y. Pomeauand G. Searby,J. Fluid Mech. 188 (1988) 436.[17] A. Clouqueurand D. d’Humières,ComplexSystems1 (1987) 584.[18] R. Cornubert,D. d’Humièresand D. Levermore,Physica,D 15 (1990) 1.[19] A. De Masi, R. Esposito,iL. Lebowitz and E. Presutti,Commun.Math. Phys. 125 (1989) 127.[20] 0. Doolen,private communication(1991).[21] B. Dubrulle,U. Frisch, M. Hénon andJ.P. Rivet, J. Stat. Phys. 59 (1990) 1187.[22] D. Enskog,The Kinetic Theoryof Phenomenain Fairly Rare Gas, Dissertation,Uppsala (1917).[23] M.H. Ernst, PhysicaD 47 (1991)198;in: Lattice GasMethodsfor PDE: Theory,Applications,andHardware,ed. G. Doolen (MIT Press,

Cambridge,1991).[24] E. Foti, S. Succi andE. Gramignani,Meccanica25 (1990) 253.[25] D. Frenkel and M.H. Ernst,Phys. Rev.Lett. 63 (1989) 2165.[26] U. Frisch, PhysicaD 47 (1991) 231; in: Lattice GasMethods:Theory,Applications,andHardware,ed. 0. Doolen(MIT Press,Cantbridge,

1991).[27] U. Frisch, B. Hasslacherand Y. Pomeau,Phys. Rev.Lett. 56 (1986)1505.[28] U. Frisch, D. d’Humières,B. Hasslacher,P. Lallemand,Y. Pomeauand J.P. Rivet, ComplexSystems1 (1987) 649.[29] U. Frisch and M. Vergassola,Europhys.Lett. 14 (1991) 439.

R.Benziet al., The lattice Boltzmannequation: theoryand applications 197

[30] R. Gatignol, ThéorieCinétique desGaza RepartitionDiscretedesVitesses,LectureNotesin Physics36 (Springer,Berlin, 1975).[31]A. Gustensen,D.H. Rothman,S. Zaleskiand 0. Zanetti, Phys.Rev.A 43 (1991) 4320.[32]i. Hardy, Y. Pomeauand0. de Pazzis,J. Math. Phys. 14 (1973) 1746.[33]J. Hardy, Y. Pomeauand 0. de Pazzis,Phys. Rev. A 13 (1976) 1949.[34]M. Hénon,Complex Systems1 (1987) 762.[35] M. Hénon, Implementationof the FCHC Lattice Gas Model on the ConnectionMachine,presentedat the NATO AdvancedResearch

Workshopon Lattice GasAutomata:Theory, Implementationand Simulations,Nice, iune25—28, 1991, J. Stat. Phys. 68 (1992) 353.[36] F. Higuera,Phys. Fluids Lett. A 2 (1990) 1049.[37] F. Higuera andi. Jimenez,Europhys.Lett. 9 (1989) 663.[38] F. Higueraand S. Succi, Europhys.Lett. 8 (1989) 517.[39]F. Higuera,S. Succi and R. Benzi,Europhys.Lett. 9 (1989) 345.[40] D. Hilbert, Math. Ann. 22 (1912) 561.[41] D. d’HumièresandP. Lallemand, CR. Acad.Sci. Paris 302 (1986) 983.[42] D. d’Humières,P. Lallemand and U. Frisch~Europhys.Lett. 2 (1986) 291.[43] D. d’Humières,Y.H. Qian and P. Lallemand,in: DiscreteKinematicTheory,Lattice GasDynamicsandFoundationsof Hydrodynamics,ed.

R. Monaco (World Scientific, Singapore,1989)pp. 102—113.[441D. d’Humières,Y.H. Qian andP. Lallemand,in: ComputationalPhysicsandCellularAutomata,edsA. Pires,D.P. LandauandH. Hermann

(World Scientific, Singapore,1990)pp. 97—115.[45] D. d’Humières,private communication(1992).[46]L.P. Kadanoffand J. Swift, Phys. Rev.A 165 (1968) 310.[47]J.M.V.A. Koelman,Europhys.Lett. 15 (1991) 603.[481A.N. Kolmogorov,Doki. Akad. Nauk SSSR30 (1941) 301.[49] R.H. Kraichnan, Phys.Fluids 10 (1967)1417.(50] L.D. LandauandEM. Lifshitz, Fluid Mechanics,2nd Ed. (PergamonPress,Oxford, 1987).[51]B. Legras,P. Santangeloand R. Benzi, Europhys.Lett. 5 (1988) 37.[52] D. LevermoreandB. Boghosian,in: CellularAutomataandModelling of ComplexPhysicalSystems,Vol. 46, edsP. Manneville,N. Boccara,

0. Vichniacand R. Bidaux (Springer,Berlin, 1989)p. 118.[53] 0. McNamaraandG. Zanetti, Phys.Rev. Lett. 61(1988)2332.[54] N. Margolus,T. Toffoli and 0. Vichniac,Phys. Rev. Lett. 56 (1986)1694.[55]D. Montgomeryand G. Doolen, Phys. Lett. A 120 (1987) 229.[56] G. Paladin andA. \~ilpiani,Phys. Rev.A 35 (1987) 1971.[57] Y.H. Qian,D. d’HumièresandP. Lallemand,in: Advancesin Kinetic TheoryandContinuousMechanics,edsR. Gatignol andSoubbaramayer

(Springer,Berlin, 1991)pp. 127—138.[58] P. Rem and i. Somers,in: DiscreteKinetic Theory, Lattice Gas DynamicsandFoundationsof Hydrodynamics,ed. R. Monaco (World

Scientific, Singapore,1989)p. 268.[59] P. Rem and J. Somers,in: Cellular Automataand Modelling of Complex Physical Systems,Vol. 46, eds P. Manneville, N. Boccara, G.

Vichniac and R. Bidaux (Springer,Berlin, 1989) p. 161.[60] P. Rem andJ. Somers,PhysicaD 47 (1991) 31; in: Lattice GasMethods: Theory,Applications,and Hardware,ed. G. Doolen(MIT Press,

Cambridge,1991).[61] D. Rothman,Geophysics53 (1988) 509.[621D. Rothmanandi. Keller, J. Stat. Phys.52 (1988)1119.[63] R. Sadournyand C. Basdevant,CR. Acad. Sci. Paris292 (1981) 1061.[64] S. Succi, E. Foti and F. Higuera,Europhys.Lett. 10 (1989) 433.[65] S. Succi, F. Higuera andF. Salenyi, ACM Proc. Int. Conf. on Supercomputing,Kreta(1989) p. 128.[66] S. Succi, M. Vergassolaand R. Benzi, Phys. Rev. A 43 (1991) 4521.[67] W. Taylor and B.M. Boghosian,Renormalizationof Lattice GasTransportCoefficients, preprint (1991).[68] B. Van Leer,J. Comput.Phys. 23 (1977) 276.[69] M. Vergassola,R. Benzi andS. Succi, Europhys.Left. 13 (1990) 411.[70] S. Wolfram,J. Stat. Phys.45 (1986) 471.[711G. Zanetti, Phys. Rev.A 40 (1989) 1539.[72] 0. Zanetti, PhysicaD 47 (1991) 30; in: Lattice GasMethods: Theory,Applications,andHardware,ed. G. Doolen(MIT Press,Cambridge,

1991).