Lattice-gas automata

McNamara G R and Zanetti G 1998 Use of the Boltzmann equation to simulate lattice-gas automata Phys. Rev. Lett. 61 2332... 

Iim88] Lim, H.A., Lattice gas automata of fluid dynamics for unsteady flow, Complex Systems 2 (1988) 45-58. 

J. R. Weimar and J. P. Boon, in Lattice Gas Automata and Pattern Formation, L. Lawniczak and R. Kapral, eds., Field Institute, Canada, 1994. 

Frisch, U., Hasslacher, B. and Pomeau, Y. (1986) Lattice-Gas Automata for the Navier-Stokes Equation. Physical Review Letters 56,14, 1505-1508. 

Patent Formation and Lattice-Gas automata, A. T. Lawniczakand, R. Kapral (Eds.), Field Institute Communications, Vol. 6, 1996. 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

Frisch U, Hasslacher B and Pomeau Y (1986). Lattice-gas automata for the Navier-Stokes equation. Phys Rev Lett, 56, 1505-1508. 

Note that when b = a, y = 1/48, as it should for a circle. Equation [27] also applies to unidirectional flow between parallel flat plates, and can be used as a model for solute dispersion in rock fractures. Setting in Eq. [5] equal to the separation distance between two plates, it can be shown that y = 1/210 (Aris, 1959a Wooding, 1960). This result has been confirmed by numerical simulations (Koplik et al 1993) and lattice-gas automata (Perea-Reeves Stockman, 1997). 

Baudet et al. (1989) used lattice gas automata to simulate flow and solute transport between two flat, parallel plates. They demonstrated good agreement between the lattice gas simulations and an analytical solution to the CDE based on the work of Aris (1959a) and Wooding (1960). 

Baudet, C., and J.P. Hulin. 1989. Lattice-gas automata A model for the simulation of dispersion phenomena. Phys. Fluids Al 507-512. 

Gutfraind, R., I. Ippolito, and A. Hansen. 1995. Study of tracer dispersion in self-affine fractures using lattice-gas automata. Phys. Fluids 7 1938-1948. 

Originated from the lattice gas automata (LGA), the LBM has been widely applied in simulating the rarefied gaseous flow in microchannel. Recently the LBM has also found success in applications to the liquid microflows, particularly the electrokinetic flows. For the continuous liquid, the Navier-Stokes equations can be recovered from the Boltzmann equation by Chapman-Enskog expansion or multiscale analysis, in which the Boltzmann equation is split into different scales for space and time variables. The lattice that an LBM operates is usually designated... 

LBM was originally proposed by McNamara and Zanetti [3] to circumvent the limitations of statistical noise that plagued lattice gas automata (EGA). LBM is a simplified kinetic (mesoscopic) and discretized approximation of the continuous Boltzmarui equation. LBM is mesoscopic in nature because the particles are not directly related to the number of molecules like in DSMC or MD but representative of a collection of molecules. Hence, the computational cost is less demanding compared with DSMC and MD. Typical LBM consists of the lattice Boltzmann equation (LBE), lattice stmcture, transformation of lattice units to physical units, and boundary conditions. 

Lutsko, J.F., Boon, J.P., and Somers, J.A., Lattice gas automata simulations of viscous fingering in a porous medium, in T.M.M. Verheggen Numerical Methods for the Simulation of Multi-Phase and Complex Flow, Springer-Verlag, Berlin, 124-135, 1992. 

Two other approaches treat a spatially distributed system as consisting of a grid or lattice. The cellular automaton technique looks at the numbers of particles, or values of some other variables, in small regions of space that interact by set rules that specify the chemistry. It is a deterministic and essentially macroscopic approach that is especially useful for studying excitable media. Lattice gas automata are mesoscopic (between microscopic and macroscopic). Like their cousins, the cellular automata, they use a fixed grid, but differ in that individual particles can move and react through probabilistic rules, making it possible to study fluctuations. 

An interesting variant of the cellular automaton approach is the lattice gas automaton. Despite the name, lattice gas automata can be used to simulate reactions in condensed as well as gaseous media. They were introduced initially as an alternative to partial differential equations (PDE) for modeling complex problems in fluid flow (Hardy et al., 1976) and have been adapted to include the effects of chemical reaction as well (Dab et al., 1990). Lattice gas automata are similar to cellular automata in that they employ a lattice, but differ because they focus on the motions of individual particles along the lattice and because they can accoimt... 

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