4.3. MODELING FLUID FLOW USING LATTICE GASES AND THE LATTICE BOLTZMANN MODEL
Even better, it turns out, again within the correct physical picture, that this
CA behaves like a real fluid (such as water) and therefore can be used as
a model for hydrodynamics. Furthermore, as the LGA rule is intrinsically
local (only nearest and next-nearest neighbor interactions) we can construct
an inherently parallel model for fluid flow.
ASSOCIATING PHYSICS WITH THE LGA-CA. Our current image of the LGACA is that of bits that first move from a cell to a neighboring cell and are then
reshuffled into another direction. Now we associate the bits in the state vector
with particles; a one-bit code for the presence of a particle, and a zero-bit
code for the absence of a particle. Assume that all particles are equal and
have a mass of 1. Step 1 in the LGA-CA is now interpreted as a streaming
of particles from one cell to another. If we also introduce a length scale, i.e.
a distance between the cells (usually the distance between nearest-neighbors
cells is taken as 1), and a time scale, i.e. a time duration for the streaming (i.e.
step 1 in the LGA-CA rule, usually a time step of 1 is assumed), then we are
able to define a velocity ci for each particle in direction i (i.e. the direction
associated with the i-th element of the state vector n) . Step 1 of the LGA-CA
is the streaming of particles with velocity ci from one cell to a neighboring
cell. The residing bits can be viewed as particles with a zero velocity, or rest
particles. Now we may imagine, as the particles meet in a cell, that they
collide. In this collision the velocity of the particles (i.e, both absolute speed
and direction) can be changed. The reshuffling of bits in step 2 of the LGA-CA
rule can be interpreted as a collision of particles.
In a real physical collision, mass, momentum, and energy are conserved.
Therefore, if we formulate the reshuffl ing such that these three conservation
laws are obeyed, we have constructed a true Lattice Gas Automaton, i.e.
a gas of particles that can have a small set of discrete velocities ci, moving
in lock-step over the links of a lattice (space is discretized) and such that all
collide with other particles arriving at a lattice point at the same time. In the
collisions, particles may be sent in other directions, in such a way that the
total mass and momentum in a lattice point is conserved.
We can now associate with each cell of the LGA-CA a density p and
momentum pu, with u the velocity of the gas:
b
p = LNi
(4.1)
i=l
b
pu = LCiNi
i=l
where N, = (ni), i.e. a statistical average of the Boolean variables; N, should
be interpreted as a particle density.
If we now let the LGA evaluate and calculate the density and momentum
as defined in (4.1) and (4.2), these quantities behave just as in a real fluid.
In Fig. 4.na and b we show an example of an LGA simulation of flow
around a cylinder. In Fig. 4.na we show the results of a single iteration of
the LGA, so in fact we have assumed that M = n.. Clearly, the resulting flow
field is very noisy. In order to arrive at smooth flow lines one should calculate
N, =(ni). Because the flow is static, we calculate N, by averag ing the Boolean
variables ni over a large number of LGAiterations or a patch of surrounding
cells. The resulting flow velocities are shown in Fig. 4.nb.
101
Even better, it turns out, again within the correct physical picture, that this
CA behaves like a real fluid (such as water) and therefore can be used as
a model for hydrodynamics. Furthermore, as the LGA rule is intrinsically
local (only nearest and next-nearest neighbor interactions) we can construct
an inherently parallel model for fluid flow.
ASSOCIATING PHYSICS WITH THE LGA-CA. Our current image of the LGACA is that of bits that first move from a cell to a neighboring cell and are then
reshuffled into another direction. Now we associate the bits in the state vector
with particles; a one-bit code for the presence of a particle, and a zero-bit
code for the absence of a particle. Assume that all particles are equal and
have a mass of 1. Step 1 in the LGA-CA is now interpreted as a streaming
of particles from one cell to another. If we also introduce a length scale, i.e.
a distance between the cells (usually the distance between nearest-neighbors
cells is taken as 1), and a time scale, i.e. a time duration for the streaming (i.e.
step 1 in the LGA-CA rule, usually a time step of 1 is assumed), then we are
able to define a velocity ci for each particle in direction i (i.e. the direction
associated with the i-th element of the state vector n) . Step 1 of the LGA-CA
is the streaming of particles with velocity ci from one cell to a neighboring
cell. The residing bits can be viewed as particles with a zero velocity, or rest
particles. Now we may imagine, as the particles meet in a cell, that they
collide. In this collision the velocity of the particles (i.e, both absolute speed
and direction) can be changed. The reshuffling of bits in step 2 of the LGA-CA
rule can be interpreted as a collision of particles.
In a real physical collision, mass, momentum, and energy are conserved.
Therefore, if we formulate the reshuffl ing such that these three conservation
laws are obeyed, we have constructed a true Lattice Gas Automaton, i.e.
a gas of particles that can have a small set of discrete velocities ci, moving
in lock-step over the links of a lattice (space is discretized) and such that all
collide with other particles arriving at a lattice point at the same time. In the
collisions, particles may be sent in other directions, in such a way that the
total mass and momentum in a lattice point is conserved.
We can now associate with each cell of the LGA-CA a density p and
momentum pu, with u the velocity of the gas:
b
p = LNi
(4.1)
i=l
b
pu = LCiNi
i=l
where N, = (ni), i.e. a statistical average of the Boolean variables; N, should
be interpreted as a particle density.
If we now let the LGA evaluate and calculate the density and momentum
as defined in (4.1) and (4.2), these quantities behave just as in a real fluid.
In Fig. 4.na and b we show an example of an LGA simulation of flow
around a cylinder. In Fig. 4.na we show the results of a single iteration of
the LGA, so in fact we have assumed that M = n.. Clearly, the resulting flow
field is very noisy. In order to arrive at smooth flow lines one should calculate
N, =(ni). Because the flow is static, we calculate N, by averag ing the Boolean
variables ni over a large number of LGAiterations or a patch of surrounding
cells. The resulting flow velocities are shown in Fig. 4.nb.
101