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, aka 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 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 Nighthawk and Light has a video doing a similar demo, link in the description. Even though our original plan was to illuminate vortex rings, I actually think the most notable part of this visual is how it sheds light on what ordinary airflow 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 nailing 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 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 is 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 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 a 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 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 times 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 quote-unquote reasonable solutions will always exist. Reasonable here means things like not blowing up 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 million 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 Andrei Komagorov. 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 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 whorls have little whorls which feed on their velocity, and little whorls have lesser whorls, 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 meter, 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 a diameter d, about how much of the fluid's total energy do they collectively carry? Is that even an answerable question? Komagorov hypothesized that the amount of energy in a turbulent flow carried by eddies with diameter d tends to be proportional to d to the power 5 thirds, at least within a specific range of length scales known fancifully as the inertial subrange. For air, this ranges 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 thirds 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 mass.

There is something ironic about talking about this energy cascade while viewing two-dimensional slices of a fluid, because it is a distinctly three-dimensional phenomenon. While fluid flow in two 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 2d slice of turbulence, it's actually very different in character from turbulence in 2d. One of the mechanisms behind this energy cascade, which could only ever happen in three 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 which 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 time scale than the vortex stretching I was just talking about. Which brings us back to Diana 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 about 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.