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# Vortex Core Algorithm Description

What is the difference between the two (eigen-analysis and vorticity) vortex core algorithms?

The two vortex core methods are outlined in the 1999 AIAA-99-3288 paper
entitled "On the Velocity Gradient Tensor and Fluid Feature Extraction"
by Robert Haimes and David Kenwright.

In brief, for both algorithms, all 3D (or 2D) elements are subdivided
into either tets (or triangles), and a unique velocity gradient tensor
V is constructed and then classified.

For the eigen-analysis algorithm, if swirling is classified using
eig(V), the direction orthogonal to the spiral plane is used as the axis
of swirl. This direction is subtracted from the nodal velocities.
These "reduced velocities" are used to see if any faces display a zero.
If so, that location on the face is marked. With two (or more) marked
faces on the tets, it is determined that the core center-line has
pierced the cell. These lines are collected and drawn to display the
core segments. This particular algorithm is described in Sujudi and
Haimes, Identification of Swirling Flow in 3-D Vector Fields, AIAA
95-1715, 1995.

For the vorticity algorithm, the issue of alignment coupled with the
linear limitations of the classification and results of the
eigen-analysis hint at an alternative technique, where by using the
eig(Omega) instead of the eig(V). The result is invariably one zero and
a complex conjugate pair of eigenvalues - always the indication of
swirling flow. The eigenvector associated with the zero eigenvalue can
be used with the face-piercing algorithm to produce core segments. This
technique has the effect of removing the bulk and shear components of
stress from the analysis. The result of this algorithm is, at times,
(depending on the dataset) longer and more contiguous core segments.
Thus, the result is that vortex cores can simply be found where Omega
aligns with the velocity vector.

Considerations

Both methods have problems finding cores of curved vortices, and fail to
predict vortex segments in regions of weak vorticies. (The Lambda2 UDMF
has been recommended here to deliniate this region.)

The eigen-analysis method (default) seems to locate swirling flow
features that are not vorticies (esp. in formation of boundary layers).
Also this method may produce incorrect results when flow is under the
influence of more than one vertex, resulting in a displaced core.

The vorticity method does not seem to produce features in formation of
boundary layers. And tends to produce longer and more contiguous cores
- at times (varies between datasets).

Verification

In both methods, we recommend validating the vortex cores by using them
as a seed for particle traces.