4.3. MODELING FLUID FLOW USING LATTICE GASES AND THE LATTICE BOLTZMANN MODEL
Equation (4.7). This means that particle densities stream from cell to cell,
and particle densities collide. This immediately solves the problem of noisy
dynamics. In a strict sense we no longer have a CA with a Boolean state
vector, but rather we can view LBM as a generalized CA. Bya clever choice of
the collision operator, the model becomes isotropic and Galilean invariant,
thus solving the second problem of LGA. Actually, a very simple collision
operator is introduced, namel y the so-called BGK collision operator which
models the collisions as a single-time relaxation towards a local equilibrium
distribution NP, i.e,
L1i BGK (N) = ! (NP - Ni)
(4.14)
r
where r is the parameter representing the relaxation time . The distributions
NP are given functions of p and u (see (4.1) and (4.2» whose expression
can be found in Rothman and Zaleski (1997) or in Chopard and Droz (1998) .
Equations (4.7) and (4.14) together with a definition of the equilibrium distributions result in the Lattice-BGK(L-BGK) model. The L-BGK model leads to
correct hydrodynamic behavior. The viscosity of a two-dimensional L-BGK
on a hexagonal lattice is given by:
v=*(r-D
(4 .15)
Note that the limit of zero viscosity (i.e. r ~ 1/2) is numerically unstable.
The L-BGK is also developed for other lattices, e.g. cubic lattices in twoor three dimensions with nearest and next-nearest neighbor interactions. The
LBM, and especially the L-BGK, has found widespread use in simulations of
fluid flow.
4.3.2 Transport, Erosion, Deposition, and Hydrodynamic Forces
Transport processes and sedimentation problems in rivers or in coastal environments are important issues for the understanding of the influence of
physical environments on the growth and form of marine sessile organisms.
For instance, tidal movement is responsible for the large amount of sand that
may deposit in places where it is locally screened by an obstacle or by an
abrupt change of the ground profile. The prediction of sediment transport in
water is also important for understanding the evolution of, for instance, river
beds. The creation of meanders is an interesting example of a pattern that is
generated in an erosion-deposition process . Similarly, it is well known that
the presence of a dike or other human construction may severely affect the
profile of a coastal line. Here again, an effective numerical simulation can be
a very useful prediction tool.
In this section we model granular material (typically sand) by a multiparticle stochastic cellular automata (Chopard and Droz 1998) in which an
arbitrary number of point particles may exist at each lattice site.
In addition to the LBM fluid introduced previously, the second important
ingredient in the erosion model is the granular particles. Suspensions are
represented by an integer n(x, t) ;::: 0 indicating how many solid particles are
present on site x at time t. Suspensions move on the same lattice as the fluid
particles and interact with them.
It is important to remember that in the present mesoscopic approach we
do not attempt to represent a specific granular material. Rather, we want to
capture the generic features of the erosion -deposition process. The existence
105
Equation (4.7). This means that particle densities stream from cell to cell,
and particle densities collide. This immediately solves the problem of noisy
dynamics. In a strict sense we no longer have a CA with a Boolean state
vector, but rather we can view LBM as a generalized CA. Bya clever choice of
the collision operator, the model becomes isotropic and Galilean invariant,
thus solving the second problem of LGA. Actually, a very simple collision
operator is introduced, namel y the so-called BGK collision operator which
models the collisions as a single-time relaxation towards a local equilibrium
distribution NP, i.e,
L1i BGK (N) = ! (NP - Ni)
(4.14)
r
where r is the parameter representing the relaxation time . The distributions
NP are given functions of p and u (see (4.1) and (4.2» whose expression
can be found in Rothman and Zaleski (1997) or in Chopard and Droz (1998) .
Equations (4.7) and (4.14) together with a definition of the equilibrium distributions result in the Lattice-BGK(L-BGK) model. The L-BGK model leads to
correct hydrodynamic behavior. The viscosity of a two-dimensional L-BGK
on a hexagonal lattice is given by:
v=*(r-D
(4 .15)
Note that the limit of zero viscosity (i.e. r ~ 1/2) is numerically unstable.
The L-BGK is also developed for other lattices, e.g. cubic lattices in twoor three dimensions with nearest and next-nearest neighbor interactions. The
LBM, and especially the L-BGK, has found widespread use in simulations of
fluid flow.
4.3.2 Transport, Erosion, Deposition, and Hydrodynamic Forces
Transport processes and sedimentation problems in rivers or in coastal environments are important issues for the understanding of the influence of
physical environments on the growth and form of marine sessile organisms.
For instance, tidal movement is responsible for the large amount of sand that
may deposit in places where it is locally screened by an obstacle or by an
abrupt change of the ground profile. The prediction of sediment transport in
water is also important for understanding the evolution of, for instance, river
beds. The creation of meanders is an interesting example of a pattern that is
generated in an erosion-deposition process . Similarly, it is well known that
the presence of a dike or other human construction may severely affect the
profile of a coastal line. Here again, an effective numerical simulation can be
a very useful prediction tool.
In this section we model granular material (typically sand) by a multiparticle stochastic cellular automata (Chopard and Droz 1998) in which an
arbitrary number of point particles may exist at each lattice site.
In addition to the LBM fluid introduced previously, the second important
ingredient in the erosion model is the granular particles. Suspensions are
represented by an integer n(x, t) ;::: 0 indicating how many solid particles are
present on site x at time t. Suspensions move on the same lattice as the fluid
particles and interact with them.
It is important to remember that in the present mesoscopic approach we
do not attempt to represent a specific granular material. Rather, we want to
capture the generic features of the erosion -deposition process. The existence
105