4.3. MODELING FLUID FLOW USING LATTICE GASES AND THE LATTICE BOLTZMANN MODEL
arriving at OtP + Oa (pu a) = 0, or
op
iit+V 'pu=o
(4·9)
which is just the equation of continuity that expresses conservation of mass
in a fluid. One can also first multiply (4.8) with ct and then sum over the
index i. In that case we arrive at
OtPUa + opiiap = 0
with
b
IIap = L CiaCipNi
;=1
The quantity IIap is the momentum density flux tensor and must be
interpreted as the flow of the a-component of the momentum into the f3direction. In order to proceed (i.e. express IIap in terms of p and u), one
must be able to find expressions for the particle densities N: This is a highly
technical matter that is described in detail in e.g, Rothman and Zaleski (1997)
and Chopard and Droz (1998). The bottom line is that one first calculates
the particle densities for a LGA in equilibrium, N?, and then substitutes
them into (4.11). This results in an equation that is almost similar to the
Euler equation, i.e, the expression of conservation of momentum for an
inviscid fluid. Next, one proceeds by taking into account small deviations
from equilibrium, resulting in viscous effects. Again, after a very technical
and lengthy derivation one is able to derive the particle densities, substitute
everything into (4.11) and derive the full expression for the momentum
conservation of the LGA, which again very closely resembles the NavierStokes equations for an incompressible fluid. The viscosity and sound speed
of the LGA are determined by its exact nature (i.e. the lattice, the interaction
list and the number of residing particles, and the exact definition of the
collision operator).
At first sight the average, macroscopic behavior of the LGA may come as
a surprise. The LGA-CA is a model that reduces a real fluid to one that consists
of particles with a very limited set of possible velocities, that live on the links
of a lattice, and all stream and collide at the same time. Yet, theoretical
analysis and a large body of simulation results show that, although the LGA
is indeed a very simple model, it certainly is a realistic model of a real fluid.
However, it is true that not all LGA behave as a real fluid. The underlying
lattice must have enough symmetry such that the resulting macroscopic
equations are isotropic, as in a real fluid. For instance, the first LGA, the socalled HPP model, which is defined on a two-dimensional square lattice with
only nearest-neighbor interactions and no rest particles, is not isotropic. The
FHP models, which have a two-dimensional hexagonal lattice, do possess
enough symmetry and their momentum conservation laws have the desired
isotropy property.
To end this section we stress once more that the LGA is an intrinsically
local CA and therefore gives us an inherently parallel model for fluid flow
simulations.
THE FHP MODEL. The FHP model, named after its discoverers Frisch,
Hasslacher, and Pomeau, was the first LGA with the correct (isotropic) hydrodynamic behavior. The FHP model is based on a two-dimensional hexagonal
\03
arriving at OtP + Oa (pu a) = 0, or
op
iit+V 'pu=o
(4·9)
which is just the equation of continuity that expresses conservation of mass
in a fluid. One can also first multiply (4.8) with ct and then sum over the
index i. In that case we arrive at
OtPUa + opiiap = 0
with
b
IIap = L CiaCipNi
;=1
The quantity IIap is the momentum density flux tensor and must be
interpreted as the flow of the a-component of the momentum into the f3direction. In order to proceed (i.e. express IIap in terms of p and u), one
must be able to find expressions for the particle densities N: This is a highly
technical matter that is described in detail in e.g, Rothman and Zaleski (1997)
and Chopard and Droz (1998). The bottom line is that one first calculates
the particle densities for a LGA in equilibrium, N?, and then substitutes
them into (4.11). This results in an equation that is almost similar to the
Euler equation, i.e, the expression of conservation of momentum for an
inviscid fluid. Next, one proceeds by taking into account small deviations
from equilibrium, resulting in viscous effects. Again, after a very technical
and lengthy derivation one is able to derive the particle densities, substitute
everything into (4.11) and derive the full expression for the momentum
conservation of the LGA, which again very closely resembles the NavierStokes equations for an incompressible fluid. The viscosity and sound speed
of the LGA are determined by its exact nature (i.e. the lattice, the interaction
list and the number of residing particles, and the exact definition of the
collision operator).
At first sight the average, macroscopic behavior of the LGA may come as
a surprise. The LGA-CA is a model that reduces a real fluid to one that consists
of particles with a very limited set of possible velocities, that live on the links
of a lattice, and all stream and collide at the same time. Yet, theoretical
analysis and a large body of simulation results show that, although the LGA
is indeed a very simple model, it certainly is a realistic model of a real fluid.
However, it is true that not all LGA behave as a real fluid. The underlying
lattice must have enough symmetry such that the resulting macroscopic
equations are isotropic, as in a real fluid. For instance, the first LGA, the socalled HPP model, which is defined on a two-dimensional square lattice with
only nearest-neighbor interactions and no rest particles, is not isotropic. The
FHP models, which have a two-dimensional hexagonal lattice, do possess
enough symmetry and their momentum conservation laws have the desired
isotropy property.
To end this section we stress once more that the LGA is an intrinsically
local CA and therefore gives us an inherently parallel model for fluid flow
simulations.
THE FHP MODEL. The FHP model, named after its discoverers Frisch,
Hasslacher, and Pomeau, was the first LGA with the correct (isotropic) hydrodynamic behavior. The FHP model is based on a two-dimensional hexagonal
\03