## 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!

## The floating light bulb: Theory vs Practice

Yes – You can go to Amazon.com today and buy one of these gimmicky toys that float a magnet in the air. Some of which will even float a circuit that can light an LED and become a floating light bulb. A floating light bulb that powers on with wireless energy? What a time to live!

A quote from Arthur C. Clarke, who wrote “The Sentinel” (which later on became the basis for the science fiction movie “2001: A Space Odyssey”), goes along the lines:

“Any sufficiently advanced technology is indistinguishable from magic.”

This is what led me to the Engineering path. Because, if the advanced technology is indistinguishable from magic, who creates the technology is a real-life wizard. Who creates the technology? The engineers and scientists all around this world. So let me complement his quote with my own thoughts:

“Any sufficiently advanced technology is indistinguishable from magic. Therefore, engineers and scientists are the true real-life wizards.

Of course, if I’m writing about it is because I went through the engineering exercise. And boy, I thought it was an “easy” project. You see these projects of floating stuff around the internet, but nobody speaks about what goes wrong. So here we’ll explore why people spend so much time tweaking their setup and what are the traps along the way.

But first, some results to motivate you to read further:

Prof. Christian Hubicki was kind enough to let me pursue this as a graduate course project in Advanced Control Systems class at FSU, so I ended up with a “project report” on it. It is in the link below:

## Full 11-page report with all the data

But if you don’t want to read all of that, here’s a list of practical traps I learned during this project:

1. DON’T try fancy control techniques if you don’t have fast and accurate hardware. This project WILL require you more than a 10-bit ADC and more than 3-5 kS/s. The dynamics are very fast because the solenoid is fast. And you want a fast solenoid to be able to control the levitating object! Unless you can have a large solenoid inductance and a rise time in the order of ~100ms, there’s no way an Arduino implementation can control this. I’m think a nice real-time DAQ controller (like the ones offered by NI) could work here. But an Arduino is just too strapped in specs to cut it! The effects of sampling and digitization are too restrictive. It MIGHT work in some specific configurations, but it is not a general solution (and certainly it didn’t work for me).
2. Analog circuits are fast – why not use them? Everyone (in the hobby electronics world) thinks Arduino is a silver bullet for everything. Don’t forget an op-amp is 100’s of times faster than a digital circuit!
3. Bang-Bang! You see many implementations in the web use a hysteresis (or bang-bang) controller. The bang-bang controller is ideal for cheap projects because it deals well with the non-linearities gracefully. But it is not bullet-proof either. It will become unstable even with high bandwidth in some cases if the non-linearity is strong enough.
4. Temperature Effects: The dynamic characteristics of your solenoid will change as it heats up (you’re dissipating power to turn it on!). So it can get very confusing if you have, say, a PID controller, to tune the gains because the gains will be different depending on the temperature of the coil. Since this effect is very slow (order 10 minutes!) it can result in you chasing your own tail because you’re tuning a plant that is changing with time!
5. The wireless TX introduces noise! This one is particular to this project: If you’re using a Hall effect sensor to sense the presence of the floating object (by its magnetic field), then your Hall sensor will also measure the solenoid strength! Apart from that, the TX is also generating a high-frequency magnetic field, which will also be measured in the Hall effect sensor signal. The effect of the TX is very small (~2mV) buy it appears in the scope. The problem is that Arduinos don’t have low-pass filtering in their ADC inputs. So anything higher than the sampling rate will appear as an “Aliased” signal, which is very nasty to deal with.
6. Make sure your solenoid can lift your object and more. This is an obvious one but I think it is easy to overlook because you need to over-design it. I designed my solenoid to lift 100 grams of weight. But in the end, I could only work with 35 grams because the controller needed a lot of space to work. So overdesign is really crucial here. I ended up shaving a lot of mass from the floating object because I couldn’t lift the original design’s mass!

I’d like to put a more complete tutorial on making this, but since I already invested a lot of time in putting the report together, I think if you put some time on reading it and the conclusions from the measurements/simulations you will be able to reproduce this design or adapt the concepts to your design. Let me know if you think this was useful or maybe if you need any help!

## Sub-microsecond Schlieren photography

(Edit: My entry on the Gallery of Fluid Motion using this technique is online!)

For the ones not introduced to the art of Schlieren photography, I can assure you it was incredibly eye-opening and fascinating to me when I learned that we can see thin air with just a few lenses (or even just one mirror as Josh The Engineer demonstrated here on a hobby setup).

For the initiated in the technique, its uses are obvious in the art and engineering of bleeding-edge aerodynamic technology. Supersonic flows are the favorites here, because the presence of shock waves that make for beautiful crisp images and help us understand and describe many kinds of fluid dynamics phenomena.

What I’m going to describe in this article is a very simple circuit published by Christian Willert here but that most likely is paywalled and might have too much formalism for someone who is just looking for some answers. Since the circuit and the electrical engineering is pretty basic, I felt I (with my hobby-level electronics knowledge) could give it a go and I think you also should. I am also publishing my EasyEDA project if you want to make your boards (Yes, EasyEDA).

But first, let’s address the elephant in the room: Why should you care? Well, if you ever tinkered with a Schlieren/shadowgraph apparatus – for scientific, engineering or artistic purposes -you might be interested in taking sharper pictures. Obtaining sharper pictures of moving stuff works exactly like in regular photography. They can be achieved by reducing the aperture of the lens, by reducing the exposure time or by using a flash. The latter is when a pulsed light source really shines (pun intended!). The great part here is that the first two options involve reducing the amount of light – whereas the last option doesn’t (necessarily).

The not-so-great part is that camera sensors are “integrators”. This means they measure the amount of photons that happened to be absorbed given an amount of time. Therefore, what really matters is the total amount of photons you sent to the camera. Of course, if you sent an insanely large amount of photons in a very short instant, you would risk burning the camera sensor – but if you’re using an LED (as we are going to here), your LED will be long gone before that happens.

So the secret for high speed photography is to have insanely large amounts of light dispensed at once. That would guarantee everything will be as sharp as your optics allow. Since we don’t live in the world of mathematical idealizations, we cannot deliver anything “instantly”, and therefore we have to live with some finite amount of time. Brief enough is relative and depends on what you want to observe. For example, if you’re taking a selfie in a party, probably tens of milliseconds is brief enough to get sharp images. For taking a picture of a tennis player doing a high speed serve, you’re probably fine with tens or hundreds of microseconds. The technical challenges begin to appear when you’re taking pictures of really fast stuff (like supersonic planes) or at larger magnifications. The picture of the jet above is challenging in both ways: its magnification level is 0.7x (meaning the physical object is as projected in the sensor at 0.7x scale) and its speed is roughly 500 meters per second. In other words, the movement of the object (the Schlieren object) is happening at roughly 63.6 million px/second, which requires a really fast shutter to have any hopes to “freeze the flow”. If you’re fond to making simple multiplications in your calculator, the equation is very simple:

$D=\frac{M*v}{s_{px}}$

Where $D$ is the object displacement in px/second, $v$ is its velocity in physical units (i.e. m/s), $M$ is the magnification achieved in the setup and $s_{px}$ is the physical pixel size of your camera (i.e. $s_{px}=5.5 \mu m$ for a Nikon D90).

I know, I know. These are very specialized applications. But who knows which kinds of high speed photography is happening right now in someone’s garage, right? The point is – getting a light source that is fast enough is very challenging. Some options, such as laser-pulsed plasma light sources, can get really expensive even if you make them yourself. But LEDs are a very well-established, reliable technology that has an incredibly fast rise time. And they can get very bright, too (well… kinda).

So what Willert and his coauthors did was very simple: Let’s overdrive a bright LED with 20 times its design current and hope they don’t explode. Spoiler alert: Some LEDs didn’t survive this intellectual journey. But they mapped the safe operational regions for overdriven LEDs of many different manufacturers. To name a few: Luminus Phlatlight CBT-120Luminus Phlatlight CBT-140Phillips LXHL-PM02, among others. These are raw LEDs, no driver included, rated for ~3.6-4V, and are incredibly expensive for an LED. The price ranges from $100 to$150, and they are usually employed in automotive applications. The powerful flash is, however, blinding. And if they do burn out, it can be harmful for the hobbyist’s pockets.

The driver circuit (which is available here) is very simple: An IRF3805 N-channel power MOSFET just connects the LED to a 24V power supply. Remembering the LED is rated for 4V – so it’s going to get a tiny bit brighter (sarcasm). Jokes apart, the LED (CBT-140) is rated for 28A continuous with very efficient heatsinking, which means we will definitely be overdriven. By how much we can measure with R2. Hooking a scope between Q1 and R2 is not harmful to the scope and allows to measure the current going through the LED (unless the currents exceed ~600A, then the voltage spike when the MOSFET turns off might be on the few tens of volts). We don’t want to operate at these currents anyways, because the LED will end up as in the figure below. There’s a trim pot (R3) that controls the MOSFET gate voltage, make sure pin 2 of U1 is giving a low voltage when tuning.

What is really happening is that C1 and C2 (C2 is optional) are being charged by the 24V power supply when the MOSFET is off. Then they discharge at the LED when the MOSFET is activated. No power supply will be able to push 200A continuously through an LED, so if the transistor turns on for too long, the power supply voltage will drop and the power supply will reset. Actually, this is one of the ways to tell if you melted the MOSFET (which happened to me once). The MOSFET needs to turn on in nanoseconds, which will require a decent amount of current (like 4-5 amps) just to charge the gate up. This means we need a driver IC – which in this case I’m using a UCC27424. Make sure to have as little resistance between the driver and the gate to minimize the time constant. The 1.5 Ohms is very close to giving 4A to the MOSFET. Since the gate capacitance is around 8nF, the MOSFET gate rise time is somewhat slow (12 ns).

Speaking about time constants, during the design I realized the time constants of the capacitor that discharges into the LED and the parasitic inductances in the path between the components will dictate the rise time of the circuit, at least for the most part. In my circuit, the time constant was measured to be 100ns, directly with a photodiode. This means we can do >1MHz photography, which is pretty amazing! Unfortunately the cameras that are capable of 1 million frames per second aren’t really accessible to mortals (except when said mortals work in a laboratory that happens to have them!).

Well, the LED driver circuit is still in development – which means I’ll keep this post updated every now and then. But for now, it’s working well enough. The BOM cost is not too intimidating (~$60 at Digikey without the LED. Add the LED and we should be at ~$200), so a hobbyist can really justify this investment if it means an equivalent amount of hours of fun! Furthermore, this circuit implements a microcontroller that monitors and displays the LED and driver’s temperature. It features an auto shut-off, which disables the MOSFET driver if the temperature exceeds an operational threshold. The thermal limits are still to be evaluated, though.

For now, I did my own independent tests,  and the results are very promising.  Below I’m showing a test rig to evaluate the illumination rise and fall times of the LED. The photodiode is a Thorlabs (forgot the model) that has a 1ns rise time if attached to a 50 ohm load. It’s internally biased, which is nice when you want to do a quick test.

The results from the illumination standpoint are rather promising. Below a series of scope traces show that the LED lights up in a very short time and reaches a pretty much constant on state. The decay time, however, seems to be controlled by a phosphorescence mechanism that is probably because this is a white LED. Nevertheless, the pulses are remarkably brief.

The good thing about having high speed cameras is that now we’re ready to roll some experiments. By far, my favorite one is shown below. I was able to use the Schlieren setup to observe ultrasonic acoustic waves at 80kHz , produced by a micro-impinging jet (the jet is 2mm in diameter). The jet is supersonic, its velocity is estimated to be 400 m/s. Just to make sure you get what is in the video: The gray rectangle above is the nozzle. The shiny white line at the bottom is the impingement surface. The jet is impinging downwards, at the center of the image. The acoustic waves are the vertically traveling lines of bright and dark pixels. I was literally able to see sound! How cool is that?

Just as a final note. You might be discouraged to know that I am one of these mortals that happen to have access to a high-speed camera. But bear in mind, these pictures could have been taken with a regular DSLR. The only difference is that the frame sequence wouldn’t look continuous, because the DSLR frame rate is not synchronized with the phenomenon. Apart from that, everything else would be the same. You should give it a try! If you do, please let me know =)