International Journal of Modern Physics C

We present a comparison between the finite-element and the lattice-Boltzmann method for simulating fluid flow in a SMRX static mixer reactor. The SMRX static mixer is a piece of equipment with excellent mixing performance and it is used in highly efficient chemical reactors for viscous systems like polymers. The complex geometry of this mixer makes such 3D simulations nontrivial. An excellent agreement between the results of the two simulation methods and experimental data was found.

Results for time-dependent, viscous, incompressible flows were investigated using the lattice-Boltzmann (BGK) automata. The decay of a synthetic turbulent flow field and the time evolution of an initial vortex were simulated for validation purposes. The focal point was the investigation of the instationary flow around a square obstacle in a two-dimensional channel for a range of Reynolds numbers between 80 and 300 and a blockage ratio of 0.125. The Strouhal number was measured for this case and found to be in the range of data given in the literature.

After a short discussion of recent discretization techniques for the lattice-Boltzmann equations we motivate and discuss some alternative approaches using implicit, nonuniform FD discretization and mesh refinement techniques. After presenting results of a stability analysis we use an implicit approach to simulate a boundary layer test problem. The numerical results compare well to the reference solution when using strongly refined meshes. Some basic ideas for a nonuniform mesh refinement (with non-cartesian mesh topology) are introduced using the standard discretization procedure of alternating collision and propagation.

The Lattice-Boltzmann method (LBM) is extended to allow incorporation of traditional turbulence models. Implementation of a two-layer mixing-length algebraic model and two versions of the k-ε two-equation model, Standard and RNG, in conjunction with a wall model, are presented. Validation studies are done for turbulent flows in a straight pipe at three Re numbers and over a backwards facing step of expansion ratio 1.5 and Re_H=44 000. All models produce good agreement with experiment for the straight pipes but the RNG k-ε model is best able to capture both the recirculation length, within 2% of experiment, and the detailed structure of the mean fluid flow for the backwards facing step.

In this paper we analyze and compare the lattice-Boltzmann equation with the beam scheme in detail. We notice the similarity and differences between the lattice Boltzmann equation and the beam scheme. We show that the accuracy of the lattice-Boltzmann equation is indeed second order in space. We discuss the advantages and limitations of the lattice-Boltzmann equation and the beam scheme. Based on our analysis, we propose an improved multi-dimensional beam scheme.

The lattice-Boltzmann (LB) method is applied to complex, moving geometries in which computational cells are partially filled with fluid. The LB algorithm is modified to include a term that depends on the percentage of the cell saturated with fluid. The method is useful for modeling suspended obstacles that do not conform to the grid. Another application is to simulations of flow through reconstructed media that are not easily segmented into solid and liquid regions. A detailed comparison is made with FIDAP simulation results for the flow about a periodic line of cylinders in a channel at a non-zero Reynolds number. Two cases are examined. In the first simulation, the cylinders are given a constant velocity along the axis of the channel, and the steady solution is acquired. The transient behavior of the system is then studied by giving the cylinders an oscillatory velocity. For both steady and oscillatory flows, the method provides excellent agreement with FIDAP simulation results, even at locations close to the surface of a cylinder. In contrast to step-like solutions produced using the "bounce-back" condition, the proposed condition gives close agreement with the smooth FIDAP predictions. Computed drag forces with the proposed condition exhibit apparent quadratic convergence with grid refinement rather than the linear convergence exhibited by other LB boundary conditions.

A lattice-gas Automata two-dimensional program was developed for analysis of single and two-phase flow behaviors, to support the development of integrated software modules for Nuclear Power Plant mechanistic simulations. The program has single-color, which includes FHP I, II, and III models, two-color (Immiscible lattice gas), and two-velocity methods including a gravity effect model. Parameter surveys have been performed for Karman vortex street, two-phase separation for understanding flow regimes, and natural circulation flow for demonstrating passive reactor safety due to the chimney structure vessel. In addition, lattice-Boltzmann Equation two-dimensional programs were also developed. For analyzing single-phase flow behavior, a lattice-Boltzmann-BGK program was developed, which has multi-block treatments. A Finite Differential lattice-Boltzmann Equation program of parallelized version was introduced to analyze boiling two-phase flow behaviors. Parameter surveys have been performed for backward facing flow, Karman vortex street, bent piping flow with/without obstacles for piping system applications, flow in the porous media for demonstrating porous debris coolability, Couette flow, and spinodal decomposition to understand basic phase separation mechanisms. Parallelization was completed by using a domain decomposition method for all of the programs. An increase in calculation speed of at least 25 times, by parallel processing on 32 processors, demonstrated high parallelization efficiency. Application fields for microscopic model simulation to hypothetical severe conditions in large plants were also discussed.

Cellular automata (CA) and lattice-Boltzmann (LB) models are two possible approaches to simulate fluid-like systems. CA models keep track of the many-body correlations and provide a description of the fluctuations. However, they lead to a noisy dynamics and impose strong restrictions on the possible viscosity values. On the other hand, LB models are numerically more efficient and offer much more flexibility to adjust the fluid parameters, but they neglect fluctuations. Here we discuss a multiparticle lattice model which reconciles both approaches. Our method is tested on Poiseuille flows and on the problem of ballistic annihilation in two dimensions for which the fluctuations are known to play an important role.

A thermohydrodynamic lattice-BGK model for the ideal gas was derived by Alexander et al. in 1993, and generalized by McNamara et al. in the same year. In these works, particular forms for the equilibrium distribution function and the transport coefficients were posited and shown to work, thereby establishing the sufficiency of the model. In this paper, we rederive the model from a minimal set of assumptions, and thereby show that the forms assumed for the shear and bulk viscosities are also necessary, but that the form assumed for the thermal conductivity is not. We derive the most general form allowable for the thermal conductivity, and the concomitant generalization of the equilibrium distribution. In this way, we show that it is possible to achieve variable (albeit density-dependent) Prandtl number even within a single-relaxation-time lattice-BGK model. We accomplish this by demanding analyticity of the third moments and traces of the fourth moments of the equilibrium distribution function. The method of derivation demonstrates that certain undesirable features of the model — such as the unphysical dependence of the viscosity coefficients on temperature — cannot be corrected within the scope of lattice-BGK models with constant relaxation time.

The progress and challenges in thermal lattice-Boltzmann modeling are discussed. In particular, momentum and energy closures schemes are contrasted. Higher order symmetric (but no longer space filling) velocity lattices are constructed for both 2D and 3D flows and shown to have superior stability properties to the standard (but lower) symmetry lattices. While this decouples the velocity lattice from the spatial grid, the interpolation required following free-streaming is just 1D. The connection between fixed lattice vectors and temperature-dependent lattice vectors (obtained in the Gauss–Hermite quadrature approach) is discussed. Some (compressible) Rayleigh–Benard simulations on the 2D octagonal lattice are presented for extended BGK collision operators that allow for arbitrary Prandtl numbers.

The model we have adopted is called "Hydrodynamic Lattice Gases", that is, a kind of lattice gas with continuous velocity. This model describes the conservation of energy naturally, using innovative collision and propagation schemes. As a result, operations of particles on the boundary become more flexible and complicated as compared with those of the FHP model. And we need to pay attention to the way of treating individual particles. Focusing on the velocity and the coordinates of particles that will return from the boundary to the system, we introduce a reasonable operation that can set temperature and flow velocity of any value required as boundary conditions. Numerical calculations show that those parameters are well controlled using a new boundary scheme.

Lattice-Boltzmann models, proposed at the end of the 1980s as the noise-free version of lattice-gas models are based on gas-kinetical-like representation of fluid flow. Their recent modifications, the lattice-BGK models, provide especially simple, effective and stable algorithms for the solution of incompressible flows. The boundary-fitting concept and local grid refinement proposed for the lattice-BGK model conserve the second order accuracy of the original algorithm for flows around complicated geometries in regions of small and moderate Reynolds numbers.

We describe a novel way based on lattice-Boltzmann representation for realizing hydrodynamic boundary conditions at a solid surface. It is shown that using this approach the resulting physics properties are independent of the position and the orientation of the surface with respect to the lattice mesh. The fluxes of mass, energy as well as both normal and tangential momenta can be accurately controlled to correspond to various fluid dynamics situations.

In this work we present the entropy functionals for dense gases that results from a maximum entropy procedure, for two levels of description. In analogy to this work, we review the case of dense mixtures. The resulting entropy functionals serve as a starting point to derive in a systematic manner, Fokker–Planck equations to model interacting Brownian particles.

The upper bound for the entropy production of a dense gas is presented and its connection with the so-called Fisher information measure is revised. The relation with dissipative particle dynamics is analyzed, which serve to identify the thermostat for the fluid system. The analogy with a turbulence approach, a probabilistic approach to dynamical sytems, detailed balance for dissipative particle dynamics and fluctuation dissipation theorems for relaxing systems is presented.

Dissipative Particle Dynamics (DPD) simulations were performed to investigate resolution or "coarse graining" effects on the simulation results. Fluid flow through a periodic array of spheres has been studied as a model for fluid filtration into a porous medium. In our model system, it appears that quantitatively correct results for the dimensionless drag can be obtained for relatively small system sizes. For higher solid volume fractions, it is necessary to increase the system size to avoid finite size and resolution effects. Simulations of colloidal spheres suspended in a DPD fluid show effective attraction between the large colloid particles, causing depletion aggregation. This effect may be expected as a consequence of the coarse-grained nature of the DPD fluid. By imposing a steady shear rate the aggregation can be suppressed. The results show that for dilute suspensions, the Brownian noise in the particle interactions causes an effective colloid polydispersity, which suppresses aggregation effects.

We present a model for treating solid boundaries of a DPD fluid. The basic idea is to model the stick boundary conditions by assuming that a layer of DPD particles is stuck on the boundary. By taking a continuum limit of this layer effective dissipative and stochastic forces on the fluid DPD particles are obtained. The boundary model is tested by a simulation of planar Couette flow which allows the performance of vicosimetric measurements. We analyze the conditions that ensure a proper stick boundary condition for an impenetrable wall, comparing with previous methods used.

We study, by means of numerical simulations, the model of dissipative particle dynamics with energy conservation for the simple case of thermal conduction. It is shown that the model displays correct equilibrium fluctuations and reproduces Fourier law. The connection between "mesoscopic coarse-graining" and "resolution" is clarified.

Spatial fluctuations in dissipative systems, such as rapid granular flows, behave very differently from those in elastic fluids. Fluctuations in the flow field drive the linear and nonlinear instability in the density field (clustering), while vortex structures appear and grow through the mechanism of noise reduction. The dynamics of the flow field on the largest space and time scales is described by diffusion equations with different diffusivities for the transverse and longitudinal flow fields. The results are obtained from analytic and simulation methods.

A model for the flow of fluids with particles in them is introduced. It relies on the coupling of discretely represented particles and a fluid pressure field that is given by a continuum description — a local Darcy law. In order to validate the qualitative behavior of the model, it is applied to simulate two different sedimentation processes. These processes include an initially homogeneous sedimenting bed of particles and a fluid bubble rising through a sedimenting field of particles. Agreement with experiments is observed on the qualitative level.

Two-phase continuum descriptions of the dynamical behavior of particulate suspensions require, among others, the formulation of a "local drag law". Such a "law" specifies the mean force f_l on particles as a function of averaged local properties, most notably, the mean difference velocity of particles and fluid and the local volume fraction Φ_l. The subscript l shall indicate the dependence of these quantities on the size l of the averaging cell. We study f_l by direct numerical simulation, solving the incompressible Navier–Stokes equation on a fixed, regular grid on a scale much smaller than the particle diameter. The particle–fluid interaction is computed by a method similar to the one proposed in [Fogelson and Peskin J. Comp. Phys.79, 50 (1988)]. We find a relation similar to the law of Richardson & Zaki, , which relates the local phase difference velocity to the local volume fraction of the particles.

We use lattice-Boltzmann simulations to examine late stage spinodal decomposition in binary fluids, showing that the scaling hypothesis for phase ordering does not hold in all cases. We also examine the effects of an applied shear on the coarsening of the spinodal pattern.

A model with a volumetric stress tensor added to the Navier–Stokes Equation is used to study two-phase fluid flows. The implementation of such an interface model into the lattice-Boltzmann equation is derived from the continuous Boltzmann BGK equation with an external force term, by using the discrete coordinate method. Numerical simulations are carried out for phase separation and "dam breaking" phenomena.

Lattice-Boltzmann (LB) models provide a systematic formulation of effective-field computational approaches to the calculation of multiphase flow by replacing the mathematical surface of separation between the vapor and liquid with a thin transition region, across which all magnitudes change continuously. Many existing multiphase models of this sort do not satisfy the rigorous hydrodynamic constitutive laws. Here, we extend the two-dimensional, seven-speed Swift et al. LB model¹ to rectangular grids (nine speeds) by using symbolic manipulation (Mathematica^TM) and compare the LB model predictions with benchmark problems, in order to evaluate its merits. Particular emphasis is placed on the stress tensor formulation. Comparison with the two-phase analogue of the Couette flow and with a flow involving shear and advection of a droplet surrounded by its vapor reveals that additional terms have to be introduced in the definition of the stress tensor in order to satisfy the Navier–Stokes equation in regions of high density gradients. The use of Mathematica obviates many of the difficulties with the calculations "by-hand," allowing at the same time more flexibility to the computational analyst to experiment with geometrical and physical parameters of the formulation.

A generalization of mesoscopic Lattice-Boltzmann models aimed at describing flows with solid/liquid phase transitions is presented. It exhibits lower computational costs with respect to the numerical schemes resulting from differential models. Moreover it is suitable to describe chaotic motions in the mushy zone.

A lattice gas with long-range interaction can simulate phase separation in the system consisting of one kind of component particle like the liquid-vapor theory of van der Waals. The generated phases are distinguished from each other by their particle density. In lattice-gas fluid with long-range interaction, the phase with high density can be observed in the phase with low density like the droplet in vapor. In this paper, the surface of the droplet in lattice-gas fluid with the long-range interaction is determined from the local density and its position is compared with that of Gibbs's dividing surface. The inside region and the outside region of the droplet are defined on the basis of the mean free path in each region. The surface tension is calculated through Laplace's formula using the droplet radius and the pressures in both regions. It is shown that the surface thickness becomes 4r where r is the distance of the long-range interaction.

We investigate a new method for simulating polymer-solvent systems which combines a lattice-Boltzmann approach for the fluid with a continuum molecular dynamics (MD) model for the polymer chain. The two parts are coupled by a friction force which is proportional to the difference of the monomer velocity and the fluid velocity at the monomer's position. The strength of the coupling can be tuned by a friction coefficient. Using this approach we examine the dynamics of one monomer immersed in the fluid, and by adding fluctuations to the fluid and the monomer, also the velocity autocorrelation function of one monomer. This results in the definition of an effective friction coefficient for the dynamics of the monomer. Furthermore we analyze the mapping of the model to an MD simulation, allowing us to compare results obtained using the new method with MD.

For the simulation of low Mach number reactive flows with significant density changes a modified lattice-BGK model in combination with the conventional convective-diffusion solvers for equations of temperature and species is proposed. Together with boundary-fitting conditions and local grid refinement the scheme enables the accurate consideration of low Mach number combustion in complex geometry as the flows around porous burners or droplets combustion.

We study model electrorheological (ER) fluids which consist of three material components in an attempt to explain recent experimental results, in which the ER effects can be promoted by adding some water. At low water concentration, the droplets tend to aggregate on the surfaces of the dispersed particles, forming coated microspheres. The ER effects are analyzed via spectral representation, and the experimental conditions for optimal ER responses are obtained. At low frequencies, it is found that by tuning the thickness and dielectric properties of the coating materials, it is possible to enhance the effective dielectric constant, thereby increasing the applicability of the ER fluids.

We study a single self-avoiding hydrophilic hydrophobic polymer chain, through Monte Carlo lattice simulations. The affinity of monomer i for water is characterized by a (scalar) charge λ_i, and the monomer-water interaction is short-ranged. Assuming incompressibility yields an effective short ranged interaction between monomer pairs (i,j), proportional to (λ_i + λ_j). In this article, we take λ_i=+1 (resp. (λ_i=-1)) for hydrophilic (resp. hydrophobic) monomers and consider a chain with (i) an equal number of hydrophilic and -phobic monomers (ii) a periodic distribution of the λ_i along the chain, with periodicity 2p. This model may be of interest in various situations (protein folding, polysoaps, …) The simulations are done on the square lattice (d=2), for various chain lengths N. There is a critical value (p_c(N)~0.07N) of the periodicity, which distinguishes between different low temperature structures. For p>p_c, the ground state corresponds to a macroscopic phase separation between a dense hydrophobic core and hydrophilic loops. For p<p_c (but not too small), one gets a microscopic (finite scale) phase separation, and the ground state corresponds to a chain or network of hydrophobic droplets, coated by hydrophilic monomers. These different cases will be explored through a Multiple Markov chain method. The results for the d=3 case (where p_c(N)~N^1/3) are similar.

We show how a lattice-Boltzmann approach can be extended to ternary fluid mixtures with the aim of modeling the diverse behavior of oil–water-surfactant systems. We model the mixture using a Ginzburg–Landau free energy with two scalar order parameters which allows us to define a lattice-Boltzmann scheme in the spirit of the Cahn–Hilliard approach to nonequilibrium dynamics. Results are presented for the spontaneous emulsification of an oil–water droplet and for spinodal decomposition in the presence of a surfactant.

We develop our existing two-dimensional lattice-gas model to simulate the flow of single phase, binary immiscible and ternary amphiphilic fluids. This involves the inclusion of fixed obstacles on the lattice, together with the inclusion of "no-slip" boundary conditions. Here we report on preliminary applications of this model to the flow of such fluids within model porous media. We also construct fluid invasion boundary conditions, and the effects of invading aqueous solutions of surfactant on oil-saturated rock during imbibition and drainage are described.

An extension to the basic lattice-BGK algorithm is presented for modeling a simulation region as a porous medium. The method recovers flow through a resistance field with arbitrary values of the resistance tensor components. Corrections to a previous algorithm are identified. Simple validation tests are performed which verify the accuracy of the method, and demonstrate that inertial effects give a deviation from Darcy's law for nominal simulation velocities.

Several results of lattice-gas and lattice-Boltzmann simulations of single-fluid flow in 2D and 3D porous media are discussed. Simulation results for the tortuosity, effective porosity and permeability of a 2D random porous medium are reported. A modified Kozeny–Carman law is suggested, which includes the concept of effective porosity. This law is found to fit well the simulated 2D permeabilities. The results for fluid flow through large 3D random fibre webs are also presented. The simulated permeabilities of these webs are found to be in good agreement with experimental data. The simulations also confirm that, for this kind of materials, permeability depends exponentially on porosity over a large porosity range.

Pore-scale simulations of fluid flow and mass transport offer a direct means to reproduce and verify laboratory measurements in porous media. We have compared lattice-Boltzmann (LB) flow simulations with the results of NMR spectroscopy from several published flow experiments. Although there is qualitative agreement, the differences highlight numerical and experimental issues, including the rate of spatial convergence, and the effect of signal attenuation near solid surfaces. For the range of Reynolds numbers relevant to groundwater investigations, the normalized distribution of fluid velocities in random sphere packings collapse onto a single curve, when scaled with the mean velocity. Random-walk particle simulations in the LB flow fields have also been performed to study the dispersion of an ideal tracer. These simulations show an encouraging degree of quantitative agreement with published NMR measurements of hydrodynamic and molecular dispersion, and the simulated dispersivities scale in accordance with published experimental and theoretical results for the Peclet number rangek 1 ≤Pe≤1500. Experience with the random-walk method indicates that the mean properties of conservative transport, such as the first and second moments of the particle displacement distribution, can be estimated with a number of particles comparable to the spatial discretization of the velocity field. However, the accurate approximation of local concentrations, at a resolution comparable to that of the velocity field, requires significantly more particles. This requirement presents a significant computational burden and hence a numerical challenge to the simulation of non-conservative transport processes.

The phenomenon of clogging in depth filtration is investigated, in which a dirty fluid is "cleaned" by the trapping of dirt particles within the pore space during flow through a porous medium. This gives rise to a self-generated gradient percolation process which exhibits a power law distribution for the density of trapped particles at downstream distance x from the input, both in idealized and lattice networks. Implications for efficient filter design are also mentioned.

In the presence of buoyancy, multiple diffusion coefficients, and porous media, the dispersion of solutes can be remarkably complex. The lattice-Boltzmann (LB) method is ideal for modeling dispersion in flow through complex geometries; yet, LB models of solute fingers or slugs can suffer from peculiar numerical conditions (e.g., denormal generation) that degrade computational performance by factors of 6 or more. Simple code optimizations recover performance and yield simulation rates up to ~3 million site updates per second on inexpensive, single-CPU systems. Two examples illustrate limits of the methods: (1) Dispersion of solute in a thin duct is often approximated with dispersion between infinite parallel plates. However, Doshi, Daiya and Gill (DDG) showed that for a smooth-walled duct, this approximation is in error by a factor of ~8. But in the presence of wall roughness (found in all real fractures), the DDG phenomenon can be diminished. (2) Double-diffusive convection drives "salt-fingering", a process for mixing of fresh-cold and warm-salty waters in many coastal regions. Fingering experiments are typically performed in Hele-Shaw cells, and can be modeled with the 2D (pseudo-3D) LB method with velocity-proportional drag forces. However, the 2D models cannot capture Taylor–Aris dispersion from the cell walls. We compare 2D and true 3D fingering models against observations from laboratory experiments.

A class of cellular automata is constructed on the basis of a set of simple rules: virtual particles obeying a full exclusion principle are subject to probabilistic dynamics with nonlocal (attractive and repulsive) interactions, and with momentum transfer memory effects. Boundary conditions can be periodic or can be set to simulate open systems, with various geometries. This model system possesses the essential elements to simulate the formation of surface aggregates. A specific application is presented for the formation and the dynamics of islands of hydroxyl groups in the catalytic reduction of NO on a Pt surface. The comparison between the simulation results and the experimental observations by Field Ion Microscopy suggests possible mechanisms for the so-far unexplained formation and dynamics of hydroxyl islands during the NO reduction processes.

The lattice-Boltzmann method (LBM) with enhanced collisions is used to calculate the flow and the temperature in an equipment used for growing GaPO₄ crystals from solution. For this purpose the lattice-Boltzmann method is extended to handle blocks with different physical properties. The results of our numerical calculations for the velocities and the temperatures in the solution are in reasonable agreement with recent experimental results.

A lattice formulation of nonrelativistic quantum mechanics is presented, based on a formal analogy with discrete kinetic theory. The method is applied to the Gross–Pitaevski equation, a specific form of self-interacting nonlinear Schrödinger equation relevant to the study of Bose–Einstein condensation.

We present a quantum lattice-gas model for a quantum computer operating with continual wavefunction collapse; entanglement of the wavefunction occurs locally over small spatial regions between nearby qubits for only a short time period. The quantum lattice-gas is a noiseless method that directly models the lattice-gas particle dynamics at the mesoscopic scale. The system behaves like a viscous Navier–Stokes fluid. Numerical simulations indicate the viscosity of the quantum lattice-gas fluid is lower than its classical lattice-gas counterpart's.

Conventional lattice-gas automata consist of particles moving discretely on a fixed lattice. While such models have been quite successful for a variety of fluid flow problems, there are other systems, e.g., flow in a flexible membrane or chemical self-assembly, in which the geometry is dynamical and coupled to the particle flow. Systems of this type seem to call for lattice gas models with dynamical geometry. We construct such a model on one-dimensional (periodic) lattices and describe some simulations illustrating its nonequilibrium dynamics.