Chapter 4 Vortex detection - Computer Graphics and Visualization

3.3. Particle tracing in -transformed grids
coordinates are defined relative to the local water elevation and depth , as
Þ
.
The top layer, where , follows the free water surface, which usually only varies
gradually. The bottom layer, where , follows the sea bed geometry, which typically
has strongly varying depths throughout the model. The layers in between have a
prescribed thickness distribution according to the -transformation. Figure 3.5 shows
one possible distribution with six layers. Figure 3.6 shows a real-life example, with
a sea bed geometry and a vertical grid slice of the Lith harbour data set, which was
produced at WL Delft Hydraulics in a simulation as part of the design of a harbour
near Lith [Meijer, 1995].
Figure 3.5: Vertical thickness distribution of a six-layer -transformed grid
In -transformed grids, a considerable number of cells may be sheared in the vertical
direction, because the number of layers is constant while the local depth varies, so
parallel vertical edges often lie at very different depths. The shearing is increased by
the cells typically being very thin in these applications: the model may be hundreds of
kilometers wide yet only tens of meters deep.
If the amount of shearing is high and the cells are very thin, the 5-decomposition
may become degenerate. Whereas in normal cells the orientation of the central tetrahedron
is as shown in Figure 3.7a, in strongly sheared cells the orientation of the central
tetrahedron is reversed (‘turned inside out’), as seen in Figure 3.7b. The top faces BEG
and DEG now lie at the bottom, while the bottom faces BDE and BDG lie on top. The
edges BD and GE have crossed each other. This is possible because the central tetrahedron
has edges spanning the entire cell.
Let us investigate the process of reversal between the normal and the degenerate
state of the central tetrahedron in more detail. Figure 3.8 shows the same cell as Fig-
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