Although several thermal lattice Boltzmann models have been proposed, this method has not yet been shown to be able to describe nonisothermal fully compressible flows in a satisfactory manner, mostly due to the presence of important deviations from the advection-diffusion macroscopic equations and also due to numerical instabilities. In this context, this paper presents a linear stability analysis for some lattice Boltzmann models that were recently derived as discrete forms of the continuous Boltzmann equation [P. C. Philippi, L. A. Hegele, Jr., L. O. E. dos Santos, and R. Surmas, Phys. Rev. E 63, 056702 (2006)], in order to investigate the sources of instability and to find, for each model, the upper and lower limits for the macroscopic variables, between which it is possible to ensure a stable behavior. The results for two-dimensional (2D) lattices with 9, 17, 25, and 37 velocities indicate that increasing the order of approximation of the lattice Boltzmann equation enhances stability. Results are also presented for an athermal 2D nine-velocity model, the accuracy of which has been improved with respect to the standard D2Q9 model, by adding third-order terms in the lattice Boltzmann equation.

The velocity discretization is a critical step in deriving the lattice Boltzmann (LBE) from the Boltzmann equation. The velocity discretization problem was considered in a recent paper (Philippi et al., From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models, Physical Review E 73: 56702, 2006) following a new approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the Boltzmann equation. As a consequence, two-dimensional lattices and their respective equilibrium distributions were derived and discussed, considering the order of approximation that was required for the LBE. In the present work, a Chapman-Enskog (CE) analysis is performed for deriving the macroscopic transport equations for the mass, momentum and energy for these lattices. The problem of describing the transfer of energy in fluids is discussed in relation with the order of approximation of the LBE model. Simulation of temperature, pressure and velocity steps are also presented to validate the CE analysis.

In this work, we present a derivation for the lattice-Boltzmann equation directly from the linearized Boltzmann equation, combining the following main features: multiple relaxation times and thermodynamic consistency in the description of non isothermal compressible flows. The method presented here is based on the discretization of increasingly order kinetic models of the Boltzmann equation. Following a Gross-Jackson procedure, the linearized collision term is developed in Hermite polynomial tensors and the resulting infinite series is diagonalized after a chosen integer N, establishing the order of approximation of the collision term. The velocity space is discretized, in accordance with a quadrature method based on prescribed abscissas (Philippi et al., Phys. Rev E 73, 056702, 2006). The problem of describing the energy transfer is discussed, in relation with the order of approximation of a two relaxation-times lattice Boltzmann model. The velocity-step, temperature-step and the shock tube problems are investigated, adopting lattices with 37, 53 and 81 velocities.

In fluid mechanics, multicomponent fluid systems are generally treated either as homogeneous solutions or as completely immiscible parts of a multiphasic system. In immiscible systems, the main task in numerical simulations is to find the location of the interface evolving over time, driven by normal and tangential surface forces. The lattice-Boltzmann method (LBM), on the other hand, is based on a mesoscopic description of the multicomponent fluid systems, and appears to be a promising framework that can lead to realistic predictions of segregation in non-ideal mixtures of partially miscible fluids. In fact, the driving forces in segregation are of a molecular nature: there is competition between the intermolecular forces and the random thermal motion of the molecules. Since these microscopic mechanisms are not accessible from a macroscopic standpoint, the LBM can provide a bridge linking the microscopic and macroscopic domains. To this end, the first purpose of this article is to present the kinetic equations in their continuum forms for the description of the mixing and segregation processes in mixtures. This paper is limited to isothermal segregation; non-isothermal segregation was discussed by Philippi et al. (Phil. Trans. R. Soc., vol. 369, 2011, pp. 2292–2300). Discretization of the kinetic equations leads to evolution equations, written in LBM variables, directly amenable for numerical simulations. Here the dynamics of the kinetic model equations is demonstrated with numerical simulations of a spinodal decomposition problem with dissolution. Finally, some simplified versions of the kinetic equations suitable for immiscible flows are discussed.

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The influence of wettability on the residual fluid saturation is analyzed for homogeneous and heterogeneous porous systems. Several simulations under different wettability, flow rate, and heterogeneity conditions were carried out using a two-component lattice-Boltzmann method. The fluid flow driving force and initial conditions were imposed using a specific methodology that allows a clear distinction between the results obtained for immiscible displacement when the porous medium is initially saturated with one fluid (called primary) and when two fluids are filling the porous spaces (called secondary). The results show that the primary sweeping process is more effective when the displaced fluid is non-wetting. We observe that the heterogeneity has an important role for the whole process since it disturbs the fluid interfaces inducing the flow in the longitudinal and transversal directions, improving considerably the effectiveness of the primary displacement when compared with ideally homogeneous cases. We noted that for oil contact angles, θo, higher than a critical value, no residual oil is found. In all homogeneous cases, the critical value is 120°. The residual fluid increases proportionally to the capillary number for primary displacements, but it also depends on the system heterogeneity and wetting conditions. For secondary displacements in heterogeneous systems, the highest residual oil saturation is found for completely oil-wet conditions, with values ranging from 29% to 41% and tending to zero for all cases when θo > 120°. The initial water–oil distribution is found to be a determining factor in the amount of trapped oil after the waterflooding process.

The lattice Boltzmann (LB) method, based on mesoscopic modeling of transport phenomena, appears to be an attractive alternative for the simulation of complex fluid flows. Examples of such complex dynamics are multiphase and multicomponent flows for which several LB models have already been proposed. However, due to theoretical or numerical reasons, some of these models may require application of high-order lattice-Boltzmann equations (LBEs) and stencils. Here, we will present a derivation of LBEs from the discrete-velocity Boltzmann equation (DVBE). By using the method of characteristics, high-order accurate equations are conveniently formulated with standard numerical methods for ordinary differential equations (ODEs). In particular, we will derive implicit LB schemes due to their stability properties. A simple algorithm is presented which enables implementation of the implicit schemes without resorting to, e.g. change of variables. Finally, some numerical experiments with high-order equations and stencils together with two specific multiphase models are presented.

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