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The lattice Boltzmann method is a discrete representation of the Boltzmann transport equation that has been employed for modeling transport of particles of different nature. In the present work, we describe the lattice Boltzmann methodology and implementation techniques for the phonon transport modeling in crystalline materials. We show that some phonon physical properties, e.g., mean free path and group velocity, should be corrected to their effective values for one- and two-dimensional simulations, if one uses the isotropic approximation. We find that use of the D2Q9 lattice for phonon transport leads to erroneous results in transient ballistic simulations, and the D2Q7 lattice should be employed for two-dimensional simulations. Furthermore, we show that at the ballistic regime, the effect of direction discretization becomes apparent in two dimensions, regardless of the lattice used. Numerical methodology, lattice structure, and implementation of initial and different boundary conditions for the D2Q7 lattice are discussed in detail.

The problem of the mixed convection in a cubic cavity is studied with lattice Boltzmann method. A multiple-relaxation-time lattice Boltzmann model for incompressible flow in the cubic cavity and another thermal lattice Boltzmann model for solving energy/temperature equation are proposed. The present models are first validated through a comparison with some available results, and then, we present a detailed parameter study on the mixed convection in the cubic cavity. The numerical results show that the flow and temperature patterns change greatly with variations of the Reynolds and Richardson numbers.

The thermodynamic consistency of kinetic models for non-ideal mixtures in non-isothermal conditions is investigated. A kinetic model is proposed that is suitable for deriving high-order lattice Boltzmann equations by an appropriate discretization of the velocity space, satisfying the Galilean invariance condition and free of spurious terms in the first moment equations.

Modelling the hydrodynamics of gas/solid flow is important for the design and scale-up of fluidized bed reactors. A novel gas/solid dual-scale model based on lattice gas cellular automata (LGCA) is proposed to describe the macroscopic behaviour through microscopic gas–solid interactions. Solid particles and gas pseudo-particles are aligned in lattices with different scales for solid and gas. In addition to basic LGCA rules, additional rules for collision and propagation are specifically designed for gas–solid systems. The solid’s evolution is then motivated by the temporal and spatial average momentum gained through solid–solid and gas–solid interactions. A statistical method, based on the similarity principle, is derived for the conversion between model parameters and hydrodynamic properties. Simulations for bubbles generated from a vertical jet in a bubbling fluidized bed based on this model agree well with experimental results, as well as with the results of two-fluid approaches and discrete particle simulations.

We present a coupled lattice Boltzmann method (LBM) for thermal viscous flows based on a two relaxation times kinetic model. The method is directly derived by projecting the coupled kinetic model equations for the distribution functions onto the tensor Hermite polynomials following Grad’s moment expansion method. By choosing a proper reference velocity, the coupling of lattice velocities and the local temperature is avoided and the resultant method is consistent with the original kinetic model. The intrinsic discrete nodes of the Gauss–Hermite quadrature provide the natural lattice velocities for the thermal LBM. Simulations of 2D natural convection flows are shown. The numerical results agree well with the previous benchmark data.

The shape and surface texture of a liquid droplet were studied in two-dimensions when a droplet impinges on a solid substrate under isothermal conditions. The lattice Boltzmann moment model was applied to simulate the fluid dynamics considering the adhesive interaction between fluid particles and surfaces. The results show the influence of wetting on the process and the drop shape. For a hydrophobic surface, the process after impinging may be divided into two stages: the spreading process driven by inertial forces and the subsequent oscillations (recoiling process) driven by surface tension forces; while for the hydrophilic surface, the droplet will only deposit on the surface and there is no the recoiling stage. In addition, the effects of the impingment speed on the shape and texture of the droplet were studied. The spreading speed and the maximum diameter of the spreading droplet increase with the rise of the impingment speed. Notation D wetting diameter of the droplet on the surface at some time D0 initial diameter of the droplet G interaction force Re Reynolds number t dimensionless time We Weber number V0 impact velocity when the droplet reaches the solid surface  the droplet density  surface tension coefficient  kinematic viscosity of the droplet. Superscripts ac after collision bc before collision eq equilibrium state

An effective matrix method is performed to build the equal degree random (EDR) lattice, and then a cellular automaton game of life on the EDR lattice is studied by Monte Carlo (MC) simulation. The standard mean field approximation (MFA) is applied, and then the density of live cells is given ρ=0.37017 by MFA, which is consistent with the result ρ=0.37±0.003 by MC simulation.

In this paper, a lattice Boltzmann single-phase model is implemented to the simulation of the process of body entering water vertically. The initial water-entry process of a two dimensional cylinder and the process of a three dimensional disk entering water vertically at constant speed is simulated. The deformation of free surface and the relationship between the relative closure depth of water-entry cavity of the disk and the Froude number is studied. It is demonstrated that the LBM single-phase method is applicable to the water-entry problems.

Lattice-gas cellular automata (LGCAs) can serve as stochastic mathematical models for collective behavior (e.g. pattern formation) emerging in populations of interacting cells. In this paper, a two-phase optimization algorithm for global parameter estimation in LGCA models is presented. In the first phase, local minima are identified through gradient-based optimization. Algorithmic differentiation is adopted to calculate the necessary gradient information. In the second phase, for global optimization of the parameter set, a multi-level single-linkage method is used. As an example, the parameter estimation algorithm is applied to a LGCA model for early in vitro angiogenic pattern formation.

A finite-volume formulation commonly employed in the well-known SIMPLE family algorithms is used to discretize the lattice Boltzmann equations on a cell-centered, non-uniform grid. The convection terms are treated by a higher-order bounded scheme to ensure accuracy and stability of solutions, especially in the simulation of turbulent flows. The source terms are linearized by a conventional method, and the resulting algebraic equations are solved by a strongly implicit procedure. A method is also presented to link the lattice Boltzmann equations and the macroscopic turbulence modeling equations in the frame of the finite-volume formulation. The method is applied to two different laminar flows and a turbulent flow. The predicted solutions are compared with the experimental data, benchmark solutions, and solutions by the conventional finite-volume method. The results of these numerical experiments for laminar flows show that the present formulation of the lattice Boltzmann method is slightly more diffusive t...

In this paper, Thermal Finite-Volume Lattice Boltzmann Method is developed. To demonstrate the temperature field, the Double Distribution Function (DDF) of thermal lattice Boltzmann equation is used. The upwind biasing factors based on pressure and temperature are defined and applied as flux corrector in the thermo-hydrodynamic lattice Boltzmann equations. A consistent open and solid boundary treatment of flow is also addressed. The unknown energy distribution at the boundary cells are decomposed into its equilibrium and non-equilibrium parts. Then the non-equilibrium part is approximated with extrapolation of the non-equilibrium part of the populations at the neighboring nodes. This treatment enlarges the domain stability and led up to faster convergence. Two test cases namely, thermo-hydrodynamic in a backward-facing step and around a circular cylinder inserted within a backward-facing step are carried out. The results are compared with the available solutions in the technical literature.

a splitting method to solve the modeled lattice Boltzmann equation. The splitting technique permits the boundary conditions for the lattice Boltzmann equation to be set as conveniently as those required for the finite difference solution of the Navier–Stokes equations. It is shown that the compressible Navier–Stokes equation can be recovered fully from this approach; however, the formulation requires the solution of a Poisson equation governing a secondorder tensor. Thus constructed, the method has no arbitrary constants. The proposed method is used to simulate thermalCouette flow,aeroacoustics,andshockstructureswithanextendedthermodynamicsmodel.Thesimulations are carried out using a high-order finite difference scheme with a two-dimensional, nine-velocity lattice. All simulations are performed using a single relaxation time and a set of constants deduced from the derivation. It is found that the finite difference lattice Boltzmann method is able to correctly replicate viscous effects in thermal Couette flows,aeroacoustics,andshockstructures.Thesolutionsobtainedareidenticaleithertoanalyticalresults,or obtained by solving the compressible Navier–Stokes equations using a direct numerical simulation technique.

The stability structure for lattice Boltzmann schemes has been introduced in Banda et al. (2006) [16], Junk and Yong (2007) [14] to analyze the stability of numerical algorithms. The first purpose of this paper is to discuss the stability structure from the perspective of matrix analysis. Its second goal is to illustrate and apply the results to different classes of lattice Boltzmann collision operators. In particular we formulate an equivalence condition–just recently also reported in Yong (2008) [18]–that guarantees the existence of a pre-stability structure. It is then illustrated by several examples, how this equivalence condition can be effectively employed for the systematic verification and construction of stable collision operators. Finally, we point out some shortcomings of the stability structure approach arising in certain cases.

A finite difference lattice Boltzmann method of second-order accuracy in time is developed based on non-oscillatory scheme with no-free-parameter dissipation (NND) difference scheme in this paper. The NND lattice Boltzmann method is used to simulate high-speed flows by constructing a new equilibrium distribution function of the lattice Boltzmann method. Compared with a variation of lattice Boltzmann method developed by Qu, et al., the present method can capture shock waves and handle oscillations of high velocity flows accurately in larger time steps and in shorter computing time. Numerical results indicate the correctness and capability of simulating shock wave interactions of the NND lattice Boltzmann method.

In this work, we focus on fluid modeling and animation via lattice methods for computer games. We consider two approaches in this area: a Lattice Gas Cellular Automata (LGCA) technique and the Lattice Boltzmann (LBM) Method. LGCAs are discrete models based on point particles that move on a lattice according to suitable and simple rules in order to mimic a fully molecular dynamics. The LBM method is based on the fundamental idea of constructing simplified kinetic models, which incorporates the essential physics of microscopic processes so that the macroscopic averaged properties satisfy macroscopic equations. In this article, we present two animating frameworks based on the above-mentioned lattice methods. Both frameworks are composed of two models: (a) a 3D fluid animation technique; and (b) a GPU surface flow animation over terrain models. In the first framework, item (a) is performed through a LGCA, while in the second one we applied the LBM. In this work we highlight the advantages of these approaches for computer game applications. In the experimental results, we emphasize the simplicity and power of the presented models when combined with efficient techniques for rendering, and compare them in terms of efficiency.

Water management is the most important subject in the development of the Polymer Electrolyte Fuel Cells (PEFC) with high performance and high reliability. Experimental and three-dimensional numerical study has been done for the serpentine channel. Two kinds of the channel width, 2mm and 8mm, were selected. It is found that the rib of each separator plays as the coverage to reduce the escaping of the water content from the GDL to the channel.

Effect of water production on the performance of the fuel cell has been studied numerically. We have done two-dimensional simulations by using the Lattice Gas Automaton (LGA) method in order to analyze the gas flow and water distributions in the different type of flow passages. Three kinds of flow passage, that is, serpentine-type, straight-type and column-type were selected and their performances were compared. The results show that the serpentine-type flow is effective for the drainage and the electric generation performance and the electrode of hydrophilic surface is effective for reducing the plugging phenomenon. It is found that the LGA method is a promising tool for the analysis of the fuel cell performance.

A lattice Boltzmann method is developed for solute transport. Proper expressions for the local equilibrium distribution functions enable the method to be formulated on rectangular lattice with the same simple procedure as that on a square lattice. This provides an additional advantage over a lattice Boltzmann method on a square lattice for problems characterized by dominant phenomenon in one direction and relatively weak in another such as solute transport in shear flow over a narrow channel, where the problems can efficiently be approached with fine and coarse meshes, respectively, resulting in more efficient algorithm. The stability conditions are also described. The proposed method on a square lattice is naturally recovered when a square lattice is used. It is verified by solving four tests and compared with the analytical/exact solutions. They are in good agreement, demonstrating that the method is simple, accurate and robust for solute transport. Copyright © 2008 John Wiley & Sons, Ltd.

A lattice Boltzmann model with higher-order accuracy for the wave motion is proposed. The new model is based on the technique of the higher-order moment of equilibrium distribution functions and a series of lattice Boltzmann equations in different time scales. The forms of moments are derived from the binary wave equation by designing the higher-order dissipation and dispersion terms. The numerical results agree well with classical ones.

Les ecoulements isothermes dans des micro-canaux sont etudies par les automates cellulaires sur reseaux et par la methode de Boltzmann sur reseaux. La geometrie est bidimensionnelle. Pour simuler ce probleme, plusieurs modeles d'automates cellulaires sur reseaux (modeles a 6-, 7- et 19-vitesses), et la methode de Boltzmann sur reseaux a 13-vitesses sont utilises. Le champ des vitesses dans l'ecoulement ainsi que les distributions de pression et de densite sont presentes. Les profils de vitesse longitudinale sont compares a ceux obtenus par simulation directe de Monte Carlo (DSMC) et a ceux theoriques de l'ecoulement de Poiseuille-Hagen avec un glissement de vitesse de Maxwell sur les parois. La methode pour obtenir la vitesse de glissement par la methode de Boltzmann sur reseaux est precisee.

In this paper a novel and simple large-eddy-based lattice Boltzmann model is proposed to simulate two-dimensional turbulence. Unlike existing lattice Boltzmann models for turbulent flow simulation, which were based on primitive-variables Navier-Stokes equations, the target macroscopic equations of the present model are vorticity-streamfunction equations. Thanks to the intrinsic features of vorticity-streamfunction equations, the present model is efficient, stable and simple for two-dimensional turbulence simulation. The advantages of the present model are validated by numerical experiments.

We show that the Chapman–Enskog expansion can be viewed as a special instance of a general expansion procedure which also encompasses other methods like the regular error expansion and multi-scale techniques and that any two expansions which properly describe the lattice Boltzmann solution necessarily coincide up to higher order terms. For a model problem, both the regular error expansion and the Chapman–Enskog expansion are carried out. It turns out that the classical Chapman–Enskog method leads to an unstable equation at super-Burnett order in a parameter regime for which the underlying lattice Boltzmann algorithm is stable. However, our approach naturally allows us to consider variants of the super-Burnett equation which do not suffer from instabilities. The article concludes with a detailed comparison of the Chapman–Enskog and the regular error expansion.

In this paper, a new lattice Boltzmann equation which is independent of time is proposed. Based on the new lattice Boltzmann equation, some steady problems can be modeled by the lattice Boltzmann method. In the further study, the Laplace equation is investigated with the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different space scales. The numerical results show that the new method is effective.

We investigate the use of various equations of state (EOS) in the single-component multiphase lattice Boltzmann model. Several EOS are explored: van der Waals, Carnahan–Starling and Kaplun–Meshalkin EOS [A.B. Kaplun, A.B. Meshalkin, Thermodynamic validation of the form of unified equation of state for liquid and gas, High Temperature 41 (3) (2003) 319–326]. The last one was modified in order to obtain the correct critical point. The Carnahan–Starling and modified Kaplun–Meshalkin EOS are in better agreement with the experimental data on coexistence curves than the van der Waals EOS.It is shown that the approximation of the gradient of special potential is crucial to obtain the correct coexistence curve, especially its low-density part. The correct method of incorporating the body forces into the lattice Boltzmann model is also very important. We propose a new scheme which allows us to obtain large density ratio (up to 109 in the stationary case) and to reproduce the coexistence curve with high accuracy. The spurious currents at vapor–liquid interface are also greatly reduced.

The Galilean invariance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar–Gross–Krook model are proposed together with their rigorous theoretical background. From the viewpoint of group invariance, recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar–Gross–Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.

In this paper, a well-known finite difference technique is combined with thermal lattice Boltzmann method to solve 2-dimensional incompressible thermal fluid flow problems. A small number of microvelocity components are applied for the calculation of temperature field. The combination of finite difference with lattice Boltzmann method is found to be an efficient and stable approach for the simulation at high Rayleigh number of natural convection in a square cavity.

The Kermack–McKendrick susceptible-infected-recovered (SIR) model describes the dynamics of epidemics in a cumulative way. This contribution compares different approaches for introducing spatial patterns into these dynamics. The applied techniques cover lattice gas cellular automata (LGCA), stochastic cellular automata (SCA) and partial differential equations (PDE). Even though these methods involve distinct types of spatial interaction, it can be shown, that consistent qualitative and quantitative model behaviour can be obtained by means of parameter adaptions and slight technical modifications. These modifications are motivated by stochastic analysis of distributed interaction (PDE, SCA) and diffusion dynamics (LGCA) as well as prevailing physical analogies. The law of large numbers permits to approximate stochastic contacts by distributed interaction. Diffusion of particles can be approximated through empiric adjustment of a Gaussian diffusion distribution.

In this paper, a new method, higher-order moment lattice Boltzmann method for one and two-dimensional Burgers’ equation is proposed. The lattice Boltzmann models presented here are based on a series of lattice Boltzmann equations in different time scales. In order to achieve higher order accuracy, we use seven and four moments of the equilibrium distribution function in one and two-dimensional models respectively. We find two kinds of strategy to seek equilibrium distribution functions for the two-dimensional model with second order accuracy. These two are equivalent when a scale factor k = 2 3 . Lastly, we provide a fine numerical result of a one-dimensional Burgers’ equation. Numerical examples show the method can be used to simulate one and two-dimensional Burgers’ equation.

An implicit lattice Boltzmann equation to simulate the locally averaged flow behavior of disperse two-phase mixtures is presented. Using a multiscale expansion technique, it is shown that these equations reduce to the widely used two-fluid model for flows of such suspensions. The viability of the lattice Boltzmann approach is demonstrated through illustrative examples. The lattice Boltzmann scheme is easy to program and it parallelizes readily. It is suggested that this approach may serve as an attractive alternative to conventional approaches to solving the two-fluid model equations.

Recently, a general theory of constructing lattice Boltzmann models as an approximation to the Boltzmann equation has been introduced [S. Chikatamarla, I. Karlin, Phys. Rev. Lett. 97 (2006) 190601]. We extend this theory to two dimensions and identify a new complete Galilean invariant lattice Boltzmann model. A general theory of constructing discrete velocity sets in higher dimensions is presented. Further implementation and optimization techniques are discussed.

Discontinuous Galerkin schemes are introduced in an isothermal Lattice Boltzmann model in one dimension. Simulation of diffusion phenomena in a two-component system shows the reduction of the numerical diffusivity.

Combining an asymptotic analysis of the lattice Boltzmann method [M. Junk, Z. Yang, Asymptotic analysis of lattice Boltzmann boundary conditions, J. Stat. Phys. 121 (2005) 3–35] with the stability estimate presented in [M. Junk, W.-A. Yong, Weighted L2 stability of the lattice Boltzmann equation, Preprint], we are able to prove some strict convergence results. The proof applies to the lattice Boltzmann method with linear collision operator both in the case of periodic domains and bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.

When lattice Boltzmann methods are used to simulate fluid–structure interaction problems, they need to be coupled with additional routines to evaluate the boundary forces without destroying the efficiency and accuracy of the original method. We use the asymptotic expansion technique to analyze one such approach, the Momentum Exchange algorithm, investigating its properties in detail, whether it can be improved and in which cases it can be successfully used. A statement regarding the accuracy is presented, together with results of numerical tests which illustrate the theoretical considerations.

A simple lattice Boltzmann model is developed for two-dimensional combustion simulations. In this model the time step and the fluid particle speed can be adjusted dynamically, depending on the “particle characteristic temperature”. The algorithm is still a simple process of hopping from one grid point to the next, the same as the standard lattice Boltzmann method. Therefore the outstanding advantages of the standard lattice Boltzmann method are retained in this model besides better numerical stability. Excellent agreement between the present results and other numerical or experimental data shows that this scheme is an efficient numerical method for practical combustion simulations.

The central moments formalism of the cascaded lattice Boltzmann automaton is a natural approach to stabilize under-resolved unsteady high Reynolds number flow simulations. On the basis of the two absolute length scales of shear flow simulations which are the physical scale η (Kolmogorov scale) and the numerical scale h (grid spacing), it is argued that higher central moments of the local momentum distribution should go much faster to zero than the explicitly traced hydrodynamic moments. The lattice Boltzmann formalism allows one to adjust these higher moments without affecting the velocity field and its gradients in the same time step. It is shown that setting the higher central moments to zero in each time step stabilizes the model. As an example vortex shedding behind an obstacle which is only one grid spacing wide is shown.

We propose an implicit-explicit finite-difference lattice Boltzmann method for compressible flows in this work. The implicit-explicit Runge–Kutta scheme, which solves the relaxation term of the discrete velocity Boltzmann equation implicitly and other terms explicitly, is adopted for the time discretization. Owing to the characteristic of the collision invariants in the lattice Boltzmann method, the implicitness can be completely eliminated, and thus no iteration is needed in practice. In this fashion, problems (no matter stiff or not) can be integrated quickly with large Courant–Friedriche–Lewy numbers. As a result, with our implicit-explicit finite-difference scheme the computational convergence rate can be significantly improved compared with previous finite-difference and standard lattice Boltzmann methods. Numerical simulations of the Riemann problem, Taylor vortex flow, Couette flow, and oscillatory compressible flows with shock waves show that our implicit-explicit finite-difference lattice Boltzmann method is accurate and efficient. In addition, it is demonstrated that with the proposed scheme non-uniform meshes can also be implemented with ease.

Invariance of the one-dimensional lattice Boltzmann model is proposed together with its rigorous theoretical background. It is demonstrated that the symmetry inherent in Navier–Stokes equations is not really recovered in the one-dimensional lattice Boltzmann equation (LBE), especially for shock calculation. Symmetry breaking may be the inherent cause for the non-physical oscillations in the vicinity of the shock for LBE calculation.

A novel coupled lattice Boltzmann model is developed for two- and three-dimensional low Mach number combustion simulations, in which the fluid density can bear sharp changes. Different to the hybrid lattice Boltzmann scheme [O. Filippova, D. Hanel, A novel lattice BGK approach for low Mach number combustion, J. Comput. Phys. 158 (2000) 139], this scheme is strictly pure lattice Boltzmann style (i.e., we solve the flow, temperature, and concentration fields using the lattice Boltzmann method only); different to the non-coupled lattice Boltzmann scheme [K. Yamamoto, X. He, G.D. Doolen, Simulation of combustion field with lattice Boltzmann method, J. Stat. Phys. 107 (2002) 367], the fluid density in this model is coupled directly with the temperature. In this model the time step and the fluid particle speed can be adjusted dynamically, depending on the “particle characteristic temperature”. And the algorithm is still a simple process of hopping from one grid point to the next, the same as the standard lattice Boltzmann method. Therefore the outstanding advantages of the standard lattice Boltzmann method are retained in this model besides better numerical stability. Excellent agreement between the present results and other numerical or experimental data shows that this scheme is an efficient numerical method for practical combustion simulations.

The spin-1 Ising model with the nearest-neighbour bilinear and biquadratic interactions and single-ion anisotropy is simulated on a cellular automaton which improved from the Creutz cellular automaton (CCA) for a simple cubic lattice. The simulations have been made for several k=K/J and d=D/J in the 0

We introduce a lattice gas model for the merging of two single-lane automobile highways. The merging rules for traffic on the two lanes are deterministic, but the inflow on both lanes is stochastic. Analysing the stationary distribution of this stochastic cellular automaton, we find a discontinuous phase transition from a free-flow phase which depends on the initial state of the road to a jammed phase where all memory of the initial state is lost. Inside the jammed phase we identify dynamical phase transitions in the approach to stationarity. Each dynamical phase is characterized by a fixed number of relaxation cycles which is decreasing as one moves deeper into the jammed phase. In each cycle step, the number of 'desperate' drivers who force their way onto the main road when they reach the end of the on-ramp increases until stationarity.

The lattice Boltzmann model for the nonlinear Schrodinger equation is proposed. The new model is based on the technique of the higher order moment of equilibrium distribution functions and a series of lattice Boltzmann equations in different time scales. The Euler equations are derived from the nonlinear Schrodinger equation by removing non-physical pressure. We have simulated two irrotational flows. These numerical results agree well with classical ones.

We present a detailed algorithm of rotationally invariant lattice Boltzmann method (RILBM). The suggested approach overcomes discrete artifacts present in the standard lattice Bhatnagar, Gross, and Krook (LBGK) model by introducing a generalized particle collision operator in arbitrarily rotated frames. We demonstrate that the Navier-Stokes equations are exactly recovered through the Chapman-Enskog expansion, and present two computational cases that show independence of numerical results relative to the lattice orientation.

The paper presents the extension of the entropic lattice Boltzmann model beyond the hydrodynamic level. Quasi-equilibrium approach is introduced to capture the higher-order dynamics in a straightforward manner. Shear decay and micro-Couette flow are simulated which gives satisfactory agreement with expected results.