 It's a pleasure to be here also because I'm a biologist and actually an ecologist by training. And this institute had a major impact on several generations of theoretical ecologists in the 80s and 90s. There was a course which was very well-attended worldwide and up to a point where many Europeans wanted to attend, I mean, among the students. So I'm glad to be here on that count as well. I'm a biologist, but I have built over the years a group which is mainly composed on physicists and sophomore physicists. Thomas Steinman, for this talk, has been key person. He's my technician. He's a fluid dynamicist, optical methods in fluid dynamics. And I will try to explain to you what we are interested in. Wave propagation at the air-water interface is pretty often a hot topic. You see big journals. You know the water strider. You might not know those ones, really gig beetles. They turn in cycles at a very high speed. Very interesting animals. We have studied the hydrodynamics of those. And so just to get known with the insects, so they are beautifully constructed, as you can see. The very small ones are a few millimeter long. Weight, half a milligram. And the adults is much larger, three centimeters and 30 milligram. If you look closely, you will see that on the legs, which are long and thin, you have all these hairs, cite, as we call those. And these hairs also have specific structures in order to be super hydrophobic. So this is one of the reasons why they can swim or walk on the water surface. Now, David and John Bush, where your PhD student, he was PhD student of John Bush, made that beautiful study who was a pioneer experimental study which triggered a whole series of work which is still continuing today, mainly on robotics more than on jerryce, where they found that, so they look at these animals from above. So it's a 2D study. And they look mainly at the horizontal plane. Later on, they did vertical as well. And they came to the conclusion that most of the energy was spent in the vortices, which you do see at the back here. And not so much in the waves, which we do see ourselves when we look not too closely at the animal. So it's actually a momentum transfer, which was mainly through vortices. I will use for the talk, or talk only on paper. We made it very clear. Momentum transfer and energy. I use it as a synonym for a reason I will explain at the very end. And this study was more or less confirmed by a person whose name is Buller. Buller works, I think, at MIT, but he is a, sorry? NYU. Exact, NYU. He is basically a applied mathematician looking at vortices. And he made a very deep study on what we should expect from an analytical point of view, and came to a conclusion which is not far from this. However, Gao and Feng, in a big paper, computational work, came to a rather different study and conclusion. So they were looking at a vertical plane and looking at the energy transfer or the momentum transfer, and came to the conclusion that two-thirds were in the wave, surface wave, and only one-third was in the vortices. So surface wave would, in that case, be the main signature of momentum transfer in the fluid. So we do have orthogonal conclusions, almost, and therefore a conflict. And part of the work we have done is to solve that conflict. Now, David's and John Bush study was also key because it meant that animals walking on the water surface were not so different from the others and using vortices due to the complexity of finding the forces and the pressure just around the body, which is very difficult to do, using the signature in the fluid, you could go back to the forces. So there was some kind of unified framework. Actually, this was news and views or commentary in the same issue of nature by Mike Dickinson and others. And so the idea is using the signature in the fluid to indirectly infer the forces used by the animal. So the aim of our work was to measure both the surface of the fluid and the within fluid movements with the hope to come to an energy balance and therefore assess this indirect inference and therefore the strength of the unified framework. Now, I will just let it go. This is the real speed, sorry, I forgot to translate. This is 1,000 times less. Well, I think you see the leg. You also see the movement of the animal, which is a bit up, does not go really horizontal. The surface waves come first and then you have the vortices we cannot see here. In other words, the question is most of the forces today at the end produce a water wave or are they within the body vortex? This needs quantification and it's a difficult question because of the interface. The interface is really something very special. So if we define the proportioned forces, we have three. We have the viscous forces, which will affect the wetting surface. We have the pressure forces and we have the curvature forces which acts only on the contact line. Now Gao and Feng in their paper came to the conclusion that the main proportioned forces was the curvature force. The two others are much smaller. How would that work? The leg is small, it produces a dimple which actually is way larger than the leg itself. If you look at the scaling, the leg diameter is 550 micron, the oar is 1,000 times more. So it's leg plus dimple, which produce these waves and vortices. So the muscular forces will stretch the water surface at the back, it also will compress ahead. This produce a vortex in the water. You have a difference of pressure, front and back, and this is the gradient of pressure which will lead to this curvature force which will be used by the animal. That's the way Gao and Feng understood the process, the exact process. So the signature is both the vortices and the surface waves. Now let me start with estimating the surface waves. We did that with the schlieren technique. I will put all the slides, I think the name. Okay, the synthetic schlieren. In the meantime, this is often used by many people. So you have a dot pattern at the back. The waves will deform the dot pattern you can see from above. You can also play around with these techniques. There are several software available. You can use actually a single camera but you have more tricks to apply or you can use different cameras. And then you can calculate the local surface elevation in real time. So you have a dynamical view of what's going on. Our system has a sensitivity of five microns. So we can measure waves of five micron high, which is useful for capillary waves. So I will show you here one example. So this is the animal moving and you see the dot pattern so you saw the wave. I can replay it maybe once more. So this is the animal. Oh, it's gone fast. So you see this feeling. So this is what the camera does and through this image analysis you can go back to the surface elevations. As so often we have seen this today we have to go through legs kinematics with all these picky details. It's all about angles. Very often asymmetrical movements of legs. We then simulated these leg movements. And then we could sum up the total surface deformation over the whole course of the leg movement. If we compare model and experiments and I use this to highlight our co-workers or collaborators, so Thomas is at the front. Mikhail Benzakim, he's a professor at Ecole Polytechnique in the meantime in hydrodynamics. He just got a chair for econophysics. So money brings money. So he's more than Jerry's on the water surface I guess. And Eli Raphael, you might know some of you. He's a theoretical physicist working with Degen and on capillary waves. The others are students of us. So we were rather happy to have a good fit even if you look at this. And we had to have a model of impulses. So a single impulse is not sufficient to produce such a wave. You have to have an addition of impulses over the course of the leg movement. We also had to change some of the theory for capillary waves for shallow waters because as you probably know some of you or all of you, depending on the height of the water, you might have changes in the capillary waves. And this is the shallow water assumption. This was not done yet quite for such purposes at small scale. So we changed that. And we calculated the surface energy over time and we came to the conclusion that a third of the kinetic energy was at the surface. So two thirds might be probably in the water body but this we did not measure. And that's, we'll need a simultaneous way of measuring water surface and within body water movement and movement of the animal. And therefore we need a more precise energy balance. That implies tomo-piv. So tomo-piv, you heard and you know most of you. Tomo-piv is more recent. It's many cameras and you have a volumetric illumination, not only a sheet of light. It's at least an order of magnitude more complex than piv. It just seems you add dimension but actually you increase the number of problems by 10, 100 times. And yeah, it's not a nightmare but almost a nightmare. In particular for us because we look at the interface. So the interface is by definition moving up and down. So when you want to do the proper calibration even with the self-calibration, you have this problem of water level moving up and down which is really tricky beyond the fact that many particles will tend to stick or not stick at the water surface. The water surface has really different physics as you know and this imply a very different way of attaching particles. Tomo-piv, this is something we did. This is not related to the insect world. It's just to show you what you can do today. So this is the movement of particle. We call it particle tracing. LaVision is probably the best company at the moment and they have a new software which is called Shake the Box, we can really go Lagrangian particle tracing. Let me replay this. I think this is just so good looking. Oh no, okay. So what we see here is the movement of the animal and we will concentrate for the timing on this red square. So very near to the leg. This is the position, the topography of the water surface. The particle will move like this. Kind of difficult to see but still we see the waves coming out. And we can reconstruct the movement of the leg and the particle and we get this. We call this the propulsion phase and we could then distinguish these vortex which David and others had seen. Now I did not enter into the details but the paper of David and John Bush proposes one type of vortex and Gawain Feng found a different type of vortex and we had to glue these things together. We can see that at some stage in the movement we will start to have the counter rotating vortices starting just below the leg. This is from above. You see the two nice vortices as David and others had shown. And in the propulsion phase, the distinction between these authors came from different point of view. The first group, David and et al, looked from the top and if you look at this, the vortex forces go to this level. But if you look at the vortex from the side, the green part here, the shape is somewhat different. It's like a sausage which would be thin in the middle. So this, of course, produce different estimation of the vortex forces. But there is a single semi-annular vortex and we could reconciliate these different opinions about how the vortices look like. Semi-annular, pinned to the surface. So you always have to have two ends. It has to be pinned on both ends. Still does not give us the answer to the relative importance of the forces. But at least we could reconciliate these two point of view. Excuse me? Do you have an idea of the magnitude of the forces? Magnitude tend to minus four, you see. Yeah. So this did require Tomo-PIV because you need to see the same object at the same time through different point of view. Now, if we look later on and also further down the animal, this is the green aspect. And we look now at different time steps. We see the waves coming. This is a plot of the waves and we have plotted in different colors the vorticity, which is of course alternating as they have to do with waves. Also Rayleigh waves on solid surface. And we have nice plots. We show the movement of the waves. Look at the time. We are brainier to the end of the propulsion, 40 millisecond. This is a plot to show you how the waves are looking in two dimension, not in 3D. The vortex comes much later. 40, almost 500 microseconds later. And this is the way the vortex looks like. So indeed, like a sausage with the two pins. So if we try to synthesize what I showed, we have first, most of the kinetic energy in the whole fluid environment is the surface waves, which are the first one to show up. Then we have the vortex arrival and we have a natural decay over the vortex much later on. So we end up by having, as a conclusion, surface waves transporting more energy than the vortices. Question, what's zero there? The time zero is when the leg starts moving or? It's I think the end of, it's when the legs leave the surface. We, yeah, that's it. No, the rowing system would need much more time. It's still, it takes some time. Now, these measurements are very difficult because they are done on real animals. You can never ask the animal to do exactly the same thing. So there is potentially a lack of reproductability. A difficulty for the animal to make, we wanted to understand the impact of the speed of the leg as well as the depth. This we cannot vary. The animal is deciding on its own. So we decided to go through mechanical systems and numerical models to test our understanding. For the mechanical system, we simplify the leg movement, which is really a 3D movement, into a 2D movement by producing a leg using a small wire of the same diameter and also the same hydrophobicity. We could play, of course, with speed and depth using, again, the laser. We will look at, sorry, this is French. At the wave and at the vortices. And for the numerical simulation, we use this scan heli-er way of modeling interfaces where you have a zone in which you have two phases of different elements and the intermingle and then the Navier-Stokes equation as well. This is the mechanical simulation. I just want to highlight that you don't even see the diameter of the leg. It shows how large the dimple is compared to the leg. And this is the same case in the animal. For the numerical simulation, we get something which is rather coherent. Look also at the vortex here. We see beautifully the vortex just in the midway here. Just so I understand, in this case, the vortex doesn't come up at full level. The sector sort of came up to the surface. Here it does as well. It's just the way we have put our laser light is just in the middle. So we look at a cross-section of something which is like a sausage at the cross-section. Yes. Then we could play with the speed. 23, 40 centimeters, 67 centimeters. And then if we do the ratio of the forces, we come to the conclusion that the ball wave, according to velocity, for the same depth is way more than the vortices. And you see the difference between experiments and measurements is not large. And then when we play for the same speed, the depth, and we have changed, we have played around, of course. You get somewhat more messy, but still the same overall message is that the ball wave takes over, has more. Ball waves implies capillary waves. That's the way which is produced by the leg, which will then detach from the leg and will continue. So what these ripples we see is our ball waves which have been detached from the object. So we have a good fit between computation and reality and what you don't see is that the waves are much bigger than the vortices. Now, why do we speak so much about energy and not momentum transfer? Is because the surface of water has no mass and that is really a problem and has been a subject of debate by three dynamicists for ages and so we came to another way of thinking in terms of energy, where you have three sources of energy. So that would be the null basis. You first have the curvature energy, which I described by this extension and pressure difference. You have the energy necessary to displace the interface up and down. This is the blue spot, the potential energy. And then you have the energy necessary to displace the fluid items. And so the first two can be combined by the energy of the interface, where the third one can be considered to be the bulk flow energy. So by playing around with the mechanical model and the simulation, we came to this plot. So as function of velocity and depth of leg, you have basically the graph is split in two half and you have, in the blue part, you have more energy spent to move the surface while in the pink part, most energy is being put into moving the fluid below yourself in the fluid body. So it means that the question of, is this more waves versus within fluid is probably misplaced or badly posed as a question. It depends. It depends on how fast the animal is moving and how deep is the leg. And of course, large animals have a lot of playroom here and they can do what they want. Small animals, like some of the animals, David and others did study, they are smaller and might not have the speed necessary to move the water surface at a good speed. That might explain the different opinions of the different people as well. So it depends. So the conflict is solved. And it's even worse than this, I think, or we think, is if you look at how much energy is put in the vortex according to how much energy is put in the whole system, the propulsive forces, you see a true linear relationship up to a point which is twice the surface tension which is as much as an animal could harvest. This is when the water almost bends itself. So you have twice the surface tension. You cannot harvest more energy than there. These are different phenomena then. This implies that the vortex might be just a side product of the energy put in the interface. Yes, that's the point where you get two sigma is those cases here. So all this is unpublished. I wanted to show you the latest, but therefore it's a bit dense. I realize I'm not over time. I'm back to my conclusions already. But that's the last talk, so I hope you don't mind. Nice weather. We have provided a reconciliation between the form and energy content in the two 3D vortices. We have a good fit between our computation, the mechanical simulation, and the data of the living organism over the whole movement of the leg. We claim that the bow wave has much more energy than the vortices, so the conflict who is winning is settled. But we mainly think that it depends what the animal wants to do. And as I said, large animals have more playroom than small ones. But I want to go back to a point which was already alluded in Gao and Feng. They say basically this distinction between energy in the vortex and surface wave is maybe a misplaced problem because the timing of the two phenomena are too disparate. And what happens is really a problem at the interface just when the animal moves for the propulsion. And then there is energy exchanges and there are different phenomena between interface and vortices, but the scales do not match. So therefore we question this unified framework because it also implied that it is pretty difficult to estimate the propulsion forces based on the vortices and on the surface waves. You need really to be clever how to think about these things. And the juries are still a peculiar case because of the interface being such a different phenomena or different place than anything else. With this, I thank you very much. We'll stop here.