4.5. GROWTH BY AGGREGATION
To obtain insight into the influence of hydrodynamics on the growth
process of sessile suspension feeders, a morphological simulation model
was developed. In the next sections (Sections 4.5.3 and 4.5.4) we will first
discuss a very simple model of the growth process of accretive organisms.
The growth process is modeled by an aggregation process , where growth is
represented by the addition of particles in a cubic lattice . In this model we use
an extension of the diffusion limited aggregation (DLA)model introduced by
Witten and Sander (1981); see also Fig. 1.10 for an example of the DLAmodel.
In Sect. 4.6 on accretive growth we will extend these ideas further in a model
of surface normal deposition processes, in which a model of the influence of
hydrodynamics is included.
In the absence of flow, the distribution of nutrients around the growth
form can be modeled as a diffusion process in a steady state: there is a source
of suspended material and the organism continuously consumes nutrients
from its environment. In general in a marine environment, there will be
a significant contribution of the hydrodynamics to the dispersion pattern of
the suspended material around the growth form . In this case the distribution
of nutrients around the organism will be determined by a combination of
flow and diffusion. The contribution of flow to the nutrient distribution can
be quantified by the Peeler number shown in (2.4). In the diffusion-limited
case Pe is very small (nearly zero), while in the flow-dominated case Pe is
large. The effect of hydrodynamics on the morphology of the aggregates is
studied by varying the Peeler number in the simulations.
4.5.2 Modeling the Nutrient Distribution
The method which we have applied to model the nutrient distributions is
the lattice Boltzmann method, which is combined with a tracer step. The
underlying method is discussed in Sect. 4.3.1, more detailed accounts of this
method can be found elsewhere (Frisch et al. 1987, Chen et al. 1992, Ladd
1994, Chopard and Droz 1998) .
In the simulations a cubic lattice is used consisting of 144
3 nodes . Each
node is connected with 18 other nodes. There are 6 links with the length of
1 lattice unit and 12 links with the length of J2 units. All parameters and
variables in the lattice Boltzmann method are expressed in lattice units. The
mean populations of particles move simultaneously from one node to one of
the 18 neighbors. The evolution of the lattice is descr ibed by the dynamical
rule shown in (4 .7) in which the NiS represent the particle densities at the
node. The momentums of the particles are changed by adding an external
force, the driving force F, to the system. One update step of the lattice now
involves a propagation step where N, particles travel from node x to node
x + ciand a collision step in which the post-collision distribution is computed.
Two types of boundary conditions are used : at the borders of the lattice
periodic boundary conditions are applied, while in the nodes adjacent to the
nodes representing the obstacle, solid boundary conditions are used. Periodic
boundary conditions can be implemented by exchanging the NiS of the links
at the borders of the lattice . Solid boundary conditions can be represented
by exchanging the NiS between the adjacent node and a neighboring fluid
node .
After the lattice Boltzmann iteration, a tracer step is applied where populations of tracer particles are released from source nodes and are absorbed
115
To obtain insight into the influence of hydrodynamics on the growth
process of sessile suspension feeders, a morphological simulation model
was developed. In the next sections (Sections 4.5.3 and 4.5.4) we will first
discuss a very simple model of the growth process of accretive organisms.
The growth process is modeled by an aggregation process , where growth is
represented by the addition of particles in a cubic lattice . In this model we use
an extension of the diffusion limited aggregation (DLA)model introduced by
Witten and Sander (1981); see also Fig. 1.10 for an example of the DLAmodel.
In Sect. 4.6 on accretive growth we will extend these ideas further in a model
of surface normal deposition processes, in which a model of the influence of
hydrodynamics is included.
In the absence of flow, the distribution of nutrients around the growth
form can be modeled as a diffusion process in a steady state: there is a source
of suspended material and the organism continuously consumes nutrients
from its environment. In general in a marine environment, there will be
a significant contribution of the hydrodynamics to the dispersion pattern of
the suspended material around the growth form . In this case the distribution
of nutrients around the organism will be determined by a combination of
flow and diffusion. The contribution of flow to the nutrient distribution can
be quantified by the Peeler number shown in (2.4). In the diffusion-limited
case Pe is very small (nearly zero), while in the flow-dominated case Pe is
large. The effect of hydrodynamics on the morphology of the aggregates is
studied by varying the Peeler number in the simulations.
4.5.2 Modeling the Nutrient Distribution
The method which we have applied to model the nutrient distributions is
the lattice Boltzmann method, which is combined with a tracer step. The
underlying method is discussed in Sect. 4.3.1, more detailed accounts of this
method can be found elsewhere (Frisch et al. 1987, Chen et al. 1992, Ladd
1994, Chopard and Droz 1998) .
In the simulations a cubic lattice is used consisting of 144
3 nodes . Each
node is connected with 18 other nodes. There are 6 links with the length of
1 lattice unit and 12 links with the length of J2 units. All parameters and
variables in the lattice Boltzmann method are expressed in lattice units. The
mean populations of particles move simultaneously from one node to one of
the 18 neighbors. The evolution of the lattice is descr ibed by the dynamical
rule shown in (4 .7) in which the NiS represent the particle densities at the
node. The momentums of the particles are changed by adding an external
force, the driving force F, to the system. One update step of the lattice now
involves a propagation step where N, particles travel from node x to node
x + ciand a collision step in which the post-collision distribution is computed.
Two types of boundary conditions are used : at the borders of the lattice
periodic boundary conditions are applied, while in the nodes adjacent to the
nodes representing the obstacle, solid boundary conditions are used. Periodic
boundary conditions can be implemented by exchanging the NiS of the links
at the borders of the lattice . Solid boundary conditions can be represented
by exchanging the NiS between the adjacent node and a neighboring fluid
node .
After the lattice Boltzmann iteration, a tracer step is applied where populations of tracer particles are released from source nodes and are absorbed
115