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. There is something ironic about talking about this energy cascade 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.]

Thanks to those who caught my speako describing Kolmogorov as a "19th-century" mathematician. Of course, I meant the 1900's. His work is quite recent and remains relevant to a number of active research efforts in a surprising breadth of fields. It's crazy to me that his name is as relevant to fluid dynamicists as to those studying machine learning.

I see 5/3 I think golden ratio. Has anyone tried to see what happens when the exponent is phi?

That 'pressure force' sure looks a hell of a lot like pressure energy.

mmmm turbulent juice

8:34 eddies* 😛

Nice Video

Hmm… 5/3 is pretty close to the golden ratio

This whole video was really rough on Youtube's compression. Apparently even video algorithms have trouble characterizing turbulence.

it's looks like patterns of gasoline on water) it's beautiful!

This is just what I needed. Thank you so much. Looking forward to more turbulence, diffusion (DLA), complexity and fractals…in mathematics 🙂

I was able to see this through my window and the sun

Can you make laminar air in room 🤔

I'm reading a book by Benoit Mandlebrot, and there is a complete chapter on turbulence. Mandlebrot studied turbulences in terms of fractal, because of the self-similarities and 5/3 non-integer dimension !

😍 Really enjoyed your video! 👏 👏 ✌

Could you talk about continuum mechanics?

With a rotating or moving laser and a high speed camera, you could map the entire volume of turbulence. You could get some very interesting and complete data with that.

Awesome video. So one question. You mentioned kinetic energy contained in an area with diameter D. If the velocity field is v=xi+yj is the kinetic energy contained double integral (x^2+y^2)/2 dxdy divided by piD^2/4?

Vape Naysh

I'm really curious where the animation at 3:45 came from. A cambered airfoil like that produces lift at zero angle of attack so shouldn't the streamlines coming of the trailing edge be angled down?

Is it also like tornadoes, like they have supercell vortexes that perpendicularly form?

That would be such a good animated background, especially if itself was random.

Can’t you not even use the Navier Stokes equations if the flow is turbulent? I assume you would use some variant, but what would that even look like?

I know a rap song where the poem by Lewis F. Richardson is quoted. Never got the reference and always thought it was weird. Finally I get it. mind=blown moment

Am I the only one who saw the wake of the vortex ring and immediately thought of stuff like gliders in The Game of Life?

Any paddlers watching?

Can someone make a background for phones and/or desktops of the images shown in this video? I think it would be amazing

great for beginners!

Since you did smoke flow visualization, a video on PIV (Particle Image Velocimetry) would be quite cool to show people how scientists actually measure real flows using the same setup you had!

Please bear in mind that every time you mention "Navier-Stokes" there is a sleeping mathematician or physicist somewhere on the planet that listens to it in his/her dreams and immediately wakes up looking for answers.

Just a thought…

Due to the fractal nature of the physical laws, could matter be nothing more than a vortex ring of energy propagating in the fabric of space time?

Also enthusiastic, with a calm intensity..

A thin slice of Superspin dimensionality.

eddies make me think of "gliders" in cellular automata simulations.

My theoretical physics Professor at University introduced the NV equation like this: there are three fundamental laws of conversation in mechanics: 1.) Energy (time symmetry), 2.) Momentum (translation symmetry) and 3.) Angular momentum (rotation symmetry). Then he said: the 0-th conservation law is: conservation of complication (not complexity) 🙂 The reason the NV is so complicated as as the term v*grad(v) is of order v² and hence non-linear. So we have a set of strongly (non-diagonal) coupled, non-linear partial equations of second order. And it is a simple fact: so far mathematics has not found a general theory of "addressing" these types of equations. Even numerical simulations are challenging due to the non-linear terms inherently causing deterministic chaos. That is: tiny approximation errors can cause of short time massive deviations (numerically instable). And apparently this complication just doesn't go away easily. There is no set of rules how to win Go! You can only win with intuition.

I would love to see a video deriving these equations, ideally with viscosity being considered being of tensor nature. Thank you for all the outstanding videos!

Thank you so much for your amazing videos!!!! I use them for wonderful visualizations to assist in teaching my children some of my deepest passions. Not only are your visualizations beautiful, they do indeed appear to assist in that 'deeper understanding' you express. I recommend your channel to many other parents and want you to know how much I appreciate the hard work you clearly put into making these videos.

5:20 mah-ass

there's nothing much about turbulence here

Cool stuff! I also did this experiment last january and have a video on my channel, albeit in lower quality. If you can shoot two consecutive rings, you can make the last ring go through the first one, which is called leap frogging. Laser planes are a very cool way to visualize flows!

"Eddies in the time space continuum"

"Ah, is he?"

Absolutly beautiful!

Big whirls have little whirls that feed on their velocity,

and little whirls have lesser whirls, and so on to viscosity – Lewis Fry Richardson 😀

this is arguably better than the fleas one

Doesn’t the 5/3 constant come from the free movement ranges of particles?

5/3 is kind of close to 1/2pi

Coincidence?

You should make a video on the Navier Stokes Equation along with Reynolds Averaging and all that other good stuff 🙂

Math, Science, and Vape.

Looks like a good way to generate a truly random number

Who is here after watching Diana’s(physics girl) video on square vortex rings. I am!!😃

Andrey Kolmogorov lived in 20th century (1903-87).

This is probably one of the best explanations I have seen of the turbulent cascade and a beautiful visualization. I actually also do fluid mechanics visualizations, and I think you might like them too – https://youtu.be/Cdx8aLoB37Q. Cheers 🙂

Just for clarity, 3:34 depicts either inviscid flow or is missing the Kutta-Condition at the trailing edge of the airfoil. The flow leaving the trailing edge of the airfoil should be parallel to the slope of the airfoil at that location.

EDIT: The stagnation point seems to be just on the upper surface of the airfoil (just barely in front of the trailing edge) In order for the stagnation point to be at this location, the flow must accelerate infinitely around the cusp of the airfoil. The Kutta Conditions ensures that the stagnation point is on the trailing edge.

What happens if you shine the laser through a fisheye lens instead of a cylindrical rod? Googled but didn't find anything informative. Just curious if you could get a 3d view.

Both yours and Dianas' videos are my favourite part of YouTube!

Kolmogorov lived in the XX century, not XIX (6:43).

or vape

I once saw a video with a 3 different laser frequencies passed through a prism and then flattened into a cone by a rotating disk. The result was 3 very nice slices of airflow all visible near each other all at once.

I really liked the video, but the clickbait-level of advertising other channels was a total deal breaker :/

That would be an awesome screensaver

watches the videohmmmmm this is really interesting i wanna learn more

clicks the vortex stretching link in the description……aaaaaaand i think i'm done here

so much quality content in just 10 mins

Please do some work on chaos theory

These people are way too smart lol

Can I know why glass rod makes razer to plane?

And this is why weather is so hard to.predict.

Can you do a video on the Reynolds number, please?

5/3… pretty much like phi.

while smoking some people used to do this by creating rings in smoke

8:40 Shouldn't it be D^{-5/3} here, and decreasing slope instead of increasing?

can you talk about taylor microscale? I never really understand that part…

Wait… Are you saying that people are looking for an equation to describe the motion of a several moles of dozens of different gasses interacting in an infinite mix of temperatures, pressures, and densities, and electrical charges, through and around various bodies of infinitely varying shapes?

Yeah… Best luck with that.

@3blue1brown I really appreciate your lexicon but every sentence i have to switch to the vocabulary to look up some word. I took one hour to crunch this eleven minutes, howsoever it was worth that. 👍

The order in chaos instantly made me think of Jordan B Peterson, nice video

👍

What you said about the kinetic energy and eddies becoming smaller eddies related to the CFD turbulent model k-epsilon. k is turbulent kinetic energy and epsilon will be turbulent dissipation rate as described in 7:35 . I still don't know how people derived the equations for those two quantities but I think it has to do with expanding the Naiver-Stokes equations. Honesty, the 5/3 proportion simplifies TKE so much so I can actually visualize relationships without actually calculating them.

You should go further and make the exit hole square but 3 dimensional, with the corners pointing inwards. To hopefully make the entire ring come out at the same time and stay square.

I was waiting through the whole video for the ‘pi’ to come in 😑. Where there is pie, there are circle =====> vortex

Would love more physics videos! You will radically change the physics education space fi you fully commit to it.

The kolmogorov theory is valid only for homogeneous isotropic turbulence which is an idealized version of real turbulence.

Amazing job you do.

@3blue1brown Record a 3D fluid dynamics environment by creating a large array of laser slices that are timed to pulse in very small time intervals effectively scanning the environment. Recording this with a high speed camera and good resolution could allow that data to create a very precise model of the fluid dynamics environment.

A few years back I thought to actually do some proper revision of the math I used to know about N-S, (because it's fun to imagine mostly), but the more I think about it the more I remember why I'm not academic material of that nature, (a bit "agricultural" a friend once described it, he was talking about a Harley Davidson, but you get the picture).

So taking the long way around to redefine the conception from first principles is Actuality-> inevitably, if you want to know why the N-S Equation is a thing of real elemental meaning. The videos are very good demos of the transverse/cross-sectioning layout of the wave-package, at this particular scale/relative spin-resonance imaging rates for which the little whirly poem applies.

Having just watched a video on the likely origins of living organisms on Earth, and seeing how this equivalent, functional transverse snapshot of bio-logical chemistry operates, the idea of math, resonances in harmonics and reciprocals for all prime numberness of multi-phase-synched moving flow of the turbulence is daunting. At best, maybe some chunking of emulated/synthetic chemistry will be a solution, (abstracted pictures within pictures), using atoms as vertices in the Quantum Fields Modulation Mechanism of QM-Time Principle In-form-ation formulae in the equivalent time-duration conception, (case by case?)

That's why Science is a process, not a thing, and all practitioners and practices are transferable.

Le suplico a cualquier alma caritativa que lo traduzca al español

This might be ignorant, but could you use a neural network to find patterns in order to solve the turbulence problem for incompressible flow?

Love u

this is also a cross-sectional view of the magnetic field of a conducting metal ring lol

Can someone explain to me the cause of the two hurricane swirls bear the beginning?

Vórtex streching is mentioned in connection with tornado formation.

physics girl sucks. watched her vertexes MUTED

rectangular vortex OSCILLATES! i see an analogy with perpendicular pendulums.

Great video explaining such an intricate subject! Note at 8:40 : Kolmogorov only really worked in real space (as opposed to Fourier space) so the energy distribution as a function of D actually scales with D^2/3 – Fourier transform that and you get the more well known k^-5/3.

Still waiting for more videos about fluid dynamics and the million dollar problem

I wish some references in this video were developed in detail in another video, the problems with the navier-stokes equationand the problems with estabilishing the exact mathematical description of turbulence .

i studied at kolmogorov's school in moscow and i am exited seeing him in your video!

Chaos is everywhere and its better

5:38 if the equation for an incompressible gas is hard to understand I don't want to imagine what's used for compressible gases.

Your videos are very very great. As I am studying aviation engineering I have to learn a lot about turbulent flow so thank you for making this video!

Good background music.

People have been making vortex rings for a long time with shisha smoke 😛

Spent the summer working with a technique called Planar Laser-Induced Fluorescence, which utilizes passive flow of a fluorescent substance to visualize scalar transport phenomena (i.e. concentration). I'm excited to see you drawing attention to this useful method. Coupled with velocimetry measurements, planar lasers are unstoppable!

I heard that a big problem with fluid dynamics is that the Navier-Stokes equations are actually incomplete. This was discovered, IIRC, less than 10 years ago.

Hello, guys. I'm a engineering student, and I need do some lasr experiments to my graduation thesis. What kind of laser do you use? Need some security glass? I will apreciate if you could help, thanks

https://paveldogreat.github.io/WebGL-Fluid-Simulation/