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Lattice gas experiments on a non-exothermic diffusionflame in a
vortex field
V. Zehnlé, G. Searby
To cite this version:V. Zehnlé, G. Searby. Lattice gas
experiments on a non-exothermic diffusion flame in a vortex
field.Journal de Physique, 1989, 50 (9), pp.1083-1097.
�10.1051/jphys:019890050090108300�. �jpa-00210979�
https://hal.archives-ouvertes.fr/jpa-00210979https://hal.archives-ouvertes.fr
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Lattice gas experiments on a non-exothermic diffusion flame ina
vortex field
V. Zehnlé and G. Searby
Laboratoire de Recherche en Combustion, Université de Provence,
Centre de Saint-Jérôme,Service 252, F-13397 Marseille Cedex 13,
France
(Reçu le 7 octobre 1988, accepté sous forme définitive le 3
janvier 1989)
Résumé. 2014 Une limitation bien connue du gaz sur réseau
provient du fait qu’il n’est pas invariantpar transformation
galiléenne. On peut remédier à ce problème, dans le cas d’un
fluideincompressible à une seule espèce, par une renormalisation du
temps, de la pression et de laviscosité. Malheureusement, cette
transformation n’est plus possible dans le cas d’un fluide forméde
plusieurs espèces de particules. Nous proposons ici une extension
du modèle collisionnel deFrisch Hasslacher et Pomeau qui permet de
restaurer une pseudo invariance galiléenne. Nousprésentons ensuite
une simulation bi-dimensionnelle d’une couche de cisaillement
réactive dans laconfiguration d’une flamme de diffusion soumise à
l’instabilité de Kelvin-Helmholtz.
Abstract. 2014 It is a known shortcoming of lattice gas models
for fluid flow that they do not possessGalilean invariancy. In the
case of a single component incompressible flow, this problem can
becompensated by a suitable rescaling of time, viscosity and
pressure. However this procedurecannot be applied to a flow
containing more than one species. We describe here an extension
ofthe Frisch Hasslacher Pomeau collision model which restores a
pseudo Galilean invariancy. Wethen present a simulation of a 2-D
reactive shear layer in the configuration of a diffusion
flamesubjected to the Kelvin-Helmholtz instability.
J. Phys. France 50 (1989) 1083-1097 1er MAI 1989,
Classification
Physics Abstracts02.70 - 47.20 - 47.60
1. Introduction.
There is an increasing interest in the use of lattice gas models
to simulate complex viscousflows at moderate Mach and Reynolds
numbers. The basic 2-D model introduced by FrischHasslacher and
Pomeau (F.H.P. model) [1] and a 3-D model [2] are now known
anddemonstrated. However these models have an inherent weakness
since they simulate a NavierStokes equation that is made non
Galilean invariant by the presence of a density-dependentfactor, g
(p ), in the non linear advection term. This leads to non physical
simulations if onetries to study flow containing two or more
species. In the first part of this paper we discuss thislack of
Galilean invariance and its consequences are illustrated by a
simple numericalexperiment. Following the ideas of d’Humières,
Lallemand and Searby [3], we then show thatan extension of the
original F.H.P. lattice gas model can restore a pseudo Galilean
invariance,at least in a reduced domain of density. In the second
section of this paper, this new model isused and adapted to a
system containing three kinds of particles, A, B and C,
reacting
Article published online by EDP Sciences and available at
http://dx.doi.org/10.1051/jphys:019890050090108300
http://www.edpsciences.orghttp://dx.doi.org/10.1051/jphys:019890050090108300
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together, i.e. A + B --. 2C. We then present results of a 2-D
simulation of a reactive mixinglayer which is submitted to the
Kelvin-Helmholtz instability. The time-evolution of the largescale
vortex structures is examined and we comment on the influence of
these structures onthe global reaction rate.
2. Galilean invariance.
2.1 POSITION OF THE PROBLEM. - The 2-D lattice gas is composed
of particles of equal massand velocity modulus (m = 1, v = 1 )
constrained to move on a regular triangular lattice. Theparticles
propagate from a lattice site to one of the six nearest neighbours
where they mayundergo a collision with other incoming particles.
There is an exclusion principle betweenparticles such that no two
particles with the same velocity vector may occupy the same
site.The state of any lattice site is thus described by a set of
six boolean variables indicating thepresence or absence of a
particle for each of the six possible directions. A seventh
variable isoften introduced to permit the presence of rest (or
immobile) particles. The particles mayeventually belong to one of
two or more species which are distinguished by « colour tags ».The
lattice gas model has natural units in which the unit of length is
the lattice spacing, theunit bf time is the particle propagation
time between lattice sites and the unit of mass is themass of a
particle. These are the units that will be used in following.
Comparison with real-world quantities is most conveniently obtained
by the use of non-dimensional numbers such asthe Reynolds number or
the Mach number. The complete details of the F.H.P. lattice
gasmodel will not be repeated here and the reader is referred to
the papers of Frisch, Hasslacherand Pomeau [1], Wolfram [4], Frisch
et al. [5] or d’Humières and Lallemand [6]. It is possibleto derive
the macroscopic conservation laws of the system from the
microscopic laws. If p is anensemble averaged mass density per
lattice cell, pu the total mass flux and c the concentrationof one
species, then in the limit of small Mach numbers, these
conservation laws can bewritten [4] :
where D is a particle diffusion coefficient, v and e are the
kinematic viscosities and w a sourceterm which describes eventual «
colour » reactions between particles. Equations (1.a) and(1.b) are
the macroscopic conservation laws for mass and species
respectively. They are adirect consequence of the corresponding
microscopic conservation laws built in to thecollision rules.
Equation (1.b) is the momemtum equation. In this small Mach
numberapproximation equation (1.b) is functionally identical to the
usual Navier Stokes equation,but the non-linear advection term
contains an additional density dependent factor,g (p ), given by
[5] :
where p m = ’-5i di - 6d is the density of moving particles (di,
i = 1, ... 6, is the density per siteof particles which move in the
ith direction of the lattice), which can be different from the
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total density, p, in the case of lattice-gas models including
rest (immobile) particles. For the 7-particle F.H.P. model, the
maximum limiting value of g is 7/12, obtained at the zero
densitylimit.The true Navier-Stokes equation is unchanged by a
Galilean transformation
(x’ = x + uo t ), however the momentum equation for the lattice
gas, equation (l.b) is notGalilean invariant because of the
presence of the factor g. The physical reason for this lack
ofinvariance is related to the existence of a privileged reference
frame : that of the hexagonallattice on which the particles are
constrained to move. In the case of a simple fluid
containingindistinguishable particles, the correct Navier Stokes
equation can be recovered in theconstant density incompressible
limit. equation (l.b) can be divided by g (which is a constantin
this limit) and absorbed by rescaling time, pressure and viscosity
but not velocity or length(t’ = t. g, P’ = P /g, v’ = v /g). In the
incompressible limit equation (1.a) reduces toV . u = 0 and is not
affected by the rescaling. Unfortunately, the diffusion equation
(l.c) doesnot contain the factor g and so for lattice gases
containing more than one species it isimpossible to find a scaling
that yields simultaneously the Navier Stokes equation and
thediffusion equation in their correct form. One physical
consequence of this is that mass andmomentum are not convected at
the same speed.To illustrate this problem we have made the
following numerical experiment. In a square
box of 256 x 256 sites, we have implemented a uniform horizontal
flow of « blue » (or A)particles with velocity ux = 0.1 (uy = 0 ).
In the middle of the box, a small vertical strip of« blue »
particles is replaced by an equal density strip of « red » (or B)
particles having thesame horizontal velocity (ux = 0.1 ) and with a
transverse velocity uy = 0.1. The fourboundaries of the domain are
made periodic. In a real-world experiment, both strips oftransverse
velocity and concentration of B particles would be advected with
the horizontalvelocity Ux and broaden in proportion to the shear
viscosity and binary diffusion coefficientsrespectively. In figure
1, we show lattice gas simulations in the situations where g ( p )
= 0.7(p = 4.3 in the model described below) and where g (p ) = -
0.5 (p = 6.5 ). It appears clearlythat these simulations lead to
non physical results. The profiles of concentration andtransverse
velocity, which are initially coincident, are well separated after
1 000 iterationsteps. Although the concentration strip is correctly
advected with velocity ux, the transversevelocity strip is advected
with velocity g . ux. This effect is particularly spectacular wheng
0 since concentration and transverse velocity are convected in
opposite directions asdepicted in figure 1b.
2.2 THE PSEUDO GALILEAN INVARIANCE. - In the case of the
standard F. H. P. model ,
involving one rest particle, it is found [6] that p = 7/6 p m
and so equation (2) implies thatIg| :5 7/12 whatever the value of
d. In order to restore Galilean invariance, it is necessary
tomodify the collision rules so that g = 1. As pointed out by
d’Humières, Lallemand and Searby[3], one way to increase g is to
enhance the factor (P/Pm). This has led them to the idea ofallowing
the existence of rest particles of mass 2 (total rest mass can be
equal to 0, 1, 2 or 3).This alone is not sufficient to obtain g (p
) = 1 and so they also relaxed the constraint of semi-detailed
balance and allowed collisions which increase the rest mass to
occur with probabilityone, while the ones that decrease it occur
with a smaller probability.
In this paper, we present a version of the above collision rules
which is extended toaccommodate the presence of up to three
different species and which includes all possiblemass and momentum
conserving collisions that change the rest mass by one unit. We
alsoconsider that the lattice site to be occupied by nr = 0, 1, 2
or 3 rest particles of identical mass.These rules maximise the
interaction between the populations of mobile and rest particles
andthus ensure the fastest possible relaxation to local
equilibrium. These optimal rules aresummarised in table 1 which
gives the positions of particles before and after collision,
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Fig. 1. - The lack of Galilean invariance. (a) The initial
configuration. (b) Left g = 0.7, rightg = - 0.5. Curves denoted by
A correspond to the concentration profiles of « red » particles and
curvesdenoted by B correspond to the transverse velocity
profile.
irrespective of the « colour » of the particule. The « colour »
information is redistributed afterthe collision by choosing one
configuration at random from the set of all possibledistributions.
The destruction of rest particles occurs with probability x, y or
z, depending onthe initial number of rest particles, np and their
creation occurs with probability(1 - x), (1 - y) or ( 1 - z )
instead of one. The optimal values for x, y and z where found to
berespectively 0.5, 0.1 and 0.1. The corresponding values of g are
given in figure 2. The factor ghas a maximum at d = 0.16 where it
takes the satisfactory value g = 1.01. Since the firstderivative of
g with density is zero for d = 0.16, g is not sensitive to local
density fluctuations.We may thus consider that the model is pseudo
Galilean at this particular density. The valuesof the binary
diffusion coefficient, D, and the kinematic viscosity, v, have been
determinedexperimentally from the time decay of an initial
sinusoidal distribution of concentration andtransverse velocity.
Figure 3 gives the measured values as a function of the density per
latticelink. For d = 0.16, we find D = 0.23 and v = 0.22, in
natural lattice gas units.
In the following, we use the collision rules defined in table 1
and perform simulations withthe « G.I » value d = 0.16. Technical
details about the initialisation of the lattice are given inthe
appendix.
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Fig. 2. - g versus density. Full line : theoretical values, (3)
experimental values obtained using thecollision rules of table
I.
Fig. 3. - Kinematic viscosity, v, and binary diffusion
coefficient, D, as a function of particle density perlattice
link.
3. Simulation of a reactive shear layer.
In the last decades, the mixing layer between two flows of
different velocities has been widelystudied. Experimental
investigations of non reactive shear flows have been carried out,
forinstance, by Roshko [7] and by Winant and Browand [8]. They have
shown the developmentof large coherent vortex structures in the
region of high velocity gradients and the merging ofthese eddies
into larger similar structures. This phenomenon has received much
attentionbecause of its frequent occurrence in many practical areas
and is particularly important in thedomain of combustion since
flames frequently develop in such flow fields. In the frameworkof
different numerical schemes, many simulations of both reactive and
non reactive flows,have given useful insight into the dynamics and
the structure of shear layers and into theinteraction between
vortices and flames (see Oran and Boris [9] and references
therein).Marble [10] has studied analytically the development of a
diffusion flame in a vortex. Heshowed that as the flame front rolls
up in the vortex flow field, the reaction surface isstretched,
enhancing the reactant consumption rate. His theoretical analysis
leads to ananalytical formula for the increase in reactant
consumption rate as compared to the simpleplanar diffusion
flame.
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Table 1. - The complete collision table. Most o f the collisions
have multiple output states. Theleft column of the distribution of
mobile output particles corresponds to the left column ofnumber o f
rest particles and to the left column of transition probabilities,
and so on.
In the following, we report on a lattice gas experiment of a 2-D
unsteady reactive shearlayer developing the Kelvin-Helmholtz
instability. This experiment is initiated as shown infigure 4. A
2-D box is filled with A and B particles lying respectively in the
upper and lowerhalves and having opposite velocities U and - U.
Periodic boundary conditions are imposed
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on the left and right boundaries of the domain. The density of
moving particles per link ischosen to be d = 0.16 everywhere, in
order to preserve Galilean invariance as discussedpreviously.
Particles fill the lattice according to equation (A.3).
Fig. 4. - The initial configuration of the reactive shear-layer
experiment.
At the interface between A and B, the irreversible reaction, A +
B - 2C, occurs as soon asA and B meet on the same site. To our
knowledge, this is the first time that three differentspecies of
particles have been used in a lattice gas simulation of unsteady
flow. The technicaldetails of the implementation of the algorithm
will be published elsewhere.
This experiment corresponds to a diffusion flame in the
following physical context :- the reaction is irreversible and
infinitely fast ;- no density change occurs with reactions (this
implies that the reaction has no influence
on the dynamics of the flow) ;- density d = 0.16 and viscosity v
= 0.22 ;- all three binary diffusion coefficient are equal to D =
0.23 ;- the Mach number is small.
3.1 MODERATE REYNOLDS NUMBER FLOW. - We have performed a first
simulation atmoderate Reynolds number. The simulation consists of a
box with 1 024 x 256 sites. On theupper and lower boundaries a
no-slip condition is imposed by specifying that particles
hittingthe wall with velocity v bounce back with velocity - v. The
uniform velocities take the valueU = ± 0.15 and the Reynolds
number, based on the velocity difference 2 U and the height ofthe
box, is Re = 667. In order to favour a rapid development of the
Kelvin-Helmholtzinstability, we have introduced a small sinusoidal
disturbance at the interface between thereactants with an amplitude
of ± 5 sites and with a wavelength À = 1 024/3 (three wavelengthsin
the box). The time evolution of this experiment is depicted in the
sequences of figure 5where we show both the iso-concentration
contours of the product « C » and the hydrodyn-amic field
associated with all the particles. We can follow the development of
the three vortexcores which grow until viscous effects slow them
down. In parallel, the deformation of theinterface by the vortex
field is shown. Most of the chemical reaction occurs at the
stagnationpoints between the co-rotating vortices where fresh
reactants are continuously draggedtowards the interface. The
reaction products are pulled out along the interface into
theviscous cores where they accumulate. Beyond t = 8 000, the flow
field has nearly vanishedand the combustion process becomes mainly
diffusion controlled.
3.2 HIGHER REYNOLDS NUMBER FLOW. - In order to reduce viscous
effects, we haveinitiated another experiment in a larger box of 1
024 x 1 024 sites with slip conditions on thehorizontal sides. The
uniform velocities are U = ± 0.15 and Re = 1 336. At t = 0,
theinterface is planar, but two small vortices, whose centers lie
on the interface between A and
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Fig. 5. - The time evolution of a reactive shear layer. Domain 1
024 x 256 sites. Re = 667. Leftfigures : iso-concentration lines of
products. In order to make the reaction zone more visible we
showconcentration contours only for values above 0.8. Right figures
: the total flow field. The macroscopicquantities are obtained by
averaging over 322 lattice sites.
B, are turned on. They are spaced by 256 sites (one quarter of
the box) and the initialtangential velocity for each is :
where ro = 12 (in lattice gas units) and the circulation r = 4
’TT. The time evolution of thisexperiment is shown in figure 6.
This time-series displays richer structures than in theprevious
example for various reasons. First of all, the forcing is much more
efficient for thedevelopment of the Kelvin-Helmholtz instability.
Secondly, the increased size of the boxallows the final vortex to
acquire a larger circulation. We can define the total circulation y
inthe domain by y = 2 vL = 300 (where L = 1 024 is the horizontal
dimension of the box). Asthe initial shear layer destabilises the
corresponding vorticity sheet is continuously redistri-
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Fig. 6. - The time evolution of a reactive shear layer. Domain 1
024 x 1 024. Re = 1 336. Left figures :iso-concentration lines of
products. In order to make the reaction zone more visible we
showconcentration contours only for values above 0.8 Right figures
: the total flow field. The macroscopicquantities are obtained by
averaging over 322 lattice sites.
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Fig. 6 (Continued).
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Fig. 6 (Continued).
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buted into the localised vortices, which eventually acquire most
of the available circulation, y.Since the diffusion coefficient, D,
is much smaller than y, the mixing process is
essentiallyconvective. We also have made the two vortices closer to
each other than to their repeatedperiodic images. As a consequence
(contrary to the equally spaced vortices in the experimentshown in
Fig. 5), the two initial vortices interact and roll around each
other (seet = 6 000) and give rise to a new vortex structure. This
well known phenomenon has beenobserved both experimentally and in
numerical splitter-plate simulations (see for instanceGhoniem and
Ng [11]). As time goes on, the new vortex develops further and the
wrapping ofthe front around the vortex centre increases. The flame
is made up of a viscous core filled with
products and of two spiral arms attached to the core. At t = 16
000 the adjacent flame sheetsare so close together that they
annihilate in a few time steps. The core is burnt and filled
withproducts. Afterwards, the structure of the front becomes more
complicated (see t = 18 000)and pockets of reactants are still
burning. At t = 20 000, the vorticity is still significant (it is«
fed » with the + U and - U uniform velocity fields) and again rolls
up the front.
3.3 EFFECT OF VORTICITY ON THE BURNING RATE. - Let us recall
that in the case of a simplediffusive flame, with no hydrodynamic
flow, the total amount of products at time t,C (t ), is given
by
where L is the length of the interface and p is the total number
density per site (in ourexperiments p = 2.34). As seen in the
previous simulation, the vortex field stronglyinfluences the shape
of the flame front and affects the values of C (t ) as compared to
thediffusion controlled values given by equation (4). We have
illustrated this fact in figure 7where we have plotted C (t) and
the burning rate dC (t )/dt corresponding to our lastexperiment
along with the value given by equation (4). As can be seen from
figure 7a, in theearly stages, the flame is diffusion controlled
but quickly departs from equation (4) as thefront rolls-up. The
first maximum in figure 7b corresponds to the roll-up around the
two smallcores between t = 0 and t = 4 000. As the cores merge, the
rate first decreases (t = 6 000 )but increases again as soon as the
new vortex starts winding up. This enhancement is due tothe
stretching of the front by the flow and to the fact that the
products are continuouslycleared away from the stagnation points
into the vortex cores, reducing therefore the diffusionlength. This
effect is maximum at t = 14 000 where dC /dt reaches a value 9
times higher than
Fig. 7. - (a) Total mass of products, C (t), as a function of
time. (b) Reaction rate dC (t )/dt.(-) Experimental values
corresponding to figures 6. (*) Diffusion controlled rate from
equation (5).(9) Reaction rate given by Marble’s analysis equation
(6).
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1095
in the case of the simple planar flame. Afterwards, the core is
rapidly consumed anddC/dt decreases until a new roll-up starts
again. Marble [10] has made a theoretical analysis of a diffusion
flame rolled-up in the flow field of
a single vortex field in an unconfined domain. Under the
restriction Y/ ( v D )1/2 > 300, heshowed that the total amount
of chemical products obeys :
We have compared (5) with our results in figure 7b (taking y =
300). Our simulation differsfrom Marble’s case in two respects.
Firstly there is a shear flow background which feeds anddeforms the
vortices. Secondly, the vortices in our system are strongly
time-dependent andthis is reflected in the rate of production which
is also found to be unsteady. However it isinteresting to note that
up to about 20 000 time steps our production rate oscillates about
thevalue obtained from Marble’s analysis. At longer times finite
size effects also affect ourresults. After 20 000 steps the
reaction products have reached the edges of the box under
thecombined effects of advection and diffusion causing a
corresponding drop in the influx of thereactants. At t - 30 000 the
reaction products account for half the total mass in the box andthe
reaction rate falls even below the diffusion controlled limit.
4. Conclusions.
In this paper, we have considered in detail the problem of
Galilean invariance of the latticegas model containing more than
one species. We show that this problem may be solved,
formulti-component flows, by an extension of the
d’Humières-Lallemand-Searby collision rules,at least in a reduced
domain of density and for low Mach numbers. This extended model
haspermitted us to perform direct simulations of a reactive shear
layer and to analyse its temporalevolution. Although performed in a
broad physical context, this simulation correctlyreproduced the
main features found with more classical simulations and presents
theadvantage of having no problems associated with the possible
instabilities of truncatednumerical schemes. Moreover the algorithm
is well structured for efficient processing on thenew generation of
computers with massive parallelism. We found it encouraging for
futureinvestigations. In future work, the model will be extended to
cover the case of a two speedlattice gas capable of reproducing the
density change associated with highly exothermicreactions such as
found in combustion.
Acknowledgments.
We thank B. Denet for his collaboration. This work was supported
in part by the D.R.E.T.under contract number 86/1359/DRET/DS/SR2.
V. Zehnlé was supported by the E.E.C.under contract N°
ST-2J-0029-1. The computations were carried out on SUN-3
workstationsfinanced in part by the E.E.C. under the same
contract.
Appendix.
In this appendix, we give some technical details about lattice
gas initialisations. We recall thatthe pressure of a lattice gas
obeys the general form [5]
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Where U is the macroscopic velocity and F is some function of pm
that depends on the detailsof the collision rules. Chen, She,
Harrison and Doolen [12] have already remarked, in thecase of the
simplest F.H.P. model, that if a non-uniform flow is initialised
with a constantdensity of mobile particles, then the velocity
dependent term is the pressure gives rise tounphysical pressure
oscillations which lead to errors in the measurement of
transportcoefficients, even at relatively low Mach numbers.
In the case of a lattice gas involving rest particles, the
equilibrium densities of the rest andmobile particles (at constant
total density) are also found to depend on the macroscopicvelocity
by a term which is also of the order of the Mach number squared
where AP is some positive constant again depending on the
details of the collision rules andpr, P m are the densities of the
rest and mobile particles respectively. 1 a non uniform flow
isinitialised at constant density, then the populations will adjust
to their equilibrium values(A.2) after a few time steps and the
near-initial pressure distribution becomes :
Comparison of these corrections shows that the àp correction
term is dominant (atp = 2.4 for instance, we found, L1p == 2.8
while F ( p ) = O ( 10- 2 )) . This correction manifestsitself in
experiments with non uniform flows by the appearance of strong
acoustic waves ofunphysical origin.The initial implementation of
particles on the lattice gas must be performed at constant
pressure. For our collision rules it turns out that, to a
reasonable approximation, it issufficient to initialise with a
constant density of mobile particles (F ( p ) is negligible) but
thedensity of the rest particles must be modulated so as to be in
local equilibrium with the mobileparticles at the local velocity.
The theoretical investigation of equilibrium populations is ahard
problem to solve because of the long list of collision events. We
have instead determinedexperimentally the equilibrium values of
rest particles at constant density of mobile particles.For d = 0.16
and up to second order in U, we found :
where ni is the density of sites filled with j rest particles
and di is the density of particlesmoving in the direction ei (i =
1, ... 6 ) of the lattice.
References
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(1986) 1505.[2] RIVET J. P., C.R. Acad. Sci. 305 (1987) 751.[3]
D’HUMIÈRES D., LALLEMAND P. and SEARBY G., Complex Syst. 1 (1987)
633.
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[4] WOLFRAM S., J. Stat. Phys. 45 (1986) 471.[5] FRISCH U.,
D’HUMIÈRES D., HASSLACHER B., LALLEMAND P., POMEAU Y. and RIVET J.
P.,
Complex Syst. 1 (1987) 649.[6] D’HUMIÈRES D. and LALLEMAND P.,
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1349.[8] WINANT C. D. and BROWAND F. K., J. Fluid Mech. 63 (1974)
237.[9] ORAN S. E. and BORIS J. P., Numerical Simulation of
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New York) 1987.[10] MARBLE F. E., Adv. Aerosp. Sci. (1984)
395.[11] GHONIEM A. F. and NG K. K., Phys. Fluids 30 (1987)
706.[12] CHEN S., SHE Z., HARRISON L. C. and DOOLEN G., to appear
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