Interdisciplinary Applied Mathematics

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• In fact, a simple modification where the nondimensional collision frequency is taken as (2/3)8 instead of exactly equal to 8, which corresponds to the standard BGK model, improves the BGK predictions significantly.

15.5 Lattice-Boltzmann Method (LBM)

The lattice-Boltzmann method (LBM) offers potentially great advantages over conventional methods for simulating microflows. It represents a “min-

FIGURE 15.18. Analogue between the BBGKY hierarchy and its lattice counterpart. Adopted from (Succi, 2001).

imal”    form    of    the Boltzmann    equation    and can be    used    for    gas    or    liquid

as well as for particulate microflows. It can handle arbitrarily complex geometries, even random geometries, in a fairly straightforward way, and it seems to be particularly effective in the regime in which microdevices operate. This method solves a simplified Boltzmann equation on a discrete lattice. Because of its intrinsic kinetic nature it can also handle the high Knudsen number regime, and it is very effective for problems where both mesoscopic dynamics and microscopic statistics are important. However, initially there was only limited use of the method in microflows, and this may have to do with its originally intended use in simulating high Reynolds number flows. In the following, we review its origin and basic idea, we compare it with Navier-Stokes solutions, and finally we present flow simulation examples, including microflows.

There are three main theoretical developments: The first one took place in the mid 1980s leading to the lattice gas methods. The second one started in the early 1990s leading to the lattice Boltzmann equation. Finally, the third main development took place in the early 2000s leading to the entropic lattice Boltmann method. A schematic representation of these methods is shown in Figure 15.18, adopted from the book by (Succi, 2001). The left column shows the classical BBGKY (Bogoliubov-Born-Green-Kirwood-Yvon) hierarchy leading from atomistic to continuum flow equations. The right column shows the corresponding approximations in the framework of lattice methods. At the atomistic level we have a description by Newton’s

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