 is the part for that. Is this good? Yes. People hear me? OK, good. OK. OK, so I'm going to talk about three different investigations of three different problems. I hope I'm not trying to cram too much in. I'm going to try and have a thread running through it, which is to say, sometimes people think that if you've done something experimentally, you do it numerically, that's good. You should reproduce methods of complementary. You should reproduce what you've done. And I don't think that. I think that for some things, they're better done experimentally. Some things, they're better done numerically. And some things, there's no point in doing both. So I'm going to try and give motivations for doing things numerically. And the first and the easiest to understand is that a numerical simulation provides a perfect experiment. It provides perfectly straight boundaries. It provides a perfectly constant temperature. If you don't want to include forces, they will just won't be there. If you want there to be no vibration, there won't be any vibration. It will just be the problem that you have defined. In addition, not only will you be conducting perfect experiment, you'll get perfect data out. You'll get all the data you want. You'll get an entire velocity field at every point. Whatever you want, you can get out. None of it's hidden to you. But that is not the only motivation for doing numerical work. Another one is that you can do things that you couldn't do experimentally. You can calculate unstable, steady states. And I'll tell you about that later. And you can calculate eigenvectors and various mathematical objects that are not accessible experimentally. That you can only approximate or hope for experimentally. And then the third reason is that you can test physical theories with it. That is to say, you don't know that your fluid is an alderoyd B fluid. Now, I don't even know what that is, but I know it's a fluid model. But you don't know that you're fluid. But you can, so that if you do something with a fluid and it doesn't do what an alderoyd fluid does, then maybe you did it wrong. Whereas you put that in your simulation, and that's what your simulation is. It's following that model, and you'll find out. And another, OK, but I don't do non-Newtonian fluid, so that's not relevant to me. But what is relevant is turbulence. Sometimes people think, you know, turbulence unsolved problem, maybe we have to go beyond Navier-Stokes. No, we don't have to go beyond Navier-Stokes. And the reason we know that is because numerical simulations of the Navier-Stokes equations give the same turbulence that you see in experiment. So there are no more terms needed. There are no, the Navier-Stokes equations are fine. It's their behavior that one doesn't understand entirely. But we see that, so you can learn physical things from the, from numerical simulations. OK, so let me first talk about the first use where you're gonna do a perfect experiment, have perfect input and perfect output. And this is something I talked about when I was here three years ago, transition to turbulence and plane-quad flow. And this is done with a time-dependent code. So it's like a perfect experiment. And let me start by showing a movie. So let's see, where's the movie? Hopefully this will just go. Ready, everybody watching the movie? OK, what is this? A plane-quad flow is the flow between two parallel plates moving in, in different, at different speeds. What you're seeing is the velocity in this direction where the fluid is, where the plates are moving, midway between the two plates. So the plate that's closest to you moves to the right, say the plate that's furthest for you moves to the left and you're right in the middle. Where if you didn't have any kind of turbulence, the flow would be zero. It's moving to the right here. It's moving to the left. In between this linear interpolation, in the middle it's zero. No fluid, no velocity, that's green. Fluid to the right, that's red. Fluid to the left, that's blue. And this is at Reynolds number of 500. Move my little cursor here, let me go. OK, so you see that at 500 there is turbulence. There's this, there's kind of little warms moving. The red is, as I say, it's fluid that used to be here where it was moving to the right that's gone to the middle. This is fluid that used to be there that's moving to the left. And now we've lowered the Reynolds number to 300 from 500, so the difference between the velocities is less. And you see that there's less turbulence, but you see that the turbulence is arranging itself. You see that in parts of it are all green, so there's no turbulence, no velocity. And you see that it's arranged in this stripe at a certain angle, do you see that? And that angle has a fixed, that stripe has a fixed angle, fixed wavelength, and really nobody knows why that happens, not really. And these are numerical simulations. And let's see, let's continue. This will just continue forever. It will just stay that way in this stripe forever and ever, which is a long time. And there, so the first thing I'm gonna talk to you about is this. Okay, so this is then simulations. And this is a reminder of what coet flow is. It's the flow between different plates. And it's a lot like telecoet flow. And I mentioned that here because of course we're doing a lot of telecoet flow in my activity and in Bruce's activity. That's not flow between two plates, but between two cylinders at different velocities. It's the same, but with curvature effects. Except it's not, for the co-rotating case it is, and you get these patterns that we've been talking about and that maybe you've been talking about Bruce, the patterns like that for the co-rotating case. But in the counter-rotating case, it's a lot like plain coet flow. Plain coet flow has no linear stability leading to, no linear instability, that is to say the basic flow where it's fast here, slower here, and then goes to the left. Is that the left for you? Yeah. That one is stable for all Reynolds numbers. And it's the same with counter-rotating telecoet flow. And yet, what you get when you do counter-rotation, this is a picture from Andrek U. Swinney, is a region of turbulence there in a spiral and a region of laminar flow. And that's from 1986. And then in a much, much larger domain done later in 2002 by a French group, one could then see that this piece of turbulence here, is there a, is this a laser pointer here? Middle, top middle? Maybe bottom? Oh, it depends maybe what you mean by top. Yes, okay, yes, okay. So this stripe here is like one little stripe here. This is a piece that, and you see that, that the angle is very regular and the wavelength is very regular. And this is not alternation between left and right or in and out, this is alternation between turbulent, not turbulent, not terminal, turbulent and so on. That's very strange. Here's a picture of the same experiment done in plain coet flow. That was one of their discoveries is that you could do this in plain coet flow as well. And this is our simulation that shows that you have turbulence and laminar flow there. And we did the simulations in a, in a domain like this tilted, tilted in the direction of variation of the pattern, the expected variation. And this is done just solving Navier-Stokes equations as the fluid would have done. And among the things that we did was to average the, in time. This is, this is now a line perpendicular. This is a line along this, in this direction, in the mid-plane, halfway between the two plates. And you can see they're forming the turbulence and the laminar region. And now you can average in time and you get something, a smooth data set. And then you can analyze it. For example, you can, here's the average in time and in the, the narrow direction. And you get the Reynolds average Navier-Stokes, these are the average Navier-Stokes equations. And from this you can get the forces that are at play. This is the non-linear term, the viscous term. And then this is the Reynolds stress. And you can get the actual exact Reynolds stress because you have every quantity that you need to do this calculation. And you can learn that the non-linear term and the viscous term are balancing each other where there is no turbulence. You can learn all these things. You can do actual force computations at every point in space, which is not something that you could do generally with experiment, not yet. When I was young you couldn't get velocity fields from experiment. And then I started to see the first, the first PIV and other kinds of experiments that actually showed the velocity fields as well as numerics. And I thought numerics was out of business. But apparently not, not yet. Perhaps someday they will be, but not at present. So you can get forces. Here's something else you can do. Our domain was in the direction of the expected and the experimentally observed wave vector. You see how this is perpendicular to the stripes? Well, you can actually put a domain that's parallel or perpendicular like that or any other direction. And you can see if there's a turbulent solution, even though this is a striped solution like this is not seen experimentally. And here's what we found. By forcing these angles we could find that there were indeed all kinds of other solutions. It can have lots of different angles and lots of different wavelengths because we could force it. Now that doesn't mean that's what's observed experimentally because these are probably unstable. But it's the kind of thing you can do numerically that you couldn't possibly do experimentally because you're making the solution do something. So here's the first, that's the end of the first example, which is to say that numerics provides a clean detailed experiment with perfect input and perfect output. And this was the example, the method was merely time integration. And what did we show? We showed that you can get the turbulent laminar banded solution in a truncated oblique domain OK, the obliqueness there's to say in a domain like this we produced this solution. The picture that you saw here, you can see that our computational domain is underneath it there. It's just repeated many times. We didn't do a computation this big. We did a computation in this box. And we showed that we could produce the pattern in this box. And that's a result by itself. And another result was that you could have many different angles, most of the patterns being unstable. And then we could calculate the balance of forces maintaining the mean flow. And all of these are things that would not be accessible to experiment. Questions so far? This is the end of story number one. Sorry? Yes, yes, they're 3D. Yes. The computational box, the box in which we did the computations has the white outline here. And then we just repeated them. You can see that here. Look at it here, perhaps. Yeah, here, sorry. You see this box? This is the box in which we did the simulations. You see that the rest is just repeated. So it's a result that you get the same flow in a small box like this with the correct wavelength and the correct angle and so on. It's a result that you get that in a small domain, tilted that way. More questions about this little story? Story number one? OK, story number two, the faraday instability. This was done by other coworkers here. And what is the faraday instability? Well, it's the oscillation of a plate of fluid, a thin layer of fluid, like this. And above a certain acceleration, a certain amplitude of oscillation, you get standing waves forming on the surface. And I think if I do this right, this will appear like this. These are square wave patterns that we have computed. As with the other thing, we did not compute in this big box. We computed in a single domain and repeated it. So this is one computation like this. And then the picture is repeated four by four times just to make clear what's going on. And so faraday observed this in 1831. And maybe other people observed it before, but he reported it scientifically. And these are the kind of patterns that you can get, stripes, and squares, and hexagons. Those are the basic patterns you get. Those are the patterns, the shapes that tile the plane. Now, although faraday did these experiments long before the experiments putting into the scientific literature Rayleigh Bernard Convection telecoette flow, which were done in the late 1800s. This was the early 1800s. The analysis of it took much longer. The linear stability analysis was not done in the 1910s and 1920s, as was the case for telecoette flow and Rayleigh Bernard Convection, but was done in 1954 by Benjamin Ersel. And the first numerical simulations were not done in the 70s, as was the case for telecoette and Rayleigh Bernard, but just recently about 10 years ago. And so what are those results? The linear stability analysis in the inviscid case was done by Benjamin Ersel in 1954. And the viscous analysis was done in 1994, 40 years later. And what is this? These are wave numbers. And this is an acceleration. And this says that with this amplitude of acceleration, the fluid becomes unstable to perturbations of wavelength, say this. This is an inverse millimeters or something. This is the wavelength to which it becomes unstable. And this is the same for the viscous case, where the threshold is higher, as is usually the case, because viscosity damps things. So inviscid analyses generally predict too low a threshold. So that's the first linear stability analysis, which was, as you can see, quite late. And the first computations, the first two-dimensional numerical simulation was done in 2000, which is really pretty recent. The first two-dimensional simulations of Rayleigh Bernard convection telecoette flow were done in the 70s. So this is pretty recent. And the first three-dimensional analysis was done 2009, the first three-dimensional calculation. Again, very recent, very late. And for reasons we don't, I don't know. I think I was just lucky. And as you can see, this is a computation, a piece of the same computation you just saw. You see the whole velocity field coming from our computations. You can see that it looks kind of like a volcano, this velocity coming in, and all that. This is the kind of information that you get from a numerical simulation that you would not get from observing the instrument in the laboratory, where you would be lucky to just see the interface. And every still, oh, the free surface code. Oh, well, OK. I cut out most of the detail about that, but there'll be a little bit more. And I can certainly tell you this is not my domain. I was saying I was lucky. And how was I lucky? Well, I was sharing an office with someone who did free surfaces. And I said, oh, you know how to do free surface simulations. I don't. I wish I could. I would then do a simulation of the Faraday instability, which has never been done. And he said, yes, I have free surface code, but what's the Faraday instability? So because we got together, we got to do this. But anyway, it requires free surface code technology, which I don't know very much about, or if anything. And I can show you as much detail as you want later. Exotic patterns were found by these researchers. Edwards and Fulv, he was at Texas, where Professor Sweeney comes from and in Pennsylvania, by forcing with two frequencies. So it's not a simple trigonometric system, but a little bit more complicated in time. They showed pretty pictures like this. The Lane Scale, yeah, yeah, they come from the Lane Scale. You know the relation of Rayleigh or Lamb, omega squared equals gk, they come from the frequency. They're not set by the depth. They're set by the frequency. So by having a two-frequency signal, you excite two different wave numbers. And that's what makes the quasi patterns. And I have a whole talk about that, which I'd be happy to show you as well. But that's right. They're excited by the nice thing about Faraday waves is that on telequette and Rayleigh, here's the cylinder. And here's the outer cylinder. And the Lane Scale is the depth. The gap are in or out. You saw the rolls. They're about like this, right? They're round. So this depth is the same as this wavelength. And the same with convection. Convection, you have two plates, hot, cold. This depth, you have rolls that are about as wide as they are tall, like that. But this is not the case for Faraday. For Faraday, the wavelength is decided by the frequency. So if you shake it fast, you get little patterns like this. And if you shake it slowly, you get long wavelengths like that. So you have much more control over the spatial wavelengths. So getting back to this interplay between experiment and numerics, before 2005, they could not even experimentally measure the height of a surface as a function of space and time. There were no quantitative measures that you would take photographs and you could do some kind of post-processing. But you could not get a map of height as a function of x, y, and t. And that changed in the middle 2000 when three methods were invented for doing that, all based on different principles. This one is based on, well, I don't know what it's based on, but it requires fluids of near equal densities and high viscosity. There's another method that uses a kind of random pattern on the bottom and calibrates the method so that you see how these random dots move under the wave motion. And then there's a third method. What's the third method? I don't know. Anyway, they were all invented around the same time, 2005 or so. So with these quantitative measurements, this experimental group showed the squares and hexagons as you increase the acceleration. And we go back here, square pattern here and hexagons. And because they now had this, the whole surface height map, they knew the whole mapping, they could then do Fourier spectrary. This is a square, the spectrum of a square is a square. You have the excited Fourier modes are on the four corners of a square. And that's the case here. And then they did for higher accelerations, they had hexagons. And there, too, the Fourier modes, it's its own dual, the Fourier transform, the two-dimensional Fourier transform of a hexagonal pattern is a hexagon also. So this is meant to show that they had enough information that they could do Fourier transform. So again, you might think that we would be out of business. But apparently not quite yet. These are some of the basics of the idea of the free surface code. I don't know much more than this, but this is what the basic slide on what they do. And you saw before these instability tongues where you have the wave number here, the wave number, and the forcing amplitude like this. And the instability occurs inside these tongues. And these are the critical accelerations. And so Nicola Perriné showed that the code reproduced the theoretical curve like that very well. And here are computations done the square one you've already seen. And here is the hexagon, the hexagonal lattice here. Let's show you again. Oh, this is a square. And then this is the hexagons. Again, this computation was just done in a single hexagonal domain. And then the picture is repeated. Yes. I can show that. The minimal domain for a hexagon, one side has to be, has a relationship, has to be a square root of three times the size of the other. Is that your question? Maybe not? To form a hexagonal pattern in a minimal domain, it has to be there's a certain length relationship. It's not a square domain. Oh, this is the domain that we computed it. We were able to produce a single hexagon in a domain that's L by square root of 3L. And then we just repeated them. That's not what's done later. I'll show you that in just a moment. This is for these first simulations. OK. Come on. Next slide. There. In fact, the information is right here. This is the minimal domain that's needed. The relationship is square root of 3 between the two lengths. And this is a minimal domain for a hexagon. And the grid that we used is about twice as dense as what's shown here. So we had this many points, 75 by 125 by 225, I think, like that, to get this one hexagon. And so as I said, not only can you get the surface which now the experiments can get, they couldn't always, we can also get the velocity fields. We see that they, we see here that when it's at its lowest point, that's when the velocity is highest. It's about to lead to the eruption of the volcano here. You can see that. And here, when you have the bump, the velocity is actually heading inwards, and it's going to make it go down. It's heading inwards, and it's going to squash it down to this. And you can see every detail of the velocity field, because it's a numerical code. We have access to all of that. And these are just nice pictures of what the hexagonal field looks like over a forcing period. But now we wanted to do more. We wanted to go to a bigger domain. So we went to a, so my colleagues, who didn't know how to write free surface codes, wrote a code that they called blue, which has the front tracking level set algorithm is what we were using in the single domain case. But this has parallelization, domain decomposition. Tests are up to 65,000 processors. Each of the processors holds a grid of 32 cube points. So you can see just how big a problem you can do here. And then we went to do larger problems. And here is one. This is the first larger one that we did. Let's see. I hope this, yes. OK, so you see this is now a large square. This is not just one square repeated, because you can see spontaneously the region divides itself into four pieces that are doing slightly different things. It's like a waffle. This one and this one are in phase, and this one is in phase with that. If you take a slice through here, this one has minima, when this one has maxima, and vice versa. And you can see that here in this picture. You can see as you take a slice here, you can see this one and that one are out of phase with one another. And it chose to do this spontaneously. It didn't want to do the same thing all throughout its domain. It wanted to divide itself into two by two squares for some reason, yes. I don't know. We haven't done a survey of this. I don't know. I don't know what it would do then. Yes, but we also did this with Dirichlet and Neumann boundary conditions, and it did about the same thing. In fact, I think this is with Neumann boundary conditions and not with periodic, I think. So this was nice. So we were excited about this. And we discovered that he had actually already been seen experimentally by Duadi and Fulf. Here's a picture that was published by Duadi here of the pattern that he saw in about 1990. We did this in around 2010. So 20 years later, we actually, we didn't know that. We didn't go to search for this pattern. We just found it as it happened. Now, what could we do that they didn't do, because that was what, that's kind of the theme of this. Well, they had a theory about what the Fourier spectrum was like, they at the time could not measure the exact surface height as a function of space and time. And they had a theory about what kind of resonances between the wave numbers led to it. And we showed that it was a different one. We showed that the wave numbers were not excited, were not exactly what they thought they were different. We still don't know really why the pattern forms. But anyway, their exact surface height measurement was not right. But basically, we were happy to see that this had already been seen. And then we went and did a different thing. That's the end of that little tiny story. Anybody want to ask any questions about that little tiny story? This is a sub-story. There's more on Faraday coming up. No? Ready? OK. Now we thought, instead of accelerating this, what if we did this with a sphere where we had a radial, everybody knows gravity, right? Gravity acts radially. Supposing the gravity were to alternate up, down, up, down, up, down, up, down, up, down, right? Radially. Not like this, but radially, a radial oscillating force. So you see the gravity that's over there. Now, in the short term, contrary to what you might think, nothing happens. After all, we have gravity here. And we know, some of you have seen this in my classroom, there's hydrostatic pressure. That is to say, pressure compensates for gravity. Even though you might think something would happen if gravity oscillated in our world here, it wouldn't actually. All that would happen would be that the hydrostatic pressure which compensates for gravity would, I mean, no. That's not true. As we ran, things would happen. But the air would not all fall to the ground because the hydrostatic pressure is what keeps the air up. And it would alternate like this. So you'd have the gravitational force and the hydrostatic pressure. And then the other half the oscillation, you'd have the hydrostatic pressure here and the gravitational force like that. So it would actually not lead to some terrible consequences. Thardet is an instability, meaning that the surface could stay flat. That's a solution the whole time. You could have a flat surface that's just oscillating up and down. It's an instability, meaning that you need an initial condition that has a wavelength like this, and then it starts doing this. And so too with the sphere. The sphere would just stay spherical. It can stay spherical, but there's an instability. And now you're going to see it. Those tongues that you saw before, now you see them in the, again, in forcing amplitude spherical wave number space here. And now you see the shapes that are excited. And I'm going to show you movies. This is one exit here. This is called L equal 2. It's oblate-prolate alternation. And you'll see it's going from a kind of football to a kind of disc. Football, disc, football, disc, football, disc. So it's conserving volume. It bulges out like this and flat, and then the other way around. It always has two axes the same, and then the third axis is different. Either the third axis is shorter than the other ones or taller. And this is the fluid drop, but it's also the velocity field. You see it all the time. So this is what it does when it's saturated. Now you're going to see something else I want you to see. You see it's on this axis here. But it's actually going to move. Come on. Keep going. You see now it's no longer straight up. You see how it's tilted? You see that it's tilted? It did that by itself. It seems to want to tilt, and now it's tilting more and more and more. And we don't know why. And we want to investigate this. We don't know why it's tilting like this. We've taken transforms, spherical harmonic transforms, and we can quantify the way it's moving around like that. And we don't know why it's doing that. Yes? I don't see the surface of the machine. Oh, you don't see the surface because it's covered with velocity arrows. It's covered with that fuzzy velocity arrows. It's in there. This is the domain here. Sorry, I should have said that. What's outside here is nothing. This is the domain, and it's the surface is the smooth part there. OK? It's kind of a bright scene. We're talking in one direction. You run your code many times, you find. Well, if you run your code many times, you always get the same thing. It's a code. You have slightly different initial conditions. You get things to be different. But you can. It is an instability. So it's not only doing this, but that it's doing at a fast time scale. But it's also doing this. It's moving for reasons that we don't know. And that perhaps is a discovery. We're not sure. And we want to look into this more. And then, let's see, oh, I did this already. OK. So that was for a certain L. This is called L equal 2. In spherical harmonics, if you know spherical harmonics, you know they're labeled with LM. This is the L equal 2 mode. And when I did this thing here, this is different L's. This was L equal 3, 4, 5, 6, 7. What you just saw was L equal 2. And now we can proceed to a different L. But before we do that, I tell you that for small L's, what you get are the platonic solids. You've heard perhaps of the platonic solids. Those are the five regular polyhedra with this many faces. And each polyhedron has a dual, a dual of a polyhedron. I didn't know this before, but each dual of a polyhedron is when you turn the vertex into a face and you turn a face into a vertex. So here you see that the dual of a cube does not have six faces. It has six vertices, which are here. 1, 2, 3, 4, 5, 6, right? And it doesn't have, whereas the cube has six vertices. 1, 2, 3, 4, 5, excuse me, no, that's not true. Hold on, it has eight vertices. 1, 2, 3, 4, 5, 6, 7, 8. And this thing here has eight faces. 1, 2, 3, 4, 5, 6, 7, 8. The faces turn into vertices, the vertices turn into faces. And you can see that when you're doing this with the drop, it becomes its dual. If you take the maximum to be a vertex and the minimum to be like a face, then it's oscillating between itself and its dual. So we can have a pattern, possibly, that's a cube and an octahedron. And the other two platonic solids are the dodecahedron and the icosahedron, which are duals of one another, which this is 12 pentagons, and this is 20 triangles. And indeed, for l equals 3, we get a tetrahedron, oscillating tetrahedron. And a tetrahedron is its own dual. For l equals 4, we get oscillation between a cube and an octagon. That one's a cube, and then it's an octagon. We're not used to seeing octagons. But octagons, you take a cube and you'd make every corner flat and every flat piece corner. So that's the alternation you see for l equal 3 and l equal 4. And again, you can control this by the frequency. You change the frequency, and that's how you do it. To go across the axis, you change, you just make the forcing have a higher and higher frequency. This is an l equal 5 pattern, which is in here. And that is something alternating between an icosahedron and a dodecahedron. And all these are predictions that you, in the pictures, of course, have details. And they have those velocity vectors. You can see those, and maybe you can see it better that it's got this fuzz of velocity vectors on top of it. What if we, oh, this is nice. This is a one pattern. What is a one pattern? Well, we didn't know. But it's where it goes, boing, boing, boing, boing, boing, boing, boing, boing, boing, boing, boing. That's to say this, what is a two pattern? The prolate and the oblate, you have two bulges and flat places. L equal 1, you have one bulge and one out and one in. And what does that mean? If you have a sphere and one place goes out and one place goes in, well, then it's going that way. And then you alternate between the bulge and the flat place, going the other way. So it does this. This is l equal 1. You're not applying a force that does this. You're applying a force that does this. But the response of the system is to go boing, boing, boing, boing, boing. Let's see if we can make it do that again. Oops. So that's l equal 1. And all this is controlled. We're doing the same thing. All that we're changing here is the forcing frequency to get these different patterns. We're changing the forcing frequency. And we get all these. In this planar case, it wouldn't be so spectacular. Because indeed, the wavelength is determined, the spatial frequency, the wavelength, is determined by the temporal frequency. Yes. So you get something that's tighter, do, do, do, do, do, or something that is a longer wavelength. That doesn't seem so different. But when you put it on a sphere, different l's do different things. As you can see, an l equal 3 is a tetrahedron. An l equal 1 is a boing, boing, boing, boing, boing. An l equal 2 is pro-late O-blade. They really have different shapes. They're not just different sizes, as would be the case in the planar case. Or you change a wave number, a planar wave number k for another one, k prime. It's just something bigger or smaller. Here it's actually very different. Yes. Most of these are subharmonics. Some are harmonic. Some are harmonic. Most of them are subharmonic. I have a detailed map of that. And for those who want to, OK, here. Now I can do my other demonstration. The faraday instability is generally subharmonic, as our colleague has pointed out. And what is subharmonic? Here is the surface. It's doing this. And here is a harmonic oscillation. You see, it has the same period as the surface. And here is a subharmonic oscillation. Up, down, up, down. It has twice the temporal period of the forcing period, as our colleague who knows about faraday seems to know. So some of these are harmonic and some are subharmonic. We can excite either one. OK. And we're almost at the end of this story here. So this is the code blue that has all these features that has this very parallel that can do a lot of stuff. And it's been, I didn't put in the coding detail, the physical details. I only put the computational ones. This is the speed up with a number of processors. This is where they show that they can take full advantage of the parallelization. And if they have 10,000 processors, then they can go 10,000 times as fast, which isn't always the case that you have to learn how to write your programs properly so as to do that. I don't know how to do that, but they do. So this is where they show that it speeds up that way. Yes? Yes? Yes? Well, there are many, many hydrodynamic things that do many things. And I'll say no there. How's that? I'll say no. This is on the surface of the sphere and it's due to, and it's an instability due to oscillatory forcing. I don't, you're talking about oscillatory forcing? Yes, for the sand quakes. The sand quakes? They are the strongest tool and they can write the program into the sphere. Okay. It's too far from my subject to say anything. Yes, you were going to say something. It is a drop surface interaction. Can you introduce a specific model that you think is going to happen? Well, fortunately, for once, this code is not my business. You'll have to ask my colleagues. Maybe they can do that. I know nothing about it. I don't think they have anything viscoelastic. They have surface tension. There is surface tension. That was there somewhere in these equations. Yeah, I don't think they've done anything like that. Yeah. Here's the surface tension over here. Where they take into account the curvature to make the surface tension. Okay, so let's see. Am I at the end of this little mini story? Not quite, I think, because, or am I at the end of this mini story? Did the L go 1? Blue, oh yes. So this code, so this does all this. And fortunately, I said I got in touch. That's how I got to be the first person to do the 3D simulation of Faraday's, because I met somebody who knew service codes. And then they were inspired and they wrote this code. And here they did some spectacular things. Since this is generally about CFD, not about pattern formation, which is what I do. But this is how they use tremendous amount of resolution here. And they did some kind of fluid through an aperture like this with engineering applications, the Omar Tahrad Imperial. And here is their simulation where they show the fluid going through the aperture. It has different viscosity. You can see it's got all these parameters just to show what they can do with their code. And I'm happy that it was inspired by working with Faraday instability. Okay, so I think now that's the end of this little mini story. Yes, so this was, again, this was the case of just doing a clean, detailed experiment. And I have these colleagues, former students, postdocs, colleagues, they did this with a free service time step encoder. What did we learn? Well, because we have a code, we have the knowledge of the free surface in time and space. Nowadays, that can be done experimentally as well, which was not the case when we first started. We also have the knowledge of the velocity field, and that every place in time and space. And that cannot yet be done experimentally. But we can do this experiment with the sphere, which experimentalists hearing about this, they want to know, how could I do this? Indeed, there are experiments with spherical drops, but they're held by some kind of piezoelectric or something. They can't be held perfectly spherically. They're held by something maybe between here and here. There's some kind of electrophoresse or something that's holding them. But there is a distinguished axis. It's hard to hold something in midair with complete radial symmetry. I don't think anybody knows how to do that, although I may be wrong. So we can actually do this experiment that nobody else can do, because you can't figure out how to hold a sphere in midair like that. We can do these Fourier modes, spherical harmonics as we did, and also this serve to help validate this new free surface code, blue. It was one of the kids, they're glad to see all this experiment, this agreement with theory and all these pretty pictures. And now I go to my third little story. And this is one that I haven't said before. This now is not just a perfect experiment. Here we also do mathematical analysis. We're going to calculate bifurcation diagram, including unstable steady states. And we're going to do linear stability analysis. People could do transient optimal growth, I'm not talking about that here. So the example is Rayleigh-Bernard Convection cylindrical container. And here we're just talking about time stepping at first, and then we will talk about Newton's method in order to compute the bifurcation diagram. So here is this story here, you have a cylinder about like a can of tuna, something like that. That's the right aspect ratio. Hot here, cold here. As I mentioned before, concerning wavelength choice, what the system likes to do is to, well, convection is when the hot fluid rises, the cold fluid falls, and it likes to make a wavelength that's about the depth. So it likes to make a pattern that's about the same as the depth. If the box is small, then it's very constrained about the pattern that it can make, and that's the case here. So we were inspired by an experimental paper, half Lucas and Mullen in 1999. This is the convection pattern seen from above, like this. The white is hot fluid rising, the black is cold fluid falling. And so you see structures that are about the same size, about this size or this size. The wavelength is about common. But you can see that the pattern itself is quite different. Here you have something that could be called a dipole here. The fact that sometimes it's black, sometimes it's white is to be expected, there's symmetry between, the Boussinesse symmetry between hot fluid rising and cold fluid falling. So this is to be expected. And it's also to be expected that you would have a white Y shape here and a black Y shape there that's normal. But why do you have a Y shape at all? They call that a Mercedes pattern, because it looks like a Mercedes logo, and then they had axisymmetric pattern. And this pattern, unlike the ones I was showing you before, these are all acquired at exactly the same parameter values. So there are multiple steady states for this, which is not surprising for fluids, but still it's always nice to see multiple steady states. And how did they get the different patterns? They did it with history. They would heat it up to a certain Rayleigh number and then go back and then heat it higher and then go back. And depending on the history of how it had been moved up and down, when they got to the same Rayleigh number, they would have a different pattern. And you see they have, some of them are duplicates because of symmetry, but there's the axisymmetric, the Y shape, the Mercedes, this one and this one are all different. There are four definitely different patterns here. Actually, this one is different too, I think. Five. One, two, three, four, five. Five completely different patterns. Experimentally, that you see the white and dark corresponding to velocities, how do you experiment with it? Uh, what's the measurement technique they use? I mean, white, black, or white. I don't know. I don't know. I don't know. Shadow graph. There, thank you. Shadow graph. Does that tell you what you need to know? I'll leave you though. I can tell you that later. Okay. Okay, so we ran a time-dependent code, which I'd written in the 1980s, but had never had enough computer space to do because computers then were, a lot of people did that. They wrote 3D codes in the 1980s, but you couldn't run 3D codes properly until 20 years later. So, fortunately, this got to be used. And we were able to reproduce a lot of these patterns and history dependence, and we saw all these different things. But this is not an actual bifurcation diagram. This is just kind of, I did this life experiment. I did this, and I saw that. I did this, and I saw that. I did this, and I saw that. It doesn't put order into it. What puts order into it is a bifurcation diagram. And this is a bifurcation diagram. This is the Rayleigh number. And these are different branches, and they're accompanied by pictures that show schematically the pattern that you get. And each of them originates through bifurcations or not. And this is very complicated. There are 21 branches here that we have calculated. But I can tell you, this is only a small fraction of the branches that exist. There exist many more branches. And I know this because there are other eigenvalue crossings, and every time there's an eigenvalue crossing, there's another branch. So this is to say that, from this, you shouldn't say, oh, this is a special problem. No, the lesson to take from this is, Navier-Stokes or Boussinesq or any of these non-linear problems, they have hundreds of solutions, thousands of solutions. Multiple solutions are normal. That is the usual situation when you're solving fluid problems or any other non-linear continuum problems. This is normal. This is just the beginning of it. And you just, they have, I won't say uncountable because it is a countable number of solutions and it's even a finite number of solutions. But it's very, very large. And how did we get all these? Well, we used Newton's method. We were able to, what you usually do when you're talking about time-dependent simulation is you're time-stepping the governing equations. This is what you do on the computer and this is also what nature does. Nature does, d-u-d-t equals f of u, where f of u is gravity, is buoyancy, is centrifugal force, is this and that. This is what nature does and what you do in the computer is often just what nature does, except more exactly. But that's not, as I was saying, that's not the only thing you can do. You can look for steady states. You can look for where f of u is zero independent of whether your system gets there or not. And so you solve these governing equations. You're not integrating them in time. You're looking for roots of f of u, f of u being the right-hand side, equilibria, even if they're unstable. Most of them are unstable. Being stable is like being, okay, being a solution is like being in a flat place in potential space, being in a more or less. And being a stable solution is you have a flat space at the bottom of a well. So you move a little, you come back, right? Being an unstable solution is like being at the top where yes, you're happy, you're flat, as long as nobody bothers you. If somebody bothers you, then you go down the hill. And that's an unstable solution. And you won't see those because things are always bothering experiments. They're also bothering numerics. They're bothering them with finite precision. They're bothering them with a truck going past. They're bothering them with the gravity of Jupiter. There's no, it's not perfect. And hence anything that is unstable, you will not observe, either numerically or experimentally. But you can compute it. You can compute it if you use Newton's method rather than doing time dependent solution. And it was by Newton's method that we found this bifurcation diagram in all these branches. And this explains, explains in some sense, these multiple observations. Let me give a tiny bit more detail about that. When you have a circle, circular symmetry, necessarily the states that appear are trigonometric. They look like this. This is dipole state. This is m equal two. This is m equal four, excuse me, m equal one, m equal two, m equal zero, m equal three. And these are the bifurcations that will take place as you increase in Rayleigh number. Did you see anything like this in those experimental pictures? Some yes, some no. You saw dipole state, sort of, you saw axis metric. But you didn't see anything like this, this two with four little dots nor something with six dots. But that's not what you saw. You saw different things. You saw that Mercedes pattern. This says m equal three and there was something with three, but that was a Y-shaped thing, which this does not look like. And indeed, if you do the bifurcation analysis and find the steady state for m equal three, this is what bifurcates as it has to. As I said, what is connected to the conductive state necessarily for mathematical reasons must have six little spots, three red, three blue, necessarily. This is what you see. This is the Mercedes pattern. And to get from here to here, we calculated all the intermediate states and this branch here, which we called Marigold. Well, here, we found another branch which we called Mitsubishi. You can see for obvious reasons, like the Mercedes. But then when we found these other states, we didn't have, there were no car logos that went with them. So we went to a flower pattern, flower names. So this one that we called Marigold. And then there's another bifurcation that breaks the symmetry between red and blue leading to these Mitsubishi states. And then there's a saddle node bifurcation for those who know what this means. But anyway, it's a transition like that to another kind of pattern that looks a little different. You see, these are bulging out and these didn't. And then another saddle node bifurcation here. And finally, you get the Mercedes. And this thick blackness here means that it's stable. None of these patterns are stable. This one, for reasons unknown, is stable. But the genesis of it, the creation of this solution has to happens through all this. It's not that the system does that. The system doesn't do anything like that. The system jumps up to here. All of these are unstable. But you wouldn't know any of this if you weren't finding the unstable steady states. And mathematically, one knows that you can't connect something that looks like this to something that is featureless. You necessarily have to go to something that has six spots. So this is the kind of thing that you can do mathematically with numerics that no experiment can do. This is making the equations do stuff. This is math. And so here I'm summarizing my story. This was in this, for Rayleigh-Bernard convection cylindrical container, we did not only time-stepping, but also bifurcation diagram and linear stability analysis. And we used time-stepping, yes. But we then also used Newton's method. And now to do Newton's method, Newton's method requires, here at a certain point, here there's this thing for multi, this is for one dimension. For multiple dimensions, this is a Jacobian. The u is a vector. It's a very long vector that describes the entire velocity field everywhere. f prime of u is a Jacobian, which is number of points by number of points. It could be, I don't know, 10,000 by 10,000 matrix. You cannot actually solve this. You have to use special numerical linear algebra to use this. You might be surprised to hear this, but the big challenges in computation are not some fancy things, but merely solving the equation ax equals b for very large a. Matrix A times vector x equals right-hand side b. So we do this in a matrix-free way and thus we're able, whoops, able to do this, find that bifurcation diagram. So what do we accomplish here? We found many different branches, almost all being unstable, and thus we were able to lose state the sequence of bifurcations leading to the observed states. Thank you.