Malkus waterwheel (real)

A waterwheel with leaky buckets undergoes chaotic motion. Our wheel is about 1 meter in diameter and was fabricated with wood in our shop. The little buckets are citronella candle holders with ¼” holes drilled out of the bottom. 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.¹

The setup

A vertical wheel similar to a Ferris Wheel. It has 8 spokes and a diameter of just under 1 meter. It’s free to rotate in either direction. A bucket with a drainage hole cut in its bottom hangs from each of the 8 spokes. There’s a water faucet directly above the wheel.

How it works

The water is turned on and starts filling the buckets. Some buckets are filled faster than they drain. Eventually the wheel becomes unbalanced and starts to rotate. The side with the fuller buckets is pulled down. The emptier side is pulled up, so it’s buckets start to get more water than the heavier side. The process is complicated by some buckets receiving water from buckets that are draining above them.

Non-periodic and unpredictable motion

“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.”²

The system is sensitive to initial conditions

Starting the process with the arms of the wheel in slightly different positions results in dramatically different patterns of motion.

The Lorenz attractor

See: notes / Lorenz attractor

“This [the waterwheels’s] apparent sensitivity to the initial conditions of the system, along with the wheel’s non-periodic motion, are hallmarks of chaotic behavior.”³

Edward Lorenz noticed the traits of sensitivity to initial conditions and non-periodic motion when working on weather modeling simulations in the late 1950s.

(This is where things get fuzzy for me.) He reasoned (proposed (?)) that the traits were due to the equations he was using to model the weather, and the varying initial conditions used when running the model.

It’s not clear to me yet how Lorenz came up with the equations. I won’t go into it here.

The Lorenz attractor differential equations:

dx/dt = σ(y - x)
dy/dt = x(ρ - z) - y
dz/dt = xy - βz

Relating the Lorenz attractor equations to the Malkus waterwheel

• $x$ is the angular velocity of the wheel. Note that angular velocity is typically referred to with the lowercase letter omega: $\omega$: LaTeX code $\omega$. It represents the rate of change of angular displacement ($\theta$) over time ($t$).

• $y$ is the first cosine mode of water distribution. (I don’t understand this yet.)

• $z$ is the first sine mode of water distribution. (I don’t understand this yet.) Ignoring the first cosine mode and first sine mode descriptions of the $y$ and $z$ values, I’ve added an intuitive explanation to notes / The relationship between the Malkus waterwheel and the Lorenz system. A somewhat satisfying explanation of first cosine mode and first sine mode

The parameters used in the equations have a straightforward relationship to the physical wheel:

• $\sigma$ (sigma): the damping coefficient (the friction in the wheel)
• $\rho$ (ro): water inflow rate
• $\beta$: water outflow (leakage from buckets)

Referring to some code I wrote last year ( https://github.com/scossar/pd-chaos/blob/master/src/lorenz~.c ), default values for the parameters that produce interesting results are:

• $\sigma = 10.0$
• $\rho = 28.0$
• $\beta = 8.0/3.0 \approx 2.67$

Is the waterwheel in the video technically a Malkus waterwheel?

In the early 1970s, Willem Malkus , Louis Howard, and Ruby Krishnamurti built a waterwheel as an example of a chaotic Lorenz attractor. In the Malkus waterwheel, the angle of the wheel is closer to vertical than horizontal. This makes it impossible for water to flow from one container into another. That detail was left out of the wheel that’s used in the video.

Related

• Malkus waterwheel simulation

References

Harvard Natural Sciences Lecture Demonstrations. “Chaotic Waterwheel.” https://sciencedemonstrations.fas.harvard.edu/presentations/chaotic-waterwheel .