Chaotic Waterwheel

What it Shows

We start with a vertical wheel—like a Ferris Wheel, but with a diameter just under 1 meter—in neutral equilibrium and free to rotate in either direction. From the ends of each of the eight spokes hang small buckets with drainage holes cut out of the bottom. Fixed directly above the center of the wheel is a faucet connected to a pump.

See our waterwheel in action on youtube https://youtu.be/Lx8gMBJBlP8

When the pump is turned on, a stream of water flows downwards from the faucet, with some of it inundating the buckets, filling them faster than they can drain out. Eventually the entire wheel will become unbalanced and start to rotate accordingly: the side with the fuller, heavier buckets will get pulled down, and the relatively empty buckets on the other side will swing up so that they can get refilled under the faucet. The situation is further complicated by buckets taking water from the ones directly above them, either via the drainage holes or because of spillage over the top.

A curious mode of motion soon becomes apparent. After several seconds of the wheel spinning steadily in one direction, it may suddenly speed up; it may slow down and start to rotate in the opposite direction; it may sometimes oscillate between clockwise and counterclockwise rotation; it may go through periods when motion in either direction is barely noticeable. Whatever the state of motion, it never survives for long. The motion is clearly non-periodic and unpredictable. (On the other hand, it is not random either, since the motion evolves continuously through certain states, determined at each instant by the physical state immediately preceding it.)

Another puzzling feature of our water wheel is observed when we pause the pump and allow all the buckets to drain completely. With the wheel in very nearly the same angular orientation as before, we resume the water flow. What we find is that, over the course of a minute or two, the complete motion of the wheel will be dramatically different than during the previous run.[2]

This apparent sensitivity to the initial conditions of the system, along with the wheel’s non-periodic motion, are hallmarks of chaotic behavior.[1]Edward Lorenz, working on computational methods of weather modeling at MIT in the late 1950s, noticed these traits when consecutive runs of his computer simulation returned dramatically different results. Since there was no programming error or physical malfunction that could have caused this divergence, he reasoned that the culprit must have been the particular set of ordinary differential equations and initial values he was using. The set of equations used by Lorenz in his simulator came to be known as the Lorenz attractor, and it remains one of the classic examples of chaos theory.[3]

graphs of omega vs time for the wheel
Graphs of omega vs time. The data is from analysis of two different video clips, with the wheel starting in nearly the same position each time. Notice how similar the graphs are for the first 20 seconds.

In nature, chaotic behavior most readily occurs in turbulent flows and in large-scale weather patterns, but to the casual observer the chaotic nature of these systems may not be very obvious, and scaling them to a laboratory or classroom setting is far from trivial.[4] The idea of building a waterwheel as a discrete, mechanical example of a chaotic Lorenz attractor was proposed and realized by Willem Malkus, Louis Howard, and Ruby Krishnamurti in the early 1970s. One of the important features of the Malkus design is that the angle of the wheel is closer to horizontal than vertical, making the water unable to flow directly from one leaky container into another. According to Lorenz, their original design “was a precision instrument, suitable for controlled laboratory experiments.” Our design is simpler and geared more towards pedagogical impact than experimental fidelity, but we think you will find it charming and instructive nonetheless.

How it Works

Our wheel was fabricated with wood in our shop, with both bearings in a fixed horizontal orientation. The sump pump was purchased from the local hardware store. A ball valve at the faucet regulates the water flow. The wheel and pump both sit in a concrete mixing tub. The little buckets are citronella candle holders with ¼” holes drilled out of the bottom.

Setting it Up

The wheel, sump pump, and plumbing all live together in the black concrete mixing tub. Everything can be set up on one of the large blue carts. The faucet will need to be well clear of the top of the wheel, so find a tall lab post to clamp it to. Below, lead bricks or some other heavy object should be used to secure the base of the wheel to the tub. This demo will splash water all over the place, so make sure there’s a mop handy!

Fill the tub with a few inches of water. Aim the faucet at the center of the axle. Start with the ball valve closed, and open it gradually to get a good flow rate. We want to have enough energy to overcome friction, but we also want to give the buckets a chance to drain out the bottom without overflowing. The main idea of the demonstration is to show chaotic—or at least, non-periodic—motion, which means we do not want to have the wheel exhibit a sustained rotation in one direction. To this end, coins and modeling clay have been used to adjust the balance of the wheel, with local regions of stability and unstability.

Comments

A fun and animated way to liven up any discussion of nonlinear systems! To make the demo a little more quantitative, have students try to keep count of revolutions and oscillations in sequential order to verify the non-periodic behavior. We also have made available four video clips of the wheel in action so that students may download them and use motion-tracking software to analyze the wheel’s motion.

References

[1] E. N. Lorenz, The Essence of Chaos (University of Washington Press, Seattle, WA, 1993).

[2] For example, suppose for the first 20 seconds of the experiment the wheel rotates steadily in the clockwise direction, then oscillates for the next 10 before starting to rotate in the counterclockwise direction for the remaining 30 seconds. After pausing and resetting the experiment, the wheel may again start by rotating in the clockwise direction, but now after 35 seconds it slows down and starts counter-rotate for 15 seconds, and then oscillates for the remaining 10 seconds. So even though the motions started out nearly the same, over time they took dramatically different courses. Since the procedure of the experiment was the same in both cases, the only thing that could account for the divergent behaviors was a slight discrepancy in some initial condition, e.g. angle and orientation of the wheel, the amount of residual water in the buckets, the internal state of the pump, etc.

[3] E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141 (1963).

[4] See Prof Paul Horowitz present an analog circuit that behaves like an attractor https://drive.google.com/file/d/1Gn2EfTQIAYLOCsPFgwH5_zbf1mZtb1tR/view?u...

K. Dreyer and F. R. Hickey, “The route to chaos in a dripping water faucet,” Am. J. Phys. 59, 619–627 (1991).

Lucas Illing, Rachel F. Fordyce, Alison M. Saunders, and Robert Ormond, "Experiments with a Malkus–Lorenz water wheel: Chaos and Synchronization," Am. J. Phys. 80, 192 (2012)

Willem V. R. Malkus, "Non-Periodic Convection at High and Low Prandtl Number," Mémoires Société Royale des Sciences de Liège, 6e série, tome IV, pp. 125-128 (1972)

Steven H. Strogatz, "Nonlinear Dynamics and Chaos" (Perseus Books, Cambridge, MA, 1994).