Lattice Gas Automata Research Articles

Lattice-Gas Automata Methods for Engineering

Modern lattice-gas automata (LGA) methods have now existed since the mid-1980s. They have been widely used in a non-engineering context to study multiphase systems, reactive flows, flow and reaction within porousmedia and turbulence. LGA methods for simple fluids have now developed to a point where they may be applied with confidence in an engineering context. Application of LGA in an engineering context to complex fluids such as immiscible mixtures and emulsions can also be contemplated with some further work. In order to assist such application, the basics of LGA are outlined here along with a brief overview of current applications. Thermal LGA methods are also discussed with the hope of stimulating their development for application in an engineering context.

Simulation of Diffusion Phenomena by Cellular Automata

This paper describes a new simulation technique for diffusion phenomena over the sea surface using Cellular Automata (CA), which is capable of making prediction of oil dispersion from tankers come into collision or stranded on rocks. Automaton is known as an abstract expression of sequencial machine in the field of information science, and CA is recognized as one of the model of finite state automata with sectional analysis domain called “cell”. CA has also been applied to demonstrate various engineering phenomena, and one of the most promising application field is flow simulation of viscous fluid, percoration, multi-phase fluid, reaction diffusion problems and so on. In this field, CA is called Lattice Gas Automata (LGA) and the solutions by LGA has been proved to coinside those by Navier-Stokes equation. In the procedure of CA, the analytical area is devided into cells and all the state variables defined on each cell are made transition by local neighbor rule and transition rule along the discrete time steps without solving the governing equation.In the present paper, CA is applied to solve a diffusion problem in which the simulation results are compared to the analytical solution of diffusion equation, and possibility is shown in predicting the oil dispersion in the accident of tanker occured in Japan Sea in January, 1997, as an example. It is shown that the simulation by CA shows good agreement with the analytical solution of diffusion equation, and that the prediction of oil dispersion in Japan Sea can be made in a short time by a personal computer.

Application of lattice gas models to plasma spraying

A two-dimensional original model simulating the deposition of droplets on to a substrate by plasma spraying has been developed. This new model is based on a lattice gas automaton which can reproduce the hydrodynamic behaviour of fluids. The model is fairly easy to apply without requiring drastic simplifying assumptions. The paper focuses on spraying experiments which were carried out to fit model parameters and to assess the success of simulations. Comparison of experimental results with model prediction is based on the study of microstructural features of plasma sprayed deposits. Using batches of nickel based alloy powders with different particles sizes and substrates of a given surface morphology, spraying was performed undervarious pressures using the innovative HPPS mode (high pressure plasma spraying) in a CAPS (controlled atmosphere plasma spraying) system. This allowed the extension of the spraying operating conditions to obtain a larger range of typical sprayed microstructures.

Automaton Simulations of Dispersion in Porous Media

A thermodynamic lattice gas (automaton) model is used to simulate dispersion in porous media. Simulations are constructed at two distinctly different scales, the pore scale at which capillary models are constructed and large scale or Darcy scale at which probabilistic collision rules are introduced. Both models allow for macroscopic (pore scale) phase separation. The pore scale models clearly show the effect of pore structure on dispersion. The large scale (mega scale) simulations indicate that when the pressure difference between the displacing phase and displaced phase is properly chosen (representing the average pressure gradient between the phases). The simulation results are consistent with both theoretical predictions and experimental observations.

Lattice Gas Model with Nonlocal Interactions

We analyze the nature of the hydrodynamic modes in a Lattice Gas Automata (LGA) model defined on a hexagonal lattice and having nonlocal interactions of attractive and repulsive type simultaneously. The model is similar in spirit to the liquid gas model of Appert and Zaleski [Phys. Rev. Lett.64, 1 (1990)]. The phase diagram for the model is computed using the kinetic pressure. The dynamics is studied with a mean field type approach in the Boltzmann approximation ignoring effects of correlated collisions. We compute the transport coefficients and the speed of sound propagation. The presence of attractive interactions show increase in the transport coefficients at intermediate densities.

Diffusion and correlations in lattice-gas automata

We present an analysis of diffusion in terms of the spontaneous density fluctuations in a non-thermal two-species fluid modeled by a lattice gas automaton. The power spectrum of the density correlation function is computed with statistical mechanical methods, analytically in the hydrodynamic limit, and numerically from a Boltzmann expression for shorter time and space scales. In particular we define an observable -- the weighted difference of the species densities -- whose fluctuation correlations yield the diffusive mode independently of the other modes so that the corresponding power spectrum provides a measure of diffusion dynamics solely. Automaton simulations are performed to obtain measurements of the spectral density over the complete range of wavelengths (from the microscopic scale to the macroscopic scale of the automaton universe). Comparison of the theoretical results with the numerical experiments data yields the following results: (i) the spectral functions of the lattice gas fluctuations are in accordance with those of a classical `non-thermal' fluid; (ii) the Landau-Placzek theory, obtained as the hydrodynamic limit of the Boltzmann theory, describes the spectra correctly in the long wavelength limit; (iii) at shorter wavelengths and at moderate densities the complete Boltzmann theory provides good agreement with the simulation data. These results offer convincing validation of lattice gas automata as a microscopic approach to diffusion phenomena in fluid systems.

Lattice gas automaton simulation of acoustic streaming in a two-dimensional pipe

The lattice gas automaton fluid modelling technique is used to study acoustic streaming phenomena arising from the interaction of sound waves with no-slip boundaries in a two-dimensional pipe. It is demonstrated that this fluid simulation tool is able to reproduce the general form of the acoustic streaming flow field observed by Andrade [Proc. R. Soc. London 175, 1 (1883)] and predicted by Lord Rayleigh [Philos. Trans. R. Soc. London 175, 1 (1884)]. The differences that are evident may be due to the restrictions on the computing resource which prohibits the use of a lattice large enough to fulfill the conditions used by Rayleigh in his derivation of the acoustic streaming flow field.

A lattice-gas study of dispersion in alveolated channels

Axial dispersion in alveolated channels was studied via lattice-gas automata (LGA), for both slug and step-change inlet conditions. There was good agreement between the effective diffusion coefficient ( D ∗ ) calculated by the LGA method, and the D ∗ predicted by the ‘stagnant pocket’ formalism developed by Aris, Turner, and Tsuda et al. The enhancement of D ∗ was dependent on the ratio of alveolar volume to central channel volume and the Peclet number. For Pe ≥ 5, D ∗ was substantially greater than the Taylor-Aris prediction for flow between parallel flat plates. For Pe < 3, D ∗ was less than the molecular diffusion coefficient, D m . In the absence of buoyancy, inlet conditions (pulse vs step-change) had little effect on the calculated D ∗ (≤ 3%). The effect of buoyancy, however, depends on the inlet condition; for an LGA corresponding to 1 mol% SF 6 tracer gas in air, D ∗ was increased up to 20% for the step-change, and decreased up to 6% for the slug.

Application of lattice gas techniques to the study of sediment erosion and transport caused by laminar sheetflow

In order to gain a deeper understanding of the dynamics of erosion and sediment transport on hillslopes, it seems important to clarify the role of some basic mechanisms involved in these processes. While there is evidence that this cannot be done using the theoretical framework of river hydraulics, the use of numerical analysis could be of considerable help. The nature of the problem requires a technique capable of solving Navier‐Stokes equations at low Reynolds number, with geometrically complex boundaries and solid particles moving inside the flow field. All these requirements make a novel method, known as lattice gas automaton LGA, a natural candidate for the study of the hydrodynamics of sheetflows. However, due to the recent introduction of this technique, there is a lack of a clear definition of its operational limits. Considering the case of a viscous sheetflow on an erodible rough boundary, we argue that by using LGA the stream Reynolds number can be increased only at the expense of a reduction of the boundary shearing stress. Accordingly, LGA cannot profitably be used to study the beginning of sediment motion and transport. On the other hand, a further evolution of LGA, known as the lattice Boltzmann method, seems highly promising for the numerical study of the erosion processes that eventually lead to drainage network evolution along hillslopes. © 1997 by John Wiley & Sons, Ltd.

A lattice gas study of retardation and dispersion in fractures: Assessment of errors from desorption kinetics and buoyancy

Lattice gas automata (LGA) were used to estimate errors in transport coefficients, as measured in laboratory experiments with Peclet numbers from 0 to 27.6 (defined relative to channel width), Damköhler numbers from 0.18 to ∞, Grashof numbers of 0 or 75, and length/width up to 180. Low Damköhler numbers yield long, low‐amplitude elution tails, which contain much of the total solute. As a consequence, at Da ≈ 0.18 and KD = 8, the solute peak travels at the same speed as the carrier fluid, yielding an apparent KD ≈ 0 after five characteristic diffusion times. Such conditions correspond, for example, to a meter‐long path through a 0.5‐cm‐wide, gas‐filled fracture. Buoyancyenhanced dispersion, found in experiments with horizontal tubes, is confirmed by the LGA analysis; however, a different mechanism is suggested for the enhancement in horizontal fractures. Both kinetic and buoyancy errors can be greatly reduced, or experiments made much smaller, if the first and second moments of a tracer pulse can be measured as functions of time.

Static and dynamic properties of dissipative particle dynamics

The algorithm for the dissipative particle dynamics (DPD) fluid, the dynamics of which is conceptually a combination of molecular dynamics, Brownian dynamics, and lattice gas automata, is designed for simulating rheological properties of complex fluids on hydrodynamic time scales. This paper calculates the equilibrium and transport properties (viscosity, self-diffusion) of the thermostated DPD fluid explicitly in terms of the system parameters. It is demonstrated that temperature gradients cannot exist, and that there is therefore no heat conductivity. Starting from the $N$-particle Fokker-Planck, or Kramers equation, we prove an $H$ theorem for the free energy, obtain hydrodynamic equations, and derive a nonlinear kinetic equation (the Fokker-Planck-Boltzmann equation) for the single-particle distribution function. This kinetic equation is solved by the Chapman-Enskog method. The analytic results are compared with numerical simulations.

Nonlinear Hydrodynamics of Lattice-Gas Automata with Semi-Detailed Balance

Equations governing the evolution of the hydrodynamic variables in a lattice-gas automaton, arbitrarily far from equilibrium, are derived from the micro-dynamical description of the automaton, under the condition that the local collision rules satisfy semi-detailed balance. This condition guarantees that a factorized local equilibrium distribution (for each node) of the Fermi–Dirac form is invariant under the collision step but not under propagation. The main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice-Boltzmann approximation; these equations have a validity domain extending beyond the region close to equilibrium. Linearization of the hydrodynamic equations derived here leads to Green–Kubo formulae for the transport coefficients.

Entropy and Correlations in Lattice-Gas Automata Without Detailed Balance

We consider lattice-gas automata where the lack of semi-detailed balance results from node occupation redistribution ruled by distant configurations; such models with nonlocal interactions are interesting because they exhibit non-ideal gas properties and can undergo phase transitions. For this class of automata, mean-field theory provides a correct evaluation of properties such as compressibility and viscosity (away from the phase transition), despite the fact that no H-theorem strictly holds. We introduce the notion of locality — necessary to define quantities accessible to measurements — by treating the coupling between nonlocal bits as a perturbation. Then if we define operationally "local" states of the automaton — whether the system is in a homogeneous or in an inhomogeneous state — we can compute an estimator of the entropy and measure the local channel occupation correlations. These considerations are applied to a simple model with nonlocal interactions.

Quantum Lattice Gases and Their Invariants

The one-particle sector of the simplest one-dimensional quantum lattice-gas automaton has been observed to simulate both the (relativistic) Dirac and (nonrelativistic) Schrödinger equations, in different continuum limits. By analyzing the discrete analogues of plane waves in this sector we find conserved quantities corresponding to energy and momentum. We show that the Klein paradox obtains so that in some regimes the model must be considered to be relativistic and the negative energy modes interpreted as positive energy modes of antiparticles. With a formally similar approach — the Bethe ansatz — we find the evolution eigenfunctions in the two-particle sector of the quantum lattice-gas automaton and conclude by discussing consequences of these calculations and their extension to more particles, additional velocities, and higher dimensions.

Velocity and density profiles of granular flow in channels using a lattice gas automaton

We have performed two-dimensional lattice-gas-automaton simulations of granular flow between two parallel planes. We find that the velocity profiles have nonparabolic distributions, while simultaneously the density profiles are nonuniform. Under nonslip boundary conditions, deviation of velocity profiles from the parabolic form of Newtonian fluids is found to be characterized solely by ratio of maximal velocity at the center to the average velocity, though the ratio depends on the model parameters in a complex manner. We also find that the maximal velocity (${\mathrm{u}}_{\mathrm{max}}$) at the center is a linear function of the driving force (g) as ${\mathrm{u}}_{\mathrm{max}}$=\ensuremath{\alpha}g-\ensuremath{\delta} with nonzero \ensuremath{\delta} in contrast with Newtonian fluids. Regarding density profiles, we observe that densities near the boundaries are higher than those in the center. The width of higher densities (above the average density) relative to the channel width is a decreasing function of a variable which scales with the driving force (g), energy dissipation parameter (\ensuremath{\epsilon}), and the width of the system (L) as ${\mathrm{g}}^{\mathrm{\ensuremath{\mu}}}$${\mathrm{L}}^{\ensuremath{\nu}}$/\ensuremath{\epsilon} with exponents \ensuremath{\mu}=1.4\ifmmode\pm\else\textpm\fi{}0.1 and \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1. A phenomenological theory based on a scaling argument is presented to interpret these findings.

Nonlinear lattice gas hydrodynamics

Starting from a microdynamical description, we derive the equations governing the evolution of the hydrodynamic variables in a lattice gas automaton. The essential features are: (i) the local collision rules satisfy semi-detailed balance; this condition guarantees that a factorized local equilibrium distribution of the Fermi-Dirac form is invariant under the collision step, but not under propagation; (ii) particles entering a collision are uncorrelated (Boltzmann hypothesis); and (iii) the system can be arbitrarily far from global equilibrium; we do not not assume linear response, as usually imposed, to obtain the dissipative contributions. Linearization of the resulting hydrodynamic equations leads to Green-Kubo formulas for the transport coefficients. The main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice Boltzmann approximation; these equations have a validity domain extending beyond the region close to equilibrium.

Theory for diffusion-limited oscillating chemical reactions

A kinetic description of lattice-gas automaton models for reaction-diffusion systems is presented. It provides corrections to the mean-field rate equations in the diffusion-limited regime. When applied to the two-species Maginu model, the theory gives an excellent quantitative prediction of the effect of slow diffusion on the periodic oscillations of the average concentrations in a spatially homogeneous state.

Microscopic model for FitzHugh-Nagumo dynamics

A microscopic reaction model with a FitzHugh-Nagumo mass action law is introduced. A Markov chain that uses a birth-death description of the reaction mechanism and a random walk model for diffusion is constructed and implemented as a lattice-gas automaton. It is shown that the local particle density probability distribution is binomial in the high diffusion limit and the average particle density is governed by the FitzHugh-Nagumo reaction-diffusion equation. The lattice-gas simulations are able to reproduce phenomena such as labyrinthine patterns and Bloch fronts predicted to exist on the basis of the reaction-diffusion equation. The effects of fluctuations on these chemical patterns, the breakdown of the mass-action and reaction-diffusion descriptions, and the existence of phase transitions in the strong reaction limit are discussed.

Applications of a lattice-gas automaton model for amphiphilic systems

We investigate surface tension and domain growth using our recently introduced hydrodynamic lattice-gas model of amphiphilic systems [1] and apply the model to the simulation of a self-replicating micellar system.

Fluctuations in diffusive lattice gas automata

A class of lattice gas automata giving rise to diffusion at the macroscopic level is investigated both analytically and numerically. We focus on the non-equilibrium properties of the automata and, in particular, on the appearance of algebraically decaying long-range fluctuation correlations when the system is constrained to evolve to a non-equilibrium steady state, arbitrarily far from equilibrium.