# Why 5/3 is a fundamental constant for turbulence

The air around you is in constant and chaotic motion,
replete with nearly impossible-to-predict swirls,
ranging from large to minuscule.
What you're looking at right now is a cross-section of the flow in a typical room,
made visible using a home demo
involving a laser, a glass rod and a fog machine.
Predicting the specifics of turbulent motion like this has long evaded mathematicians and physicists.
But we are steadily getting closer to understanding some consistent patterns in this chaos.
And in a minute, I'll share with you one specific quantitative result
describing a certain self-similarity to this motion.
To back up a bit, I was recently in San Diego
and spent a day with Diana Cowern a.k.a. Physics Girl, and her frequent collaborator, Dan Walsh,
playing around with vortex rings.
This is a really surprising fluid flow phenomenon,
where a donut-shaped region of fluid stays surprisingly stable as it moves through space.
If you take some open container with a lip, and you fill it with smoke (or fog),
you can use this to actually see the otherwise-invisible ring.
Diana just published a video over on her channel showing much more of that particular demo,
including a genuinely fascinating observation
about what happens when you change the shape of the opening.
The story for you and me starts when her friend Dan had the clever idea
to visualize a slice of what's going on with these vortex rings,
using a planar laser.
So, you know how if you shine a laser through the fog,
photons will occasionally bounce off of the particles in the fog along that beam,
towards your eye, thereby revealing the beam of the laser?
Well, Dan's thought was to refract that light through a glass rod,
so that it was relatively evenly spread across an entire plane.
Then, the same phenomenon would reveal the laser light along a thin plane through that fog.
The result was awesome!
The cross-section of such a smoke ring looks like two hurricanes rotating next to each other,
and this makes abundantly clear how the surface of these rings rotates as they travel,
and also, how chaotic they leave the air behind them.
And, as an added bonus, the setup doubles as a great 'death eater'-themed Halloween decoration.
If you do want to try this at home, I should say:
Be super careful with the laser! Make sure not to point it near anyone's eyes.
This concern is especially relevant when the laser is spread along a full plane.
Basically, treat it like a gun.
Also (credit where credit is due), I'd like to point out that after we did this,
we found that the channel NightHawkInLight (great channel)
has a video doing a similar demo. (Link in the description.)
Even though our original plan was to illuminate vortex rings,
I actually think that the most notable part of this visual is how it sheds light
on what ordinary air flow in a room looks like, in all of its intricacy and detail.
We call this chaotic flow 'turbulence', and just as vortex rings give an example of
unexpected order in the otherwise-messy world of fluid dynamics,
I'd like to share with you a more subtle instance of order amidst chaos in the math of turbulence.
First off, what exactly is turbulence?
The term is familiar to many of us as that annoying thing that makes plane rides bumpy,
but naming down a specific definition is a little tricky.
It's easiest to describe qualitatively.
Turbulence involves many swirling eddies, it's chaotic and it mixes things together.
One approach here would be to describe turbulence based on what it's *not*:
laminar flow.
This term stems from the same Latin word that 'lamination' does:
'lamina', meaning 'a thin layer of a material',
and it refers to smooth flow in a fluid, where the moving particles stay largely confined to distinct layers.
Turbulence, in contrast, contains many eddies: points of some vorticity,
also known as positive curl, also known as a 'high swirly-swirly factor',
breaking down the notion of distinct layers.
However, vorticity does not necessarily imply that a flow is turbulent.
Patterns like whirlpools, or even smoke rings, have high vorticity (since the fluid is rotating),
but can nevertheless be smooth and predictable.
Instead, turbulence further characterized as being chaotic,
meaning small changes to the initial conditions result in large changes to the ensuing patterns.
It's also diffusive, in the sense of mixing together different parts of the fluid,
and also diffusing the energy and momentum from isolated parts of the fluid to the rest.
Notice how in this clip, over time,
the image shifts from having a crisp delineation between fog and air,
to instead being murky mixture of both of them.
As to something more mathematically precise,
there's not really a single, widely agreed-upon, clear-cut criterion
the way that there is for most other terms in math.
The intricacy of the patterns you're seeing is mirrored by a difficulty to parse the physics describing all of this,
and that can make the notion of a rigorous definition somewhat slippery.
You see, the fundamental equations describing fluid dynamics,
the Navier-Stokes equations,
are famously challenging to understand.
We won't go through the full details here, but if you're curious,
the main equation is essentially a form of Newton's second law:
that the acceleration of a body times its mass equals the sum of the forces acting on it.
It's just that writing "mass × acceleration" looks a bit more complicated in this context,
and the force is broken down into the different types of forces acting on a fluid,
which again, can look a bit intimidating in the context of continuum dynamics.
Not only are these hard to solve in the sense of
feeding in some initial state of a fluid and figuring out how the equations predict that fluid will evolve,
there are several unsolved problems around a much more modest task,
of understanding whether or not "reasonable" solutions will always exist.
"Reasonable" here means things like
not blowing to a point of having infinite kinetic energy,
and that smooth initial states yield smooth solutions,
where the word 'smooth' carries with it a very precise meaning in this context.
The questions formalizing the idea of these equations predicting reasonable behavior
actually have a \$1,000,000 prize associated with them.
And all of that is just for the case of incompressible fluid flow,
where something compressible, like air, makes things trickier still.
And the heart of the difficulty,
both for the specific solutions and the general theoretical results surrounding them,
is that tricky-to-pin-down phenomenon of turbulence.
But we're not completely in the dark!
The hard work of a lot of smart people throughout history
has led us to understanding some of the patterns underlying this chaos,
and I'd like to share with you one found by the 19th century mathematician Andrey Kolmogorov.
It has to do with how kinetic energy in turbulent motion is distributed at different length scales.
In simpler-to-think-about physics, we often think about kinetic energy at two different length scales:
a macroscale, say the energy carried by your moving car,
or a molecular scale, which we call 'heat'.
As you apply your brakes, energy is transferred more-or-less directly
from that macroscale motion to the molecular-scale motion,
as your brakes and the surrounding air heats up—
meaning all of their molecules start jiggling even faster.
Turbulence, on the other hand, is characterized by kinetic energy at a whole spectrum of length scales,
from the movement of large eddies,
to smaller ones, and smaller ones, and smaller ones still.
Moreover, this energy tends to cascade down the spectrum, where what I mean by that
is that the energy of large eddies gets converted into that of smaller eddies,
which in turn, bring about smaller eddies still.
This goes on until it's small enough that the energy dissipates directly to heat in the fluid
(which is to say, molecular-scale jiggling)
due to the fluid's viscosity (which is to say, how much the particles tug at each other).
Or, as this was all phrased in a poem by Lewis F. Richardson:
"Big whirls have little whirls which feed on their velocity,"
"And little whirls have lesser whirls And so on to viscosity."
Now you might wonder, whether more of the kinetic energy of this fluid
is carried by all of the larger eddies (say, all those with diameter 1 m),
or by all of the smaller ones (say, all those with diameter 1 cm, counted together).
Or more generally, if you were to look at all of the swirls with diameter D,
about how much of the fluid's total energy do they collectively carry?
Is that even an answerable question?
Kolmogorov hypothesized that the amount of energy in a turbulent flow carried by eddies of diameter D
tends to be proportional to D^(5/3),
at least within a specific range of length scales, known fancifully as the "inertial subrange".
For air, this range is from about 0.1 cm up to 1 km.
This fact has since been verified by experiment many times over.
It would appear that 5/3 is a sort of fundamental constant of turbulence.
It's an oddly specific fact, I know, but what I love about the existence of a constant like this
is that it suggests there's some predictability, however slight, to this whole mess.
while viewing 2-dimensional slices of a fluid,
because it is a distinctly 3-dimensional phenomenon.
While fluid flow in 2 dimensions can have a sort of turbulence,
this energy transfer actually tends to go the other way: from the small scales up to larger ones.
So keep in mind, while you're looking at this 2-D slice of turbulence,
it's actually very different in character from turbulence in 2-D.
One of the mechanisms behind this energy cascade (which could only ever happen in 3 dimensions)
is a process known as 'vortex stretching':
a rotating part of the fluid will tend to stretch out, perpendicular to the plane of rotation,
resulting in smaller eddies, spinning faster.
This transition from energy held in a large vortex to instead being held in smaller vortices
would be impossible if there weren't another dimension to stretch in.
Or, if this vortex were bent around to meet itself in a ring shape,
in a way, it's like a vortex that is blocking itself from stretching out this way.
And, as mentioned earlier, this is indeed a surprisingly stable configuration for a fluid;
order amidst chaos.
Interestingly though, when we made these vortex rings in practice
and followed them over a long period of time,
they do have a tendency to slowly stretch out
(albeit at a much longer timescale than the vortex stretching I was just talking about).
Which brings us back to Dianna and Dan.
Huge thanks to the both of them, for getting so much footage and making all of this happen.
[Make sure to hop over to Physics Girl now to see some of the vortex ring demos,]
[and as I said, you'll also get to learn something that happens]
[when you change the shape of the hole in this vortex cannon.]
[The result and its specifics certainly surprised me,]
[and you'll get to hear it through Diana's typical (and infectious) superhuman level of enthusiasm.]