Finding Vortex Cores from PIV fields with Gamma 1

Vortex core tracking is a rather niche task in fluid mechanics that is somewhat daunting for the uninitiated in data analysis. The Matlab implementation by Sebastian Endrikat (thanks!), which can be found here, inspired me to dive a little deeper. His implementation is based on the paper “Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows” by Laurent Graftieaux, which was probably one of the first to perform vortex tracking from realistic PIV fields. The challenge is that when PIV is used, noise is introduced in the velocity fields due to the uncertainties related to the cross-correlation algorithm that tracks the particles. This noise, added to the fine-scale turbulence inherent to any realistic flow field encountered in experiments, makes vortex tracking through derivative-based techniques (such as λ2, Q criterion and vorticity) pretty much impossible.

Computational results are less prone to this effect of the noise and usually are tamer in regards to vortex tracking, though fine-scale turbulence can also be a problem. The three-dimensionality of flow fields doesn’t help. But many relevant flow fields can be “deemed” vortex dominated, where an obvious vortex core is present in the mean. Wingtip vortices are a great example of these vortex-dominated flow fields, though there are many other examples in research from pretty much any lift-generating surface.

As part of my PhD research I’m performing high speed PIV (Particle Image Velocimetry) on the wake of a cylinder with a slanted back (maybe a post later about that?). This geometry has a flow field that shares similarities with military cargo aircraft, but is far enough from the application to be used in publicly-available academic research. The cool part is that it forms a vortex pair, which is known to “wander”. The beauty of having bleeding-edge research equipment is that we can visualize these vortices experimentally in a wind tunnel. But how to turn that into actual data and understanding?

That’s where the Gamma 1 tracking comes into play. Gamma 1 is great because it’s an integral quantity. It is also very simple to describe and understand: If I have a vector field and I’m at the vortex core, I can define a vector from me to any point in this vector field (this vector is called by Graftieaux, “PM“). The angle between this vector and the velocity vector at that arbitrary point would be exactly 90º if the vortex was ideal and I was at the vortex core. Otherwise, it would be another angle. So if I just look at many vectors around me I just need to find the mean of the sine of these two vectors. This quantity should peak at the vortex core. That’s Gamma 1, brilliant!

Sebastian Endrikat did a pretty good job at implementing Graftieaux’s results, and I used his code a lot. But since each run I have has at least 5000 velocity fields, his code was taking waaaay too long. Each field would take 4.5 seconds to parse in a pretty decent machine! So I decided to look back at the math. And I realized that the same task can be accomplished by two convolutions after some juggling. A write-up of that is below:

PDF File with the math

The result, though, is really impressive. Now each field takes 5 milliseconds (3 orders of magnitude better!) to parse in the same machine. So good I made a video of the vortex core. Here it is:

I’m really thankful amazing people like Graftieaux and Endrikat are in the academic community publishing this stuff. Standing over the shoulders of giants!