 Is this working? I guess. Is it? Morning. So I'm going to be talking about simulation of RMI in the presence of thermal fluctuations using fluctuating hydrodynamics. And thanks to the first speaker who gave a pretty nice introduction to RMI, so it saves me a lot of time. So this is also, I'm from KAUST, from the Fluids and Plasma Simulation Lab. And I work with Ravi Santane. We are classically a, I mean, traditionally we've been a fluid dynamics group. So this is sort of a first venture into anything that's stochastic. And so this talk I had to pick an audience to direct it at. So I'm going to pick a CFD audience. I understand there's people from the stochastic community as well in this conference and some of the details might be mundane to you. And for that I apologize. So the reason to include thermal fluctuations into your usual macro scale deterministic system is that they do become significant in small volumes. So if you look at the relative magnitude of the density fluctuations, for example, in this Hartz-Pierre model, then they scale as one by the volume. And so one example of this, one illustration of it, rather is a molecular dynamic simulation where it's a nano scale channel. And then if you look at the fluid center of mass under the imposition of a forcing, they follow like a stochastic trajectory. And if you look at the PDF of the center of masses, they're actually Gaussian. The other situation is under microgravity. So this is an experiment they did in space and then said they took a mixture of two solutions, toluene and polystyrene. And then when they impose a temperature gradient, they saw that there were these huge fluctuations correlated in space and time that were developing. And this is about four millimeter by four millimeter and then 1.5 mm thick. So they are actually large in magnitude. So, and then the other stipulation is that thermal fluctuations might play a key role in triggering instability. So you wanna capture, if you're studying onset in super detail, then you wanna capture these. So this is the initial condition, a CFD result and MD result which captures these perturbations and then a fluctuating hydrodynamic system which does exactly the same. Just to show that, if you include fluctuations in your hydrodynamic model, you're able to do this. So that said, the overarching goal for me was to, this work was to develop a two fluid model and the numerical methods had to be developed from scratch and then we wanted a code which had a equation of state hook because we wanted to do this for liquids. The objectives for this particular talk would be for this set of work is to verify the equilibrium solution and then use it for our my simulations and then can we say something about what happens to the post shock growth rate in the presence of thermal fluctuations. The governing equations, we have our classic compressible navier stokes with the deterministic terms and then there's a divergence of a stochastic flux on the right hand side which models thermal fluctuations. So we want to extend our traditional CFD methods to stochastic systems. This is a divergence of a noise. It's not a noise which is external by itself. So we're not using collocation method and so on. So the equations look like this. This is your solution vector with the two fluid. That's the other species there and then your adeptive terms, the diffusion term and then there's a stochastic term. These SNQs are stochastic stress and heat vectors. We'll talk more about these later. The closure for the mean values is this and then the fluctuating terms themselves and magnitudes are given by these where these ends are basically your noise terms. And we look at how we set these magnitudes. But that's what the governing equation looks like. So a little bit more about the noise term. So how can we understand this better? I mean, is it even based on some sort of solid theory? Yes. So one answer which is also true for nonlinear fluctuations around a mean value is given by coarse-graining microscopic dynamics and then you will see if you go, if you crank through it that you can get the magnitude of the noise as well as the form of the governing equations which turns out to be exactly similar. Now, I have the details of this in my backup slides and if anybody wants to go into it, we can. The easy explanation which to illustrate is valid for linear fluctuations is Landau's framework which I'll just quickly show here. So what, so if you take a single fluid then your dissipation function is always a conjugate, a dissipative force, a dissipative flux times a force conjugate contractions of these. So then we have our Q and then temperature gradient and shear stress and then grad U. So Landau's postulate was, it was completely intuitive at that time which was later proved that dissipative fluxes are macroscopic manifestations of the microscopic degrees of freedom and therefore they have to be made fluctuating quantities. So then if you have our J as the fluxes here, you basically say everything is equal to a mean part plus a fluctuating term. In this presentation my delta tau and delta Q were denoted by S and Q respectively. So that was the key contribution of Landau's framework which works for linear fluctuations and how do we get the exact form of noise which is something people here might be familiar with. There are FDTs for this sort of thing. So basically what that says is a physical way of coming into peace with this is that if you have any thermodynamic flux which is a mean plus a fluctuating part then this is your standard phenomenological coefficient based relation which says how non-small disturbances relax and then random math here will give you this correlation function and so what the FDT says is it says these, the regression of this is related to these guys. So physical meaning of course on such as statement gives that the regression of the microscopic thermal fluctuations at equilibrium follows the macroscopic law of relaxation of small non-equilibrium disturbances. So preservation of this thing is kind of important when you design numerical methods. Okay this is again just exactly an illustration of how we come up with our equations. So you can, we know Fourier's law of heat conduction and then the constitutive model for a new tune in fluid so you can crank out your M alpha betas and so you can get your exact form of the noise where these z's like we saw actually absorb these noise fields here. So again this was aimed at a CFD community so if you recall these were the noise terms and there was a typo in the previous slide here, okay? So a Brownian motion is, if you're gonna look at how to, I'm gonna have a one slide about how to go about discretizing Brownian motion so it has probability of zero, a probability of one at t equal to zero and then increments are basically independent from each other, right? So if we had a function of a noise which was something like that then you would have to discretize your time in increments where each increment is given by that where this is the Gaussian noise and then so you will have several instances and then the average would of course give you the expected value of the function. Why we were looking into that because now in addition to your space discretization you also have to discretize the noise. So one D results for a single fluid and we were looking at equilibrium problem so mean flow is zero. We started with the fourth order method for reasons we thought we'd get better spectrum across mid-wave numbers. So we have our after method of lines discretization we have this thing, the numerical fluxes we used fourth order methods and if you take just the divergence of the stochastic flux at the term looks like this and where this cell centered stochastic stress and heat flux looks like that where this is your incremental noise term here that goes into it. So when we looked at the fluctuation spectrum for zero mean flow and we normalized them by the theoretical values and we were looking at the, okay this happened from, this was a Mac from Windows thing. So basically these are squared errors from the, and the x-axis is gone as well, okay. Squared errors from the theoretical values. So this is the row U, this is the row row and then this is the U, U. So we were doing and all the values are gone okay basically this was like 5%, that's I think comparable to this, these two are comparable, that's a shame. So I mean doing in the density fluctuation being within 15% is considered okay. So we were sort of convinced that our method was validated, I'm verified, sorry. So we looked at weak convergence with respect to expected values of the structure factors, this sucks, sorry. There's nothing I can do. Okay so these are the first order and second order slopes and we have for a few different delta T's which is like one picosecond and so on, which are really small and then what we found was that there was no gain with respect to weak convergence from the higher order method which was sort of obvious because I mean once you add stochasticity in your order of accuracy is gonna drop down and I hope that doesn't happen to the other plots. So then we moved on to RMI simulations for the two fluid case and we decided to drop our accuracies down to save compute time so we just decided to go with second order methods and then your cell centered, again cell centered, stochastic stress and heat flux looks like that. So okay so all my data values or axis values are gone. So we decided to first verify our code for the non fluctuating case against experimental values and so we picked up this experiment from Air SF6 case we matched the domain size, the wavelength, the amplitude of the perturbation, everything one on one even the location of the shock and the interface and so we looked at that's the simulation, the red line is the experiment and this is the interface location and then the non dimensionalized evolution of the amplitude and this is just a simulation result of the growth rate which is sort of standard to the linear theory that the first speaker just said and then qualitatively we can see time wise there's a pretty good match in terms of features and this is green cause the density value is not rescaled to show it as red in the simulations but otherwise everything was pretty nice. So we were pretty happy with our deterministic code with the stochastic code, thank God the axis is visible here, with fluctuations turned on then we picked up a problem which was smaller so that we thought there would be some sort of business, some sort of rational or reason or effect produced from including thermal fluctuations so we picked up a problem from this paper where they use DSMC to study RMI and so we again matched the system dimensions and the mach numbers and locations of shocks and so on and then we were of course we couldn't match the perturbation amplitude exactly cause they set up their perturbations in a different way and we saw this was their DSMC result and we are so we are pretty happy with and without fluctuation the fluctuations at this scale are not supposed to have any effect actually you have to go to like really you have to go to like one micron sized dimensions to see an effect, again so with the amplitude as well we are doing fine, we would do finer if we matched their A naught exactly so and then we see that the ensemble average fluctuating result is pretty close to the deterministic result which is what is expected so at least we have done things right in terms of including fluctuations into the model now so the future work, I mean I had some reserves but there was, so what we did is we shrank the size of the channel to a micron and so we are seeing some effects of the fluctuations but I just basically took the slides out cause we want to make sure what we are seeing is okay and then so what happens is the shock gets slowed down and so the interaction between the shock and the interface starts at a slightly later time and then the growth also changes and we want to make sure that that's actually a solid result cause it's sort of new and then we don't want to be putting something on there which is not, which doesn't, it's not at least properly explained so that's why we stand and I acknowledge the chaos supercomputing co-lab and then thank you, we didn't because the fluctuations coming from, so they did the DSMC study cause they wanted to show that DSMC can be used for RMI and then it gives the same universal behavior the fluctuations arising in the DSMC solution is because of statistical noise so it's not, yeah it doesn't make sense to compare, yeah had they sampled more which they were limited by computing time they would have gotten it down, yeah? Comparison with Yako's, oh yes, this one. Could you explain, seems to me not small difference but some future of the difference of the shape? Of the shape, yes, of course so if that's, some people have done like say ninth order compact difference and so on you'll get a better feature-based comparison so of course, I mean this is a fourth order and then if you dial down it to second order it becomes I mean the resolution of the features obviously I mean the quality of the features goes down so that's just, if you use a high order method you can definitely get good features but we were just looking for verification of these quantitative because once we, we're not chasing features per se we wanna, you know, build an FHD code, so.