Galton Turbulence

This idea came to me when I built a wooden Galton-Board ("Bean Machine"), which is is a device invented by Sir Francis Galton used to explain the normal distribution.

The main question in turbulence (for example in water flowing in a canal) is why smooth current (laminar flow) is starting to have eddies/whirlpools/swirls (turbulent flow).

Here I will explain the mechanism of how this happens, based on the model of the Galton Board.

A hard question for God

According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. His reply was:

"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."

A similar witticism has been attributed to Horace Lamb (who had published a noted text book on Hydrodynamics)—

his choice being quantum electrodynamics (instead of relativity) and turbulence.

Lamb was quoted as saying in a speech to the British Association for the Advancement of Science:

"I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment.

One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic." For a cool very interesting and easy to follow introduction to turbulence - watch this short video clip by 3Blue1Brown



Computer Simulation



My idea is based on the 2 minutes long simulation in this video:
Galtonboard / Galtonbrett Simulation (or Bean machine or quincunx or Galton box) by petabyte99 We will focus on how the large balls behave and how the small balls behave.

Large balls - laminar flow

Most of the particles continue to flow straight down the stream, just like most of the large balls fall to the middle sections of the model.

Small balls - turbulent flow

(I'm quoting the explanation from a comment on that video by Mr. David Bulger) :

Watching the small balls fall, you can see that they tend to gather momentum and run in diagonal paths to the left or right —

that is, they don't change direction much.

This illustrates the importance of the independence assumption in the central limit theorem:

if the individual random variables are not independent, then their sum may not tend to a normal distribution.


In the model this means they go more and more to the side until they hit a wall,

but in real flow they go more and more to the side which creates a swirl or eddy.

In a real flow this happens in 3-D.

The pegs (the static "pins") are the slower particles, the balls are the faster particles.

In a real flow the "large" and small" is some other parameter, which still needs to be found,

that "triggers" turbulent behavior.

Maybe some critical threshold ratio like distance between "pegs" to speed of "balls"?



By the way, I built at home a Galton board out of a wooden board and metal ball bearings and this effect also happens in reality:

Small balls sometimes go fast to the side and "ignore" the middle.