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A thread of streamlets
To come up with a useful technique of locally enhanced stream
lines, we propose a model for the generation of a
thread of streamlets. Near a
stream line of interest
(the base
trajectory) many
short streamlets are placed. Thereby a continuous representation
of the system's behavior in the vicinity of the base trajectory
is approximated.
Figure 7.1:
Relation between streamlet density (
),
streamlet integration length (
), and streamlet
instantiation interval (
).
|
Using constant flow as a reference model - stream lines are
straight lines in this case - the thread of
streamlets
is defined as
follows: Any
cross-section perpendicular to base
trajectory
is
pierced by a constant number (
)
of streamlets. Using
integration time t as parameterization of the base trajectory
(
),
streamlets
are
instantiated at time
and integrated over
the time interval
.
See Fig. 7.1 for an illustration of the relationship
between
,
,
and
,
i.e.,
.
Seed points
of newly
instantiated streamlets are randomly chosen within a perpendicular
cross-section through
(reference location on the base trajectory) corresponding
to a probability distribution function (PDF)
(see
Eq. 7.1 and Fig. 7.2).
|
(7.1) |
PDF
is defined by parameters R (the maximal
distance between
and
)
and q[0,1). The latter
parameter is used to define PDF d as a truncated cone. This
shape provides the fade-out characteristic of the
streamlet placement procedure with respect to the distance
from
.
To guarantee that d is a PDF,
must equal to
1, i.e., the volume of the truncated cone must be 1. This
constraint can be expressed as specification for parameter D:
In other words,
- many streamlets are positioned around a certain base
trajectory
in a circular fashion. Thus,
polar coordinates (r and )
where used to describe the
seed points of the streamlets.
- Through PDF d the generated streamlet distribution is
uniform within a certain radius (qR) and fades out linearly
outside radius qR. This way of instantiating streamlets
emphasizes the flow near base trajectory
.
Computing a thread of streamlets for the reference model
(
=
const.),
a bunch of line segments
(streamlets
)
of
equal spatial length
(
)
is
generated.
It this case of constant flow the streamlets are parallel
to the base trajectory which is a straight line itself.
The seed points of the
streamlets, i.e.,
=
,
are determined according to the PDF
.
For any time t the
cross-section perpendicular through
is
pierced by exactly
=
streamlets.
Figure 7.2:
Probability density function
for the
instantiation of streamlets based on a perpendicular
cross-section through the base trajectory.
|
Applying this model to real (usually non-constant) flow data, local flow
characteristics are
visualized through the following variations from the constant flow
reference setup:
- the shape of the streamlets directly visualizes the
flow locally to the base trajectory. Local
convergence/divergence or rotational behavior with respect to the
base trajectory is intuitively depicted. Since local variations
are significant in the area of (partial) degeneracies of the flow,
characteristic trajectories are especially well suited to be
chosen as base trajectories.
- the streamlet length is a direct visualization of flow
velocities near the base trajectory.
Flow velocity, for example, can be depicted very good. Compared
to color coding which is often used for velocity visualization the
use of streamlets is more effective.
Taking a linear node repellor with eigenvalues 1, 10, and 100 as
example, the flow
characteristics in the vicinity of this critical point can be
visualized in different ways (see Fig. 7.3).
Using threads of streamlets for a visualization of the
characteristic trajectories - those which are aligned with the
eigenvectors of the critical point's Jacobian matrix - a dense and
intuitive representation of the 3D flow near the critical point is
generated. Through the threads of streamlets
(Fig. 7.3(b)) the flow next to the
characteristic trajectories is visualized. A purely abstract
notation (Fig. 7.3(a)) encodes the
eigenvectors of the Jacobian matrix and the magnitudes of the
associated eigenvalues. No direct information about the vicinity of the
characteristic trajectories is provided.
Next: Rendering
Up: Visualizing characteristic trajectories
Previous: Introduction
Helwig Löffelmann, November 1998, mailto:helwig@cg.tuwien.ac.at.