8.3.2 The Lattice Boltzmann Method

Problem with HPP and FHP: Numerical noise $\Longrightarrow$ coarse-graining needed to smooth out the results.
Reason: at each grid point, each discrete velocity $\mbox{$\bf c$}_{i}$ may be taken by just one or no particle.

Improvement: Relax the "zero or one" rule; use a floating point number to describe the degree to which each $(\mbox{$\bf r$},\mbox{$\bf c$})$-cell is filled.
$\Longrightarrow$Floating point arithmetic again, but digital noise reduced.

Let $f_{i}(\mbox{$\bf r$},t)$ be the density, at time $t$, at position $\mbox{$\bf r$}$ and velocity $\mbox{$\bf c$}_{i}$.
Let the velocity vectors point to each of the nearest neighbours on the lattice, with magnitudes such that after one time step ($\Delta t=1$) each particle arrives at that neighbouring site.

Example 1: 2D square lattice; 8 neighbour sites, one "rest" status ($i=0$) $\Longrightarrow$9 numbers $[f_{i}(\mbox{$\bf r$},t)\, , \; i=0,1, \dots 8]$ at each grid point.
Speeds: $\vert\mbox{$\bf c$}_{0}\vert=0$, $\vert\mbox{$\bf c$}_{i}\vert=1$ along the 4 grid axes, $\vert\mbox{$\bf c$}_{i}\vert=\sqrt{2}$ along the 4 diagonal directions.

Example 2: 2D hexagonal (FHP) lattice; 6 nearest neighbours, 1 rest particle.

Example 2: 3D models; invoke the method introduced by [D'HUMI`ERES] for lattice gas cellular automata (4D FCHC lattice plus down projection.)

Procedure: Translation and collision are included in the propagation formula

$$f_{i}(\mbox{$\bf r$}+\mbox{$\bf c$}_{i},t+1)= f_{i}(\mbox{$\bf r$},t) + \Delta_{i}(f)$$ (8.84)

where $\Delta_{i}(f)$ denotes the increase or decrease of $f_{i}$ due to the collision process.

Collision term: Originally, boolean operators were invoked as in the HPP and FHP schemes. Later the LB model was regarded as a representation of the NS equations, independent of the lattice gas model.([QUIAN 92,CHEN 91])
$\Longrightarrow$New approximations for the collision term: the single time relaxation expression

$$\Delta_{i}(f)=- \frac{f_{i}-f_{i}^{eq}}{\tau}$$ (8.85)

was found to be sufficient to reproduce Navier-Stokes dynamics. Here, $1/\tau$ is a relaxation rate, and $f_{i}^{eq}$ denotes an equilibrium distribution. For the 2-D hexagonal lattice this distribution is [QUIAN 95,CHEN 94]

$$f_{i=1 \dots 6}^{eq} = \frac{\rho}{12} +\frac{\rho}{3}\, \mbox{$\bf e$}_{i}\cdot\mbox{$\... ...e$}_{i}\cdot\mbox{$\bf v$}\right)^{2} -\frac{\rho}{6}\, \cdot\mbox{$\bf v$}^{2}$$ (8.86)

$$f_{i=0}^{eq} = \frac{\rho}{2} - \rho \mbox{$\bf v$}^{2}$$ (8.87)

where $\mbox{$\bf e$}_{i}$ is a unit vector along $\mbox{$\bf c$}_{i}$, and $\rho$ and $\mbox{$\bf v$}$ are the hydrodynamic density and flow velocity, respectively:

$$\rho(\mbox{$\bf r$},t) = \sum_{i} f_{i}(\mbox{$\bf r$},t) \;... ...bf r$},t) = \sum_{i} \mbox{$\bf c$}_{i}f_{i}(\mbox{$\bf r$},t)$$ (8.88)

Applications:
- Compressible and incompressible flow
- Basic studies of the dynamics of vortices
- Applied studies of turbulent channel flow
- Oil recovery from sandstone
- etc.

Recent survey: [QUIAN 95].