Lattice gas experiments on a non-exothermic diffusion flame in a vortex field

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Lattice gas experiments on a non-exothermic diffusion flame 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�

Lattice _gas experiments ^{on a} non-exothermic diffusion flame in

a 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.

Une limitation bien connue du gaz ^sur réseau provient ^{du fait} qu’il n’est pas invariant par transformation galiléenne. ^On peut remédier à ce problème, ^{dans le cas} d’un fluide

incompressible ^{à une seule} espèce, par ^une renormalisation du temps, ^{de la} pression ^{et de la}

viscosité. Malheureusement, ^cette transformation n’est plus possible ^{dans le cas} d’un fluide formé de plusieurs espèces ^de particules. ^Nous proposons ici ^une extension du modèle collisionnel de Frisch Hasslacher et Pomeau qui permet ^{de restaurer une} pseudo invariance galiléenne. ^Nous présentons ^{ensuite une} simulation bi-dimensionnelle d’une couche de cisaillement réactive dans la

configuration d’une flamme de diffusion soumise à l’instabilité de Kelvin-Helmholtz.

Abstract.

It is a known shortcoming of lattice gas models for fluid flow that they ^{do not} possess Galilean invariancy. ^{In the case of a} single component incompressible flow, ^this problem ^{can be} compensated by ^{a suitable} rescaling ^{of time,} viscosity and pressure. However this procedure

cannot be applied ^{to a flow} containing ^{more than one} species. We describe here an extension of the Frisch Hasslacher Pomeau collision model which restores a pseudo ^Galilean invariancy. ^We

then present ^a simulation of a 2-D reactive shear layer ^{in the} configuration ^{of a} diffusion flame

subjected ^to the Kelvin-Helmholtz instability.

Classification

Physics ^Abstracts

02.70

47.20

47.60

1. Introduction.

There is an increasing interest in the use of lattice gas models ^to simulate complex ^viscous

flows at moderate Mach and Reynolds numbers. The basic 2-D model introduced by ^Frisch

Hasslacher and Pomeau (F.H.P. model) [1] ^{and a} 3-D model [2] ^{are now known and}

demonstrated. However these models have an inherent weakness since they ^{simulate a Navier}

Stokes equation that is made non Galilean invariant by the presence of ^a density-dependent

factor, g (p ), ^{in the non} linear advection term. This leads to non physical simulations if one

tries to study ^flow containing ^{two or more} species. ^{In the first} part of this paper ^we discuss this lack of Galilean invariance and its consequences ^are illustrated by ^a simple ^numerical experiment. Following the ideas of d’Humières, ^{Lallemand and} Searby [3], ^we then show that

an 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 is} used 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

together, ^{i.e. A + B --.} 2C. We then present results of a 2-D simulation of a reactive mixing layer ^{which is submitted to} the Kelvin-Helmholtz instability. The time-evolution of the large

scale vortex structures is examined and we comment on the influence of these structures on

the global ^{reaction rate.}

2. Galilean invariance.

2.1 POSITION OF THE PROBLEM. - The 2-D lattice gas is composed ^of particles ^of equal ^mass

and velocity ^modulus (m ^{= 1, v} = 1 ) constrained to move on a regular triangular lattice. The

particles propagate ^{from a} lattice site to one of the six nearest neighbours ^where they may

undergo ^a collision with other incoming particles. ^{There is an exclusion} principle ^between particles ^{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 ^the

presence ^or absence of a particle for each of the six possible directions. A seventh variable is often introduced to permit the presence of ^rest (or immobile) particles. ^The particles may

eventually 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, ^the

unit bf time is the particle propagation time between lattice sites and the unit of ^mass is the

mass 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 as

the Reynolds ^{number or the Mach} number. The complete details of the F.H.P. lattice gas model will not be repeated here and the reader is referred to the papers of Frisch, ^Hasslacher

and Pomeau [1], ^Wolfram [4], ^{Frisch et al.} [5] ^or d’Humières and Lallemand [6]. ^{It is} possible

to derive the macroscopic conservation laws of the system ^{from the} microscopic ^{laws. If} p is ^an

ensemble averaged ^mass density per lattice cell, pu the total mass flux and ^c the concentration of one species, then in the limit of small Mach numbers, these conservation laws can be written [4] :

where D is a particle ^diffusion coefficient, ^v and e ^are the kinematic viscosities and ^{w a source} term 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 a}

direct consequence of the corresponding microscopic conservation laws built in to the collision rules. Equation (1.b) ^{is the momemtum} equation. In this small Mach number

approximation 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 site

of particles ^{which move} in the ith direction of the lattice), ^{which can} be different from the

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} density

limit.

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 not}

Galilean invariant because of the presence of the factor g. The physical ^reason for this lack of invariance is related to the existence of a privileged reference frame : that of the hexagonal

lattice on which the particles ^are constrained to move. In the case of a simple ^fluid containing indistinguishable particles, ^{the correct} Navier Stokes equation ^can be recovered in the constant density incompressible ^limit. equation (l.b) ^can be divided by g (which ^{is a constant}

in 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 to}

V . u

0 and is not affected by ^the rescaling. Unfortunately, the diffusion equation (l.c) ^does

not contain the factor g ^{and so for} lattice gases containing ^{more than one} species ^{it is} impossible ^{to find a} scaling ^that yields simultaneously the Navier Stokes equation ^{and the}

diffusion equation ^{in their correct form. One} physical consequence of this is that ^mass and momentum 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 ^the

same horizontal velocity (ux

0.1 ) ^{and with a} transverse velocity _uy

0.1. The four boundaries of the domain are made periodic. ^{In a} real-world experiment, ^both strips ^of

transverse velocity and concentration of B particles would be advected with the horizontal

velocity Ux and broaden in proportion ^{to the shear} viscosity ^and binary diffusion coefficients

respectively. ^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 clearly

that these simulations lead to non physical results. The profiles of concentration and transverse velocity, ^{which are} initially coincident, ^{are well} separated ^after 1 000 iteration steps. Although the concentration strip ^is correctly advected with velocity ux, the ^transverse

velocity strip is advected with velocity g . ux. This effect is particularly spectacular ^when

g 0 since concentration and transverse velocity ^{are convected in} opposite directions as

depicted 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 ^that Ig| :5 7/12 whatever the value of d. In order to restore Galilean invariance, it is necessary ^to

modify ^{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 of

allowing 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 probability

one, 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 to accommodate the presence of up ^to three different species and which includes all possible

mass and momentum conserving collisions that change ^{the rest mass} by ^one unit. We also consider 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 ^and

thus ensure the fastest possible relaxation to local equilibrium. ^These optimal ^{rules are}

summarised in table 1 which gives ^the positions ^of particles before and after collision,

Fig. ^1.

The lack of Galilean invariance. (a) The initial configuration. (b) Left g

0.7, right

g

0.5. Curves denoted by ^A correspond ^to the concentration profiles ^{of « red »} particles ^{and curves}

denoted by ^B correspond ^{to the} transverse velocity profile.

irrespective ^{of the « colour » of the} particule. ^{The « colour »} information is redistributed after the collision by choosing ^one configuration ^{at random from the set of all} possible

distributions. The destruction of rest particles ^{occurs with} probability x, y ^{or z,} depending ^on

the 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 be

respectively ^{0.5, 0.1 and 0.1. The} corresponding ^values of g are given ⁱⁿ figure ^{2. The} factor g

has a maximum at d

0.16 where it takes the satisfactory value g

^{1.01. Since the first}

derivative 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 values}

of the binary ^diffusion coefficient, D, and the kinematic viscosity, ^{v, have been determined} experimentally from the time decay ^{of an} initial sinusoidal distribution of concentration and transverse velocity. Figure ³ gives ^{the measured values as a} function of the density per lattice link. 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 with the ^« G.I ^» value d

0.16. Technical details about the initialisation of the lattice are given ⁱⁿ

the appendix.

Fig. 2. - g ^versus density. Full line : theoretical values, (3) experimental values obtained using ^the

collision rules of table I.

Fig. ^3.

^Kinematic viscosity, v, ^and binary ^diffusion coefficient, D, ^{as a} function of particle density per lattice link.

3. Simulation of a reactive shear layer.

In the last decades, ^the mixing layer ^{between two} flows of different velocities has been widely

studied. Experimental investigations ^{of non} reactive shear flows have been carried out, for

instance, by ^Roshko [7] ^and by Winant and Browand [8]. They have shown the development

of large ^coherent vortex structures in the region ^of high velocity gradients ^{and the} merging ^of

these eddies into larger ^similar structures. This phenomenon has received much attention because of its frequent occurrence in many practical ^{areas and is} particularly important ^{in the}

domain of combustion since flames frequently develop in such flow fields. In the framework of 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 the interaction 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. He

showed that as the flame front rolls up in the ^vortex flow field, the reaction surface is

stretched, enhancing ^{the reactant} consumption ^rate. His theoretical analysis ^{leads to an}

analytical formula for the increase in reactant consumption ^{rate as} compared ^{to the} simple

planar diffusion flame.

Table 1.

The complete ^collision table. Most o f ^the collisions have multiple output ^{states. The} left ^column of ^the distribution of ^mobile output particles corresponds ^{to the} left ^column of

number 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 shear

layer developing the Kelvin-Helmholtz instability. ^This experiment is initiated as shown in

figure ^{4. A} 2-D box is filled with A and B particles lying respectively in the upper and lower

halves and having opposite velocities U and - U. Periodic boundary conditions are imposed

on the left and right boundaries of the domain. The density ^of moving particles per link is chosen to be d

0.16 everywhere, ^{in order to} preserve Galilean invariance ^as discussed

previously. 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 as

A and B meet on the ^same site. To our knowledge, this is the first time that three different

species ^of particles ^have been used in a lattice gas simulation of unsteady flow. The technical details 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 at moderate Reynolds number. The simulation consists of a box with 1 024 x 256 sites. On the upper and lower boundaries ^a no-slip condition is imposed by specifying ^that particles hitting

the wall with velocity ^v bounce back with velocity - ^{v. The} uniform velocities take the value U

± 0.15 and the Reynolds number, ^{based on the} velocity difference 2 U and the height ^of

the box, ^{is Re}

667. In order to favour a rapid development of the Kelvin-Helmholtz

instability, ^we have introduced a small sinusoidal disturbance at the interface between the reactants with an amplitude ^{of ±} 5 sites and with a wavelength À

^{1 024/3} (three wavelengths

in the box). The time evolution of this experiment ^is depicted in the sequences of figure ⁵

where 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 vortex cores which grow until viscous effects slow them down. In parallel, the deformation of the interface by ^{the vortex} field is shown. Most of the chemical reaction occurs at the stagnation points between the co-rotating vortices where fresh reactants are continuously dragged

towards the interface. The reaction products ^are pulled ^out along the interface into the viscous cores where they accumulate. Beyond t

⁸ 000, the flow field has nearly ^vanished

and the combustion process becomes mainly diffusion controlled.

3.2 HIGHER REYNOLDS NUMBER FLOW. - In order to reduce viscous effects, ^{we have}

initiated another experiment ^{in a} larger box of 1 024 x 1 024 sites with slip conditions ^on the horizontal sides. The uniform velocities ^are U = ± 0.15 and Re = 1 336. At t = 0, ^the

interface is planar, ^{but two small vortices, whose centers lie on} the interface between A and

Fig. ^5.

^The time evolution of a reactive shear layer. Domain 1 024 x 256 sites. Re

667. Left

figures : iso-concentration lines of products. ^{In order to make} the reaction zone more visible we show concentration contours only ^{for values} above 0.8. Right figures : the total flow field. The macroscopic quantities ^{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 initial

tangential velocity for each is :

where ro = 12 (in lattice gas units) and the circulation r

4 ’TT. The time evolution of this

experiment is shown in figure 6. This time-series displays ^richer structures than in the

previous example for various reasons. First of all, ^the forcing ^{is much more} efficient for the

development of the Kelvin-Helmholtz instability. Secondly, the increased size of the box allows the final vortex to acquire ^a larger circulation. We ^can define the total circulation y in

the domain by ^y

^{2 vL}

³⁰⁰ (where L = 1 024 is the horizontal dimension of the box). ^As

the initial shear layer destabilises the corresponding vorticity ^{sheet is} continuously ^redistri-

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 show

concentration contours only for values above 0.8 Right figures : the total flow field. The macroscopic

quantities ^{are obtained} by averaging ^over 322 lattice sites.

Fig. ⁶ (Continued).

Fig. ⁶ (Continued).

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 essentially

convective. We also have made the two vortices closer to each other than to their repeated periodic images. ^{As a} consequence (contrary ^{to the} equally spaced vortices in the experiment

shown in Fig. 5), ^{the two} initial vortices interact and roll around each other (see

t

6 000) ^and give ^{rise to a new} vortex structure. This well known phenomenon ^{has been}

observed both experimentally and in numerical splitter-plate simulations (see for instance Ghoniem and Ng [11]). As time goes ^on, the new vortex develops further and the wrapping ^of

the 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 sheets}

are so close together ^that they annihilate in a few time steps. ^{The core} is burnt and filled with

products. 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 simple

diffusive 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 ^our experiments ^p

2.34). ^{As seen in the} previous simulation, ^{the vortex field} strongly

influences the shape of the flame front and affects the values of C (t ) ^as compared ^{to the}

diffusion controlled values given by equation (4). ^We have illustrated this fact in figure ⁷

where we have plotted C (t) ^{and the} burning ^{rate dC} (t )/dt corresponding ^{to our last} experiment along with the value given by equation (4). ^{As can be seen from} figure 7a, ^{in the} early stages, the flame is diffusion controlled but quickly departs ^from equation (4) ^{as the}

front rolls-up. The first maximum in figure ^7b corresponds ^{to the} roll-up around the two small

cores 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 ^to the stretching of the front by the flow and to the fact that the products ^are continuously

cleared away from the stagnation points ^{into the vortex cores,} reducing therefore the diffusion

length. 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).

in the ^case of the simple planar ^flame. Afterwards, ^{the core is} rapidly consumed and

dC/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, ^he

showed that the total amount of chemical products obeys :

We have compared (5) ^{with our} results in figure ^7b (taking ^y

300). Our simulation differs from Marble’s case in two respects. Firstly ^{there is a shear flow} background which feeds and deforms the vortices. Secondly, the vortices in our system ^are strongly time-dependent ^and

this is reflected in the rate of production which is also found to be unsteady. ^{However it is} interesting ^{to note} that up ^to about 20 000 time steps ^our production ^rate oscillates about the value obtained from Marble’s analysis. ^At longer times finite size effects also affect our

results. After 20 000 steps the reaction products have reached the edges of the box under the combined effects of advection and diffusion causing ^a corresponding drop in the influx of the reactants. At t - 30 000 the reaction products ^account for half the total mass in the box and the 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 lattice gas model containing ^{more than one} species. We show that this problem may be solved, ^for multi-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 has

permitted ^{us to} perform direct simulations of a reactive shear layer ^{and to} analyse ^its temporal

evolution. Although performed ^{in a broad} physical context, this simulation correctly reproduced the main features found with more classical simulations and presents ^the advantage ^of having ^no problems associated with the possible instabilities of truncated numerical schemes. Moreover the algorithm is well structured for efficient processing ^{on the}

new generation ^of computers with massive parallelism. ^{We found it} encouraging ^{for future} investigations. ^{In future} work, the model will be extended to cover the case of a two speed

lattice gas capable ^of reproducing ^the density change associated with highly ^exothermic

reactions 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 workstations financed 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 that

the pressure of ^a lattice gas obeys ^the general ^form [5]

Where U is the macroscopic velocity ^{and F is some} function of _pm that depends ^on the details of the collision rules. Chen, She, Harrison and Doolen [12] ^have already remarked, ^{in the}

case of the simplest ^F.H.P. model, ^{that if a} non-uniform flow is initialised with a constant

density ^{of mobile} particles, ^{then the} velocity dependent ^term is the pressure gives ^{rise to} unphysical pressure oscillations which lead ^{to errors} in the measurement of transport coefficients, ^{even at} relatively low Mach numbers.

In the case of ^a lattice gas involving ^rest particles, ^the equilibrium densities of the rest and mobile particles (at ^{constant total} density) ^are also found to depend ^{on the} macroscopic velocity 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 and pr, P m ^are the densities of the rest and mobile particles respectively. ^{1 a non} uniform flow is initialised 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 (at

p = 2.4 for instance, ^we found, L1p == ^{2.8 while F ( p )}

O ( 10- 2 )) . This correction manifests itself in experiments ^{with non} uniform flows by the appearance of strong ^{acoustic waves of} unphysical 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 is}

sufficient to initialise with a constant density ^{of mobile} particles (F ( p ) ^is negligible) ^{but the} density ^{of the rest} particles ^must be modulated ^{so as} to be in local equilibrium with the mobile

particles ^{at the local} velocity. The theoretical investigation ^of equilibrium populations ^{is a}

hard problem ^to solve because of the long ^{list of collision events. We} have instead determined

experimentally ^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 particles moving in the direction ei (i

^1,

6 ) of the lattice.

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