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8.3.2 The Lattice Boltzmann Method
Problem with HPP and FHP: Numerical noise
coarse-graining needed to smooth out the results.
Reason: at each grid point, each discrete velocity
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
-cell is filled.
Floating point arithmetic again, but digital noise reduced.
Let
be the density, at time , at position
and
velocity
.
Let the velocity vectors point to each of the nearest neighbours on the
lattice, with magnitudes such that after one time step
() each particle arrives at that neighbouring site.
Example 1: 2D square lattice; 8 neighbour sites, one "rest" status
()
9 numbers
at each grid point.
Speeds:
,
along the 4 grid axes,
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
|
(8.84) |
where denotes the increase or decrease of 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])
New approximations for the collision term:
the single time relaxation expression
|
(8.85) |
was found to be sufficient to reproduce Navier-Stokes dynamics.
Here, is a relaxation rate, and
denotes an equilibrium distribution.
For the 2-D hexagonal lattice this distribution is [QUIAN 95,CHEN 94]
where
is a unit vector along
, and
and
are the hydrodynamic density and
flow velocity, respectively:
|
(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].
Next: 8.4 Direct Simulation Monte
Up: 8.3 Lattice Gas Models
Previous: 8.3.1 Lattice Gas Cellular
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001