 So, good morning everyone and thanks you all for being here for my presentation. As it was already said, I will talk about the simulation of Rishmaio Meshkov's turbulent mixing zone using a PDF model. I'm a second year PhD student at the CEA Laboratory in France. My advisor is Olivier Soulard who works in the same lab and my PhD directors are Vladimir Sabelnikov from the UNERA and Serge Simoise from the LMFA of Ecole Centrale de Lume. So, I know we already had many presentations about this topic but I wanted to give you a short reminder about the Rishmaio Meshkov instability. So, this is an instability that occurs when a shockwave interacts with the interface between two frids of different densities. And once the shock is finer from the interface, there only remains an isotropic freely decaying and diffusing turbulent mixing zone, which is represented here. And lastly, if the initial density contrast between the two fluids is small enough, this decaying TMZ obeys the Wusinesk approximation. So, the velocity is incompressible and the concentration is a passive scalar. So, now concerning the evolution of this TMZ, it was shown that in Rishmaio Meshkov flows, the TMZ will reach a self-similar state and in this self-similar state, turbulent quantities such as the width of the mixing zone will obey a parallel with an exponent theta, which is the growth rate of the TMZ. And it depends on the initial conditions of turbulence through the following relation where S0 is the initial infrared slope of the energy spectrum. So, this relation is due to the principle of permanence of the large scales, which implies that the growth of the TMZ will be related to this large scale. So, now our goal here consists in modeling the evolution of this TMZ. So, first of all, most engineering models that are used for the Rishmaio Meshkov configuration attempt to capture this self-similar state with the correct value of the growth rate theta. These models include, for example, the K-epsilon, the Reynolds stress and the B-fluid models. And for the two first one kind of models, they apply a first gradient closure for the turbulent advection term. But this modeling is a very simple modeling and has been shown to present several limits. By contrast, the one-point probability density function models, or PDF models, which are based on the calculation of PDF, give a PDF of the velocity fluctuations that contain every one-point statistic of the velocity fluctuation. That's why, more particularly, they do not require any closure for the calculation of the third-order velocity correlations, which are involved in the turbulent advection term. So, in this way, they allow a better description of the transport in the flow, and they are not limited to the diffusion regime, as it's the case for the K-epsilon and the Reynolds stress models. However, these PDF models were never used or calibrated for Rishma and Mashkov flows, and their behavior has to be analyzed, so this is the purpose of this work. My presentation will be divided into two parts. The main one, which is the first one, will focus on the analysis of the properties of a given PDF model, which is called the Simplified Langevin model. And on the second one, I will give you the first comparisons we were able to make between the results given by the model and the results from larger simulations. So, as I said before, we want to model the evolution of the PDF of the velocity fluctuations. The evolution equation of this PDF can be derived from the incompressible Navier-Stokes equations, and is given below. So, I recall that in this equation, we said that advection will be treated exactly, but the boxed terms, which correspond to the molecular mixing and the turbulent acceleration, have to be closed. So, this is why we use our Simplified Langevin model, which is a particular PDF model. So, this Simplified Langevin model applied to our turbulent mixing zone, resulting from the Rischwein-Meschkoff instability, gives the following transport equation for the PDF, where we can see our model terms, which are return to the mean term and dissipation term. And this model includes four closure coefficients, so two for the equation of the PDF and two for the equation of dissipation. So, these coefficients are C1, C0, which is related to C1, and C epsilon and C omega. So, we are going to solve this equation using a Lagrangian Monte Carlo method, which consists in considering a given number of particles, and for each one of these particles, instead of solving the transport equation, we will solve the following stochastic ordinary differential equation system, which is statistically equivalent to the original one. So, we see the appearance of the stochastic component here in our equation system, with this term, which is a vinyl process, sorry. So, now that we settled the basis of the Simplified Langevin model, we wanted to focus on its properties when applied to a Rischwein-Meschkoff flow. So, first of all, the zero-G analysis of this model predicts self-similar behavior for our TMZ. So, once again, we will have the evolutions of turbulent quantities given by porolos. So, this is the case for the width of the TMZ, the maximum of the kinetic energy and of the dissipation. And the exponent of this porolo will be given depending on one of the closure coefficients of the model C omega. So, this coefficient, of course, can be adjusted in order to fit to the correct value of theta. And, thanks to this, we were able to verify numerically this analytical result. This is what's shown here on this figure. We plotted the ratios of the quantities calculated by the simulation over the ones calculated analytically, thanks to this relation. And we showed that they remain close to one, meaning that the self-similar behavior is verified numerically. Again, we focused on the PDF, on the shapes of the PDF given by the model. And what we can see is that they're slightly different from a Gaussian. And if we go further and divide the PDF into its symmetric and anti-symmetric parts, we can see that this deviation from Gaussianity is mainly due to the anti-symmetric part of the PDF, with the symmetric one almost superposed to a Gaussian. So, this observation raises two main questions. What's responsible for this deviation? And are we able to express it analytically in order to identify it clearly? To do so, we use the fact that in the center of the TMZ, the gradients of the velocity variances are weak. So, this allows us to use the assumption of local weekly in homogenous turbulence, or local quasi-homogeneous turbulence. Thanks to this assumption, we're about to make an assentical development of the PDF in the center of the TMZ. And by conserving the two first orders of this development, we get the following analytical expression for our PDF. So, what you can see on this figure is the comparison between the anti-symmetric parts of the PDF given by our analytical expression with the one given by the numerical simulation. And we can see that they fit quite satisfyingly, which is encouraging for our expression. Now, if we focus on this expression, we can see that the shape of the PDF is a Gaussian, a classical Gaussian with this term, but corrected by this term, which is third order Hermite Polinom. And we can see that the coefficient of this polinom is related to the gradient of the kinetic energy, which means that we have a diffusion type correction and that the deviation from Gaussianity is due to the homogeneity in the flow. Now, we can use this analytical development to go further in the investigation because we can use it to have an estimation of this product because it's directly calculable from the expression of the PDF. And this calculation led us to the formation of a diffusion term in the equation of the turbulent kinetic energy. So, what's important here is that we can see that in the center of the TMZ, our PDF model behaves like a standard K-epsilon model. And in particular, the turbulent transport will be given on first order by a diffusion term, which coefficient CK here depends directly on the closures of the PDF model. However, I remind you that this property is not true on the edges of the TMZ, but only in the center. Now, another important thing that must be known is that the previous results were obtained for a value of C1 of 4.15. This is the usual value used for the simplified Langevin model. But we can note that C1 plays an important role in the shape of the PDF because it controls its return to gaussianity. To show this, we plotted here several shapes of PDF given by several values of C1. And we can see that the closer C1 tends to its limit value of 1, the more the division from gaussianity increases with the extremal shape given by this curve for a value of C1 of 1.2. And a second observation we can make is that the shape of the PDF is really different from the one predicted by our analytical development. So in order to understand why we have such a difference, we made two hypotheses. First of all, maybe this difference was due to our quasi-homogeneity assumption on which our development is based, which could be wrong. Or maybe this is due to the truncation of the limited development we made because we only kept the two first orders. And maybe the higher orders of the development were more important than we thought. So to verify this assumption, we first decided to compare the coefficients of the third order Hermite polynomial given by the analysis, by our analytical expression, to the ones that we extracted from our numerical results. And what we can see on this figure is that they stay really close in the entire mixing zone. So this leads us to think that the quasi-homogeneity limit is valid, is not wrong, and really allows us to understand the PDF behavior in the center of the TMZ. And in order to convert this idea, we also chose to extend our analytical development to include the fourth and the fifth orders Hermite polynomials and to see what was their influence on the shapes of the PDF. And what is shown here is that the same curve as before except for the dashed one which corresponds to our analytical prediction and we can see that it's significantly different from the one we obtained before. So this confuses in the idea that the difference between our numerical and analytical results was due to the higher orders of Hermite polynomials in the limited development. So now that we know a little more about our SLM properties, Simplified Lunge Variable Properties, we can give a quite short debriefing. First of all, I recall that in PDF models we bring no hypothesis about the turbulent eviction term in the flow, even for the simplest PDF models, because the one we used is really a simple model. Then we showed that the asymptotic development of the PDF allowed us to show the existence of a diffusion limit for the evolution of the turbulent transport and this diffusion limit is similar to the one given by a K-epsilon model. However, this is valid only in the center of the TMZ, as we said before, and only, of course, applied for the Reshmayan-Meshkov configuration. And finally, we showed that in this limit, the diffusion coefficient CK depends on the closure coefficient C1 of the model. And one can note that this coefficient is the only coefficient of the model if we don't consider the coefficients applied to the dissipation equation. And it plays many roles in the model. For instance, it controls the return to isotropy and to gushanity for the shape of the PDF. And moreover, we've shown in the equation of the turbulent kinetic energy that it also controls the turbulent transport in the TMZ. So this will be a key point in our understanding of the properties of the simplified large-band law. Now that we have analyzed these properties, we wanted also to compare the SLM results to the ones written by larger dissimulations in order to have an idea of its restitution of the evolution of the TMZ. So we proceeded to implicit larger dissimulations. We used the compressible Heuler code and we initialized our simulations by choosing an initial perturbation spectrum at the interface and then applying the linear theory of the Reshmayan-Meshkov instability. We were able to get a velocity field corresponding to the post-shock phase of the Reshmayan-Meshkov instability. And we used this to initialize the simulation. We also had two key parameters to fix for the ALS, the Atwin number, which had to be small to stick to the business approximation and the initial infrared slope of the energy spectrum because, as we said, it controls the growth rates of the TMZ. We did the same for our model. We initialized it by extracting results from the ALS and using them to initialize the particles of the Lagrangian-Tecalo code. And once again, we had key parameters to fix, which are obviously the coefficients of the model. We fixed the value of C1 to be the same as the one used in Reshmayan-Meshkov models generally applied to Reshmayan-Meshkov flows. And since we aim to get the same growth rates with the model and with the larger dissimulations, we fixed the value of the coefficient C omega depending on the value of the infrared slope of the energy spectrum. So, I'm going to be quick on the results. What's important here to see is that we compared the results given by the ALS and by the model, the ALS in red and the model in blue. And what we can see is that the temporal evolutions such as the evolution of the maximum of the terminal kinetic energy or the evolution of the width of the mixing zone are quite similar and it's the same for the spatial profiles like the profile of the kinetic energy at the final time of the simulation. And finally, concerning the shapes of the PDF, we can also compare the shapes given by the ALS and by the model. And what we can see is that in the center of the TMZ, so here we are at the center of the TMZ, here at the edge and this is an intermediate point. In the center, the model is really satisfying towards the results given by the ALS, but a significant difference appears at the edges. So, this will be the leading point for our future works to try to understand why there is this difference. So, as a conclusion, we developed a Lagrangian Monte Carlo code for solving the transport equation for the PDF. We also determined an analytical expression for this PDF resulting from the SLM and applied to the Reshmayer-Meshkov TMZ. This allowed us to show the existence of a different limit for the turbine transport in the center of the TMZ and to identify the key roles played by the coefficient C1 of the model. And finally, we compared the SLM with results given by the ALS and although the first comparisons were promising, we saw that there are still differences that exist and that must be corrected and this is one of the perspectives of our work to identify the physics missing in the model to get closer to the ALS and then to do same work but for the Reshmayer-Meshkov TMZ instability. Thanks for your attention and sorry for being late.