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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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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

[1] FRISCH U., HASSLACHER B. and POMEAU Y., Phys. Rev. Lett. 56 (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., Complex Syst. 1 (1987) 598.[7] ROSHKO A., AIAA J. 14 (1976) 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 Reactive Flow (Elsevier Sci. Publish. Co.,

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 in Complex Syst.