Chapter 4 Vortex detection - Computer Graphics and Visualization

4.4. The enhanced curvature centre method
caused by shear flow or interacting vortices. This is illustrated by two analytical flows:
a circular and an elliptic flow.
circular flow
The ideal test case for the curvature centres method is a circular flow. Since the streamlines
are perfect concentric circles, all the curvature centres should coincide. Let the
velocity field Ú Ü Ý ÚÜ Ü Ý ÚÝ Ü Ý of the circular flow be defined by
Ú Ü Ý Ý Ü (4.13)
Figure 4.13a depicts streamlines of this velocity field. The centripetal acceleration
Ü Ý ÜÝ is given by
Ü Ý ÚÜ
Ý
Ú
Ü
which results in the curvature centre Ô :
Ú
ÚÝ
Ý
Ü
(4.14)
Ü Ý (4.15)
Ô Ô Ü Ý Ü Ý (4.16)
which proves that for perfect, circular streamlines, the curvature centres coincide in
the origin (0,0), as Figure 4.13b shows.
elliptic flow
In the case that the streamlines are not perfect circles, but deformed to ellipses, the
velocity field is given by:
Ú Ü Ý Ý Ü (4.17)
with and . Figure 4.13c shows an example with .
Then, the centripetal acceleration ÜÝ is given by
Ü Ý ÚÜ
which results in the curvature centres Ô :
Ú Ú
ÚÝ
Ü Ý
Ý Ü
Ü
Ü
Ý
Ý
(4.18)
(4.19)
(4.20)
Ô Ô Ü Ý Ü Ý Ü Ý (4.21)
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