We are in the Tokyo bathroom in northern California, near San Francisco, on Stanford
campus. Something that was discovered between my son who's just turned four, but he was two at the
time, and me two summers ago; we used to go to a swimming pool and he really liked it, I would of
course cuddle him and go into the pool and he was very very happy about the cold water and warm sun
and so on. And we took a little ball with us and we played with this but then the following
experiment was suggested by my son: he wanted me to push the ball which floats in water into the
water and then release. And of course it bobs up and then down and stays on on surface of the
water. And then he said, well release deeper. And when I release a little deeper it pops up
clear above the water, it actually jumps into the air.
And so we thought, well, if I go deeper and deeper it will jump higher
higher but watch what happens when I release really deep: it just wobbles up to the surface
and you know stays there, it doesn't clear the surface. So if you think of the height to which
it pops as a function of depending on how deep you release, it's not monotonic, it's not that,
you know, the shallower the higher or deeper the higher - actually there's a maximum around what
you might call the sweet spot or sweet interval. If a good distance below the surface, well
it pops rather nicely. There has been some literature on this I discovered subsequently,
and people seem to talk about pop up, popping, which I have been doing but
I'd like to introduce a new terminology. I think that it should be called porpoising,
as in the animal porpoise, because you know, those sea creatures; porpoises,
dolphins, and seals are able to leap out of the water right. And of course these creatures have
very good muscles and hydrodynamic shapes and so forth, but it would be rather surprising if they
didn't take advantage of this mechanism where, for free, just by starting from a sweet depth,
you can leap into the air free of charge right just like that. So it would be a good thing that
porpoises are doing so that's why it should be called porpoising. And besides, then we can ask
questions like what's the purpose of porpoising and so on so. I'd like to make a suggestion as
to what's going on and which, you know, my son more of less said the same thing in his babbling language.
And so when you dip very very shallowly of course there isn't really much you know buoyancy, and and
there isn't much time or distance over which the buoyancy acts, the floating force, acts so you know
it doesn't go very far and that's understandable and that's intuitive for everyone else. When you
sink them a little deeper, then the buoyancy - as I said the floating force - has sufficient time
and sufficient distance over which to act so it pulls it up and then you know there is enough momentum
where as it breaches the surface of the water to go up, so that's again believable. You know, and details
can be calculated by classical hydrodynamics and so on. But the mystery is, why if you
give it a really deep release it doesn't pop up? And I think the answer is already suggested
by what you're seeing: when I release from there do you see? It doesn't go straight up, it actually
moves sideways, it oscillates sideways. And it's, you know, wasting a lot of energy, a lot of motion
and when it comes to the surface that's where it stops. Yeah so it goes sideways yeah and
that wasn't really convincing but, yeah, that was maybe a little more convincing so I'm going to
let the water settle. - (Brady: Yeah, it kind of has a-) - Sideways motion, okay?
(Yeah yeah okay, very nice.) - All right okay. So, at this stage I'm going to take up the brown paper.
So this ping pong ball is moving forward like that okay? Now as it moves forward it has to get the
water around it out of the way, so what's going to happen? For example, here is a little blob of water,
and then this water has to get around the ball so it rolls rolls rolls like that and then comes off.
So the result is that it sheds what is called a spinning pattern of water which is called a vortex
just behind like this. And then a little earlier you are shedding some vortex like that,
and vortex like that, and so on and so on. As it goes up it starts shedding vortices, it's creating
some kind of vortical pattern, spinning pattern, around it. On this side and on this side in opposite ways,
and of course the picture is three-dimensional so it's shedding vortices all round and so forth.
Now, this kind of spinning pattern destabilizes the ball and so when it sheds the vortex like this
it kind of pulls this sideways and when it sheds the vortex on this side sideways and so on; and it
is has been noticed that for some range of motion, speed, size, and density of the water and especially
viscosity, this shedding is in alternating patterns. So one side, the other side, one side-
so it starts oscillating. Whereas if you start from too shallow then this mechanism doesn't have
time to kick in, so it just goes straight up and then that's very very effective because you come
straight up and you pop straight up and porpoise straight up; whereas here if you start
too deep you go sideways and then by the time you breach, you reach the surface water, you don't
really get anywhere. So that's what's- what's going on. - (Professor I imagine millions of these)
(are happening per second though, why aren't they sort of all cancelling each other out)
(and allowing the ball to go straight up?) - That is- that is a very good question but the reason is, as I said,
for this range of motion- you know, if it's very very fast, very very slow, that's a different
story; but for this kind of daily motion of that size and magnitude and this shedding it turns out
happens in alternating fashion. So, you know, there is no cancellation because when it's shedding-
being being shed over here, the previous vortex is already far away and the previous one before
that is even further and so forth. So you go one side, the other side, one side, the other side; and
that's why it wobbles hither and thither. If everything was happening much much faster and everything-
and then you get something called turbulence, this is a huge mess of vortices that are kind of
clouded together and so forth, then you know all bets are off and you just get a messy motion.
(And Professor, why does this mechanism stop the ball from leaping high? Is it because it's it's)
(wasted its energy?) - That's right so it's wasting a lot of energy behind and we shall come back
to that; but also, you know, because it goes sideways that's you know part of the energy that should be
moved- used to move upward and being wasted sideways and so it doesn't really have anything
left by the time it reaches the reach the surface. Okay, this is zero, above water, and it had some
strange maximum and then goes down. So the question was what is this maximum? Well we understood that
in fact what was going on is that somewhere here, and somewhere here, the motion of the
movement of the ball was very different. So
here you just, you know, bobbed up and down. Here it kind of went
hither and thither and wasted motion sideways and so it didn't go; but here it cleared the water
very nicely. By the way, another thing that we didn't really emphasize when we saw it is
that, in this case because it's kind of going sideways it also tends to splash quite a lot
of water and that's more wastage if you like in the in the motion. But anyway, so because of this
there's some kind of strange maximum. I would say that here, when there is this separation between
this region and this region, is about one diameter deep when you release; and here when there is some
loss that starts because of this oscillatory motion, ultimately because of the vortex shedding
as we explained, is maybe two or three diameters deep, yeah. So that's more or less the picture.
I'd like to show you the vortex patterns that are left behind or in the wake of the moving ball, but
in order to see that best you would like to have a shallower water so I'm going to let some water out.
That's good, that's good, okay. The vortex pattern that is left behind in the wake of a moving object,
or they flow past and then you know in the wake that is formed, can be seen in the bathtub if
you have it at the suitable depth. So here you can move a finger across in your bath, or in this case
let's move a familiar ping pong ball. When I move it across, do you see at the bottom of the bath you see some
shadows of those black spots? Lots of black spots on this side, on this side, and in some
kind of very alternating pattern, it's not so clean of course, but you can see it right? And
that's- those are the shadows of the vortices that are being left behind. And this pattern,
the alternating pattern of vertices that are being shed at all times, and which cause if you like
some sideways oscillation and so on - and and so on will come to come back to that later - was
discovered by a number of people but it's now known as von Kármán Vortex Street.
And this is really a very beautiful pattern that arises behind some fluid motions in the wake.
This pattern is used in rather clever, and in fact miraculous fashion, by swimming creatures
such as fish. So have you ever wondered why not only fish, but you know seals and dolphins, and
porpoises indeed that I mentioned before, have you ever wondered- I mean you have certainly been to
see some seal and so on and you know they don't seem to be doing anything, they just wiggle their
body a little bit and off they go right? Whereas human champions, the best of the swimmers,
have to do so much work and splash all over the place and then they don't go that fast do they?
I mean you know, we can all outrun these people on the on the ground. So how come those creatures
just by wiggling their body goes so fast and so efficiently in terms of energy and everything?
That is the mystery that needs to be explained and this is how it works, to a good approximation.
So let's say that water is coming from front to back, or this creature is moving from back to front
like this like that; and let's say in the first instance that this is a submarine, so it's a
rigid body so no flexibility there, there's no animal mystery there. Well as I explained,
well in the two-dimensional model, what happens is that on this side the water has to roll along the
body like this. There is something called no-slip boundary condition which was noticed by Stokes in
the 19th century. Anyway so it rolls and rolls and rolls and then behind it, it leaves a vortex like
that. On the other side water rolls and rolls like this, so it's rolling in the opposite direction; and
then in the back it leaves a vortex like that. And so on, it keeps shedding vortices as we say.
Okay so this is the pattern that is left behind of vortices which we saw some rough representation of
in our experiment in the bathtub. Remember this is completely rigid. Now what's going on? You can see
that this vortex for example is kind of swirling and stirring the water around it, so it induces
a flow like that around it which then pushes those things forward right? Because it's kind of
inducing the flow like this. And on the other side, this vortex for example is spinning like that
and induces a flow- circular flow like that, and it pushes those vortices forward as well. So this row of
vortices and this row of vortices are helping each other and they're pushing each other forward.
(Almost back to where they came from?) - What they came from exactly. And that's a forward momentum yeah
of this vortex train- street, but this creature or this submarine then is leaving a forward momentum
in the wake. Well that's, you know, for the momentum that it really wanted to keep for itself to keep
move forward that it has wasted by leaving it in the wake, so that's why it suffers.
(You're saying this is a bad thing?) - That's right, because it's all left behind. You know, this submarine wants
to keep all the forward momentum to itself but instead it's giving the forward momentum in the
wake to the fluid that it is leaving behind. So you know, you are ejecting this forward momentum just if
like, well I don't want to have any important analogy, but anyway so you know it's all wasted.
So that's why it suffers and that's why it feels that, well short of forward momentum,
he feels a sort of backward force if you like, and that's called a drag, you know, it can't really move
forward so easily. That was a rigid submarine, which of course has to have huge motors
and so on in order to keep moving forward and it has to expend a lot of energy. Now what does a
fish do? Or one of those creatures do? It does more or less the same thing because of course the motion
of the fluid is exactly the same, water keeps rotating like this, rolling on the surface of the
fish or this swimming creature like that is the same. But a swimming creature is flexible so it
flexes its body like that, yeah? And it's the flapping motion of the tail. And by doing so it guides this
vortex on this side and then flicks it and leaves it on this side. And the vortex that started growing as
the fluid kept rolling on this side, again you know after that it flexes its tail like that,
and flick it leaves on this side. So it keeps leaving this kind of vortices on this side and
this kind of vortices on this side, okay. And there, there, and there. It may not be so apparent but this
row used to be over here, in the case of submarine, and this row used to be over here in the case of
submarine. But what the fish did by flicking its tail and placing the vortex on the other side
was to exchange those two rows with respect to submarine, okay. With what consequence? Well
this vortex again you know induces circular motion like that; but look in this case, it's pushing
the partner vortices backward yeah? And this side is inducing a circular motion and it's
pushing the circular and the partner vortices backward. In other words, those two rows of vortices
um of the vortex street are actually helping each other to push backward, that's backward momentum.
Whereas in the case of a submarine we had a forward momentum that was left behind. Well this creature
then, the fish, is leaving backward momentum in the wake. Well that's exactly like a jet engine. You
see, you're ejecting something behind which is going that way very fast by recoil, you
go forward. So it gains forward momentum, yeah, because it's leaving a lot of backward momentum,
well that helps and that's how it can go so fast. Because it just exchanged those rows of
vortices that it's leaving behind, thanks to a judicious flexibility of deflection of its tail.
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...so that's version number two. What's going to happen if I pull the ends of the paper? Well
you know, somebody not observant might say, oh what's the difference in here
here? Well probably it doesn't matter, well, but you remember that version
number one ended up having the rubber band hanging from the paper ribbon...