How do fish swim so quickly?

Featuring Tadashi Tokieda... Get 60% off your first box from MEL Science at melscience.com/sBEz/ and use code NUMBERPHILE60 (episode sponsor)
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Tadashi Tokieda is a professor of mathematics at Stanford University, specialising in applied mathematics.
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[@numberphile]

Get 60% off your first box from MEL Science at melscience.com/sBEz/ and use code NUMBERPHILE60 (episode sponsor) More videos with Tadashi: bit.ly/tadashi_vids

[@ImranAli-xj1on]

Ooooo

[@Omegashotgun]

ah! this man gave me the physical metaphors to understand alan turing's unfinished work! i adore seeing this sort of content :)

[@annaclarafenyo8185]

When you have a video detailing an original discovery, rather than exposition, I think you should mention it somewhere. Is this idea for how fish swim new?

[@Bibibosh]

English please. Sorry my fan was on I thought he was speaking japaneese !Japenese . Japanise japan eeeeezeee

[@markruffing3461]

B

[@bighugejake]

This is the most easily-digestible fluid dynamics lesson I've ever watched. Probably because of how perfect the demos and explanations are. Tadashi is awesome.

[@caniggiaful]

Help a layman. I don't quite get why the momentum of each vortex is not zero. Why pushing "in the middle" in a sense, more important than, on the outside. I mean, he is clearly right, but what am I missing?

[@foobargorch]

@@caniggiaful the explanation at 12:00 was a bit confusing to me as well, but i think it's mainly about how the vortices dissipate the energy. the inner region of the wake can have net momentum which i think it what is described as backwards and forwards momentum, and even though on the whole there is net zero momentum, the question is as a particle gains and loses energy between when it is first displaced and when it finally settles into its new location (ignoring things like Brownian motion), as the vortices dissipate the overall path taken by such a particle does have a net displacement, and the key is how the vortices interact with each other to affect this average path. the more net displacement in the direction opposite to the motion of the organism (at opposed to e.g. to the sides) the less energy the organism has to expend imparting momentum on to particles in the fluid. intuitively maybe it helps to think of where did a particle originates from when settling in the wake. in both cases particles in front of the organism are pushed aside, and others rush in to you the void. but if these particles intact in such a way that on the whole it's more likely that a particle ends up with a net displacement along the path of the organism then that is less net momentum propelling the organism forward. momentum is conserved either way of course, but the question is what proportion of the energy end up propelling particles in random directions. the cleaner the exchange in location between the organism and the fluid particles it displaces, the less energy is lost, and if the vortices reinforce each other so that the inner part of the wake tends to flow backwards, the closer the overall path length traveled by a particle is to its net displacement

[@benp2291]

@@caniggiaful This wasn’t the easiest thing for me to realise when I first learnt about vortex momentum. Firstly you have to remember that the vortices aren’t fixed in position, so they are free to move. Then, because of the flow induced by the vortices it will drag the others with it. So we don’t need to worry about what it’s doing on the outside because it doesn’t affect the set up. I seem to remember doing calculations of n vortices arranged in an n sided polygon and lots of things cancelled out. But I do clearly remember having two vortices which spun in the same direction, and ones which spun in opposite direction. In the former case, because the vortex would drag it around they end up circling each other. The latter is the one in which we observed and would work together to push each other along

[@kindlin]

@@caniggiaful I think a simpler explanation has to do with the water in-between the vortices, not the vortices themselves. The vortices have inherent momentum due to their rotation (Angular Momentum), but each vortex is paired with an opposite, so their net affect on the system is, I believe, relatively small. Now, take his setup at 11:46 (submarine), and look at the water in between the vortices. Due to the rotation of the vortices, the middle water is being pushed forwards, with the forward motion of the sub. So, the engines in the submarine is not only pushing the submarine itself through the water, but also the large mass of all the water that it encounters along the way. Granted, the sub is going maybe 12 knots, and the water in it's wake is going at most a knot or 2 forward, but that is a huge mass of water to be constantly accelerated forward. All of the water also moving forward is sapping the forward momentum that could otherwise go to the submarine. For a fish (setup @ 12:21), the water in between the vortices is actually going backwards, against the forward travel of motion, so the total forward power of the fish is not only from the forces it's personally creating from quickly slapping it's tail back and forth (think doggy paddle), but also the net momentum of all the water it's leaving behind that's now going in the direction the fish came from. It's like an elastic vs a plastic collision, which is like a bouncy ball vs a ball of play-dough. If you threw a ball of playdough at a scale and added up all the force it feels over the duration of the playdough squashing into the scale, you would be directly measuring the momentum of the hunk of playdough that you through. Now, take the bouncy ball, it not only contacts the scale and comes to a complete stop, rather quickly I might add, but it then pushes off the scale with a nearly identical force (minus friction), and bounces back at you; with a perfectly elastic collision, you get double the momentum transfer.

[@X22GJP]

Sorry, but the explanation is flawed/not complete, and no, I'm not that well versed in fluid dynamics! It dumbs it down to explain why they observe this alternating pattern by showing vortices rolling off one side, then the other side...rinse and repeat. Anybody with half a brain can understand that water would be "rolling" off the surface a full 360 degrees around it. In a hypothetically perfectly stable and symmetrical environment where the vortices and net displacement all around are absolutely identical, the net effect would be zero and the ball would go straight up. But, we don't live in such a perfect, stable world, so tiny deviations and movements in the water cause these "butterfly effects" and result in more turbulent behaviour.

[@soupappreciator8977]

Tokieda must be a great parent, how he refers to his son as thinking about this together with him. These are precious connections.

[@vigilantcosmicpenguin8721]

He knows how to teach and learn from his child.

[@pabloagsutinnavavieyra2308]

Actually is fascinating what kids can perceive and understand when given the opportunity. I don't have a doubt that they came to it, obviously having Tadashi the knowledge to theorise on this common insights.

[@NoPrefect]

It's the sign of a genuinely curious mind that loves his family and his field. Discovery is often a very intimate experience, and having a heart that is open helps us to perceive the world around us with fresh eyes.

[@bhzucker]

Professor Tokeida is my favorite Numberphile guest, a supergiant among stars. So glad to see him back after all these years!

[@LeoStaley]

He's one of the few that hold a candle to Cliff Stoll, in my book.

[@zh84]

You have often interviewed a hydrodynamicist (Tom Crawford), who could have explained this in terms of the Navier-Stokes equations, but I doubt it would have occurred to him to use a cuddly toy fish and paper discs with arrows on them.

[@punpckldqd4322]

Personally, I think it would be great to see Tom's explanation as well. Now I kind of wish Numberfile would make a follow up video with Tom. In general, it can be quite interesting to see different ways and methods of exploring some topic.

[@vigilantcosmicpenguin8721]

Both very valid ways of explaining the same concept.

[@romanski5811]

I hope you're vaccinated, zh84.

[@davidchapman7667]

"The following experiment was suggested by my son." Love It! It's the old adage that the only difference between an experiment and playing around is documenting your results! It is with this line of thinking by this son's father, that this boy will grow up knowing HOW to learn. Kudos to Dad!

[@Syntax753]

That would be ridiculous - he's actually 4 so that of course explains it :p

[@toolbaggers]

This is how little boys that torture insects become serial killers. Constant guidance at a young age is crucial to human development.

[@whitcwa]

Young children are scientists. They are forming hypotheses about how the world works and perform experiments to test them. When a baby drops their food onto the floor, they are testing a theory of physics and of behavioral science as they watch their parent's reaction.

[@martijn8554]

@@whitcwa and the whole time the kids are thinking: how can I (ab)use this to my advantage :)

[@christianbarnay2499]

@@Syntax753 He was 2 at the time he first conceived the experiment and started working on the subject. Then it took 2 years of collaborative effort to reach those conclusions. Now he has one year left to finalize and publish his PhD thesis.

[@mateuscarvalho5959]

I have a PhD in aeronautics and I'm amazed by this video. A lecture on science dissemination, congratulations!

[@Hahahahaaahaahaa]

I would watch Tadashi explain literally anything.

[@johnchessant3012]

Can't believe I never thought about how fish are able to swim like that until now. We have unlimited knowledge on the internet which makes asking the right questions even more important. Meanwhile I foresee great things in Tadashi's son's future.

[@pansepot1490]

Instead I have often wondered about that. Watched a ton of videos of dolphins, orcas and whales and I was amazed at the things, the jumps especially, they can do. The BBC has amazing clips from their documentaries of dolphins surfing at high speed in shallow waters and almonds beaching themselves to herd and catch fish. I could have thought it was unbelievable if I wasn’t watching actual footage. This video goes a long way to explain how they can do those incredible acrobatics.

[@vigilantcosmicpenguin8721]

And the crazy thing is, the fish don't even think about it at all!

[@christianbarnay2499]

@@vigilantcosmicpenguin8721 That you don't know. Maybe they use a lot of brain power to solve Navier-Stokes and fine-tune their wiggling to perfectly adjust to the local temperature, pressure, viscosity, etc.

[@Kycilak]

@@christianbarnay2499 Well, their brain has to do something so they could swim. But one does not exactly have to think about walking, so it is reasonable to assume that fish don't think about swimming.

[@christianbarnay2499]

@@Kycilak When you walk you solve linear equations in a 2D space. When you swim or fly you solve differential equations in a 3D space.

[@LeoStaley]

I'm a simple man. I.see Tadashi Tokieda and I click and watch the whole thing. He basically makes Numberphile into Physicsphile

[@grayaj23]

My other current KZbin passion is Formula 1 racing, which has a very different "porpoising" problem right now. Cars develop so much ground force that they start oscillating / bouncing up and down. So I hear "Porpoising" and think someone is making a joke until I realize this has nothing to do with F1.

[@user26912]

Is it another force, or also related to vortexes but then in the air?

[@dnikiforovskii]

@@user26912 It is an oscillation of the ground clearance caused by the ground effect (Venturi effect) and car suspension acting against each other. They should cancel each other, but due to dynamic nature of racing (and bad engineering) this situation leads to oscillation. It is called porpoising due to visual similarity of the car going up and down and porpoises jumping out of the water.

[@user26912]

@@dnikiforovskii thanks!

[@Erowens98]

​@@dnikiforovskiiits mainly an issue with the dampers being optimized for a specific level of downforce at a specific speed. Active damping could potentially solve the issue but regulations are in the way. That said, the control system required for such active damping would be quite difficult technically.

[@CaptainHandsome]

I like to think that he was treating it like a proper experiment. Like he was all "I'm going to adjust these variables and see what happens" "Goo goo ga ga" "An interesting hypothesis to be sure..."

[@vigilantcosmicpenguin8721]

Gotta start them off young.

[@woodfur00]

Hey, some two-year-olds speak in full sentences, even if his son wasn't at that level I doubt it was pure babbling.

[@frankhooper7871]

​@@woodfur00Yep - according to my Mum, I was relatively coherent at just over 2. A lot depends on how parents communicate with their children; speak to your child using typical "baby talk" and expect slower development; speak to them using proper language (and frequently!) and voilà.

[@Stormorbiter]

This, the submarine model, is also how drafting works in cycling! It goes beyond just hiding from the air resistance; it actually helps propel you forward

[@VAXHeadroom]

I never though about the tail motion swapping the vortices. I also love the bathtub demo with the shallow water. Great demonstrations!

[@aceman0000099]

I wonder why modern boats don't seem to make use of this property, it could make them more efficient it seems

[@metallsnubben]

@@aceman0000099 I feel like the big problem is first of all making the boat flexible without breaking/wearing out too fast, and also that you want to be able to have people inside that don't get insanely seasick from wiggling lol

[@Triantalex]

??

[@oakspines7171]

No wonder the tail side flexing did not seem to be that vigorous to produce thrust at all. It turns out to be a vortices guide rather.

[@oakspines7171]

We divers and snorkelers use the up and down flexing of the legs together and that seems to do samething.

[@Ankzar13]

"Porpoising" as a Formula One fan, seems this word is following me everywhere I go lol

[@integra097]

haha I was thinking the same thing!

[@omgawesomeomg]

I had to do a double take when he said that. I have never heard this word in my life and now it's everywhere.

[@fletcherb4728]

Bono, my vortexes are reversed.

[@slavanossar]

@@omgawesomeomg You are experiencing what is known as the "frequency illusion" (more commonly known as the Baader-Meinhof phenomenon). It's the tendency for newly learned terms or concepts to suddenly appear everywhere, due to a combination of how our brains filter relevant information, plus a little bit of confirmation bias.

[@sheilakijawani2526]

@@omgawesomeomg that's a syndrome idk the name but its like when you're thinking of buying a car (some model) from a certain company..then on roads you are suddenly start to observe more of that same car you thought of buying.

[@condor6222]

Just when I thought I'm moving on from F1, this man brings up porpoising

[@R.S.R.S.]

I was searching for a comment about that efect. ✌

[@user26912]

Haha first thing I found when I googled "porpoising". So it's also a factor at play in the aerodynamics of Formula 1 cars. Maybe they need moving fins.

[@condor6222]

@@user26912 F1 banned moving aerodynamic parts, but porpoising recently started appearing since they've introduced the ground effect on the floor of cars. Really interesting stuff if you're interested in the physics

[@Oscee613]

@@user26912 Just a note, it's a different effect on the racecars. But it results in (unwanted) bouncing of the car, hence the name porpoising.

[@NerdBowtie]

Tadashi!!! I missed him so much. His videos are always the best

[@ResidentEvil9922]

The return of the most enthusiastic (and lovable) mathematician.

[@johnlewis2930]

I think Cliff Stoll just edges him for enthusiasm but it’s close

[@minte1972]

@@johnlewis2930 That's not really fair, Cliff edges most people for enthusiasm about anything.

[@mastershooter64]

that's cliff stoll!!!

[@Arikayx13]

I worry if the calm energy of Tokeida and the manic energy of Stoll were to ever meet, a singularity would form and the universe end.

[@minte1972]

@@Arikayx13 The true ending

[@timharris3292]

The Professor's home seems a beautiful, happy place. Full of toys and family and books and a piano. He's one of my favourite numberphile presenters.

[@Liero_]

I experienced this effect when I was bouncing a baloon with friends back in the day. When I tried to throw it as hard as I could, it would wiggle in the air and stop very quickly covering little distance. Now I know why it happened :D

[@hemanthsai4584]

oh fark.... didn't saw that coming 🤔🔥

[@bugglest0n]

My favorite episodes are always the ones with more than just the brown paper. Don't get me wrong, the pure paper videos are great too, but the real props and demonstrations bring it to another level. Fascinating!

[@vigilantcosmicpenguin8721]

It's some real applied mathematics.

[@charstringetje]

Cliff Stoll's pizza, or conic bread. We use maths so much to describe the real world. Every once in a while the real world has to return the favour. And sometimes it comes full circle, like when minutephysics uses a mechanical device to show Lorentz transformations, which in turn show what happens in special relativity.

[@gasdive]

That's a great explanation. I can feel that vortex shedding when I swim with fins. There's a little flick I give with each finstroke. It feels almost like I'm standing on a hard bit of water rather than having the water just move out of the way of my fins.

[@cuhy3406]

I feel that fins because of the bigger surface area, make the water feel more solid and more force exerting on resulting in more propulsion like our legs pushing on ground.

[@gasdive]

@@cuhy3406 it's more than that.

[@leonardm6858]

Reminds me of how Richard Feynman once spent hours analyzing the motion of breaking pasta sticks.

[@AylaTheQueenIdk]

I think one of the many reasons Tokieda is such a great teacher is because there is clear motivations behind processes and concepts. Amazing video as usual

[@thehappyhobbyist8980]

The legend is back!

[@boRegah]

I find it's a nice touch that he explained the physics in his bathtub and on the tiny children's table.

[@cubingtubing8172]

Another one featuring Takashi Tokieda! I love the episodes with him!

[@nicodemosvarnava2520]

These are complex phenomena and yet he has this clear and concise framing. I love his explanations, simply beautiful! he awakes that childhood wonder!

[@esce69]

Wow, one of your most fascinating contributions lately. Thank you professor, for explaining this mechanism so clearly. I often wondered how fish move so fast with seemingly little effort. If you didn't know, now you know!

[@annaclarafenyo8185]

I don't think he is "explaining" anything, I think this is an original discovery.

[@Bluhbear]

@@annaclarafenyo8185 I'm not sure what you're disputing. Even if it's an original discovery, he's still definitely explaining it throughout the whole video. 🤔

[@annaclarafenyo8185]

@@Bluhbear It's not a dispute, I am trying to give him credit. This is a pretty big deal in fluids, on par with Einstein's paper on the circulation of water in a teacup and riverbed sedimentation. It's really tough to produce qualitative insight in this field. Most people just do exposition.

[@douche8980]

Didn't watch video but some things in life are easy to understand by observing the factors behind why such happens which in this case is that fish weigh very little and experience very little resistance in water as opposed to land animals.

[@Bluhbear]

@@annaclarafenyo8185 Ok, fair. I just construed it as combative, but couldn't figure out why. 😅 Tbf, I feel like the OP is praising him pretty well, though.

[@Leeisateam]

Do the fish deliberately time their tail flicks to be in sync with the created vortices? Or do the tail flicks occur in sync due to the physical properties of the fish, like the flexibility and length of the tail? Are larger fish like sharks able to exploit this effect too, or are their tails too big and heavy to flick at a fast enough frequency?

[@bananatassium7009]

wow, that's really interesting! as someone who has never studied how fluids work, this introduced an interesting question I never thought about and then gave it a fascinating answer. great explanation!

[@therocknrollmillennial535]

This was the most interesting Numberphile video I've watched, and I've been a loyal viewer for several years now. Nothing against any of the other brilliant minds that have appeared but I was just absolutely taken by this description. And when I stop to think about it, it makes so much sense. Thank you, Numberphile, for helping this twentysomething regain a curiosity and love for the world that I'd lost for a really long time.

[@jamebo9390]

This is the most excited I've ever been for a numberphile video - Tadashi is really wonderful. ありがとう、先生!

[@curtiswfranks]

This sounds like a valuable lesson for submarine (and aerospace, and even car/bus/train) engineers, and Olympic swimmers.

[@jim_jim1674]

Just an amazing demo of how science was established, we live, we try something, we observe, we try and error, we think it seriously, try and error again, then we have theory to explain what amazes us.

[@frogenthusiast235]

This was such a beautiful video Brady, Tadashi is a brilliant communicator. Thank you for making it!

[@chaoslab]

Tadashi, you have such a wonderful and compelling way of speaking, very relaxing to listen to.

[@emy5845]

07:00 This curve reminds me of the Poisson distribution! It may not be a coincidence! "poisson" in French is "fish" in English for those who don't know 😅

[@snookerkingexe]

The comeback of the great Tadashi Tokieda!! Knew it would be just a matter of time!

[@theherk]

Brady, thank you for all the years of wonderful videos on each of the channels. I have learned so many fascinating things from so many charismatic, intelligent people.

[@Harlequin_3141]

I am so happy to see this gentleman return! More please, and thank you!

[@mli7361]

that's so cool, does it mean fish needs to sweep their tails at an optimal frequency to match the alternating frequency of the vortexes?

[@DavidOfWhitehills]

Likely.

[@Mongooosa]

These are the videos I’ve been waiting for

[@SKyrim190]

Always a pleasure to hear Tokieda-sensei!

[@rickseiden1]

Dr. Tokieda came across as a really great father in this video. Swimming with his son, helping his son explore, using his son's toys. It made me smile.

[@tommihommi1]

put the porposing in the thumbnail and title to catch all the formula 1 fans!

[@Ishanaroya]

I didn't know I wanted to know this, but now I do and it is great! Thank you for this!

[@geiger21]

Isn't it almost the same phenomenon that causes tall chimneys to sway in the wind? That's why some of them have helical strips to break the air current and stop the vortices from forming.

[@JimGobetz]

Tokieda san is just awesome in how he explains concepts and you can feel the joy he has for the subjects. Thanks for this content, it's always thought provoking, can't ask for more than that!

[@MetalicAtheist]

Except on the exterior of the vortex street, the boundary of the turbulent wake, the opposite conclusion can be made. In the usual case, the vortex velocity is along the main stream direction, allowing more fluid to enter the wake. In the swapped case, the reverse is true, and the velocities at the wake boundary are lower, preventing as much water from the main stream from entering. I think this is the real explanation. Preventing the main stream fluid from entering the turbulent wake actually increases the mass transfer past the body, reducing the drag. The more fluid that enters the wake, the more fluid converts its energy into rotational rather than translational energy, and the ultimate effect is a transfer of momentum along the main stream into the body. That's just a guess, though. I don't think the flow lines along the axis of the body within the wake are as important as those on the wake boundary.

[@7177YT]

Finally! I missed him and his quirky little musings very much indeed. (:

[@HeavenlyTennyo]

I’ve never been so early to a Numberphile release! And how joyously to a Tokeida-sensei episode!! He is so engaging and has such playful curiosity.

[@dudewaldo4]

Tadashi!!! Thank you for meeting with him again, I love this guy

[@MrDaraghkinch]

I love how his strong accent belies his exceptional articulacy. He's such a brilliant communicator.

[@LeoStaley]

Years ago I commented that listening to him for a while I begin to feel like his way of pronouncing things is the natural, native way to speak this language, since he is so effortlessly articulate. He's also a professor of philology, in addition to math and physics. He's a veritable Renaissance man.

[@positivegradient]

Great comment

[@andres6868]

great Tadashi is back...we were missing him

[@triciaamheiser785]

I love how Prof Tokeida explains the focus of his passion. Gifted communicator. Thank you for posting him again. By far my favourite.

It's great seeing Tadashi Tokieda again. Love his videos.

[@Marconius6]

Porpoising is (as I just found out) a term in Formula 1, and a new one at that, having become a thing just this year. It apparently refers to the cars bouncing up and down on the suspension due to aerodynamic effects, so it's kind of related to this video too.

[@danhenson]

It has become relevant again only this year, but the term had existed many years ago in F1, the last time the cars were allowed to utilise ground effect as they do this year.

Gotta tell you, this time I'm not convinced. There seems to be a break in causality between the backward momentum gained by the vortices, which results from their own interaction AFTER the tail movement, and the forward momentum gained by the body.

[@kalle123]

Agree, I don't think it is simple like that ....

[@ncikb2093]

There's a major issue with this theory. Von Karman streets only develop behind blunt bodies; bricks, buses, poles. Submarines and fish are not blunt bodies, they're very streamlined, such that von Karman streets probably don't develop. @numberphile need to address this in a subsequent video, maybe even append another viewpoint to this video.

[@badabing3391]

its not telling you entirely whats going, but the video is essentially using momentum conservation to describe the system, instead of the mechanisms directly applying force to the fish. For that, i imagine youd have to do some math comparing how the vortex being slid laterally applies some forwards force

[@craftycurate]

Yay! I've missed Tadashi's videos - great to see you back :)

[@karenwilcox2191]

Wonderful. Love the use of the bathtub, the ping pong ball, the toys- familiar objects- to demonstrate hydraulic principles. Because I already have an understanding of those objects, it gives me a head start on understanding the concepts. This is a characteristic (rare) of the BEST teachers. Thank you!

[@trewq398]

Oh, nice to see him again :)

[@WAMTAT]

I'm glad there are future generations of Mathematicians that will eventually start producing content for this channel (and adding to mathematical knowledge as well I suppose)

[@matrixate]

Most of them are too busy watching TikTok though...

[@ArtJorgensen]

I just stumbled across this video and of the 11,278 videos (more or less) that I've watched, this one ranks at the very top. Tadashi Tokieda has got to be the best teacher I've ever run across. He made everything so clear I'm sure his young son would have understood. I was entranced the entire 17 minutes. I'll be searching for more videos by Tadashi. A genius!

[@james240878]

Love a good Tadashi video. This was fascinating!

[@thenefariousnerd7910]

Always love Prof. Tokieda's demonstrations. I'm a bit confused on the explanation of momentum being lost to/gained from the vortex street, though. In the case of the submarine (around 12:30), for instance, there should be an equal amount of backward momentum left behind outside of the vortex street as forward momentum left behind on the inside of the vortex street. On the other hand, the interior of the vortex street makes direct contact with the back of the submarine whereas the exterior does not (sort of by definition). So is it fair to say that we only care about the momentum on the interior of the vortex street because that is what can interact directly with the submarine? It's possible that this is an implied part of the explanation that I just didn't pick up on.

[@andrewmatas6984]

(Warning: I am not a fluid mechanics expert!) I believe he mentioned that the vortices on the right side of the vortex street will push the vortices on the left side of the vortex street forward, and the left vortices will also push the right vortices forward. So I think the inner part of the street is relevant because this is what controls the interactions between the vortices. I think the net forward motion of the vortices, driven by this interaction, is what ultimately leads to the forward momentum of the wake which leads to the drag force on the submarine.

[@killerbee.13]

The water outside the vortex street isn't moving much, so it doesn't contribute much momentum. If you're a ways off to the side of something moving (at a medium speed, through water, because fast things in air behave differently due to turbulence) past you, you won't get pushed backwards due to drag. You won't feel much of anything, but if you do it will probably be in the direction of the movement. Basically, the rigid body pulls a column of water behind it, and the vortices wrap around that column and essentially roll around it to stop it from dragging on the water around it. You could imagine a cylindrical object inside of a tunnel, with ring-shaped rollers surrounding it. If the tube is moved forward, the rings (vortices) will be dragged along with it at half the speed, and by rolling they reduce friction against the walls of the tunnel, allowing those walls to stay put while the object moves. Most of the shed momentum is contained within the wake and the rollers that fall off (which in this example would slowly roll to a stop, but real vortices act a little bit different), rather than being put into the walls. In the case of the fish, the vortices also serve to reduce the drag between the column of water behind the fish and the surrounding water, but the water is moving in the opposite direction within the vortices and the wake now. The surrounding water still moves very little in response. Eventually those vortices spread out and get lost in turbulence, dissipating the energy in the wake, which will impart a slight amount of momentum to the whole body of water, but that can happen very far behind the object that created that wake.

[@MetalicAtheist]

In the first case, the fluid velocity in the wake boundary is moving opposite the direction of the body. This creates a sharp velocity gradient that tends to pull fluid from outside of the wake into the wake region. Work has to be done on that fluid to accelerate and rotate it, so there is excess drag on the body to make up for this. In the latter case, the fluid in the wake boundary layer might be in the same direction as the body, so the velocity gradients aren't as large and the pressure builds up on the boundary. My guess is that this actually changes the shape of the wake slightly, and ultimately it means less water from outside the wake is dragged into it, undoing the drag effect previously mentioned.

[@Cliff86]

I'm assuming that vortices like these are the same mechanism as a knuckleball pitch in baseball I'd label the graph Tokieda drew with Bobbing, Porpoising, and Knuckling

[@HalSchirmer]

One of the most hypnotizing things you'll see- are vortex shadows and bright dots at the shallow end of a swimming pool. I grew up with a 1960s era "swimming pool in the back yard; it only had one outlet pipe, so the entire body of water would eventually spin counter-clockwise; because it was "free form" at the curves the water turns turbulent and separated from laminar flow- you'd get dark spots or bright spots, depending on how deep the meniscus of the vortex was. Another great image is Von-Karman shedding when a beach-ball is trapped in the current by the suspended rope that separates the shallow end from the deep end.

[@iluvtacos1231]

Yay!!! I've been waiting for more Tadashi!

[@LudosErgoSum]

This man need to start an ASMR channel were he reads out math equations slowly!!!

[@gordonwiley2006]

You know I am here for Tadashi. Fluid dynamics always makes my head hurt.

[@soaringvulture]

Then you're doing it right. It's supposed to hurt.

[@shortshortshort]

Ah numberphile, you bring me joy and knowledge! Thank you numberphile!

[@MrSerbianOrthodox]

I love your animations!

[@arpyzero]

I find it interesting that in theory, this effect should be seen in birds as well, but my guess is that it isn't nearly as present because water is more viscous and denser than air. The reason why for the alternating vortices initially form wasn't particularly clear, but I guess that as the rising ball collides with randomly moving water, vortices become inevitable and then promote their own creation (in an alternating way, of course).

[@APando93]

You’re very close. The viscosity of the water means that there is a significant velocity gradient between water close to the fish and water far from it, and this tends to create vortices.

[@killerbee.13]

Von Kármán vortex streets can occur in air too, but they require slower movement than any birds could fly with (I think for small objects in air to create vortex streets, they have to be slower than a walking pace, and you can usually only see them in smoky air. Physics Girl and 3blue1brown once did a couple collaborative videos looking at vortices in air though, I recommend looking them up). At the speed birds go, it's just turbulence left behind. You can however see vortex streets left in clouds when wind passes over mountain islands in some cases. They show up regularly in satellite imagery and are very cool. I think that the mountains being much larger allows them to be created even when the wind is a bit faster, but don't quote me on that.

[@wolfie54321]

@@killerbee.13 von Karman streets can occur at relatively high speeds in air. In fact it's very hard to avoid forming Karman or Karman-like vortices when something moves through the air, as Karman shedding occurs over a very large range of Reynolds numbers. If you follow a heavy vehicle when it's raining or even better snowing, watch the particles or moisture/snow on the wake of the vehicle, they'll whip left and right, that's caused by a Karman-like vortex shedding. Karman vortices can be turbulent, they need not be laminar. I haven't done the calcs, but I would imagine fish don't actually use "Karman" vortices, rather the motion of their body suppresses the natural Karman vortices and they generate a lower frequency vortex street that is in the opposite direction to a Karman street. I'm actually not convinced on his vortex explanation for the ball bouncing either, as spheres actually have higher drag prior to the onset of Karman shedding. If you asked me why this phenomenon happens without giving me any other information, I probably would have guessed that "added mass" was a contributing factor. "Added mass" is the idea that the ball isn't just accelerating itself, it's also accelerating a portion of surrounding fluid, and since water is much denser than the ball that will be significant, and the closer to the surface the ball is then the less added mass there's likely to be... but maybe it is a Karman vortex thing, his description didn't really convince me though, perhaps there's more literature out there on it.

[@jousis_]

F1 ✅ Math ✅

[@benjaminlehmann]

😲 Love those moments when you get a sudden flash of understanding. Great video! Such a simple idea hidden in the depths of hydrodynamics. Thanks!

[@davidgillies620]

The Kármán vortex street is why telephone wires or guy wires on a flag pole "sing" at certain wind velocities. It's also why you sometimes see helical baffles on sheet metal chimneys: to stop the alternating vortices from hitting a resonance, which could cause the structure to fail.

[@DavidOfWhitehills]

I read that the Burj khalifa is asymetrically stepped to prevent resonance.

[@roger_isaksson]

The ball actually moves in a spiral according to the Coriolis force. Close to the surface the steady state “radius” as defined by the force balance of the Coriolis and hydrodynamics isn’t yet developed.

[@Shadow4707]

In which direction does the spiral go? Is it a chaotic system where the direction depends on some tiny parameters or is there more to it like with the Lorentz force?

[@nickpatella1525]

By Coriolis do you mean the rotating of the Earth? That doesn't affect things on small scales. It's been debunked hundreds of times.

[@rtyuik7]

i love how a primary reason to call this 'pop-up' Porpoising, is to make the pun "what is the Purpose of Porpoising?"

[@KillianDefaoite]

Perfect timing! I am just finishing up my thesis on low Reynolds number flows past a circular obstacle.

[@zerosiii]

Huh... I wonder if you can devise a swimming style utilising this technique, making you swim faster than before. Or if not, then engineer a contraption to emulate this

[@charstringetje]

The flippers that divers use, do this.

[@Pfhorrest]

Could we thus build much faster and more efficient submarines (and boats?) just by adding a waggling tail to them? The tail wouldn't have to provide all of the propulsion, just swap the vortex trails, which then reverses drag and converts it into additional propulsion.

[@JimGobetz]

I thought due to water being a progressive resistance medium it wouldn't scale up but then I thought, Blue Whale, maybe it's the efficiency of muscles tendon and bone that makes it hard to reproduce in a mechanism. Otherwise it would have been put in use I would think.

[@SgtSupaman]

@@JimGobetz , it probably wouldn't be difficult to reproduce, but I'm guessing the workings of that mechanism would be so inefficient as to be pointless. Submarines typically travel faster than whales.

[@friendlybane]

Submarines have different structural requirements since they are not fish and are very large. You could build something that flaps it's "tail" like a fish but the structural weaknesses created by that make it useless as a submarine. You'd rather just build a regular submarine and power it with a nuclear reactor. You lose more than you gain so it's not worth it.

[@Disco_J]

absolutely spectacular demonstration and a perfect example of how a comprehensive understanding of material enables an excellent lay explanation

[@stco2426]

Excellent. I worked in a research group that studied fish exercise physiology and swimming but I’ve never heard such an elegant explanation of the vortex switching as here. Many thanks.

[@crunchytoast6007]

Ze Frank raid incoming!

[@norma4792]

Me, a young postdoc: F*ck, my phenotype has already been described in the literature. This old guy: This phenomenon is called "popping" in the literature but I'm introducing a new terminology...

[@keithbecker3142]

One the best numberphile videos and a great explanation of drag forces!

[@harroman42]

One of the best Numberphile videos I've seen. A superb teacher.

[@BurningmonkeyGTR]

There's actually another phenomenon called porpoising found in ground effect race cars, when they generate enough downforce to touch the ground they stop generating downforce because they choke their own airflow and they lift rapidly off the ground, then because the airflow is no longer choked they begin generating the lost downforce again and got the ground again, causing a constant bouncing effect

[@marcusmangelsdorf9211]

Simply wholesome. Thank you!

[@Taverius]

Always up for more Tokieda-san! 😎

[@christopherblevins1968]

I watched this video because of a suggestion by Ze Frank. He found it while researching for his own upcoming video. I enjoyed this so much that I immediately subscribed!

[@ingledont]

this reminds me of conducting small experiments with my father as a kid. i love to see Tadashi's encouragement for exploration and thought !!

[@tanmaya1111]

This is such a beautiful explanation! Thank you!

[@95triforce]

This is so fascinating. An observation I’ve been making my whole life, now I know why it happens. Great video Brady!

[@esmenhamaire6398]

Lovely to see A video froTadashi again - I love his style of explanation!

[@mr.cheese5697]

3:18 tension & pressure of water above. It's obvious that this effect is dependent on at least on: depth, pressure on the ball, liquid, temperature of liquid, density of ball, tension of materials interacting with liquid, shape, gass pressure, wast complexity of fluid dynamics. No big wonder one of cases is more complicated than straight connection.

[@eswing2153]

This was amazing. I love how this professor can explain things so well.

[@housellama]

Tadashi is the best. I was so jazzed to see another video with him

[@ianardeant]

We need more and more from this great professor. What a great teacher!