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4. SIMULATING GROWTH AND FORM
of universal behaviors in systems with many interacting particles is common
in many areas ofscience and there are numerous examples where the macroscopic behavior depends very little on the microscopic details of the system.
For this reason it is expected that, to first approximation, this dynamics of
fictitious particles produces the same deposition patterns as real systems,
even if not all the parameters are of the correct order of magnitude.
TRANSPORT RULE FOR SUSPENSIONS. We describe briefly the rule of motion
for solid particles. After each time step, the particles jump to a nearestneighbor site, under the action of the local fluid flow and gravity force. Gravity
is taken into account by imposing a falling speed "fall on the particles. Here,
suspensions are passive particles since their presence does not modify the
flow field, except when they form a solid deposit. However, it would be quite
easy to modify the fluid properties so as to make the relaxation time T vary
according to the local density of transported particles since, in real systems,
it is observed that the fluid viscosity depends on the local concentration of
the suspension.
If the local fluid velocity at site x is u(x), the particles located at that
site will move to site x + Ts(U + UfaU) where r, is the time unit associated with
the motion of the granular particles. Unfortunately, this new location usually
does not coincide with a lattice site. The solution to this problem is then to
consider a stochastic motion: each of the n(x) particles jumps to neighbor
x + Cj with a probability proportional to the projection of Ts(U + UfalI) on lattice
direction ci. The quantity T s is adjusted so as to maximize the probability of
motion, while ensuring that the jumps are always smaller than a lattice
constant (see Chopard and Droz 1998, Masselot and Chopard 1998, for more
information).
This stochastic cellular automata rule produces a particle motion with
the correct average trajectory and a variance which can be interpreted as a local diffusive behavior (Masselot 2000). Note that in this model no specific rule
is needed to split the transport among creeping, saltation, and suspension,
as is usually done in traditional numerical models.
DEPOSITION RULE. The next aspect of the particle dynamics is the deposition rule. Under the effect of gravity, the particles keep moving downward
until they land on a solid site (e.g, the bottom of the system or the top of
the deposition layer). On such deposition sites, when motion is no longer
possible, particles start piling up. In our model, up to Nthres particles can
accumulate on a given site (Nthres gives a way to specify the space scale of
the granular particles with respect to the fluid system). When this limit is
reached, the site solidifies and new incoming particles pile up on the site
directly above. The solid sites formed in this way represent obstacles over
which the fluid particles bounce back from where they came. Thus, this solidification process implies a dynamically changing boundary condition for
the fluid.
Note that before solidification, the fluid is not affected by the presence
of the rest particles piling up on top of a solid site. Also, these rest particles
are no longer subject to the suspension transport rule. Only the erosion
mechanism discussed below can move them away.
In case of particles with high cohesion, such as snow under some conditions, one can keep piling particles without worrying about a possible
4. SIMULATING GROWTH AND FORM
of universal behaviors in systems with many interacting particles is common
in many areas ofscience and there are numerous examples where the macroscopic behavior depends very little on the microscopic details of the system.
For this reason it is expected that, to first approximation, this dynamics of
fictitious particles produces the same deposition patterns as real systems,
even if not all the parameters are of the correct order of magnitude.
TRANSPORT RULE FOR SUSPENSIONS. We describe briefly the rule of motion
for solid particles. After each time step, the particles jump to a nearestneighbor site, under the action of the local fluid flow and gravity force. Gravity
is taken into account by imposing a falling speed "fall on the particles. Here,
suspensions are passive particles since their presence does not modify the
flow field, except when they form a solid deposit. However, it would be quite
easy to modify the fluid properties so as to make the relaxation time T vary
according to the local density of transported particles since, in real systems,
it is observed that the fluid viscosity depends on the local concentration of
the suspension.
If the local fluid velocity at site x is u(x), the particles located at that
site will move to site x + Ts(U + UfaU) where r, is the time unit associated with
the motion of the granular particles. Unfortunately, this new location usually
does not coincide with a lattice site. The solution to this problem is then to
consider a stochastic motion: each of the n(x) particles jumps to neighbor
x + Cj with a probability proportional to the projection of Ts(U + UfalI) on lattice
direction ci. The quantity T s is adjusted so as to maximize the probability of
motion, while ensuring that the jumps are always smaller than a lattice
constant (see Chopard and Droz 1998, Masselot and Chopard 1998, for more
information).
This stochastic cellular automata rule produces a particle motion with
the correct average trajectory and a variance which can be interpreted as a local diffusive behavior (Masselot 2000). Note that in this model no specific rule
is needed to split the transport among creeping, saltation, and suspension,
as is usually done in traditional numerical models.
DEPOSITION RULE. The next aspect of the particle dynamics is the deposition rule. Under the effect of gravity, the particles keep moving downward
until they land on a solid site (e.g, the bottom of the system or the top of
the deposition layer). On such deposition sites, when motion is no longer
possible, particles start piling up. In our model, up to Nthres particles can
accumulate on a given site (Nthres gives a way to specify the space scale of
the granular particles with respect to the fluid system). When this limit is
reached, the site solidifies and new incoming particles pile up on the site
directly above. The solid sites formed in this way represent obstacles over
which the fluid particles bounce back from where they came. Thus, this solidification process implies a dynamically changing boundary condition for
the fluid.
Note that before solidification, the fluid is not affected by the presence
of the rest particles piling up on top of a solid site. Also, these rest particles
are no longer subject to the suspension transport rule. Only the erosion
mechanism discussed below can move them away.
In case of particles with high cohesion, such as snow under some conditions, one can keep piling particles without worrying about a possible