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Chapter 13: Lattice Boltzmann Method
translation. A simple solution is to invert the directions of all vectors on the wall
(for both the carrier fluid and each tracer); this approach is called the “bounce-back”
boundary, and it assures that there is a no-slip velocity boundary near the wall (Chen
etal., 1996). Section13.4willdiscusstheconsequencesofbounce-backinmoredetail.
If a solid boundary is placed at the ends of the automaton, it automatically
supercedes the wrap condition, because the solid wall deflects all outward-pointing
vectors back into the automaton.
Self Study Example:
Wewillsimulatedispersioninalongduct, withsquarecross-section, bythemethod
described in Section 13.2.2. The initial duct contains 24 nodes in the x-direction,
and 26 in each of the y- and z-directions, and is open in the x-direction, with the
other 4 sides closed off with solid walls, as shown below:

y x
z

Flow 24 lu

A x-direction body force is placed on the fluid, causing the fluid to accelerate
2
along the x-axis. The viscosity ν is 0.01 lu /ts, and the average speed is limited to
0.01 lu/ts. The initial calculation begins with one 19-vector carrier fluid, to calcu-
late the fluid velocity field. After 60000 ts, the memory from the flow calculation
is reclaimed, the flow field is “frozen in,” and the channel is cloned 47 additional
times in the x-direction. The resultant (48 × 24) × 26 × 26 channel is used for a
dispersion study with a single 6-vector tracer. A slug of tracer is injected into the
left end of the channel, and is followed as it disperses in the flow; this part of the
calculation proceeds for additional 60000 ts.
(1a) Are 60000 ts adequate to reach viscous equilibrium?
(1b) On your computer, the LB program processes a single 19-vector compo-
nent at a rate of 9 million site-updates per second (MUPS), and processes a single
6-vector component (in dispersion-only mode) at 35 MUPS. How many seconds
will be needed to complete the first (19-vector) and last (6-vector) parts of the
calculation?
(1c) The calculation is performed in single precision, and each vector is repre-
sented by a 32-bit (4 byte) floating-point number. How much memory is required
for the first (19-vector) and second (6-vector) parts of the calculation?