 Hi, can you all hear me? Yes? OK. Today I'm going to be talking about really Taylor unstable flames. So really Taylor unstable flames play an important role in type 1A supernovae. So here's one possible scenario for a type 1A supernovae. There are two stars here. There's a red giant star, and there's a white dwarf. The red giant is losing material, which is falling down onto the white dwarf star. As the white dwarf is compressed, the center starts to heat up. You start to have convection, and eventually you start to have this huge bubble that rises out of the star. The edge of the bubble is burning. The inside of the bubble is the products of fusion, and the outside of the bubble is what's being consumed. So this thin surface on the top of the bubble is called a flame. So the flame is really Taylor unstable because the fuel that the flame is consuming is more dense than the ash, and the flame is propagating upward against the direction of gravity. So if you want to understand type 1A supernovae, one of the best things you can do is try to do a simulation of the entire star. These simulations are very large, difficult, and computationally expensive. And one of the really hard things about them is actually simulating the flame itself. And the reason for that is because the flame is extremely thin. It's only about one centimeter wide, but the star is big. It's about the size of the Earth. So if you think about a simulation, you can never resolve both of those things in the same simulation. That means what you need is you need a subgrid model for how fast the flame is going to move at small scales. So there's two types of subgrid models that have been used. One in one, the flame speed is set by the Rayleigh-Taylor instability. And the other, the flame speed is set by turbulence. So both these models models have been considered in the past. They both have problems. The Rayleigh-Taylor model is sort of slightly better. It fits well when the Rayleigh-Taylor instability is weak, but it under-predicts the flame speed when the Rayleigh-Taylor instability is strong. So a better subgrid model is definitely needed for this, for these kinds of simulations for this problem. Let me tell you a little bit more about the Rayleigh-Taylor subgrid model. So Rayleigh-Taylor subgrid model is based on the premise, the physical premise that the Rayleigh-Taylor instability is controlling the flame and the flame speed. So in this model, the flame speed at the unresolved scale is just going to be equal to the linear growth rate for the Rayleigh-Taylor instability at this scale. So physically, what this ends up meaning is that the flame has to be both self-similar and self-regulating. So let me talk a little bit more about self-regulation. Here's the visualization of the flame. I'm just gonna sort of explain what you're seeing here. So if you imagine first you have a completely flat flame, a laminar flame that's not being disturbed by the Rayleigh-Taylor instability or by turbulence or anything, you turn on gravity. You start to get these sort of like bubbles that are rising, Rayleigh-Taylor bubbles that are rising, and then spikes that are falling. And on the edge of the bubbles and spikes you may see some secondary Kelvin-Hummels. So this is what you're seeing in this visualization of the flame surface here. So the gravity is pointing downward. The blue is the cold side of the flame and the yellow is the hot side of the flame. And so you can see some bubbles here. You can see some spikes here. One of the things you'll probably notice is that there are a lot of areas where there's these really pocket shapes. These are called cusps. These are areas where possibly some really intense burning could take place. So what is self-regulation? So self-regulation is this idea that there's this competition between the Rayleigh-Taylor instability creating surface area. So Rayleigh-Taylor instability is making the flame have more surface so it moves faster and then burning, which is burning out these sort of cusp shapes and that's making the flame speed, the flame moves slower because it's decreasing the surface area. So this idea works pretty well, except for the fact that it assumes that the flame speed is always proportional to the flame surface area. But in fact, it's been shown recently for some other situations that these cusps, these pockets of burning can actually have a higher local flame speed than you would expect. So the question is, what does that sort of do in this situation? So we want a new sub-grid model. What are we gonna do? So first we'll build off the Rayleigh-Taylor sub-grid model, hopefully. But then the next thing to do is to take into account these local effects and figure out how local changes in the flame structure and in the flame speed are going to affect the global flame speed. So one very simple thing that you could do is just take the curvature of the flame surface and into account, that's a local effect. So today I'm gonna be talking about whether or not that works. So what I'm gonna be showing you are the results from some generic numerical simulations. These are the equations that I'm simulating. They're Boussines, the most important part is the way that I'm adding the reaction in is I have a reaction term here, which I'm just using a simple model reaction. As a result of the non-dimensionalization I'm using, I have three dimensional parameters. There's a non-dimensional gravity, there's a random number which I just equal set equal to one and then I have a non-dimensional box size, which is just the box width divided by the flame width, which is either 32 or 64. Okay, let me show you what one of these Rayleigh-Taylor flames looks like. Here we go. Okay, so I'm showing you the temperature field, the vorticity field and the velocity field for a flame. So the flame's going up, this is the fuel, this is the ashes, gravity is pointing downward. So you can see that the flame has a very dynamic structure. There's a lot of bubbles and spikes kind of things that are happening and the simulations have periodic boundary conditions. So let me just go back to the presentation. Okay, so if I carry out some simulations, for each simulation the flame speed is going to vary with time. So what I can do is I can measure the average flame speed for the entire simulation for each simulation and then that's sort of what I'm showing here. So I'm showing on the y-axis, I'm showing the average flame speed for an entire simulation. So each one of these points represents a different simulation and then on the x-axis I'm showing you the strength of the Rayleigh-Taylor instability. So down here when GL is small, the Rayleigh-Taylor instability is weak and when GL is large, the Rayleigh-Taylor instability is strong. And then I'm showing you the prediction of the Rayleigh-Taylor flame speed model in here in red. This is just what I was telling you earlier, that this model works well when Rayleigh-Taylor instability is weak but it starts really not working as well when the Rayleigh-Taylor instability is strong. So the question is, where does this extra flame speed come from? So the first thing I wanna talk about is, so I've been telling you, there's these pockets of really strong burning and maybe the flame speed is higher in those pockets and maybe that's where all this excess flame speed is coming from. But the first thing I really need to show you is, that implies that the flame structure is different, right? So I need to show you that the flame structure of these flames is actually different than a laminar flame structure. So let me just talk about a laminar flame. So this is the flame that's not disturbed by any kind of instability. If you just look at the temperature equation that I'm evolving here, it's just an advection diffusion reaction equation. So the change in temperature at any point is just given by advection diffusion and reaction. For the laminar flame, there is in the laboratory frame, there is no advection. So diffusion and balance and reaction are just balancing and the flame is just propagating with a fixed shape, which you can see here. So here I'm showing sort of space and this is the fuel. This is the ashes. So the fuel is at a temperature of zero and the ashes are at a temperature of one and this is just the shape that the flame has. Then if I plot the reaction term, that's the term here in red. So you can see that it peaks at around a temperature of about T equals two thirds. The diffusion term is this term here in blue. So the hot part of the flame is losing temperature to the front part of the flame, which is gaining temperature. And then I'm also plotting here just this entire right-hand side of the equation and that is this black line. So ideally what I'd like to do is compare this laminar flame to my Rayleigh-Taylor unstable flame. So this is the 1D profile and the Rayleigh-Taylor unstable flames are 3D. So this is a really hard comparison to do. So actually there's a better way to think about it, even though it's a little more complicated. So what I'm showing here is I'm showing the amount of a certain quantity, for example, reaction that's volume integrated for all the parts of the flame that have a temperature less than the temperature shown here on the x-axis. So for example here, for a temperature of one, the total amount of integration of the reaction term is one. So that just means the flame speed of the laminar flame is one, basically. The thing that I want you to see here is really just the shape of these curves. So just notice the right-hand side. When I integrate it, it gives me a straight line and then I've got sort of this curved up shape and then this curved down shape like this. Because now I'm gonna show you one snapshot from a Rayleigh-Taylor unstable flame. So you can see it's different. It's not the same as the laminar profiles I was just showing you. So instead of this right-hand side being a straight line, it's kind of curved up and then there's not as much as you would expect here and then the diffusion curve is kind of like shifted this way. This is a sign that the Rayleigh-Taylor unstable flames just don't really have the structure that the laminar flames do. So here's another way to look at it. At each point in the flame, I can compare the value of the diffusion at that point to the value that a point in the laminar flame with the same temperature would have. And then I could say, if there's a difference, I'm going to call that sort of an extra diffusion and I'm gonna plot it. So here, the red parts, you can see the flame here. The red parts are places where they're really gaining temperature a lot faster than you would expect. So there's diffusive focusing of temperature. And then the blue parts are losing temperature faster than you would expect. So you can see that the diffusive focusing is really happening in these cusp and pocket regions. So that works out well with what I've been saying. And also, the diffusive focusing is stronger than this diffusive loss. So now for each simulation, I can go and I can integrate over the entire box an average over the entire simulation to find out sort of the average extra diffusion for that simulation. So that's what I'm showing here. So the most important thing about this plot is just that when Rayleigh-Taylor instability is weak, the flame is pretty much diffusively similar to a laminar flame. But when Rayleigh-Taylor instability is strong, there's actually starts to be quite a difference. And this difference is sort of about the same amount as you would expect that would explain that sort of extra flame speed. And so here's another sign that this is sort of along the right track. So if I plot, I'm finding the extra diffusion here on the x-axis and the flame speed on the y-axis. And this is just for one simulation. And you can see that there's a correlation between the extra diffusion and the flame speed. So at times there's extra diffusion, there's also more flame speed. So let me just sum up what I've told you so far. So I've told you that the Rayleigh-Taylor flame structure is not like a laminar flame structure. It varies a lot with time. And it shows these areas are very strong diffusive focusing, which means that we expect that the flame speed for the Rayleigh-Taylor unstable flame should deviate from the flame speed that you would expect from models that just treat the flame as a thin surface. So the question is, what is altering the flame structure? And can these changes be modeled by accounting for curvature? OK, let me just review curvature really briefly. You probably all remember this. But if you have a 1D plane curve and you want to know it's curvature, you just nestle a circle in at the point that you want the curvature at, measure the radius of the circle and then whatever that is, the curvature, right? OK. For 2D, it's a little more complicated. You pick the point on the surface that you want to measure the curvature at. You have planes normal to the surface. You cut through all the possible normal planes, measure the maximum and minimum radius of curvature, and then you can get the two principal curvatures, K1 and K2. Now, K1 and K2 can be combined to make two other kind of curvatures. There's the Gaussian curvature, and that just tells you whether or not the flame is sort of whether or not the shape is bowl shaped or saddle shaped. And then there's the mean curvature, which tells you whether or not the surface is minimal. And that's the one that I'm going to be concerned with today. So let me just show you a measurement of the curvature of one of the Rayleigh-Taylor unstable flames. So here is the temperature of the flame. So you can see what the flame surface looks like. So you can sort of compare it to this over here. This is the mean curvature. So the white areas have a negative curvature. The red areas have a positive curvature. So what this means is if you think about a curvature vector, the red has a curvature vector that points into the fuel, and the white parts have a curvature vector that point into the ashes. So you can see that sort of the bubbles tend to be negatively curved, and the spikes tend to be positively curved. So how does curvature affect the flame speed? It's well known the curvature changes the flame speed. Usually positively curved flames, they generally burn faster. And negatively curved flames generally burn slower. And this is how the flame speed depends on the curvature. So if the curvature is small, the dependence is just linear. So here's an example. Here's an inrally burning circle. So I have this is the fuel in the middle. It's blue, and the ashes out here are yellow. The circle is burning inward. And what I'm going to do is just measure the flame speed as the circle burns in. So that's what I'm showing here. A laminar flame would move at a speed of equal to 1. So you can see even to begin with when the circle has a radius of 16, the flame speed is already a little higher than you would expect for a laminar flame. And then as the flame burns in, the flame speed goes up, and then goes down as the flame burns out. So this doesn't may not seem like a lot, but keep in mind for the Rayleigh-Taylor unstable flames, I'm looking for differences, flame excesses of maybe 10% or 20%. So this is kind of maybe on the order of what you would need to explain that. So now I'm going to talk about the Rayleigh-Taylor unstable flames and ask this question. So I've told you that there's positive regions of the flame, negative regions of the flame. Positive regions should burn slower. Negative regions should burn faster. So what it should say is do regions of negative KM, outweigh regions of positive KM, and enhance burning. So I guess it should be positive. Yeah, positive and negative. I'm sorry. It was right the way it was written. So now I'm showing you a plot of the amount of the flame speed that's generated in areas of high curvature. So for example, let's say that I'm also showing it to you for six different simulations. So the smallest gravity ones are is black all the way up to the dotted blue line, which is the strongest gravity one. And that's more Rayleigh-Taylor unstable. So what we have is, so for example, let's say that I go to KM of equal to 1. And I'm looking at the G equals 32 simulation. And I would like to know how much of the flame speed is generated in regions where the mean curvature is greater than 1. And so the answer is about 5%. And then it works the same way for the negative side, but for how much is generated below that curvature. So there's a few things I want you to notice here. So first of all, as I go to simulations with higher G, more of the flame speed is being generated in regions of high curvature. So higher G flames are more curved. But then the second thing I want you to notice is that as I go to higher and higher G, in order for this explanation, I've been telling you where regions of positive curvature outweigh regions of negative curvature to really work, there'd have to be a lot more on this side of the plot than on this side of the plot. So the plot should look asymmetrical. That was true, right? But what you can see is especially, let's take this G equals 32 simulation, the top dark blue curve dotted curve, you can see that actually the two sides are pretty equal. So there's not a lot of support for the explanation that I was just looking at, that maybe the positive curvature is outweighing the negative curvature. Moreover, if I'm looking at just time slices from one simulation, and I'm looking at the average mean curvature per point in that simulation and comparing it now with the flame speed, there is no correlation between the two. Which is also not good for this explanation. So what's going on here? Okay, I wanted to point out something very important, which is that curvature isn't actually necessary for there to be extra burning produced. So here I'm showing you two completely laminar flames and they're colliding with each other. But as they collide, their structure starts to change and there starts to be extra burning here. So the flame speed here is gonna be greater than one. But there's no curvature at all, right? Now if I'm now going back to the Rayleigh Taylor problem, let's say that I'm looking at a Rayleigh Taylor cusp forming. So here's just a cartoon. So you have like some sort of a mushroom shape. It starts to form these like sort of cusp areas. So this whole thing should have enhanced burning. But you can see like this area is positively curved. These areas are negatively curved. So the curvature doesn't actually seem to have a lot to do with like this structure forming. Okay, so here's what I think is happening basically. First question was, is the structure of Rayleigh Taylor unstable flames different from a laminar flame? And the answer is yes, it varies with time. And very importantly, there's these areas of strong disease of focusing, which suggests that the formation of these cusps, it could be from Rayleigh Taylor, Kelvin-Homeholz, even from turbulence, is increasing the flame speed over the Rayleigh Taylor model's prediction. I think that's okay. But then the second question is, is positive curvature, are weighing negative curvature and producing enhanced burning? And there's just not a lot of evidence for that. But what there is evidence for is the fact that the more unstable flames are more highly curved, but there's just kind of equal amounts of positive and negative curvature. So this is what I think, I wouldn't think it's happening now maybe, is that the global flame speed is higher than you would expect because the flame is very densely packed. So what's happening is the Rayleigh Taylor instability is producing all this very densely packed flame, which is, as a side effect, highly curved, but the curvature is not what's changing the structure of the flame so much. It's the fact that there's just all these flame sheets that are hitting each other. So that means that traditional curvature-based flame speed models are not likely to be effective in this situation, but if you were to design a model that takes into account the density of the flame sheets and how tightly they're packed together, that that kind of model will probably work. That's it. Both sides of the interface on the front were the same. You assumed they were the same. Did the spectra or the rudeness where it's the same on both sides? You assumed that. Yes or no? I'm not sure. I'm gonna have to think about it, yeah. Oh yeah, I think you assumed it, but you have to revise Turner 1960 experiment and check the long-term data. So why did you make the assumption that the thermal conductivity was constant? Basically the reason I made that assumption is because I'm looking at a very, very, very simple situation here. The flame in the star is a lot more complicated. First of all, it's got like actual fusion, I mean it's got like a real nuclear chain, right? So basically what I'm doing here is I'm trying to get to the simplest situation possible and just do the simplest model and try to understand that first before I try to make all the more complicated assumptions. So I mean, that's the reason for that, yeah. It's just as simple as you can get, so. I feel the same as you do again. Thank you.