 Good morning. So my name is Santiago Lopez Castano. And today's presentation is about shear wind driven air, shallow water, turbulent boundary layers under the large eddy simulation framework. As a motivation for this work, amongst other things, we can think about climate change. Let's say you have a thermal boundary layer developing above a lake. What is that? Developing above a lake. And one is interested to study the small scales interfacial dynamics between the air and the water's face. And obviously, the inherited turbulence that comes with the increased mixing. One could ask whether this thermal boundary layer has some effect on the fish population in that lake. One question begs. One could ask, how can I model efficiently this? Or whether I can implement some parametrization for low order models, say GCMs, or for very large scale CFD models. So the outline of this presentation is the following. We're going to be talking briefly about the mathematical methods used to model interfacial fluid dynamics between highly stratified fluids. It's coupling the scaling of the archetype cases. And afterwards, we're going to dedicate more to the results. And obviously, since we're interested in the near-interface turbulence, so we're going to do some quadrant analysis first and second order models. And we will propose a way to separate a strong from what we call random realizations of the flow in order to do the quadrant analysis. Now, for the mathematical model, it's just the space filter navier-stokes equations. The unresolved scales are modeled via dynamic Lagrangian-Smagorinsky model. As for the coupling, we use what is called between the highly stratified faces, say, air and water. In this case, we use what is called a Picard iteration technique. Basically, under a certain time step, we divide the time step in two, in which on one of the fluid faces, we impose the dynamic condition. And on the other one, we impose the kinematic condition for velocities. Now, in order to, let's say, study any system, we have to make records of the most archetype case. The archetype case that we chose in this case for this simulation was basically a co-current Couette-Quassette. Basically, think of wind-driven, a wind-driven flow that is applying some stress in a channel. Now, the thing is that this is, when you try to scale an archetype that is combined by two boundary layers, or in this case, three boundary layers, you have to come up with the, you have to make some, yeah, you have to accept, let's say, some constraints or complications. In this case, we wanted to model a case in which the two highly stratified fluids, the flows, had the same Reynolds number. So when we do the scaling, this assumption of equal Reynolds number, meaning equal viscous lens scales, impose the following complication. This term has to be equal to 1. The implication of that complication is the following, that if I want to transport my information from a purely, let's say, fundamental study in a real case scenario, the assumption is that you have an ambient temperature, let's say, of air-water interface of around 320 Kelvin. That's a very hot summer in the Adriatic Sea, let's say. Now, let's start with the results. Studying the water subdomain, we find the Couette profile. This Couette profile, nonetheless, is not symmetric, as one would assume when you have traction on the top. This profile, when plotted in inner scales, shows a big difference, a non-symmetry in the interface side of the flow, meaning in this part. In this part, in this first part, the profile is no longer a wall-like boundary layer, but it's a thinner boundary layer, meaning that as later we will show, it means increased mixing. Now, as for the RMS for both the wall in the water subdomain, close to the wall and close to the interface, we find some difference with classical profiles that you would find in literature. Especially, and very important, is the span-wise RMS fluctuations that guarantee this increased mixing that I was talking about. Notice that all this increased mixing and peaks of maxima and minima, low side of maxima and minima, occur around the Y plus of around 10. Have that value in mind. Now, when you move to the air side, statistics are pretty much channel flow, very classical. Now, when you move out into the, into study how is this interface increased mixing on the water side affecting the structural improvements, you can do an adoption technique on the water side close to the interface, and you find that you have, that the streaks in the water side are shorter, and you have interface-connected vertices, and obviously the smaller vertices that are born because of the shear interaction or the fluid itself. When you move to the air domain, you find that the structures and streaks are longer, are more elongated even that classical channel flows. We'll speak about that later, and well, they're less chaotic than the previous picture, and well, obviously shear-born vertices prevail over the interface-connected ones. Now, what is this about interface-connected vertices and shear-born vertices? If you do a vortex analysis of the fluctuations a la Antonia anuin, you find the following description for the air side and the water side. For the air side, you find that the angle, the horizontal angle of the vertices, is pretty much straight, rounding in zero. It's pretty much expected. You don't expect for the wind or the pressure gradient driving these structures to be deviating much. Also, the same happens for the distribution of the vertical angle of these streaks. For quite a long time, we know that in channel flows, this vertical angle of the streaks or of these vortices is around eight degrees, between five and eight degrees. We find kind of the same values, around five, we find the peak at five. But when we study the boundary layer, when we study the boundary layer close to the, on the boundary layer on the water side, we find a quite different behavior. This quite different behavior shows three peaks. These two peaks, around 90 and 225, 270, represents what we call interface-connected vortices. Obviously, this leaves the center one, the ones that are generated by the act of share. You have the share in certain directions, so you should have all the streaks going on that, more or less, same direction. Interestingly enough, consider the fact that 90 degrees in this scenario, represents a vertical angle going down. A vertical angle going down. Then 260 would represent, 70 would represent the opposite case. And the angles in between represent certain, yeah, this specific quadrant. Now, one curious thing, though, is that even though all these differences that we have shown, when you do the quadrant, the classical quadrant analysis of Wallace, you find a pretty much a wall bounded, kind of quadrant analysis. It doesn't show many features that one would expect or differences, except maybe in the heights or the actual values that you get. Other interesting thing is that for the Q2, Q4 quadrants, which means ejections and which means sweeps, they also happen to cross at a value y plus approximately 10. So in this specific work, we wanted to study, do a more precise quadrant analysis of turbulence at that height, 10, which would correspond in wall bounded channels to what we call the viscous region or the top of the viscous region, at least for these very low Reynolds numbers. So the proposal is, okay, you have the complete set of signals. Why don't we, of all those signals that you could get near the viscous region on either side of the interface, we take the particular ones that we call strong. So the technique is called with the shear and with the interface fluctuation shear. So again, so what we do is whenever these three probes happen to be in the same quadrant, we record that strong event and we record the velocity fluctuations at y plus equal to 10, and we also record for the sake of analysis, the shear stress, the fluctuating shear stress on the interface. And then, well, after we detect that a strong event, we go further and we define what is called a strong couple event. So as I was saying, as I was saying, as I was saying, as I was saying, this is an event, a realization in the flow that is not necessarily strong, that usually in quadrant analysis you take it. In the strong quadrant analysis, what you do, the red box represents a strong event on either side of the interface and the blue represents a coupled strong event. Now, let's start defining the strong events without necessarily talking about the coupling. So in this case, we find that the strong events, the strong quadrant analysis on the air side represent 27% of the total realizations on the viscous layer, 27%. We find that for the ejections, sorry, for the sweeps, the sweeps generate a negative stress fluctuation in the mean, whereas the positive, whereas the sweeps generate a positive normalized shear fluctuation events. That's what you should expect in the mean, whereas the interaction, the interaction events, well, their mean is approximately zero. On the water, on the other side, these strong events only represent 3% of the total of events near the interface. The interesting thing about this would, in some sense, would explain the disorganization of the more chaotic behavior that you have and the increased mixing that you have on the water side for the interface boundary layer. Interestingly enough, you have the Q2 and the Q4 events, around 30% and 26%, whereas for the air side, you have a stronger presence of sweeps and ejections. Well, especially for sweeps, for the Q2. Now, we move on and we try to study the PDFs of these fluctuations, meaning the velocities that we take at y plus equal to 10 when you have a strong event. We find that for the water side, you have some bimodality, but obviously because of the convection of the interface, this bimodality is moved towards the positive side of the u-velocity, whereas for the air side, since the interface is more rigid because of the higher inertia of water, you can find the bimodality more clear. One could speculate that this bimodality behavior is due to the interface-connected vortices on one side and to the vortices that are generated because of the shear interaction in the no and the near viscous region. If you do a covariance integrant study, meaning quadrant analysis on that specific point, you find that on the water side, the Q2 and Q4 events are not as strongly present. In fact, the interaction events which are not necessarily turbulence are even stronger in value and wider in spectrum. On the air side, on the other hand, you find a more classical behavior of turbulence, which corresponds roughly, obviously, differently but roughly to all bounded flows. Now, we talk about strong events. Now, we move to a study of strong coupled events on the interface. Strong coupled events, meaning the whole combinations along the Q analysis, meaning that you have a total combination of 16, Q1, Q2, Q1, Q3, and then all these events, you can divide them basically in high probability events and low probability events. Interestingly enough, we find that sweep ejections and ejection sweeps are on the high probability side. Well, we should expect that, we're starting turbulence. And the total of the high probability events correspond to 9% of the strong coupled events, whereas the weak events that are mainly interaction events, not much related to turbulence, only represent 9% in total. Ejection sweeps and vice versa on the converse event, meaning sweep ejection, take 10% and 9% respectively of the total coupled events. Now, if we further analyze these coupled events and we only take the velocity realizations that correspond to these coupled, the strong events, we find the following. In the water, again, we have this B modality. We find it obviously weaker, but we have it. And this B modality roughly corresponds to a Q2, Q4 behavior of turbulence. So in the water side, sorry, in the air side, I'm talking about the air side, I'm sorry about this. What I wanted to say is that on the water, we know that with this we can conclude that the water side, in fact, is the one forcing motions on the air side. It could be counter-intuitive on the beginning because you could say, but why if it is water that is giving tension into the surface that is giving stress? Well, yes, it's giving stress, but the thing is that water has a way higher inertia than air. So a punch given by the air to the water is not as felt as a punch given by the water to the air. So basically, what this plot means is that air motions in the near-interface mixing are dominated by water events because you have the same wall behavior on coupled events. Whereas in the case of the interfacial Reynolds stresses and motions on the water, they are again weak, they are very weak. The probability is not as high, but these coupled events show a structure of turbulence, a turbulent structure, Q2, Q4 behavior. Final remarks, just a summary. Again, this b-modality is thought to be connected to the kind of vortices that are generated in the near-interface between air and water or any highly stratified fluid that respects that Reynolds number and the density ratio. Interface-connected vortices have angles between 60 degrees and 90 degrees, 90 being the peak. Which means vortices moving along the flow with this angle where at least 225 to 270 being 70 the peak are counter, let's say counter-vortices moving in the vertical direction. And these, well, this is basically uncoupled strong events, make very little for the overall events. And then of all the 16 coupled events, only Q2, Q4, which are the most important ones, are highly probable. This comes with a little bit of difference with the previous works in DNS. Not our exact case, but they detected different, very different highly probable events except for the Q2, Q4, and Q4, Q2. And b-modality remains. So, that's all. Yes, it's that. Well, I mean, if you're doing, if the idea, the initial idea of this model was in order to calibrate large-scale models that usually do not represent or do not deform the grid in order to represent waves. Usually waves, let's say in GCM models, are represented using potential theory, especially gravitational ones. That's one reason. The second reason is more of a computational constraint. Well, if you think of a very large-scale system that is only driven by wind, you expect in the mean the water surface just to have an angle. Sure, but, I mean, putting, you could say that also instead of doing, let's say, LES for such a low-range or Reynolds number, is also insufficient for DNS, because if you, as I say, if you put the additional constraint of putting the pressure fluctuations does deform in the surface, you're putting, I mean, you have to separate the scale analysis of the fluctuations of Reynolds stresses and remove the part that comes from back because of the waves. That makes the analysis a little bit harder. And again, our final objective is to, well, basically we just pass the horizontal stresses and the horizontal fluxes, and fluxes of velocity. We just set V equal to zero, the velocity, the vertical velocity. Thank you. Yeah.