Visualizing turbulence

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<div id="description" class="style-scope ytd-video-secondary-info-renderer">Here we look at some of the order amidst chaos in turbulence. Vortex rings with Physics Girl: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/watch?v=N7d_RWyOv20">https://youtu.be/N7d_RWyOv20</a> Brought to you by you: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?redir_token=Kb_fnkdZ3sxnfNlgfRwhr4I_OS98MTU3MDkzNTY4N0AxNTcwODQ5Mjg3&v=UoTTq651dE&q=http%3A%2F%2F3b1b.co%2Fturbulence-thanks&event=video_description" target="blank" rel="nofollow">http://3b1b.co/turbulence-thanks</a> Home page: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?redir_token=Kb_fnkdZ3sxnfNlgfRwhr4I_OS98MTU3MDkzNTY4N0AxNTcwODQ5Mjg3&v=UoTTq651dE&q=https%3A%2F%2Fwww.3blue1brown.com&event=video_description" target="blank" rel="nofollow">https://www.3blue1brown.com</a> Thanks to Dan Walsh for many great ideas, and thanks to Mike Hansen for many helpful conversations. Error correction: I meant to describe Kolmogorov as a “20th-century mathematician” not “19th-century”. Whoops! I think during the narration I must have made the classic 1900s vs. 19th-century mix up. Anyone aware of his work is more than aware of what century he lived in, which apparently applies to quite a few commenters. Introduction to turbulence: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?redir_token=Kb_fnkdZ3sxnfNlgfRwhr4I_OS98MTU3MDkzNTY4N0AxNTcwODQ5Mjg3&v=UoTTq651dE&q=http%3A%2F%2Fwww.astronomy.ohio-state.edu%2F%7Eryden%2Fast825%2Fch7.pdf&event=video_description" target="blank" rel="nofollow">http://www.astronomy.ohio-state.edu/_...</a> More details on vortex stretching: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?redir_token=Kb_fnkdZ3sxnfNlgfRwhr4I_OS98MTU3MDkzNTY4N0AxNTcwODQ5Mjg3&v=UoTTq651dE&q=https%3A%2F%2Fwww.math.nyu.edu%2Ffaculty%2Ftabak%2Fvorticity.pdf&event=video_description" target="blank" rel="nofollow">https://www.math.nyu.edu/faculty/taba...</a> Video on NightHawkInLight with a similar demo: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/watch?v=K94Cc21KEIA">https://youtu.be/K94Cc21KEIA</a> Music by Vincent Rubinetti: <a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/redirect?redir_token=Kb_fnkdZ3sxnfNlgfRwhr4I_OS98MTU3MDkzNTY4N0AxNTcwODQ5Mjg3&v=UoTTq651dE&q=https%3A%2F%2Fvincerubinetti.bandcamp.com%2Falbum%2Fthe-music-of-3blue1brown&event=video_description" target="blank" rel="nofollow">https://vincerubinetti.bandcamp.com/a...</a> ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. 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<div class="cue style-scope ytd-transcript-body-renderer">The air around you is in constant and chaotic motion,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">replete with nearly impossible-to-predict swirls,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">ranging from large to minuscule.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">What you're looking at right now is a cross-section of the flow in a typical room,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">made visible using a home demo</div>

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<div class="cue style-scope ytd-transcript-body-renderer">involving a laser, a glass rod and a fog machine.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Predicting the specifics of turbulent motion like this has long evaded mathematicians and physicists.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But we are steadily getting closer to understanding some consistent patterns in this chaos.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And in a minute, I'll share with you one specific quantitative result</div>

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<div class="cue style-scope ytd-transcript-body-renderer">describing a certain self-similarity to this motion.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">To back up a bit, I was recently in San Diego</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and spent a day with Diana Cowern a.k.a. Physics Girl, and her frequent collaborator, Dan Walsh,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">playing around with vortex rings.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is a really surprising fluid flow phenomenon,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where a donut-shaped region of fluid stays surprisingly stable as it moves through space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If you take some open container with a lip, and you fill it with smoke (or fog),</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you can use this to actually see the otherwise-invisible ring.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Diana just published a video over on her channel showing much more of that particular demo,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">including a genuinely fascinating observation</div>

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<div class="cue style-scope ytd-transcript-body-renderer">about what happens when you change the shape of the opening.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The story for you and me starts when her friend Dan had the clever idea</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to visualize a slice of what's going on with these vortex rings,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">using a planar laser.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So, you know how if you shine a laser through the fog,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">photons will occasionally bounce off of the particles in the fog along that beam,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">towards your eye, thereby revealing the beam of the laser?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Well, Dan's thought was to refract that light through a glass rod,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">so that it was relatively evenly spread across an entire plane.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Then, the same phenomenon would reveal the laser light along a thin plane through that fog.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The result was awesome!</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The cross-section of such a smoke ring looks like two hurricanes rotating next to each other,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and this makes abundantly clear how the surface of these rings rotates as they travel,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and also, how chaotic they leave the air behind them.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And, as an added bonus, the setup doubles as a great 'death eater'-themed Halloween decoration.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If you do want to try this at home, I should say:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Be super careful with the laser! Make sure not to point it near anyone's eyes.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This concern is especially relevant when the laser is spread along a full plane.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Basically, treat it like a gun.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Also (credit where credit is due), I'd like to point out that after we did this,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we found that the channel NightHawkInLight (great channel)</div>

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<div class="cue style-scope ytd-transcript-body-renderer">has a video doing a similar demo. (Link in the description.)</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Even though our original plan was to illuminate vortex rings,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I actually think that the most notable part of this visual is how it sheds light</div>

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<div class="cue style-scope ytd-transcript-body-renderer">on what ordinary air flow in a room looks like, in all of its intricacy and detail.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">We call this chaotic flow 'turbulence', and just as vortex rings give an example of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">unexpected order in the otherwise-messy world of fluid dynamics,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I'd like to share with you a more subtle instance of order amidst chaos in the math of turbulence.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">First off, what exactly is turbulence?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The term is familiar to many of us as that annoying thing that makes plane rides bumpy,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but naming down a specific definition is a little tricky.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It's easiest to describe qualitatively.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Turbulence involves many swirling eddies, it's chaotic and it mixes things together.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">One approach here would be to describe turbulence based on what it's not:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">laminar flow.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This term stems from the same Latin word that 'lamination' does:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">'lamina', meaning 'a thin layer of a material',</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and it refers to smooth flow in a fluid, where the moving particles stay largely confined to distinct layers.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Turbulence, in contrast, contains many eddies: points of some vorticity,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">also known as positive curl, also known as a 'high swirly-swirly factor',</div>

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<div class="cue style-scope ytd-transcript-body-renderer">breaking down the notion of distinct layers.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">However, vorticity does not necessarily imply that a flow is turbulent.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Patterns like whirlpools, or even smoke rings, have high vorticity (since the fluid is rotating),</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but can nevertheless be smooth and predictable.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Instead, turbulence further characterized as being chaotic,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">meaning small changes to the initial conditions result in large changes to the ensuing patterns.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It's also diffusive, in the sense of mixing together different parts of the fluid,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and also diffusing the energy and momentum from isolated parts of the fluid to the rest.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Notice how in this clip, over time,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the image shifts from having a crisp delineation between fog and air,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to instead being murky mixture of both of them.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">As to something more mathematically precise,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">there's not really a single, widely agreed-upon, clear-cut criterion</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the way that there is for most other terms in math.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The intricacy of the patterns you're seeing is mirrored by a difficulty to parse the physics describing all of this,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and that can make the notion of a rigorous definition somewhat slippery.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">You see, the fundamental equations describing fluid dynamics,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the Navier-Stokes equations,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">are famously challenging to understand.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">We won't go through the full details here, but if you're curious,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the main equation is essentially a form of Newton's second law:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that the acceleration of a body times its mass equals the sum of the forces acting on it.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It's just that writing "mass × acceleration" looks a bit more complicated in this context,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and the force is broken down into the different types of forces acting on a fluid,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which again, can look a bit intimidating in the context of continuum dynamics.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Not only are these hard to solve in the sense of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">feeding in some initial state of a fluid and figuring out how the equations predict that fluid will evolve,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">there are several unsolved problems around a much more modest task,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of understanding whether or not "reasonable" solutions will always exist.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">"Reasonable" here means things like</div>

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<div class="cue style-scope ytd-transcript-body-renderer">not blowing to a point of having infinite kinetic energy,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and that smooth initial states yield smooth solutions,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where the word 'smooth' carries with it a very precise meaning in this context.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The questions formalizing the idea of these equations predicting reasonable behavior</div>

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<div class="cue style-scope ytd-transcript-body-renderer">actually have a $1,000,000 prize associated with them.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And all of that is just for the case of incompressible fluid flow,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where something compressible, like air, makes things trickier still.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And the heart of the difficulty,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">both for the specific solutions and the general theoretical results surrounding them,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is that tricky-to-pin-down phenomenon of turbulence.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But we're not completely in the dark!</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The hard work of a lot of smart people throughout history</div>

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<div class="cue style-scope ytd-transcript-body-renderer">has led us to understanding some of the patterns underlying this chaos,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and I'd like to share with you one found by the 19th century mathematician Andrey Kolmogorov.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It has to do with how kinetic energy in turbulent motion is distributed at different length scales.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In simpler-to-think-about physics, we often think about kinetic energy at two different length scales:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a macroscale, say the energy carried by your moving car,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">or a molecular scale, which we call 'heat'.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">As you apply your brakes, energy is transferred more-or-less directly</div>

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<div class="cue style-scope ytd-transcript-body-renderer">from that macroscale motion to the molecular-scale motion,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">as your brakes and the surrounding air heats up—</div>

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<div class="cue style-scope ytd-transcript-body-renderer">meaning all of their molecules start jiggling even faster.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Turbulence, on the other hand, is characterized by kinetic energy at a whole spectrum of length scales,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">from the movement of large eddies,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to smaller ones, and smaller ones, and smaller ones still.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Moreover, this energy tends to cascade down the spectrum, where what I mean by that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is that the energy of large eddies gets converted into that of smaller eddies,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which in turn, bring about smaller eddies still.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This goes on until it's small enough that the energy dissipates directly to heat in the fluid</div>

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<div class="cue style-scope ytd-transcript-body-renderer">(which is to say, molecular-scale jiggling)</div>

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<div class="cue style-scope ytd-transcript-body-renderer">due to the fluid's viscosity (which is to say, how much the particles tug at each other).</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Or, as this was all phrased in a poem by Lewis F. Richardson:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">"Big whirls have little whirls which feed on their velocity,"</div>

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<div class="cue style-scope ytd-transcript-body-renderer">"And little whirls have lesser whirls And so on to viscosity."</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now you might wonder, whether more of the kinetic energy of this fluid</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is carried by all of the larger eddies (say, all those with diameter 1 m),</div>

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<div class="cue style-scope ytd-transcript-body-renderer">or by all of the smaller ones (say, all those with diameter 1 cm, counted together).</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Or more generally, if you were to look at all of the swirls with diameter D,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">about how much of the fluid's total energy do they collectively carry?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Is that even an answerable question?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Kolmogorov hypothesized that the amount of energy in a turbulent flow carried by eddies of diameter D</div>

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<div class="cue style-scope ytd-transcript-body-renderer">tends to be proportional to D^(5/3),</div>

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<div class="cue style-scope ytd-transcript-body-renderer">at least within a specific range of length scales, known fancifully as the "inertial subrange".</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For air, this range is from about 0.1 cm up to 1 km.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This fact has since been verified by experiment many times over.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It would appear that 5/3 is a sort of fundamental constant of turbulence.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It's an oddly specific fact, I know, but what I love about the existence of a constant like this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is that it suggests there's some predictability, however slight, to this whole mess.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">There is something ironic about talking about this energy cascade</div>

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<div class="cue style-scope ytd-transcript-body-renderer">while viewing 2-dimensional slices of a fluid,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">because it is a distinctly 3-dimensional phenomenon.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">While fluid flow in 2 dimensions can have a sort of turbulence,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this energy transfer actually tends to go the other way: from the small scales up to larger ones.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So keep in mind, while you're looking at this 2-D slice of turbulence,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it's actually very different in character from turbulence in 2-D.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">One of the mechanisms behind this energy cascade (which could only ever happen in 3 dimensions)</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is a process known as 'vortex stretching':</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a rotating part of the fluid will tend to stretch out, perpendicular to the plane of rotation,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">resulting in smaller eddies, spinning faster.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This transition from energy held in a large vortex to instead being held in smaller vortices</div>

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<div class="cue style-scope ytd-transcript-body-renderer">would be impossible if there weren't another dimension to stretch in.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Or, if this vortex were bent around to meet itself in a ring shape,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in a way, it's like a vortex that is blocking itself from stretching out this way.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And, as mentioned earlier, this is indeed a surprisingly stable configuration for a fluid;</div>

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<div class="cue style-scope ytd-transcript-body-renderer">order amidst chaos.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Interestingly though, when we made these vortex rings in practice</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and followed them over a long period of time,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">they do have a tendency to slowly stretch out</div>

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<div class="cue style-scope ytd-transcript-body-renderer">(albeit at a much longer timescale than the vortex stretching I was just talking about).</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Which brings us back to Dianna and Dan.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Huge thanks to the both of them, for getting so much footage and making all of this happen.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">[Make sure to hop over to Physics Girl now to see some of the vortex ring demos,]</div>

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<div class="cue style-scope ytd-transcript-body-renderer">[and as I said, you'll also get to learn something that happens]</div>

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<div class="cue style-scope ytd-transcript-body-renderer">[when you change the shape of the hole in this vortex cannon.]</div>

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<div class="cue style-scope ytd-transcript-body-renderer">[The result and its specifics certainly surprised me,]</div>

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<div class="cue style-scope ytd-transcript-body-renderer">[and you'll get to hear it through Diana's typical (and infectious) superhuman level of enthusiasm.]</div>

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<div id="label-text" class="style-scope yt-dropdown-menu">English</div>

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Visualizing turbulence

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