Analytical solutions of the lattice Boltzmann BGK model

Analytical solutions of the two dimensional triangular and square lattice Boltzmann BGK models have been obtained for the plain Poiseuille flow and the plain Couette flow. The analytical solutions are written in terms of the characteristic velocity of the flow, the single relaxation time $\tau$ and the lattice spacing. The analytic solutions are the exact representation of these two flows without any approximation.Comment: 10 pages, no postscript figure provide

Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media

Numerical micropermeametry is performed on three dimensional porous samples having a linear size of approximately 3 mm and a resolution of 7.5 $\mu$m. One of the samples is a microtomographic image of Fontainebleau sandstone. Two of the samples are stochastic reconstructions with the same porosity, specific surface area, and two-point correlation function as the Fontainebleau sample. The fourth sample is a physical model which mimics the processes of sedimentation, compaction and diagenesis of Fontainebleau sandstone. The permeabilities of these samples are determined by numerically solving at low Reynolds numbers the appropriate Stokes equations in the pore spaces of the samples. The physical diagenesis model appears to reproduce the permeability of the real sandstone sample quite accurately, while the permeabilities of the stochastic reconstructions deviate from the latter by at least an order of magnitude. This finding confirms earlier qualitative predictions based on local porosity theory. Two numerical algorithms were used in these simulations. One is based on the lattice-Boltzmann method, and the other on conventional finite-difference techniques. The accuracy of these two methods is discussed and compared, also with experiment.Comment: to appear in: Phys.Rev.E (2002), 32 pages, Latex, 1 Figur

Simulating Three-Dimensional Hydrodynamics on a Cellular-Automata Machine

We demonstrate how three-dimensional fluid flow simulations can be carried out on the Cellular Automata Machine 8 (CAM-8), a special-purpose computer for cellular-automata computations. The principal algorithmic innovation is the use of a lattice-gas model with a 16-bit collision operator that is specially adapted to the machine architecture. It is shown how the collision rules can be optimized to obtain a low viscosity of the fluid. Predictions of the viscosity based on a Boltzmann approximation agree well with measurements of the viscosity made on CAM-8. Several test simulations of flows in simple geometries -- channels, pipes, and a cubic array of spheres -- are carried out. Measurements of average flux in these geometries compare well with theoretical predictions.Comment: 19 pages, REVTeX and epsf macros require

Boundary flow condition analysis for the three-dimensional lattice Boltzmann model

In the continuum limit, the velocity of a Newtonian fluid should vanish at a solid wall. This condition is studied for the FCHC lattice Boltzmann model with rest particles. This goal is achieved by expanding the mean populations up to the second order in terms of the ratio $\varepsilon$ between the lattice unit and a characteristic overall size of the medium. This expansion is applied to two extreme flow situations. In Poiseuille flow, the second eigenvalue of the collision matrix can be chosen so that velocity vanishes at the solid walls with errors smaller than $\varepsilon^2$ ; however the choice depends on the angle between the channel walls and the axes of the lattice. In a plane stagnation flow, the tangential and normal velocities do not vanish at the same point, except for particular choices of the parameters of the model ; this point does not coincide with the solid wall. It is concluded that the boundary conditions are as a matter of fact imposed with errors of second order

An accurate curved boundary treatment in the lattice Boltzmann method

The lattice Boltzmann equation (LBE) is an alternative kinetic method capable of solving hydrodynamics for various systems. Major advantages of the method are due to the fact that the solution for the particle distribution functions is explicit, easy to implement, and natural to parallelize. Because the method often uses uniform regular Cartesian lattices in space, curved boundaries are often approximated by a series of stairs that leads to reduction in computational accuracy. In this work, a second-order accurate treatment of the boundary condition in the LBE method is developed for a curved boundary. The proposed treatment of the curved boundaries is an improvement of a scheme due to O. Filippova and D. Hänel (1998, J. Comput. Phys. 147, 219). The proposed treatment for curved boundaries is tested against several flow problems: 2-D channel flows with constant and oscillating pressure gradients for which analytic solutions are known, flow due to an impulsively started wall, lid-driven square cavity flow, and uniform flow over a column of circular cylinders. The second-order accuracy is observed with a solid boundary arbitrarily placed between lattice nodes. The proposed boundary condition has well-behaved stability characteristics when the relaxation time is close to 1/2, the zero limit of viscosity. The improvement can make a substantial contribution toward simulating practical fluid flow problems using the lattice Boltzmann method. c ○ 1999 Academic Press I