So you’ve probably already seen demos on Youtube showing this really weird “camera effect” where they stick a hose to a subwoofer and get the water to look like it’s being sucked back to the hose, seemingly against gravity.

I personally love this effect. In the case of the subwoofer, the effect is due to what is technically called “aliasing”. Aliasing is an effect important to all sorts of fields, from data analysis to telecommunications to image processing. In technical jargon, you get aliasing when you don’t satisfy the Nyquist criterion when sampling your signal. This might not be accessible to everyone, so I’ll explain it differently.

In the case of the hose stuck to the subwoofer, the speaker shakes the hose back and forth with a single frequency (a single tone) and generates a snaking/spiraling effect on the water stream. If the water stream is slow enough (that is, if its Reynolds number is low enough for the flow to be laminar), then no weird stuff (non-linear effects) occur, and we get a simple, single-toned spatial wave in the water jet. That being the case, the cycles repeat very nicely, becoming indistinguishable from each other. If you can fulfill this criterion, then aliasing also occurs in a nice manner, that is, if you happen to fail to satisfy the Nyquist criterion, you don’t get a jumbled mess but a nicely backward or forward motion that looks like it’s in slow motion.

It is a simple thing to do, but there’s some beautiful fluid dynamics on it. Generating repeatable patterns and laminar flows is not that simple, especially when you are engineering a device. If you attempt to reproduce the video linked, you’ll find yourself suffering through a parametric search of flow rate/shake amplitude until you get the right combination that displays a nice effect.

Here, I’ll discuss a different device, though – I’ll talk about the piddlers of Dr. Edgerton, that inspired awe in many people around the world – including myself. I have never personally seen one. But I understood what it was and that the effect was not just a camera artifact, but something that could be seen with the naked eye because they use a stroboscopic light to show the effect to the viewer. I have not – as of 2018 – found any instructions on how to make these. And since it turns out it’s quite simple, I think it should be popularized. Here’s my take:

## the piddler design considerations no one talks about in practice

The water piddler is a device that generates a jet of water droplets. Water jets are naturally prone to breaking down into droplets – if you’re a man you know it! But on a more scientific tone: Water jets are subject to a fluid dynamic instability called the Rayleigh-Plateau instability. This document here is an incredible source that enables the prediction of what are the conditions for this unstable behavior without hassling you with all the complicated math behind. The Rayleigh-Plateau instability looks like shown below:

It’s beautiful – but it is also not really a single frequency. There seems to be some level of repeatability to it, but not enough to make the strobe light trick work. The reason for this non-repeatability is the following curve:

This is the dispersion relationship, extracted from equation (23) of Breslouer’s work. It corresponds to an axisymmetric disturbance in the radius of the jet – , where is a small disturbance in the radius and is the streamwise coordinate. gives us a wavenumber of the ripple in the jet, and gives us a frequency of this disturbance. The dispersion relation normalizes the wavenumber by the original radius of the stream, . When , we get an exponentially growing disturbance in the radius, which eventually makes the jet break down into droplets. So the black curve in the chart shows that any disturbance between will grow, but higher frequency disturbances will not – they simply oscillate. Disturbances closer to the peak at will grow faster, which is an important design guideline when we want to break down the jet into a stream of droplets.

The problem, though, is that any other frequencies around the peak also grow. The peak is somewhat smooth, so there will be a lot of space for non-uniformity, especially when the disturbances themselves start at different amplitudes.

So what would be a good design procedure? Well, first, we need to make sure the jet will be laminar. One way to guarantee that is to make the Reynolds number of the nozzle that makes the stream lower than 2000. That guarantees the pipe flow is laminar, which in turn makes the stream laminar. Of course, this is a little limiting to us because we can only work with small jet diameters. You can try to push this harder, since the flow inside the stream tends to relaminarize as the stream exits the nozzle due to the removal of the no-slip condition generated by the nozzle wall.

The other constraint has to do with reasonable frequencies for strobing. You don’t want to use too low of a strobe frequency, because that is rather unbearable to watch. Strobe frequencies must be above 30Hz to be reasonably acceptable, but they only become imperceptible to the human eye about 60Hz. We get a design chart, then:

The chart shows the growth rates (real part of ) for combinations of realistic jet diameters and velocities, which are the actual design variables. The line of constant Reynolds number looks like a 1/x curve in this space. The white line shows the upper limit for laminar pipe flow. You want to be under the white line, as well as in the growing region, which is about the region enclosed by the dashed line. For higher frequencies, the slope of the black boundary decreases, meaning you need smaller diameters to make the strobe light work. For lower frequencies, the slope increases, improving the available parameter space, but too low frequencies will be uncomfortable to watch. In case you want to develop your own piddler, a Matlab implementation to generate the colorful chart above is here.

It is actually rather remarkable that the parameter space looks like this, because feasible diameter/frequency combinations actually will break down into droplets if excited with 60Hz – the line frequency in the US. Say, for example, for 3mm jet diameter and 1m/s speed, we have a high growth rate and the piddler will produce a nice effect. At 6mm, 0.5m/s, we still have laminar flow but the instabilities won’t grow at 60Hz (lower frequency instabilities grow instead). Thus, you’ll not get a good piddler out of that combination. You might be able, for example, to push the bounds a bit (which I did) and make the jet diameter 4.75mm and the jet speed about 1.2m/s. In that case, the Reynolds number is about 3200, which still makes a reasonably repeatable piddler pattern.

Another thing you can attempt (I did) is to try to use a more viscous fluid. More viscous fluids will increase the viable diameter/velocity combinations where the Reynolds number is still low by pushing the white line up and to the right. For example, propyleneglycol allows us to approximately double the diameter of the pipe. The problem, obviously, is that it’s incredibly messy!

## now to the real world

The design map is a good guideline to start this, but there are a few tricks that no one really describes in the internet. I’ll save you weeks of suffering: The best way to generate the disturbance is to use a coffee machine pump. **Yes, a coffee machine pump**! It accomplishes the two tasks for this device: Recirculating the fluid and generating a strong enough, single-frequency disturbance such that you don’t really need to trust the Rayleigh-Plateau instability alone to generate the droplets.

This is the basic implementation of the piddler I built. The coffee machine pump is a ULKA-branded one (like this one). I believe it doesn’t really matter which model you use, since mine had too much flow rate and I had to reduce the flow with the flow control valve indicated in the schematic. These pumps are vibratory pumps. They function by vibrating a magnetic piston back and forth with a big electromagnet. The piston pumps a small slug of fluid, that goes through a check valve inside the pump. When the piston returns, the suction shuts off the check valve, preventing back-flow. Suction fills the piston cavity of new fluid, and the cycle repeats.

Since the piston is pumping the fluid in tiny slugs at the correct frequency (assuming you have a viable Rayleigh-Plateau instability on your design), an acoustic wave will go through the water until the nozzle, generating the intended velocity disturbance in the mean flow. It will not be a choppy flow, but an oscillating one, due to the really strong surface tension forces in water. The figure in the left shows that there’s a full wave of instability before the stream breaks down into droplets.

Now that we discussed the design, let’s go for a little demo. In this video, I’ll also go through the z-Transform, which is a cool mathematical tool for modeling discrete-time control systems. I used this piddler as a prop for the lecture!

I just came here from Hackaday. This was a really good explanation and was a great refresher for me on the Z-transform. Good luck on the PhD!

-Sam Ellicott

Soli Deo Gloria

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