Chapter 4 Vortex detection - Computer Graphics and Visualization

2.3. Surfaces
¯ hyperstreamline: a tangent curve of an eigenvector (usually with the largest magnitude)
of a tensor field [Delmarcelle & Hesselink, 1993].
In a steady flow (also called stationary or time-independent flow), where the velocity
field is constant in time, pathlines, streamlines, and streak lines are identical [Strid
et al., 1989]. In an unsteady flow (also called instationary or time-dependent flow),
these curves may all be different. Streamlines, vorticity lines, and hyperstreamlines
are mathematical abstractions, but they are all based on the idea of field lines. Many
of these curves have been inspired by experimental visualization [Post & van Walsum,
1993].
Most of the curves are based on the notion of particle advection. They can be generated
in a straightforward way, by integrating the vector field, which results in an
integral curve. Particle tracing is a visualization method which simulates the release
of massless particles, calculates their paths, and then renders them in an animation at
a constant frame rate. Particle tracing is a common way of visualizing the results of
CFD flow simulations; an extension for turbulent flow was introduced in [Hin & Post,
1993; Hin, 1994].
The choice of release points for particles or flow curves is critical. If flow curves
are released too sparsely, or in the wrong regions, important flow features may be
missed. If they are released too densely, cluttering may occur. An important technique
for an even distribution of the lines is described in [Turk & Banks, 1996]. Recently, this
technique was extended to curvilinear grids [Mao et al., 1998].
Effectively rendering 3D curves is much more difficult than rendering 2D curves,
due to occlusion, and the lack of direction information and depth cues. In [Fuhrmann
&Gröller, 1998], some solutions are proposed, such as an algorithm for an even distribution
of curves, and the use of texture for depth cues.
Our focus in this thesis is on the calculation of pathlines, or particle tracing in special
grid types: in structured curvilinear grids (see Section 3.2), in -transformed grids,
which are common in hydrodynamic applications (see Section 3.3), and in unstructured
grids, which are more common in aerodynamics (see Section 3.4).
2.3 Surfaces
Curves are easy to visualize, but they do not carry much spatial information, and are
therefore difficult to locate visually in space [Post & van Walsum, 1993]. Surfaces are
much better for visualization, because lighting and shading are effective cues for perception
of 3D shapes by a human observer.
Our focus in this thesis is on the generation of flow surfaces of three kinds. The
first kind is adaptive isosurfaces, which can be used to approximate local isosurfaces
with the desired accuracy specified by the user (see Section 5.5.1). The second kind is
separation surfaces (see Section 5.5.2), which are used to find recirculation zones. The
third kind is vortex tubes (see Section 6.4). For completeness, a brief introduction to
isosurfaces, stream surfaces, and implicit surfaces is given below.
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