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lattice-gas model

Youzou Imagawa

School of Information Science,

Japan Advanced Institute of Science and Technology

February 15, 2001

Keywords: lattice-gas model, continuous-velocity lattice-gas model,

Maxwell-Boltzmanndistribution, Benard convection.

Background and Purpose

In generalnumericalmethods, we obtainthe numericalsolusionby solvingthe systems

(Navier-Stokes equation,etc.) which govern the uid ow. As another analysis method,

there are molecularmethods to analyze the uid dynamics by simulating the movement

of molecularwhich constructs the uid material. These methods modelthe macroscopic

uid dynamics by microscopic molecularmovement.

Ifwecompletelysimulatetheuidowbyusingthemolecularmethod,ahugenumber

of molecularis needed. It is necessary to simulate the uid dynamics by the movement

of sample molecular.

The lattice gas model isone of the molecularmethod toanalyze the uid ow.

This model has the followingproperty:

Particle position, velocity, space, and time are discrete.

Particle movement consists of two section, streaming section and collision section.

Particle moves at its velocity per unit time in streaming section. Particles change

their momentum and velocity in collisionsection.

Collisionoccurs on latticepoint. Collision rule isexpressed by Boolean operation.

Local physical value is calculated by spacial average, according to circumstances,

time average.

Copyrightc 2001byYouzouImagawa

However, inlattice-gasmodel, itis necessary toestablishthe collisionrule and lattice

shapetosatisfythe isotropy ofparticlemovement. Moreover, becausethe collisionrule is

expressedbybooleanoperator,exclusiverulemustbeapplied,andthenumberofparticles

onlattice pointis limited. Andwhen wemakea spacediscrete, it isnecessary touse the

hexagonal lattice (which is called FHP lattice) instead of square lattice. And, because

the number of state of particle momentum is very few, we cannot simulate the thermal

ow by using lattice gas model.

Recently,continuous-velocity lattice-gas model was developed. In this model, space

and time are discrete. Discrete space is composed of square lattices. However, particle

velocity is real number. As particle collision occurs on lattice point, particle position

in real number is transferred at lattice point by using the probability process which is

decided by particle position and velocity. On particle collision, particles on the lattice

pointexchange theirmomentumand velocitybyrotatingeachvectorswhichindicatethe

dierenceofthe velocityofcentre ofmass ofthe collidingparticles. Bythis collisionrule,

momentum andkineticenergy ofparticlesonthelattice pointareconserved, andthere is

nolimitofthe numberof particlesonlatticepoint. Moreover, the distributionofvelocity

atequilibriumstateisMaxwell-Boltzmanndistributionwhichissimilartogeneralparticle

movement. So, it may be possible tosimulate the moregeneral physicalphenomenon by

continuous-velocity lattice-gas modelthan by lattice-gas model.

There is no limit of the number of state of particle momentum and energy equation

is producedinthis model. Itis supposed that wecan analyze the heatow phenomenon

by continuous-velocity lattice-gas model as we set the suitable boundary condition.

However,theresearchforconcretecomputationofheatowphenomenonbycontinuous-

velocity latticegas modelhas not stillstudied. This research aims atthe simulationand

observation of heat ow problemby the continuous-velocity lattice-gasmodel.

Result

I simulatedthe2D couette ow, and itwasrealizedthatthe boundarycondition which

eliminatesthe paralleland vertical components of particle velocity was expressed by the

non-slipboundarycondition.

Isimulatedthe2Dcavityow, andIcomparedtheresultbythismodelwithnumerical

solution. It wasrealized thatthis modelis availablefor the owof hundreds ofReynolds

number. It wasconrmed that 2D thermalcavity owcould be simulatedand boundary

condition with temperatureproperty was suitable.

I simulatedthe2DBenard convection,itwasrealizedthadbythe inuencesofgravity

and thermal boundary condition, thermal convection generates, and temperature and

density were transported by thermalconvection.

As the speed of sound and viscosity coeÆcient are proportional to temperature, it is

necessary toincreasethe latticenumberforsimulatingtheowoflargeReynoldsnumber.

And, continuous-velocitylattice-gasmodelhasexcellentpropertythatitiseasytoextend

to the 3D ow simulation. However, 3D ow simulation needs much number of lattice

point and particles than 2D ow simulation. In continuous-velocity lattice-gas model,

because computation time increases in proportion to the number of lattice points and

particles, I consider that it needs much more time to simulate the 3D ow. However,

becausethecomputationofthis modelisexplicit,weconsiderthatitispossibletoreduce

the computation time by the parallel computation.

The square lattice has been used in continuous-velocity lattice-gas model. In this

research, computation space issquare or rectangle space. However, actually,the compu-

tationof uid owaroundthecomplicated objectis needed. Inlattice-gas model, curved

surfaces and curved lines are expressed by using the ne lattice. However, computation

cost is huge. We should consider how we set the boundary condition onthe complicated

surface.