 I'm going on. So here's my talk. OK. So I'll come down and listen to you. As long as you make a sense out of where you think you are. I hope you are hearing my voice. OK. It's not easy for me to give you a talk after Professor Germanos talk. And of course, after this nice discussion, so I don't really want to interrupt, sorry for that. And thank you for having your time, by the way. It's a happy time. I'll keep it short. So my name is Elias Yilmaz. I'm from Stamble Beach University. I'm coming from Stamble, Turkey. As you understand from the long, long, long title, I'm on the numerical side of turbulent mixing phenomenon. So what I do is to develop numerical methods with some nice features to analyze the flows with, let's say, challenging physics. Recently I proposed an LDS algorithm. Yes, it's an LDS algorithm. Which as I said, it has some nice features. Oops. Sorry. So here it's non-dissipative, fully implicit, discrete, kinetic energy conserving at the discrete level. And it doesn't need to use Bosnian-Oberwech assumption. It doesn't rely on this assumption. So maybe, I mean, theoretically it makes it more accurate physically compared to the other numerical methods. And of course, these features make it suitable for some kind of flows. For example, low mach number flows or variable density and viscosity flows and thermally and buoyancy affected flows. I'm sorry, driven flows. Radiotrader instability. These type of flows are mainly hydrodynamic instabilities. So the radiotrader instability is one of them I studied. I actually solved radiotrader instability using this method at low to moderate at-foot numbers. Now I am trying to show that in this study. The method I propose can handle the high at-foot numbers radiotrader instability problem. So you simply, yes, yes, yes. You simply limit the changes in the density into the buoyancy term. Yeah, it's called a Boussinesse approximation for the rheumastrous to make it an end of this constant. Yeah, it's, yeah. There's something, yeah, there's some confusing there. I'll come to that point. Okay, so another aim of this work is to analyze the, I'm sorry, I want to show the physics of high at-foot number captured or reproduced by this algorithm, by this LDS algorithm. And as a secondary aim, I want to observe the effects of high at-foot numbers or large density ratios on the radiotrader instability. So, sorry, here is the equations solved by the algorithm. They're from compressive nervous ducts equations in conservative form with buoyancy term. And I made them non-dimensional. Look at the pressure. The way of making pressure non-dimensional is a special trick that allows you to study low Mach number flows. It's called incompressible scaling. And of course, this is an LDS algorithm. I need to apply far filtering. This is the result of far filtering. As you mentioned about, this is the added viscosity assumption or business assumption, which is different than the business approximation I mentioned before. I had to employ a sub-scale model. How do you use that? Yeah, I'm coming. I'm coming. So, the sub-scale model I employed here is the veiler model. It's one of the most efficient, well-known models. To my knowledge, to the best of my knowledge, this is the first time the veiler model is applied to high at-foot number, radiotrader instability. High at-foot number, multiple radiotrader instability. To the best of my knowledge, please correct me if I'm wrong. This is some other details of discretization. I'm skipping the details because of the time limit. I also developed an LDS solver. I gave it a name, I gave a name I-LDS, which is different than ILS or miles, please, because the implicit nature is coming from its discretization. It's not coming from its dissipation. So, implicit LDS, it's not an implicit LDS or it's not a monotone integrated LDS. This is an in-house solver, as you understand. I made it fully parallel with Patsy. It shares the same features with the numerical method. Additionally, I solved the linear systems arising from the discretization using incomplete LE preconditioned GMRES, which is an efficient approach. Make it to test everything. The term here represents its fully implicit nature coming from the discretization. The term used here, implicit LDS or monotone integrated LDS, it comes from the dissipation. So, let's focus on the problem. This is, as I mentioned, a very relative mixing problem. It happens when a heavy fluid on top is supported by the light fluid at the bottom against the gravitational collapse. Initially, the system is in hydrostatic equilibrium. It's the upside-down bottle problem. Okay. Thank you. But when it's subject to perturbations, then equilibrium is lost. And with the help of baroconic vorticity contribution, flow evolves into a fully nonlinear region as a state of turbulent. At late times, bubbles and spikes are formed. Bubbles of light fluid rise up through the heavy fluid and spikes of heavy fluid falling down through the light fluid into the light fluid, right? So, this motion is controlled by the at-foot number. Here is the famous at-foot number. I'm sorry. Again, please. You said it was kind of like turbulence. Yeah, it's a state of turbulence. What do you mean by turbulence? Yeah, it really doesn't satisfy the connoisseur's five-thirds rule. It's very close, but not really satisfied. That's what I'm saying, it's a state of turbulence. Again, please. No, it's not the constant density. Initially, you have two fruits. But for high density, for high variability density, we have a substantial growth. And no, it's... If you think, for example, like you do yet, this is in relation only to constant density. Pressure is not linear. Pressure is not linear in the compressible fluid. Okay. I set the simulation... No, get the steady state. Can the steady state have a compressible fluid? Yeah, but this is the trick here, actually. I set this pressure like this, okay, and then let them flow, evolve. Okay, so it's not constant density here. Just initial value, initial density value. It's 1.2 as a low density, and let's say, depending on the output number... Not constant density. Roger, why is constant density? Can you give me some time? Because I'm coming at some points. Come to that point. Well, at the density... So how do you get the y-part in? I didn't assume that the density was constant. So you've got to suggest the other... So the rho g y... Yeah. So that gives you the pressure? Yeah, it's a distribution, pressure distribution. Well, if you integrated the equation, then the integral of the rho is there. But you're missing... Okay, but I see... So you're going to do a series of expansions of rho there. No, but that would then be the derivative of rho with respect to y. So that... Second, is this a Taylor series expansion of the pressure? Again, please. So you've got... Yeah, yeah, yeah. So did you just do a Taylor series expansion of the pressure and then just kept the first terms? Yeah, it was done here. Okay, I missed it. Here, it was done here. Okay, it was done here. So I'll take two questions. If you don't mind, I'm going to go fast because, you know, I'm taking your time. It makes me stress. So here is the example. So for the Taylor instability, the pharmaceutical physics from nuclear experiments, industry, and many other natural flows. If you notice, you have to treat body force properly. You have to handle turbulent mixing, diffusion, and interface capturing in the problem. Okay, so many of these examples occur at large density ratios. So there is an interest to solve a high output number relative to instability as this is one of my motivations, as I mentioned. Here is the details of the simulation. The length scale, the velocity scale, and this is the time scale of the problem. Okay, this is important quantity omega. At the initial, you have to provide a kinetic viscosity to the system, I mean to the method or algorithm or simulation, whatever. So it depends on output number and your resolution, of course. It is scaled by this term. So you have to choose this term. You have to choose this term correctly to obtain reasonable, physical or reasonable results. It takes some time to determine this scaling factor. Okay, here is the... Actually, I have lots of results, by the way. I completed lots of results, but there's no time. So I'm gonna show you just two of them. Okay, I'm skipping many others. By the way, I should say that some of the results are completed again for the first time for the high output number, multi-mode relative to instability. Okay, including some turbulent statistics. So here is the evolution of the flow when the output number is... By the way, I forgot to mention here, the output number I'm working with starts from 0.5 to 0.9. In some papers, this is called ultra-high output number region. I don't know. Missable, of course. Okay. Here is the evolution of the local mole fraction, Chi or he in Greek. I don't know. It's a measure of mixing. It takes value between 0 and 1. It shows the development of instability. And as you see here, for the output number 0.5, I set the local mole fraction value to 0.5 again. This time increases from left to right. Okay, for the 0.5, here's the evolution of the local mole fraction. Initially, there is nothing, but in time, the bubbles and spikes are formed, merged, and penetrate into the flow. It starts with a smaller size and they're getting larger and larger. Okay. Let me see. For this case, here is this value. Okay. Let's compare the different effect of high output numbers. Let's compare the local mole fraction evolution of high output numbers. Here, I'm sorry. This is the initial, actually not initial. The time is 1 in terms of non-dimensional units. There's nothing. Here is the time is 3. The difference is, yeah, it's now visible. Here is the... Okay. This is a non-dimension time, 6.5 at foot number, 0.6789. Actually, for the 0.9 at foot number, the non-dimension time is 5, because the flow has already reached the end of the domain. Again? Yeah, a spec ratio, you mean. I'll come down. Okay, just wait for a second. I have to say some details. By the way, the spec ratio is 3 and the 2 pi length is resolved by 48 grids, which is even for an LES. Really, really coarse, but there is... If you look at the successful results, let's say, I mean, this coarse grid resolution doesn't affect the performance of the algorithm, so it's a kind of measure for the strength of the algorithm. Grid is very coarse, but results are fine. I think. My at number based on density. I'm really sorry, my English is broken, so can you tell me using broken English so that I can understand you? Yeah? No, no, no. So it's the top one? Yeah, yeah, yeah. Half of the domain, the upper half of the domain is filled with high density fluid and the bottom half of the domain is filled with low density fluid at the initial. So 2, yeah. No, 3. For the 0.5 output number, for the output number 0.9 here, the density ratio is 0.19, it's quite high. It will increase the output number. Okay. Here. Yeah. We can see that the high output number in stability develops faster than the others and the forming bubbles and spikes are larger than the others. So you see this 0.5 developing. When time is 9 in terms of local mole fraction. So this is the last result I will show you. If you take the average of the local mole fraction, this will give you the penetration length for bubble and spike. Okay. And I take the ratio here, HS divided by HP. So if you look at there are two findings here actually. First of all the 0.5 and 0.6 output numbers have already reached the strongly nonlinear or self-similar region. But 0.7 6 and 9 haven't reached yet. So this says that for the to analyze the high output numbers correctly or properly, you need higher aspect ratio domains. So this is the first finding. The second one if you look at their values the ratio increases with an increasing output number. So if you look at the ratio the increase of the penetration length for spike is much more greater than the increase of the penetration length for the bubble. So this brings you into the flow a strong asymmetry. In other words, spikes develop faster than bubbles as output number increasing. Yes there is only two papers in the literature I confirm their results by the way I mean these results confirm their findings. So in order to provide a scaling between the ratios of penetration lengths and the density ratios I have to run on a longer domain and of course a longer time then I can find their plateau and then I can give a scaling factor this is usually given by the papers to emphasize the asymmetry. I'll do that in the full manuscript by the way the mixing efficiency I'm skipping this the tailor micro scales and the Reynolds number based on I'm skipping this Reynolds stress terms I should say something here you see the asymmetry of the Reynolds stress terms with an increasing output number anisotropic tensors once you get the Reynolds stress terms you can easily compute the anisotropic tensor another measure for the large scale anisotropy the most affected component is the B22 is a vertical component there is no significant change in the others so I also compute the kinetics and potential energies and compare them there is no big I expect that there will be no difference between among their values at the end of simulation 0.4 0.5 I'm skipping this I also plotted the sub-scale dynamic viscosity and auto-scale Reynolds number I'm finishing the slide so what I can conclude is the algorithm proposed can handle the high output number multimode radiator instability physics of high output number multimode radiator instability this is the first time to the best of my knowledge the way the model is applied to this kind of problem results are fine and will be better if I do that on a longer domain and in terms of radiator mixing I can say that I also computed some of the quantities diagnostics for the first time for that problem and all are consistent with the previous data I'm summarizing there are two findings of this work that confirm the other results the high output number need longer domains and times due to the faster development of flow structures especially spikes and this comes with larger asymmetry and this brings large asymmetry flow structures I will provide a scaling factor at the end of my whole simulations whole work 0.1 Mach number but it's not shown here it's non-dimensional version yeah but linear pressure it's not steady state at all okay it's my mistake okay I'll tell you because I tried to be fast so I missed some details here is the initial perturbation if you look at the slope equation you put a gel castle dpdz or ogg you put the initial condition as a thermal for example g is gone so you have very drastic difference and then you put your interface so your output number is actually and your stratification could be thousands by making it linear you're putting two problems at once number one you don't start with steady state solution you start with a solution which is not steady so your interface will develop exponential number one so your mixing two problems would be I think you need to work with change initial conditions change what? because if you solve the steady state equation it's not linear because it's exponential because you have to solve dpdz or ogg I'm solved I'm steady state you mean? steady state but my equations are not steady state you start with the classical way you get steady state solution that's actually what I did steady state is not linear how did you decide that? you set the ideal gas law go to equation okay it's not shown here ideal gas law is also non-dimensional make it non-dimensional so yes I mean all these problems are gone yeah please enlight me because I'm let's assume it is y dvdy whatever you do 1 over fruit yeah so now you assume initial condition but you are thinking in terms of business assumption I don't need I don't apply the business assumption I'm doing low machna approximation so then you do I guess you're vasiliated I read your papers many times so then you do if you give me a minute I'm gonna thank you thank you for thank you for having your time I'm sorry and thank you again okay bye bye good luck