ttice-Boltzmann - Federation of American ScientistsThe Lattice-Boltzmann Method In the...

Number 22 1994

Los Alamos Science 9998 Los Alamos Science Number 22 1994

Lattice-Boltzmanna versatile tool for multiphase

Fluid Dynamicsand other complicated flows

Shiyi Chen, Gary D. Doolen, and Kenneth G. Eggert

Long-term research efforts at theLaboratory often produce resultsthat prove highly valuable to

U.S. industry. The example describedhere is a new approach to modeling theflow of multiphase fluid mixtures, suchas oil and water, through very compli-cated geometries. This new modelingtechnique, called the lattice-Boltzmannmethod, evolved out of ideas that havebeen intensely investigated since 1985,when Laboratory scientists and othersdiscovered that very simple models ofdiscrete particles confined to a latticecan be used to solve very complicatedflow problems.

This lattice method can be regardedas one of the simplest microscopic, orparticle, approaches to modeling macro-scopic dynamics. It is based on theBoltzmann transport equation for thetime rate of change of the particle dis-tribution function in a particular state.The Boltzmann equation simply saysthat the rate of change is the number ofparticles scattered into that state minusthe number scattered out of that state.

The method is fully parallel (thesame calculations are performed atevery lattice site) and local (only near-by particles interact with each other).It is therefore easily programmed andruns efficiently on parallel machines.Complex boundary conditions are in-corporated in a straightforward way andcause the calculational speed to de-crease by only a few percent. The

method yields a good approximation tothe standard equations of fluid flow, theNavier-Stokes equations, in the limit oflong wavelengths and low frequencies.Also, recent generalizations of themethod have extended its applicabilityto multiphase flows, chemically react-ing flows, diffusion and thermohydro-dynamics.

United States oil companies haveexpressed considerable interest in thelattice-Boltzmann method for address-ing problems in oil recovery. For ex-ample, we are collaborating with theMobil Exploration and Producing Tech-nical Center on using the lattice-Boltz-mann method to simulate the flow ofoil and water through oil-bearing sand-stone at a scale and an accuracy neverbefore possible. Mobil provided LosAlamos with 5-micron-resolution sand-stone geometries for use in the simula-tions. (Typical pore diameters are tensof microns.) The goal of the work is tosimulate, at the scale of individualpores in the rock, what happens whenwater is pumped though the rock toforce out oil. Viscosities, surface ten-sions, contact angles, and surfactant ef-fects can be varied to determine howwell this method of oil recovery can bemade to work in specific circumstances.The collaboration is described by Mobiland Los Alamos scientists in the com-panion article, “Toward Improved Pre-diction of Reservoir Flow Perfor-mance—Simulating Oil and WaterFlows at the Pore Scale.” The workhas included the development of soft-ware to calculate relative permeabilitiesof oil and water in complex geometries.The software received an R&D-100award for 1993.

The lattice-Boltzmann work is a su-perb example of applying the extraordi-nary computer power and expertise incomputational physics available at LosAlamos to problems in petroleum pro-duction. The Laboratory’s success at

projects of this type helped motivatethe oil companies to ask that Congresssubstantially increase DOE support foroil and gas research and make thosefunds available for significant new col-laborations between the oil and gas in-dustry and the National Laboratories.The result is ACTI—the AdvancedComputational Technology Initiative—an initiative that will receive $30–50million in funding in 1995

.In this article we will first sketch the

basic ideas of the lattice-Boltzmannmethod and its general applicability.We will then explain how the lattice-gas and lattice-Boltzmann methods areadapted to describe the dynamics at theinterfaces between two immiscible flu-ids such as oil and water and presentsome simulations of phase separation.Finally, we will mention a few of thenew directions into which lattice-Boltz-mann research is moving. Specific ap-plications to the flows in oil reservoirsare discussed in the companion article.

Lattice Methods for ModelingContinuum Dynamics

Lattice methods, including the lat-tice-gas method and its derivative, thelattice-Boltzmann method, present pow-erful alternatives to the standard “top-down” and “bottom-up” approaches tomodeling the behavior of physical sys-tems. The “top-down” approach beginswith a continuum description of macro-scopic phenomena provided by partialdifferential equations. The Navier-Stokes equations for incompressiblefluid flows are an example. Numericaltechniques, such as finite-difference andfinite-element methods, are then used totransform the continuum descriptioninto a discrete one in order to solve theequations numerically on a computer.

The “bottom-up” approach is basedon the microscopic, particle description

Lattice-Boltzmann Fluid Dynamics

100 Los Alamos Science Number 22 1994

Illustration on previous spread: A lattice-Boltz-mann simulation of flow past a slab shows thecomplicated flow patterns that can be modeledwith this technique. The lattice is 1024

3 256sites; the thickness of the slab is 32 sites, whichgives the flow a Reynolds number of 960. Wind-tunnel boundary conditions are used—the flowvelocity at the entrance (left), the top, and thebottom boundaries is fixed. The top image showscontours of equal vorticity; the bottom imageshows the pressure distribution of the flow. Thesimulation was done using the Connection Ma-chine 2 at the Advanced Computing Laboratory.

provided by the equations of moleculardynamics; here the position and veloci-ty of each atom or molecule in the sys-tem are closely followed by solvingNewton’s equations of motion. Thismicroscopic description is straightfor-ward to program on a computer butsimulations using the largest computerspresently available are limited to verysmall systems (ten million particles)and very short times (a few picosec-onds). Therefore molecular dynamicssimulations are more suitable for under-standing the fundamental interactionsthat underlie macroscopic materialproperties than for modeling macro-scopic dynamics.

Intermediate between the twoschemes are the lattice-Boltzmann andlattice-gas methods, which might beconsidered mesoscopic approaches.The lattice methods begin from a parti-cle description of matter: A gas of par-ticles exists on a set of discrete pointsthat are spaced at regular intervals toform a lattice. Time is also dividedinto discrete timesteps, and during eachtimestep particles jump to the next lat-tice site and then scatter according tosimple kinetic rules that conserve mass,momentum, and energy. This simpli-fied molecular dynamics is very care-fully crafted to include the essentials ofthe real microscopic processes. Conse-quently the macroscopic, or averaged,properties of lattice simulations obey,to a good approximation, the desiredcontinuum equations. Despite their ori-gin in a particle description, latticemethods are essentially numericalschemes for studying averaged macro-scopic behavior. They nevertheless re-tain the advantages of a particle descrip-tion, including clear physical insight,easy implementation of boundary condi-tions, and fully parallel algorithms.

Because lattice methods are entirelylocal, they are extremely fast. A simu-lation of a simple lattice-gas model can

attain a speed of 20 gigaflops on a 512-processor CM-5 Connection Machine.Typical simulations of flow throughporous media include 100 million lat-tice sites and run for 5000 timestepsand therefore require only a few hourson that machine. When applied to peri-odic geometries, the three-dimensionallattice-Boltzmann model is able toachieve the same spatial resolution asconventional methods in half the time.Also, because boundary conditions—even complex ones—are imposed local-ly, lattice methods simulate flows inboth simple and complex geometrieswith almost the same speed and effi-ciency. They are therefore suitable forsimulating flows in the extremely com-plex geometries of porous mediawhereas conventional methods are not.Finally, developing code for latticemethods is considerably faster and easi-er than for traditional schemes.

General lattice models already existfor solving the equations of fluid flow,the wave equation, and the diffusionequation. Special versions of thesemodels have been developed for simulat-ing flow through porous media, turbulentflows, phase transitions, multiphaseflows, chemically reacting flows, ther-

mohydrodynamics, magnetohydrody-namics, the dynamics of liquid crystalsand the design of semiconductors.These models are validated by precisecomparisons with other numerical algo-rithms, analytic results, and experiments.

The Lattice-Boltzmann Method

In the lattice-Boltzmann method,space is divided into a regular lattice(for example, a simple cubic lattice)and real numbers at each lattice siterepresent the single-particle distributionfunction at that site, which is equal tothe expected number of identical parti-cles in each of the available particlestates i. In the simplest model, eachparticle state i is defined by a particlevelocity, which is limited to a discreteset of allowed velocities. During eachdiscrete timestep of the simulation, par-ticles move, or hop, to the nearest lat-tice site along their direction of motion,where they “collide” with other parti-cles that arrive at the same site. Theoutcome of the collision is determinedby solving the kinetic (Boltzmann)equation for the new particle-distribu-tion function at that site and the particle


Number 22 1994 Los Alamos Science 101

lattice1.adb 7/26/94

Figure 1. Allowed Velocities at a Lattice SiteThe arrows indicate the magnitudes and directions of the allowed velocities ei at a lat-

tice site in a three-dimensional lattice-Boltzmann simulation. The lattice has a cubic

structure. Six arrows point to nearest-neighbor sites. Eight arrows point along body

diagonals. The sphere at the lattice site represents zero velocity, e0

5 0. Lattice-Boltz-

mann simulations based on a proper equilibrium particle distribution and this minimum

set of velocities preserve the desired isotropy of fluid properties.

distribution function is updated.More specifically, the single-particle

distribution function at a single site in asimple cubic lattice is represented by aset of real numbers, fi(x,t),the expectednumber of particles at lattice site x andtime t moving along the lattice vectorei, where each value of the index ispecifies one of the allowed directionsof motion. Figure 1 shows these direc-tions. Six lattice vectors point to thesix nearest-neighbor sites. Eight latticevectors point along the body diagonalsto the next sites along those diagonals.Thus particles can travel in fourteen di-rections from each lattice site. The ballin the center denotes the vectore0,which is equal to 0 and representsparticles that are not moving. In total,fifteen real numbers describe the parti-cle distribution function at a site.

The first operation in each timestepDt of the calculation is to advance the

particles to the next lattice site alongtheir directions of motion. Since speedequals the distance traveled divided byDt, this model has only three speeds,zero for particles at rest, c for particlesmoving to nearest-neighbor sites and

Ï3wc for particles moving to the nextsites along body diagonals. Usually theunits are chosen such that the distanceto nearest neighbors and Dt are unity,so that c 5 1 and the lattice vector ei isnumerically equal to the velocity of theparticles moving in direction i. If wealso set the mass of each particle equalto unity, the momentum in direction i atsite x and time t is just fi(x,t) ei. Figure2 illustrates sample momentum distribu-tions in two dimensions.

The second operation is to simulateparticle collisions, which cause the par-ticles at each lattice site to scatter intodifferent directions . The collisionrules are chosen to leave the sum of the

fi’s unchanged. (No particles are lost.)The rules are also selected to conservethe total energy and momentum at eachlattice site. To ensure that the particleshave zero average velocity at bound-aries (both perpendicular and parallel tothe walls), one normally imposes“bounce-back” boundary conditions:Any flux of particles that hits a bound-ary simply reverses its velocity so thatthe average velocity at the boundary isautomatically zero, as observed experi-mentally.

The outcome of collisions is verysimply approximated by assuming thatthe momenta of the interacting particleswill be redistributed at some constantrate toward an equilibrium distributionfi

eq. This simplification is called thesingle-time-relaxation approximation.In mathematical terms, the time evolu-tion of the single-particle distribution isgiven by

fi(x 1 ei, t 1 1) 5

fi(x,t)

− ,

where i = 0, 1,…, 14. The second termon the right-hand side is the simplifiedcollision operator Vi. The rate ofchange toward equilibrium is 1/t, theinverse of the relaxation time, and ischosen to produce the desired value ofthe fluid viscosity.

Figure 3 illustrates the single-time-relaxation approximation in two dimen-sions. The net momentum of the in-coming state is zero. In that case, if noexternal forces exist, the equilibriumdistribution consists simply of equalamounts of particle momentum in eachof the allowed directions of motion.The collision rules will therefore forcethe larger fi’s at the site to decrease andthe smaller fi’s to increase so that theparticle distribution is closer to theequilibrium distribution after the colli-sion than before. External forces (grav-

fi(x,t) 2 fieq(x,t)

}}t




f0e0

A

B

f4 e4A

f3 e3A

f7 e7A

f2 e2A

f1 e1A

f5 e5A

f6 e6A f8 e8

A

Total momentum at lattice site A = 8

i=1

AΣ f i e i

c

Figure 2. Momentum Distributions in Two DimensionsThe single-particle distribution function at each lattice site, fi ( x,t), equals the expected

number of identical particles in each of the available particle states i. Particle veloci-

ties ei are limited to a discrete set defined by the geometry of the lattice, and the mo-

mentum distribution at a lattice site is equal to fi (x,t)ei. On a two-dimensional square

lattice, eight directions of motion are possible, so i 5 0, 1,…, 8. Each arrow indicates

the momentum in one of the allowed directions of motion. The circles represent the

rest particles. The figure shows several examples of momentum distributions in two

dimensions. The distribution at site B has a net momentum in direction 3. The net

momentum is equal to zero at sites A and C.

ity or electromagnetic forces) can beadded to the model and make the fi’sgrow in the direction of the net forceand shrink in the opposite direction.

Although lattice models consist of avery simple set of rules, those ruleslead to very complicated flow patterns.The opening pages of this article showa snapshot from a two-dimensional lat-tice-Boltzmann simulation performedby Lishi Luo here at the Laboratory.The simulation models the flow of air

past a rectangular plate. Periodicboundary conditions are imposed per-pendicular to the plate; wind-tunnelboundary conditions (air flowing infrom the left at higher pressure and outat the right at lower pressure) are im-posed in the flow direction. Attachedvortices and vortex shedding are evi-dent, showing the complicated flow pat-terns that can be modeled with lattice-Boltzmann methods. Detailed,quantitative comparisons with other

methods and with experiment have ver-ified the accuracy of this method. Ingeneral, lattice-Boltzmann simulationsagree with exact solutions to theNavier-Stokes equations to secondorder in the lattice spacing and thetimestep.

The lattice-Boltzmann method is anoutgrowth of lattice-gas models, provid-ing something akin to an ensemble av-erage of many realizations of a latticegas but without the unphysical effects.



Particle momenta that will collide at site A


τ = 1

τ = 2

A A A

A A

Incoming momentum distribution

Outgoing momentum distribution

Equilibrium distribution

Halfway between incoming and equilibrium distributions

Step 1—Hopping Step 2—Collision

(a) Single-time relaxation with τ = 1

(b) Single-time relaxation with τ = 2

A

Figure 3. The Single-Time-Relaxation ProcessIn the single-time-relaxation approximation, the momentum distribution at each lattice site is forced toward the equilibrium distribu-

tion at each timestep. In the absence of external forces, the equilibrium distribution of a state with zero net momentum is just equal

amounts of momentum in each direction. The figure illustrates the two calculational steps that occur during each timestep. First

the incoming momentum distribution assembles at a lattice site as particles at the neighboring sites hop along their directions of

motion to that site. Second, the incoming distribution changes, according to the single-time-relaxation collision rule, to an outgoing

distribution that is closer to the equilibrium distribution. When t 5 1, the incoming momentum distribution changes to the equilibri-

um distribution in one time step; when t 5 2, the outgoing momentum distribution is halfway between the incoming and the equilibri-

um distributions.

Lattice-gas models differ from lattice-Boltzmann models in that the formerfollow the motion of actual particleswhereas the latter utilize the expectedvalues of the particle-distribution func-tion. The gas is composed of identicalparticles that obey an exclusion princi-ple: At most one particle can occupy agiven particle state at a given time.Since Ni ( x,t), the number of particlesthat occupy state i at a lattice site, is ei-ther 0 or 1, only single-digit binaryarithmetic is required to update the par-ticle numbers following each collision.Consequently lattice-gas calculationsare extremely fast. Collision rulesspecify the possible outcomes of two-particle and three-particle collisions andthe explicit choice at each site is imple-mented by random sampling. The lat-tice gas is therefore very noisy, andspatial and temporal averaging are re-quired to obtain macroscopic quantities.Also the method has several majordrawbacks, including restriction to lowReynolds number, lack of Galilean in-variance (convective flow velocitiesdon’t add properly unless the system isat nearly constant density), and an un-physical equation of state in which thepressure depends on the local velocityin addition to the usual dependence onthe local density.

In contrast, the lattice-Boltzmannmethod uses real (continuous) numbersto describe a particle distribution ateach site; in this sense it approaches acontinuum description. Although onepays a penalty in that floating-pointarithmetic operations are now requiredto carry out the simulations, the contin-uum feature eliminates most of thenoise associated with lattice-gas simula-tions. In addition, the local equilibriumparticle distribution fi

eq that appears inthe collision operator can be chosen toeliminate the unphysical features of thelattice gas while including dependenceon the local fluid variables only and

leading to the appropriate macroscopicequations. In other words, although lat-tice-Boltzmann simulations ignore par-ticle-particle correlations and use sim-plified collision rules, the collision rulescan be tailored to reproduce the correctevolution of the macroscopic behaviorof a system in a wide variety of cir-cumstances. The relevant equations arepresented in the sidebar, “Equations ofthe Lattice-Boltzmann Method.”

Strategies for Modeling theFlow of Two Fluid Phases

Modeling the flow of two immisci-ble fluids, such as oil and water, pre-sents the difficulty of how to treat thedynamics at the interfaces between thetwo. These dynamics control phenome-na such as the flow of oil and waterthrough porous media, the developmentof viscous fingering, which occurswhen water pushes oil, and dendriteformation (such as the growth ofsnowflakes). Traditional finite-differ-ence and finite-element schemes can beused to model two-phase flow in simpleflow geometries, but they are difficultto apply in the complicated geometriesof porous media. Here we describe lat-tice methods for complex two-phasephenomena; these methods are beingsuccessfully applied to problems in oilrecovery.

To extend lattice methods to theflow of two immiscible fluids (say redand blue fluids), one must double theinformation at each lattice site. In thelattice-Boltzmann approach this is doneby postulating a single-particle distribu-tion for each of the fluids. It is alsonecessary to create new scattering rulesthat will cause the two fluids to sepa-rate at an interface and to emulate theeffects of surface tension. The essentialidea is to compute a force at each lat-tice site that depends on, say, the red-

fluid density at the neighboring sitesand that pushes the red fluid in the di-rection of increasing red-fluid density.The effect of the force must be addedto the collision operator in such a waythat it does not change the total mo-mentum of the two fluids. The size ofthe force must be chosen to produce thedesired surface tension between the twofluids. Under these rules, an initiallyrandom mixture of the red and blue flu-ids should unmix and form an equilibri-um distribution in which a spherical in-terface separates the two fluids, asexpected physically in mixtures of oiland water.

Lattice-gas simulations of immisci-ble fluids. The extension of latticemethods to two immiscible fluids wasdone originally in the context of latticegases. Our particular method imple-ments the effects of surface tension in apurely local manner—through the intro-duction of two species of colored holes.A hole acts like a massless “ghost” par-ticle that moves freely on the lattice butcarries no momentum. A hole of onecolor is created at a lattice site when anincoming particle of that color is scat-tered into a new direction; the new holemoves in the incoming direction of thescattered particle and thereby carries amemory of that particle. The hole isannihilated, or disappears, when itmeets a particle of the same color trav-eling in the same direction.

To create surface tension, a new col-lision rule is added to the usual onesthat causes the colored particles at asite to respond to the presence of col-ored holes by moving in the oppositedirection. The overall effect is like thatof a short-range attractive potentialamong particles of the same color. Al-though the “color potential” has a rangeof more than one lattice spacing, onlylocal information is needed to computethe results of collision at each lattice



site; consequently the colored-holescheme is much faster for modeling theeffects of surface tension than methodsrequiring the calculation of nearest-neighbor color gradients.

The colored-hole scheme has beenused successfully to simulate phase sep-aration, surface tension between phases,and contact angles between fluids andsolid walls. Figure 4 shows the resultsof a lattice-gas simulation of red andblue fluids: Red drops have formed inblue fluid starting from a random initialconfiguration in which the density ofred particles was equal to one-fifth thedensity of blue particles. Figure 5demonstrates that the results shown inFigure 4 obey the Laplace formula forsurface tension. That is, the pressuredrop Dp across the boundary of eachdrop is given by

Dp 5 pred 2 pblue 5 ,

where pred is the pressure of red parti-cles inside the drop, pblue is the pres-sure of blue particles outside the drop,s is the surface-tension coefficient, andR is the radius of the drop. Figure 5ashows the pressure versus the distance rfrom the center of a drop; the pressurechange at the red-blue interface demon-strates the existence of surface tension.Figure 5b shows Dp, the pressurechange across the interface of the dropfor drops of different radii R. As re-quired by the Laplace formula, thepressure change is directly proportionalto 1/R.

In addition to surface tension atfluid-fluid interfaces, the two-fluid lat-tice-gas model must also simulate inter-actions between the fluids and thewalls. Typically, when the interfacebetween two fluids intersects a solidwall, the angle of intersection is fixed.Figure 6 illustrates the two contact an-

gles, one for each fluid, formed at thepoint of intersection. By definition, thesum of the two contact angles is equalto 1808.

The contact angle of a fluid decreas-es as the affinity, or preference, of thewall for that fluid increases. That pref-erence is called wettability. On astrongly water-wet surface, waterdroplets in a background of oil willtend to spread out on the surface andthe contact angle of water will be closeto 08. On a strongly oil-wet surface,those water droplets will bead up andthe contact angle will be close to 1808.In our lattice-gas simulations, wettabili-ty is controlled by a special rule forcollisions of colored holes with thewall: With a certain probability P, ahole of either color bounces back andbecomes, say, a red hole. This rule canproduce contact angles for the red fluidof between 208 and 1808.

Figure 7 shows the results of colored-hole two-fluid simulations in which dif-ferent values of the probability P forthe upper and lower surfaces of a wall(white) have led to marked differencesin the contact angles of the red fluid.The results shown were obtained by av-eraging 12 realizations of the latticegas.

Lattice-Boltzmann model for twofluids. Lattice-gas simulations producerealistic surface phenomena for immis-cible fluids, but they do so only whentemporal and spatial averaging are usedto eliminate the noise induced by parti-cle fluctuations. A more convenient ap-proach is to extend the lattice-Boltz-mann method to colored fluids and usecolor gradients, or nearest-neighbor in-teractions, for modeling interfacial dy-namics. In this approach, each coloredfluid is described by a single-particledistribution function, and each is as-signed a number density as well as acharacteristic relaxation time, which

s}R



Figure 4. Droplet Formation inTwo-Phase Lattice-Gas MixturesThe figure shows the results of a lattice-

gas simulation in which the dynamics of

the interface between red and blue phas-

es was modeled by using the colored-

hole scheme summarized in the text. The

simulation produced this distribution of

red droplets in a blue background after

17,200 time steps, starting from a random

initial distribution of red and blue parti-

cles on 512 3 512 lattice sites. The total

particle density is 0.4 and the ratio of red

to blue particles is 1/5. Because the lat-

tice gas is very noisy, the droplets have

very rough surfaces, in contrast to the

smooth surfaces expected at fluid-fluid

interfaces. Consequently the results of

the lattice-gas simulations must be aver-

aged spatially and temporally to be com-

pared with the macroscopic description

of immiscible fluids (see Figure 5). This

figure should also be compared with

Figure 8, which shows that the lattice-

Boltzmann method produces smoother,

more realistic droplets.

determines its viscosity. The collisionoperator now has two parts. The firstpart of Vi induces single-time relax-ation to equilibrium, as was done forthe single phase. Again each fluidtends to relax to a local equilibrium dis-tribution that depends on the local den-sity and velocity and is analogous tothe single-phase equilibrium distribu-tion presented in the sidebar. The massof each fluid is conserved and the totalmomentum of the system is conserved.

The second part of the collision op-erator Vi depends on the red and bluedensities at nearest neighbors and deter-

mines the dynamics at the fluid-fluid in-terfaces. It is given by

F3 2 1/24,

where k denotes red or blue, Ak is afree parameter that controls the surfacetension, and F is the local color gradi-ent, defined as

F(x) 5i

ei [rr(x 1 ei) 2rb(x 1 ei)].

(ei?F)2

}F2

Ak}2



1.2

1.15

1.1

0.01

0.005

0

p∆

p

0 20 40 60 80

0 0.01 0.02 0.03 0.04

lattice5.adb 7/26/94(a) Pressure profile of red droplet

(b) Pressure difference across red-blue interface for red droplets of different radii (R )

r (lattice units)

1/R (lattice units)

∆p ~ σR

∆p

MixedRed Blue

Figure 5. Verification of the LaplaceFormula for Surface Tension inTwo-Phase Lattice-Gas Simulations For real droplets in a mixture of immisci-

ble fluids, a pressure difference across

the interface balances the surface tension

created by the mutual attraction of parti-

cles of like kind. The relation is given by

the Laplace formula, Dp 5 s/R where s is

the surface-tension coefficient and R is

the radius of the droplet. To compare the

lattice-gas results of Figure 4 with that

macroscopic description, the pressure is

calculated from the local particle-number

density, p 5 n/3, and the results are aver-

aged temporally over 1000 timesteps and

spatially in the circumferential direction

around the center of each droplet.

(a) That average pressure is plotted as a

function of the distance r from the center

of a droplet. The pressure difference be-

tween the red phase and the blue phase

demonstrates the existence of surface

tension in these two-phase simulations.

(b) The pressure difference across the

droplet surface is plotted as a function of

1/R, where R is the radius of the droplet.

The linear relation shows that the surface

tension coefficient s is a constant, as re-

quired by the Laplace formula. Thus the

averaged results of the colored-hole, two-

phase lattice-gas simulations reproduce

the correct macroscopic interfacial dy-

namics.

In the incompressible limit, the densityin a single-phase region is uniform andtherefore the local color gradient iszero. Thus this second part of the colli-sion operator does not contribute. In amixed-phase region, this part of the col-lision operator maintains the interfacesbetween fluids by forcing the momen-tum of the red fluid, jr 5 Si fi

rei, toalign with the direction of the localcolor gradient. In other words, the reddensity at an interface is redistributedto maximize the quantity 2( j?F ). Theblue-particle distribution can then beobtained by applying mass conservationalong each direction: fi

b 5 fi 2 fir.

Using a scaling and expansion pro-cedure similar to that used to derive themacroscopic behavior of a single phase(see sidebar), one can rigorously provethat each fluid will obey the Navier-Stokes equations and that the pressuredifference across fluid-fluid interfaceswill obey the Laplace formula. Resultsof two-phase lattice-Boltzmann simula-tions have also demonstrated that thesurface-tension coefficient s is a con-stant independent of the radius of adrop.

Over the past three years, severalLos Alamos scientists and several sci-entists from Mobil Oil Company havedeveloped a collaboration, applying thetwo-phase model to study flows throughporous media. The results of simula-tions in two and three dimensions areshown in the companion article. Thegoal of the collaboration is to develop agreater understanding of the fundamen-tal physics related to enhanced oil re-covery, and to predict relative perme-abilities and other bulk properties ofporous media from basic properties ofthe fluid-fluid and fluid-rock interac-tions. So far, lattice-Boltzmann simula-tions look promising: Initial resultscompare well with experimental mea-surements of flow patterns and relativepermeabilities.



θblue

θred θred

θblue

Red fluid

Wall

Blue fluid

θred + θblue = 180°

lattice6.adb• 7/26/94

Figure 6 . The Definition of Contact AngleThe contact angle is defined as the angle between a two-fluid interface and a solid sur-

face. As shown in the figure, each fluid has its own contact angle and the sum of the

two must equal 1808. The wetting fluid (the fluid that tends to wet the surface) has a

contact angle of less than 908, and the nonwetting fluid (the fluid that has less affinity

for the solid surface) has a contact angle of greater than 908.

Figure 7. Contact Angles in Two-Phase Lattice-Gas Simulations This simulation demonstrates the ability of lattice-gas simulations to model different

contact angles, or different wettability conditions, as required in studies of porous-

media flow. The simulation conditions are identical to those described in Figure 4 ex-

cept that a solid horizontal wall (white) divides the computational box into an upper

and a lower region. The upper surface of the wall has a greater affinity for the blue

fluid than for the red fluid, whereas the opposite is true of the lower surface of the

wall. The affinity, or preference, of the wall for one fluid over the other is modeled by

a special rule for collisions between a hole and the wall that controls the color of the

hole flux at the walls. The rule specifies that when a hole of either color collides with

a wall, there is a probability P that it will bounce back and become a red hole. In the

simulation shown the upper surface has P = 0.2; that choice produces a surface that is

strongly blue-wet, and the contact angle of the red fluid is greater than 908. The lower

surface has P = 0.8, which results in a strongly red-wet surface and a contact angle for

the red fluid of close to 08.

Applications and Extensions

Lattice multiphase models have beenused for investigating the details of in-terface dynamics, including the well-known phenomena Rayleigh-Taylor in-stability, Saffman-Taylor instability,and domain growth in the separation oftwo immiscible fluids (see Figure 8).The two-phase lattice-Boltzmann modelhas been extended to three and fourphases to study critical and off-criticalquenches and phase transitions.

Features of interest in biologicalproblems have also been added to lat-tice-Boltzmann models. For example,the addition of a surfactant fluid to two-phase flows is being used to study theformation of lipid bilayers. If one endof the surfactant prefers the red fluidand the other end of the surfactantprefers the blue fluid, the fluids are

forced to have a much more extensiveinterface. This approach is being usedfor studying the self-assembly of bio-logical membranes, surfactant effects intwo- and three-dimensional multiphasemodels, the formation of membranes,and self-assembly of amphiphilic struc-tures at aqueous organic interfaces.

Simple reaction-diffusion systems arealso being modeled with lattice-Boltz-mann methods. Recent work has shownthat these simple systems can exhibitextremely complicated and unexpectedbehavior. For example, solutions havebeen found that behave like biologicalcells; that is, concentrations of chemicaldensities within well-defined regions ofthe solution divide as cells divide.

The lattice-Boltzmann method, likeany new method, is going through anexploratory stage to determine its limi-tations and optimal areas of application.

Recently, stability analyses have beencompleted, and the conditions underwhich the method becomes unstable (asdo all finite-difference schemes) havebeen determined. When the allowednumber of speeds is increased fromthree to many, a thermohydrodynamicversion of the lattice-Boltzmann methodemerges. Such a model has been devel-oped and tested by several scientists.The current lattice models for reactingsystems are based on simple isothermalmodels, but they can easily be extendedto include temperature effects and tomodel phase transitions in which the re-action rate depends on local temperature.

Clearly, the lattice-Boltzmannmethod is still in the developmentalstages. Because it is relatively straight-forward to introduce new physics intothe model, we expect it will be appliedto many additional physical phenomena.



Figure 8. Lattice-Boltzmann Simulations of Phase SeparationEach of the three figures is a snapshot of a lattice-Boltzmann simulation after 6000 timesteps, starting from an initial random distrib-

ution of the phases present. (a) The red and blue densities are 0.4 and 0.6, respectively. Since the density of the red fluid is lower

than that of the blue, red drops form in a blue background. At each site the lattice-Boltzmann simulation produces, in essence, an

ensemble average of lattice-gas simulations, so the boundaries between the red and blue fluids are much smoother in this simula-

tion than in the corresponding lattice-gas simulation shown in Figure 4. (b) The densities of the red and blue fluids are 0.5 and 0.5.

Phase separation again takes place readily but produces a different pattern than in (a) because the densities of the two phases are

equal. (c) The model for immiscible fluids can be applied to any number of phases. Here there are three phases, red, blue, and

green, and each has a density equal to 0.33. Again the three phases separate from an initial random distribution.

Acknowledgements

Members of the Lattice-Boltzmann Re-search Effort: Francis Alexander,Mario Ancona, Daryl Grunau, BroslHasslacher, Shuling Hou, Nathan Kreis-berg, Turab Lookman, Lishi Luo,Daniel Martinez, Guy R. McNamara,Balu Nadiga, Silvina Ponce-Dawson,Xiaowen Shan, James Sterling, andQisu Zou.

Further Reading

U. Frisch, B. Hasslacher, and Y. Pomeau. 1986.Lattice-gas automata for the Navier-Stokes equa-tion. Physical Review Letters 56: 1505.

S. Wolfram. 1986. Cellular automata fluids I:Basic theory. Journal of Statistical Physics45: 471–526.

G. McNamara and G. Zanetti. 1988. Use of theBoltzmann equation to simulate lattice-gas au-tomata. Physical Review Letters 61: 2332.

D. H. Rothman and J. M. Keller. 1988. Immis-cible cellular-automaton fluids. Journal of Statis-tical Physics 52: 1119.

S. Succi, P. Santangelo, and R. Benzi. 1988.High-resolution lattice-gas simulation of two-di-mensional turbulence. Physical Review Letters61: 2738–2740.

S. Chen, H. Chen, D. Martínez, and W. H.Matthaeus. 1991. Lattice Boltzmann model forsimulation of magnetohydrodynamics. PhysicalReview Letters 67: 3776.

S. Chen, G. D. Doolen, K. Eggert, D. Grunau,and E. Y. Loh. 1991. Local lattice-gas modelfor immiscible fluids. Physical Review A43: 7053.

A. K. Gunstensen, D. H. Rothman, S. Zaleski, andG. Zanetti. 1991. Lattice Boltzmann model forimmiscible fluids. Physical Review A 43: 4320.

Y. H. Qian, D. d’Humieres, and P. Lallemand.1992. Lattice BGK models for Navier-Stokesequation. Europhysics Letters 17: 479.

F. J. Alexander, S. Chen, and D. W. Grunau.1993. Hydrodynamic spinodal decomposition:Growth kinetics and scaling functions. PhysicalReview B 48: 634–637.

F. J. Alexander, S. Chen, and J. D. Sterling.1993. Lattice-Boltzmann thermohydrodynamics.Physical Review E 47: 2249–2252.

R. Benzi, S. Succi, and M. Vergassola. 1993.The lattice Boltzmann equation: Theory and ap-plications. Physics Reports 222: 145–197.

G. D. Doolen, editor. 1993. Lattice Gas Meth-ods for PDE’s: Theory, Application and Hard-ware. Physica D 47.

D. Grunau, S. Chen, and K. Eggert. 1993. A lat-tice Boltzmann model for multi-phase fluid flows.Physics of Fluids A 5: 2557.

W. E. Soll, S. Y. Chen, K. G. Eggert, D. W.Grunau, and D. R. Janecky. July, 1994. Appli-cation of the lattice-Boltzmann/lattice gas tech-nique to multi-fluid flow in porous media. InProceedings of Computational Methods in WaterResources.



Seated at the monitor, Shiyi Chen; standing, left to right, Gary D. Doolen, Kenneth G. Eggert, andWendy E. Soll. [Soll is a co-author of “Toward Improved Prediction of Reservoir Flow Perfor-mance.”]

Shiyi Chen first joined the Laboratory as a postdoctoral fellow in 1987; he was an Oppenheimer Fel-low from 1990 to 1992. In 1992 he became a staff member in the Complex Systems Group of theTheoretical Division. His research interests include lattice-gas automata and lattice-Boltzmann compu-tational methods, the statistical theory of fluid turbulence, and two-phase and compressible fluid flows.A native of the People’s Republic of China, Chen earned his B.S. in 1981 from Zhejing University andhis M.S. and Ph.D. from Beijing University in 1984 and 1987.

Gary D. Doolen joined the Laboratory’s Applied Theoretical Physics Group in 1975. His research in-terests include nuclear physics, plasma physics, magnetohydrodynamics, neural networks, and nonlinearmathematics. Doolen was with the Center for Nonlinear Studies from 1988 through 1993. He is nowthe leader of the Complex Systems Group. He received his B.S. in engineering science in 1961 and hisM.S. and Ph.D. in physics in 1964 and 1967 from Purdue University.

Kenneth G. Eggert joined the Laboratory’s Geoanalysis Group as a section leader in 1987 and becamegroup leader in 1992. His research interests include use of simulation in the management of surfaceand subsurface hydrologic systems, physics of multiphase flow, and parallel computation in the earthsciences. From 1984 to 1987 he was a physicist in the Earth Sciences Department at Lawrence Liver-more Laboratory where he participated in the Yucca Mountain Project. Eggert received his B.S. inaerospace engineering from Purdue University in 1969 and his M.S. and Ph.D. in civil engineeringfrom Colorado State University at Fort Collins in 1976 and 1980. Eggert is a member of the Societyof Petroleum Engineers and the American Geophysical Union.

The Boltzmann equation for any lattice model is an equation for the timeevolution of fi (x,t), the single-particle distribution at lattice site x:

fi (x 1 ei, t 1 Dt) 5 fi (x,t) 1 Vi( f(x,t)),

where Vi 5 Vi( f(x,t)) is the local collision operator at that site, i 5 0,1,…,14, andDt is assigned a value of unity. Since the usual aim of lattice methods is to modelmacroscopic dynamics, the “exact” collision operator is unnecessarily complex andtherefore numerically inefficient. Two groups (see Chen et al. and Qian et al. inthe Further Reading) nearly simultaneously suggested that the collision operator beapproximated by a single-time-relaxation process in which relaxation to some ap-propriately chosen equilibrium distribution occurs at some constant rate. In particu-lar the collision term, V( f ), is replaced by the single-time-relaxation approximation,

Vi( f(x,t)) 5 2 .

The appropriately chosen equilibrium distribution, denoted by f eq, depends on thelocal fluid variables, and 1/t is the rate of approach to this equilibrium. The rela-tions Si Vi

5 0 and Sie

iV

i5 0 must be true to conserve mass and momentum, re-

spectively. In order for the fluid to have Galilean-invariant convection and a pres-sure that does not depend on velocity, an appropriate equilibrium distribution, fi

eq,must be assumed. For a two-dimensional hexagonal lattice, the formula is:

fieq 5 1 ei?v 1 (ei?v)2 2 v2

and

f0eq 5 ar 2 rv2.

(The corresponding formulas for the cubic lattice appear in the article by Alexan-der, Chen, and Grunau listed in the Further Reading.) In these equations the den-sity r (x,t) 5 mn(x,t) (where m is the mass of each particle), the number densityn(x,t) 5 Si fi(x,t), and a is a free parameter related to the sound speed as shownbelow. For the lattice-Boltzmann method, the particle distribution does not havean upper bound. The only requirement is that fi $ 0.

To derive the macroscopic equations obeyed by this model, one performs a Taylorexpansion in time and space and takes the long-wavelength and low-frequencylimit of the lattice-Boltzmann equation for the single-particle distribution. The re-sult is a continuum form of the Boltzmann equation correct to second order in thelattice spacing and the timestep. A scaling expansion argument, the assumption ofsingle-time relaxation, and the neglect of higher-order terms lead to the followingfinal form of the macroscopic equations obeyed by the simulated system

r}6

2r}3

r}3

r(1 2 a)}

6

fi(x,t) 2 fieq(x,t)

}}t


Equations of the Lattice-Boltzmann Method

1 5 0, the equation of mass continuity;

the equation of momentum conservation; and

p 5 r.

the equation of state.

In the above equations, vb is the component of the velocity in the b-direction; p isthe pressure; and the sound speed, cs, is Ï(1w 2w jw)/w2w, where j is a free parameter.The shear viscosity, m, and the bulk viscosity, l, are given by

m 5 r

and

l 5 r.

The above equations converge to the exact incompressible Navier-Stokes equa-tions only when the derivatives of the number density in the second viscosity termon the right-hand side of the equation are small. Since the gradients of the densityare O(v2), the unphysical terms in the momentum-conservation equation are cor-rect to order (v3). Thus, although the physics of the lattice-Boltzmann method con-tains compressibility effects, one may come arbitrarily close to solving the incom-pressible Navier-Stokes equations by reducing the Mach number (through thechoice of a) and thereby reducing the simulated flow to very low speed. (Never-theless the compressibility effects in the lattice-Boltzmann approach are physicaland the method can also be used to simulate compressible fluids.)

Traditional methods for solving incompressible flows, such as finite-difference orfinite-element, require solution of a Poisson equation for the pressure term, whichis induced by the mass-continuity equation and the momentum-conservation equa-tion. In the lattice-Boltzmann approach, this time-consuming step is avoided be-cause the incompressibility requirement has been relaxed and the effects of pres-sure changes are controlled by an equation of state rather than a Poisson equation.It can be argued that the conventional methods most closely related to the lattice-Boltzmann method are the pseudocompressible algorithms for solving incompress-ible fluid flows.

(t 2 1/2)(2j2 1)}}

4

2t 2 1}

8

1 2 j}

2

∂(rvb)}

∂xb

∂r}∂t

Equations of the Lattice-Boltzmann Method


1 rvb 5 2 1 3 1 d ab1 va 1 vb 24 1 3m1 1 24,∂va}∂xb

∂vb}∂xa

∂}∂xb

∂r}∂xa

∂r}∂xb

∂rvg}∂xg

l}r

∂}∂xb

∂p}∂xa

∂va}∂xb

r∂va}

∂t

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