Numerical Simulations of Hydrodynamics with Lattice Gas Automata in Two Dimensions

We present results of numerical simulat ions of the Frisch, Hasslacher, and Porneau lattice gas model and of som e of its variants. Equilib rium distributions and several linear and nonlinear hydrodynamics flows ar e presented . We sho w that interesting phenomena can be studied with this class of mod els, even for lattices of limited sizes . 1 . Introdu ction Since Frisch , Hasslacher, and Pomeau II] have shown that particles moving on a triangu lar lattice with very simple collisions on the nodes of the lat t ice obey the Navier-Stokes equation, th e use of lat t ice gas models to study hydrodynamics has received considerable interest during th e last two years 12]. However, the Navier-Stokes equat ion is recovered only in the limit of large systems and for incompressible flows. More theoretical analysis remains to be done to bound the errors of the lattice gas numerical scheme for finite lat t ice sizes and velocities. At pr esent, some partial answers to this question can be obtained by numerical simulat ions of the dynamical behavior of the triangular lat t ice gases an d by compar ison of t hese results with classical hydrodyn amics. In section 2, we will describ e precisely the models we have studied and recall briefly their theoretical properties obtained with the Boltzmann approximation [2,3]. Section 3 will be devoted to a detailed description of how these models can be simu lated on general-purpose computers. In section 4, numerical evidence of the Fermi-Dirac distribution for equilibrium will be presented. The measurement of the transport coefficients will be given in section 5. Finally, we will report in section 6 some examples of nonlinear flows either for stationary or nonstationary situations for moderate Reynolds numbers. © 1987 Complex Syst em s Publications, Inc. 600 Dominique d'Humieres and Pierre Lallemand 2. The models We cons ider particles moving on a triangular lattice with unit velocity c, in dir ection i between a node and one of its six neighbors (i = 1, ... ,6). At each t ime st ep , part icles incoming on a node interact together according to collision laws assumed to conserve the number of particles and the total linear momentum on the nod e. The particles then propagate according to the ir new velocity. In addit ion, there is an exclus ion principle such that no two particles with the same velocity may occupy the same node at the same time (0 or 1 particle per cell, as defined in reference 2. Thus, each node of the la t t ice can be described by a six-b it word whose ones represent particles moving with the velocit ies assoc iated with their bi t posit ions within the wor d . We have also used variants of these six-b it models which allow addit ional particles with zero velocity at each site (i = 0 for notational purposes) . T hus ,·we can introduce seven-bit models with at most one "res t particle" per lattice node, or eight-bit models with up to three rest particles per nod e.' In the eight-bit models, two bits are used to code the presence of rest part icles, one for mass-one particles and one for a new kind of rest particle of mass two , equivalent to two rest particles of mass one . To handle the case of particl es with zero velocity and different masses , the theoretical results of reference 2 need some modifications, given in Appendix A. Here, we will give t he general resul ts for the case of bm moving particles with un it mass and b,. rest particles with mass m.l: = 2.1: , k E {O, . . . , b,. I} . The macroscopic quantities: density p and momentum pu, are related to the local average populat ions Ni.l: of particles with mass 2.1: and velocity c., by p = L2.1: Ni.l: ;pu = L2i .l:Ci ' i, k i ,k (2.1) T hese popu lations are given by the following Fermi-Dirac d ist ributions 1 N,« = ---~""':---~7 1 + exp (2'(h + q ' eil) (2.2) where h and q are nonlinear functions of p and u. When u = 0, the average density is t he same for all the particl es with same mass 2 and will be denoted d.l:; taking the mass of the lightest particles as un it mass and do = d, one gets (2.3) d" d. = <p' + (1 dl" Thus, when particles with mass greater than one are added, the density is re lated by a nonlinear law to the average density of particles of mass one, IThese particles ma.y be considered to have an int ern al energy to satisfy energy conservation whi ch is undist inguishable from mass conservation. Numerical Simulat ions of Hydro dynamics 601 which will be called "density per cell" in what follows, as was done for the case where all the particles have the same mass . This nonlinear relat ion implies that all the expans ions around equilibrium cannot take the simple expressions used in reference 2. The density of moving and rest particles , Pm and P" respectively, can be defined as br-l Pm = bmd and P" = L: 2 with P = Pm+ p,, ; .1:=0 For small u , the expans ion of (2.2) up to first order gives (2.4) N " ;0 N " 0' 2p d(1 + -c; . u), Pm d,. (2.5) The speed of sound is given by 2 bm d(1 d) ( ) C, = 2(bmd(1 d) + Z~::0'4'd, (1 _ d, )) 2.6 and, up to second-order terms in velocity and gradients, the lattice gas dynamics is described by2 a,p + div(pu) = 0