International Journal of Modern Physics C

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.

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.

We present an outline description of some fundamental theoretical properties in the Digital Physics lattice-gas algorithm.

Digital Physics is a new extension of the lattice-gas concept for the simulation of fluid flow which removes disadvantages that prevented practical application of the original method. The extensions are summarized. Simulation results for a three-speed model demonstrate the absence of artifacts and significant reduction in viscosity. Also, simulation results for 2D and 3D lid-driven cavities for a range of Reynolds numbers and geometries are compared with experiment, CFD and the lattice-Boltzmann BGK method. Accurate results are obtained with the new method.

We developed a new LGA model which has the applicability for simulation of immiscible two phases with wide difference in density. We introduced long-range interparticle forces into the Rothman and Keller's ILG model to represent density difference between phases. We attempted some simulations of phase separation using our new model. Two-phase interfaces are stably made with density distribution coinciding with particle color distribution. Furthermore, the two-phase interface is clearer than that obtained by the Appert and Zaleski's LG model.

A general class of discrete unitary models are described whose behavior in the continuum limit corresponds to a many-body Schrödinger equation. On a quantum computer, these models could be used to simulate quantum many-body systems with an exponential speedup over analogous simulations on classical computers. On a classical computer, these models give an explicitly unitary and local prescription for discretizing the Schrödinger equation. It is shown that models of this type can be constructed for an arbitrary number of particles moving in an arbitrary number of dimensions with an arbitrary interparticle interaction.

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.

The numerical error of the interpolation-supplemented lattice-Boltzmann equation (ISLBE) scheme is theoretically analyzed. The ISLBE is proved to recover the Navier–Stokes equation as long as a second order interpolation scheme is used and the flow domain is appropriately discretized. Using a linear interpolation scheme will inevitably generate the numerical diffusivity and viscosity, making the ISLBE scheme unable to simulate the Navier–Stokes equation correctly. The analysis is consistent with previous numerical experiments.

The accuracy of the lattice-Boltzmann method (LBM) is moderated by several factors, including Mach number, spatial resolution, boundary conditions, and the lattice mean free path. Results obtained with 3D lattices suggest that the accuracy of certain two-dimensional (2D) flows, such as Poiseuille and Couette flow, persist even when the mean free path between collisions is large, but that of the 3D duct flow deteriorates markedly when the mean free path exceeds the lattice spacing. Accuracy in general decreases with Knudsen number and Mach number, and the product of these two quantities is a useful index for the applicability of LBM to 3D low-Reynolds-number flow. The influence of boundary representations on LBM accuracy is captured by the proposed index, when the accuracy of the prescribed boundary conditions is consistent with that of LBM.

Lattice-based models have been attracting much interest in recent years and have been applied to many complex systems. The derivation of large scale dynamical equations of lattice-gas models as well as lattice-Boltzmann models was based on the belief that only the physically interesting quantities (mass, momentum and energy) are conserved. Staggered invariants in lattice-gas models were found in 1988 and there have been no efficient methods to eliminate these invariants. In this paper, we will first discuss the existence of staggered invariants, then we propose to use fractional propagation as an effective way of suppressing these undesired invariants. Numerical simulations will be used to confirm the theory and to show the improvement of computations.

In this paper, we present numerical results of finite size effect in lattice-BGK models applied to liquid-gas phase transitions in one dimension. Several "pseudo-potentials" are used in simulations and the lattice size varies from 4 lattice sites to 4096 sites. Power law scaling is found and simple mean-field theory is used to explain the numerical finding.

In this article we use a lattice-Boltzmann simulation to examine the effects of shear flow on an equilibrium droplet in a phase separated binary mixture. We find that large drops break up as the shear is increased but small drops dissolve. We also show how the tip-streaming, observed for deformed drops, leads to a state of dynamic equilibrium.

We show that by including thermodynamic functions derived from a chosen free energy in a lattice-Boltzmann simulation of fluid flow it is possible to ensure that the fluid relaxes to a well-defined equlilibrium corresponding to the minimum of the input free energy. Two examples are given of phase separation in a binary fluid: bulk two-phase coexistence and a lamellar phase stabilised by a competition between negative surface tension and positive curvature energy. The lattice-Boltzmann framework simulates the Navier–Stokes equations of fluid flow and hence allows investigation of the effects of hydrodynamics on the kinetics of phase separation and on the rheology of the ordered structures.

The lattice-Bhatnagar-Gross-Krook (BGK) method has been used to simulate fluid flow in the nearly incompressible limit. But for the completely incompressible flows, two special approaches should be applied to the general model, for the steady and unsteady cases, respectively. Introduced by Zou et al.,¹ the method for steady incompressible flows will be described briefly in this paper. For the unsteady case, we will show, using a simple numerical example, the need to solve a Poisson equation for pressure.

The classical lattice-Boltzmann scheme is extended in an attempt to represent visco-elastic fluids in two dimensions. At each lattice site, two new quantities are added. A suitable coupling of these quantities with the viscous stress tensor leads to a nonzero shear modulus and visco-elastic effects. A Chapman–Enskog expansion gives us the equilibrium populations and conditions for isotropy of the model. A finite wave vector analysis is needed to study the relaxation of sound waves and to determine the dependence of the transport coefficients upon the frequency.

Tracer dispersion is governed by the velocity fluctuations that the particles are subjected to during their movement. The fluctuation of particle velocity is due to deviations from the mean velocity in the flow field and also to the change of the streamline caused by diffusion. The lattice-BGK method is a good tool to investigate the interaction of both of them, because it models the flow field in detail with even small flow structures. A serious drawback of direct simulations are the requirements in computer time and memory. For spatially periodic media, this can be overcome by using the generalized Taylor-dispersion method to calculate the asymptotic effective dispersion from a solution in an elementary cell. This solution is obtained by simulations with an FHP-BGK-lattice gas. Joining the two methods yields a tool to study the effective dispersion constant of a given periodic geometry.

We present a new lattice-Boltzmann method for hydrodynamic simulations, which is capable of handling very large density and temperature gradients. Unlike other LBGK models, the discrete velocities we used center at the local mean flow velocity, and their values vary according to the local temperature. The adiabatic index of the gas can be easily controlled by a parameter.

A two-dimensional lattice-Boltzmann model with a hexagonal lattice is developed to simulate a boiling two-phase flow microscopically. Liquid-gas phase transition and bubble dynamics, including bubble formation, growth and deformation, are modeled by using an interparticle potential based on the van der Waals equation of state. Thermohydrodynamics is incorporated into the model by adding extra velocities to define temperature. The lattice-Boltzmann model is solved by a finite difference scheme so that numerical stability can be ensured at large discontinuity across the liquid-gas phase boundary and the narrow phase interface thickness can be attained. It is shown from numerical simulations that the model has the ability to reproduce phase transition, bubble dynamics and thermohydrodynamics while assuring numerical instability and narrow phase interface.

In this paper, a numerical study of nonlinear flow phenomena in two-dimensional symmetric channels using the lattice-Boltzmann equation method is presented. The results are compared with both experimental results and other numerical results using some traditional methods. Comparisons are found to be quantitatively accurate.

High resolution lattice-Boltzmann simulations of turbulent channel flow on the Quadrics parallel machine are presented. The parallel performance is discussed together with some preliminary results concerning the vorticity structures which appear near the wall layer and their influence on the scaling laws.

A lattice-Boltzmann scheme for natural convection in porous media is developed and applied to the heat transfer problem of a 1000 kg potato packaging. The scheme has features new to the field of LB schemes. It is mapped on a orthorhombic lattice instead of the traditional cubic lattice. Furthermore the boundary conditions are formulated with a single paradigm based upon the particle fluxes. Our scheme is well able to reproduce (1) the analytical solutions of simple model problems and (2) the results from cooling down experiments with potato packagings.

The application of lattice-Boltzmann methods to electroviscous transport problems is discussed, generalising the moment propagation method for convective-diffusion problems. As a simple application, electro-osmotic flow in a parallel-sided slit is analysed, and the results compared favourably with available analytic solutions for this geometry.

We propose a coarse-graining procedure for a fluid system that allows us to discuss from a conceptual point of view different "mesoscopic" approaches to hydrodynamic problems. Dissipative Particle Dynamics (DPD) and Smoothed Particle Dynamics (SPS) are discussed simultaneously within this framework. In particular, we give physical meaning to the weight function used in SPD. The close analogy between DPD and SPD suggests a synthesis of both approaches that overcomes the conceptual shortcomings of both.

We investigate the domain growth and phase separation of two-dimensional binary immiscible fluid systems using dissipative particle dynamics. Our results are compared with similar simulations using other techniques, and we conclude that dissipative particle dynamics is a promising method for simulating these systems.

Dissipative Particle Dynamics (DPD) is a new simulation technique that focuses on intermediate time and length scales. With this technique it is possible to simulate the essential aspects of the rheological behavior of polymeric liquids quite efficiently. Model studies show that DPD reveals the expected shear thinning and normal stress effects. We also show that the effects of thermodynamic solvent quality on the configurations and rheological behavior of dissolved polymers can be studied with the DPD model.

Polymers in high density 2D melts are believed to segregate into compact disks. This is in contrast to the entanglement and interpenetration characteristic of 3D melts. We investigate this problem using the two-space algorithm, which is both a Cellular Automaton and a Monte Carlo algorithm for polymer structure and dynamics. Our simulations of high density melts in 2D show that contrary to expectations polymers do not completely segregate at high density — there is significant interpenetration. We show that the characteristic size of a polymer in the high density limit is intermediate between the size of a compact disk and a random walk. We then introduce a "shape factor" that measures the ratio of the polymer circumference squared to the area. The shape factor increases with increasing melt density, clearly indicating the observed interpenetration.

Ballistic Deposition (BD) is a prototypical model for interface growth and for dynamic scaling behavior in non-equilibrium systems. BD is typically investigated with computer simulations where randomness is replaced by the use of deterministic Pseudo Random Number Generators (PRNGs). In this study of BD, several results discrepant with the prevailing paradigm, were observed. First, the value of the roughness exponent, χ, obtained is below the value for a random walk (i.e., χ < 1/2). The value χ=1/2 is predicted by the KPZ equation, and many models of growth obtain this exponent. Second, height fluctuations of the growing interface appear not to satisfy simple scaling. Third, a decrease in the surface roughness is observed in a conjectured steady state regime. Computer implementations of BD may be responsible for the discrepancies. A coupling between the BD algorithm and a PRNG algorithm is identified, and statistically discrepant results are obtained for an implementation with a different PRNG.

An initially homogeneous freely evolving fluid of inelastic hard spheres develops inhomogeneities in the flow field u(r, t) (vortices) and in the density field n (r, t)(clusters), driven by unstable fluctuations, δa = {δn, δu}. Their spatial correlations, <δa(r, t)δa(r′,t)>, as measured in molecular dynamics simulations, exhibit long range correlations; the mean vortex diameter grows as ; there occur transitions to macroscopic shearing states, etc.

The Cahn–Hilliard theory of spinodal decomposition offers a qualitative understanding and quantitative estimates of the observed phenomena. When intrinsic length scales are of the order of the system size, effects of physical boundaries and periodic boundaries (finite size effects in simulations) are important.

We present a formulation and numerical solution procedure for heterogeneous atomistic-continuum representations of fluid flows. The ingredients from atomistic and continuum perspectives are non-equilibrium molecular dynamics and spectral element, respectively; the matching is provided by a classical procedure, the Schwarz alternating method with overlapping subdomains. The technique is applied to microscale flow of a dense fluid (supercritical argon) in a complex two-dimensional channel.

We apply the lattice-gas method to study droplet behavior. The ascending velocity and deformation of a drop under buoyancy are numerically analyzed. We develop an algorithm to adjust surface tension by changing collision rules, and deformation of a droplet is calculated with different surface tensions. The deformation of a drop driven by the pressure gradient under gravity is also investigated.

The translation and rotation of non-spherical particles, such as ellipsoidal, cylindric or disk-like pigment particles, in a Couette flow system similar to a blade coating system in the paper industry⁶ have been successfully simulated by using the lattice-Boltzmann method combined with Newtonian dynamic simulations. Hydrodynamic forces and torques are obtained by the use of boundary conditions which match the moving surface of solid particles. Then Euler equations have been integrated to include three-dimensional rotations of the suspensions by using four quaternion parameters as generalized coordinates. The three-dimensional rotations have been clearly observed. Consequently, the motion of the particles suspended in fluids of both low-Reynolds-number and finite-Reynolds-number, up to several hundreds, has been studied. It appears that the 3D translation and rotation of the non-spherical particles are more clearly observed in a high-Reynolds-number fluid than in a low-Reynolds-number fluid.

We present a numerical study of the density fluctuations in granular flows in the hydrodynamic limit. By simulating the continuum equations posed for granular flows,^1,2 we study the onset of the density-cluster formation. Clusters are observed when the restitution coefficient of granular collision is sufficiently small. However, unlike the clusters seen in molecular dynamics simulations,^3,4 these macroscopic clusters appear to be transient. In the long time limit, they are damped by the viscous dissipation.