Chapter 4 Vortex detection - Computer Graphics and Visualization

2.3. Surfaces
in a local coordinate frame at each point on the streamline. The ribbon is constructed
by weaving a strip of triangles between the points.
Both methods for creating ribbons have advantages [Pagendarm & Post, 1997]. The
first method can show vortical behaviour of the flow, but can also show other effects as
well. For instance, it will also show divergence, through the varying width of the ribbon.
The second method shows purely local vortical behaviour on the central streamline.
In both cases, the surface is not an exact stream surface, as the tangency condition
is only true for the constructing streamlines.
A general stream surface can be constructed by generating streamlines from each
of a number of points on an initial line segment or rake. If for all these streamlines a
single constant time step is used, then the lines connecting points of equal time on all
streamlines are time lines. Streamlines and time lines thus make a quadrangular mesh
(see Figure 2.2a), which can be easily divided into triangles for visualization.
rake
time line
streamline
(a) Mesh for stream surface, with
streamlines from a rake, and timelines
Figure 2.2: Stream surface generation
(b) Divergent flow
splits a stream surface
This standard algorithm has some disadvantages. If the flow is strongly divergent,
adjacent streamlines will move too far apart. If there is an object in the flow, the surface
must be split, and this is problematic with the standard algorithm (see Figure 2.2b). Finally,
if there are high velocity gradients in the flow direction, the mesh will be strongly
distorted and unequal-sized and poorly-shaped triangles will result.
To solve these problems, an ‘advancing front’ algorithm has been proposed
[Hultquist, 1992]. The surface is generated in the transverse direction by adding a
strip of triangles to the front. Using adaptive time steps to compensate the gradients
in the flow direction, all points at the front will move forward by about the same distance.
Also, if two adjacent points on the front move too far apart by divergence, a new
streamline is started at the midpoint between them. Conversely, if two points move
too close together, one streamline is terminated. If an object in the flow is detected, the
front can be split, and the two parts can move on separately.
Stream surfaces can also be used to generate streamlines [Kenwright & Mallinson,
1992a]. Two local stream surfaces are determined from dual stream functions defined
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