 So, let me recall first, so first via outline of a talk. After a brief introduction, I will talk about incompatible model on the limit on the derivation from the full Navier-Stokes equation. And then I will talk about free numerical simulation, free DNS, direct numerical simulation. One with Boussinesq and two in fully Navier-Stokes equations. So, just to recall the various configurations of relative instability. I mean, we are talking about this instability for a week and for years. But there are several configurations, this unsratified configuration. So, the density as this profile, for example, for a Boussinesq model for liquid. And I will talk also about stratified configuration. So, I mean by this two pieces of stably stratified field with density jump at the interface. So, this is an example of relative instability, a single model, the concentration. And here is the gravity, is the, is the vorticity, the horizontal component of the vorticity. And one is positive and the other one is negative. So, I mean, this flow are described by the full Navier-Stokes equation for a binary mixture. And this is a full Navier-Stokes equation, an equation for the energy. And in the heat flux, I take into account the antalpic contribution diffusion, which is here, this contribution in delta, and for a perfect gas EOS. So, this is, I dimensionally analyzed with four dimensionless numbers. And classical Reynolds, Schmidt and Prandt number. And I'm using stratification. So, this is the ratio of gravity by a typical length divided by the square root of, the square of a sound speed, sorry. And this is also close to a Mach number. But for a static configuration, I think it's better to use a stratification than a Mach number. So, we know that when we are dealing with a full Navier-Stokes equation, we have stratification and also acoustic wave. And this is a problem with numerical simulation because the time step usually is very weak. And so, we classically use incompressible limit. And there are several models to describe this flow in the limit of incompressibility. And we have derived four models. The anelastic quasi-sobaric model, the so-called Sandoval model, which has been derived by Sandoval in 1995. And the businesque model for RT flows. So, how to derive this model? We use a formal expansion, asymptotic expansion in terms of a Mach number. So, we choose a Mach number, a formal Mach number of a flow, and we perform expansion. We introduce this in the full Navier-Stokes equation. And then, by taking limit of dimensionless parameter, we get several models. For stratification of the other one, we get the classic anelastic model. And for vanishing stratification, in first case, we get a quasi-sobaric model. And then, if you take a limit of a larger gamma and very small value of print number, we get the Sandoval model. These two models are quasi incompressible since the velocity divergence is not exactly zero. They are a source term. In the quasi-sobaric model, this source term depends on the radiance of concentration, of temperature, too. Since in this model, we have an equation for the temperature. Then, we take the limit of small print number. So, the energy equation disappears. We so get the Sandoval model. And then, for vanishing Atwood number, we get the so-called Boussinesque model. So, in the following... So, this is what I was saying. I mean, anelastic, which allows only weak stratification, a weak Atwood number. So, actually, it's not very useful for our art instability. The quasi-sobaric model, there is zero stratification. And so far, it has not been used for RT flows. It has been derived by Paoloche for thermal convection. So, the Sandoval model, zero stratification, non-zero velocity divergence. It has been used by several investigators since Cook and Dimotakis 2001. And then, the Boussinesque model, zero stratification, zero Mach number, no energy in the equation. So, we have investigated, first, the anelastic model for stratified flow. The constraint is of a form of velocity divergence. Divergence of a momentum is actually zero. But, as I was saying, this model is valid only for a very weak stratification on the Atwood number. And if you increase this parameter, the limit of validity is quickly reached. So, it's not very useful, at least when you are dealing looking for turbulence. So, we are going to focus on Boussinesque and compressible approximation. And for this, we have done several DNS by using pseudo-spectral code, 3D, Chebyshev Fourier Fourier. And this is an auto-adaptive multi-domain method, at least in one direction, in the vertical direction, of course. The horizontal direction are homogeneous, so we can use Fourier expansion. So, the Boussinesque DNS is started with intermediate resolution with 250 million of collocation points. And then it is increasing during the simulation up to more than 800 million of collocation points. And compressible, we have done also two compressible DNS. The first one, the older, with low resolution. And the second one started with 200 million, up to 900 million of collocation points. So, the first thing to do is to look at the stability analysis. And the dispersion curve has been obtained with stability code based on normal mode analysis. And the numerics is also Chebyshev, multi-domain auto-adaptive method, Chebyshev method. And so, we have two different Reynolds numbers. So, we see the difference, I mean, on the stability. And when you increase the Reynolds number, of course, we increase the number of unstable mode. But the stratification does not influence much of this dispersion curve. So, the stability does not depend strongly on the stratification. The dashed line represents the initial condition for this free simulation. So, first, the Boussinesque model, so I said it's a zero Mach number. And this model with which we can get a self-similar behavior of the turbulence layer. And, for example, it is well known that the mixing length is increasing as t squared, gt squared, exactly. The main dissipation as t and the astrophy as the square root of t. So, I think this is only the sole model, which is able to reproduce power law in time, self-similar behavior. Because in any other model, we have, we introduce length scale, density gradient length scale, for example, or more time scales in such a way that it will break down this self-similar behavior. For example, for the dissipation, we get, by using the assumption of a self-similar behavior, we can build the scaling law of this dissipation. So, we introduce the self-similar variable, xx, which is over from z divided by t squared and the behavior of the length scale on the sole length scale of the problem. And then we decompose the dissipation with a structure function on the pre-factor, which depends only on t. And by calculating the average of this dissipation, we obtain that it has to grow as t as in many turbulent flow, for example, in isotropic turbulence. And we check it in our simulation, first in time, the mean dissipation, horizontal and vertical contribution on the total dissipation, which follows this law. And in space, in terms of a self-similar variable, z over h, we see the convergence as time goes on. So, of course, we can perform a data analysis of these results. And we get, for example, for the statistic, for the mixing, for the concentration fluctuation, we get a Gaussian for the concentrations at large scale. So, large scale our random Gaussian process. And we see the convergence if you use values, thickness inside of a mixing layer. So, now if you look at the gradient, the gradient of the consultation, either of the horizontal gradient or vertical. So, the horizontal gradient are symmetric but non-Gaussian, non-Gaussian as it should be. And the vertical one is non-Gaussian and strongly asymmetric. So, it means that random at large scale and the order at intermediate and small scales. We have the same kind of results with vorticity. PDF of vorticity show exponential tail as it should be. So, we can see that the Boussinesque Rayleigh-Teller turbulent mixing layer is a quite classical turbulent flow. Since if you look at the spectrum, PDF and so on, and we recover the classic features of a turbulent flow. However, there are some differences. And for example, the isotropy of intermediate scale, which has been noted by other authors. And so we have, here we have the deviation in the spectral space of velocity. So, it means that near the intermediate scale, near the Taylor scale, the velocity are almost isotropic. And small scales are strongly anisotropic. So, now the opposite limit, after the simplest model, Boussinesque model, the fully compressible flow. So, as I said, the anelastic model is not very valid for large stratification and large atwood number. So, for this regime we have to choose the full Navier-Stokes equation. So, we have to resolve both turbulence and the acoustic. So, it turns out that even if you are using strongly stratified configuration, the velocity divergence remains small, the turbulent Mach number remains small, and acoustic waves are present. So, this is the configuration we are using to stably stratify the configuration separated by a density jump. And along the time here, we can see that the density profile is smooth by the turbulence. So, we start at T equals 0 in blue. The last one is in magenta, where the density jump has been erased by turbulence. So, the behavior of concentration, density, and temperature profile are following. At the beginning, we have a linear density profile across the mixing layer. I mean, after a while, the instability, the flow is not instable anymore since the density jump has been erased. So, production, there is no production of turbulence anymore, but the turbulence inside the mixing layer flattens the concentration profile. On the other side, for the mean temperature, we see that along the time, the light fluid is heated and the heavy fluid is cooled at the same time. So, if you look at the evolution of the mixing layer, here I put the result of two simulations. Simulation with Reynolds number 3, 10 to the 4, and 6, 10 to the 4. And these two simulations are quite comparable. Instead of the initial condition, the initial condition of a small result with a small resolution is much larger. So, it results in a difference in the chronometry. So, we see clearly the saturation after a while. So, here we don't have any more of a g-square law for the evolution of the mixing length. And indeed, the density gradient length scales are of the order of the mixing length themselves. So, if you look at this in log screen scales, we don't have t-square behavior except on very small, only for the light fluid on a very small interval of time. So, it has been defined two atwood Reynolds, two time-dependent atwood Reynolds, a larger scale atwood Reynolds, which is defined by the minimum of density, maximum and minimum of density, and each time calculated on the mean profile of the density. And we have also defined, following Cook and Melado, a small scale atwood number by the RMS value of a density. So, we see that the large scale atwood number is decreasing with the same time scale than the density profile is flattening. And then, instead, the small scales atwood number is increasing as the turbulence develops. And so, it means that at small scales, we have atwood number, we have density jump, and so we have buoyancy beyond the stability of a large scale atwood number. So, this is related to the anisotropy of small scales. So, the Reynolds number in comparison between the compressive Boussinesque, which the Reynolds number grows as t to the power 3, and the Reynolds number, the Taylor Reynolds number, of the two simulations at Reynolds number 3, 10 to the power 4, and 6 to 10 to the power 4. And we see two bumps. The first one corresponds to the evolution of the large scale atwood number, and the second one corresponds to the acoustic wave. So, the system, the instability creates a strong system of acoustic waves, which creates turbulence also. So, we have performed a full data analysis of this simulation. And for example, here is kinetic energy, which follows the evolution of the Reynolds number with two bumps. Production of energy, which follows the large scale behavior. The dissipation here, and we see also the influence of the acoustic wave in the second part. And we also look at, for example, at the disequilibrium of a mixing layer. So, the ratio of production term, baroclinic and so on, divided by the dissipation. And as opposed to a shear layer, here, there is no equilibrium. Instead, there is a strong disequilibrium all along the phenomenon. Since the flow is compressible, it is of interest to separate contribution So, this is the so-called Kovacner decomposition. So, we decompose between vorticity, entropy and acoustic contribution. And we assume first that the acoustic part is given by the fluctuation of pressure. And then the acoustic part of the density is simply given by the pressure, acoustic pressure divided by the sound speed. And the acoustic part of fluctuating temperature is given by an adiabatic process. And so, the anthropic part is obtained by subtraction from the full contribution. So, here we saw that we plotted the amplitude mean value of this quantity. So, in blue, this is the density anthropic mode. And in red, this is the temperature anthropic mode. And they are strongly dominant. See, it's logarithmic scale. So, we have two order of magnitude between acoustic and anthropic contribution. And we see that the maximum, there is a significant shift between maximum of anthropic part of density on temperature. We also look at the correlation between fluctuation of density and temperature. And we define this, we calculate this coefficient here. And we obtain such a result as a function of time. I mean, this is plotted as function of z, the vertical coordinate, at three different times. So, this quantity is not defined outside the mixing layer. So, this explains this shape in red here at the preliminary instance. And then the turbulence developed. So, and we get this curve. It means that inside the mixing layer, density and temperature are correlated. And just on the boundary of the mixing layer, which are quite sharp, they are anti-correlated. So, we also look at statistics of various quantity, dissipation, vorticity, of course, and so on. And for example, density and temperature here. So, density themselves and temperature are Gaussian. They are not represented here. But we represented instead density gradient, pdf of density gradient, horizontal and vertical ones. And the same for the temperature. And we see that they are quite equivalent, if not identical. So, this is one of the result of this simulation is that temperature behave very close to a density. So, we said that temperature is the slave of density. So, we look at also at the anisotropy in this simulation. And it's less clear here the isotropy of intermediate scale. But they are more isotropic on the small scales. And this is probably due to the acoustic wave. And maybe also to the resolution. Because of the acoustic wave, the resolution is not perfect. The simulation is slightly under resolved. So, another output of this simulation, if you look at the mean temperature equation, we can look at the main contribution. And it turns out that the main contribution are the enthalpy part, this one. And of course, the turbulent heat flux. But the difference between these two is not so large, which is related to a moderate Reynolds number. I mean, we expected that this term be very weak in a large Reynolds number, RT mixing layer. I mean, the turbulent heat flux should dominate more these source terms. This is not anisotropy, this is a mistake. This is actually the mass flux. And we look at the contribution of source term of the mass flux. So, this is a classical equation obtained from the father of the regime. And we have free source term proportional to the gradient of density, vertical velocity and temperature and pressure. And this one is strongly dominant in this case. And we have the equivalent in the physical space in the z-coordinate here, where we see that this term is strongly dominant. We recover the sharp boundary of the mixing layer. And here is, at the final time, the reproduction of the acoustic waves. Which are going back and forth in the computational domain. So, I'm going to end this presentation by looking at the PDF of the cosine between the vorticity and the scalars. And we have looked at the cosine of the angle between vorticity of gradient of temperature, density and concentration. And we see that they are superimposed and they are strongly localized around zero. Which means that these two vectors are strongly correlated here. And so, vorticity is correlated to the gradient of temperature. And it also means that gradient of temperature are correlated to the gradient of density. And which is not necessarily the case for other quantities. For example, if you are looking at the product of pressure gradient and density gradient, they are not correlated here. They are almost a flat distribution in terms of cosine, except on the boundary here. So it means that this gradient of density and temperature are a little bit correlated or anti-correlated, but essentially not correlated. So here is a list of the output that we get from this analysis. Molecular fraction in the compressible case or in the Boussines case, which is a classical value. And in the Boussines case and in the compressible case, sorry, it reached almost one. Because turbulence is not produced anymore, but it, I mean, turbulence keeps mixing the two components. And so molecular mixing increases to very large value. We also look at the elasticity and we have shown that it grows during the RT regime. We have shown that RT turbulence is in disequilibrium. Vertical baroflinic vorticity production in compressible flow is non-zero, but much smaller than the horizontal. We have seen also the influence of the acoustic wave of turbulent kinetic energy and production. The anisotropy in terms of turbulent kinetic energy at the maximum, it's 88% at the maximum of turbulent kinetic energy. It is 64, so around 2.3. So we have also seen that mass flux sources are dominated by pressure gradient. RMS density are dominated by density gradient. And temperature is the slave of mixing. Thermal and mixing length scales are equal. But if you look at spectra, anisotropy and density, they are very close. However, there is a significant time line between the evolution of these two scalars. So the general conclusion. So we have derived four incompressible models to study the RT instability. We have also shown that for large stratification, we have to use the full Navier-Sox equation. It is now clear that the RT instability is close to a classical turbulence, not far away from anisotropic turbulence. But still there are some anisotropies as we have seen. If you like PDF, spectra and termitancy. However, the differences are, for example, strong acoustic production in the compressible case, the small scales anisotropy. And in this case, we have shown also that the temperature field is the slave of a mixing. So I think for the future, the problem will be to understand better the relationship between mixing and temperature. And for this, the quasi-sobaric model will be well suited because it contains an energy equation, an equation of state. But there is no acoustic wave. They have been filtered out by the asymptotic expansion. So I think this is the right model to continue in this way. About the compressible case, of course, the influence of the US boundary condition, temperature profile, and so on. And the Boussinesq model is also of interest. You have to reach a higher Reynolds number. Okay, thank you.