 Good morning. I'm Arun from Indian Institute of Technology in Metros. First of all, I would like to thank the organizing committee for giving me the opportunity to present my research work here on compressibility of X on the evolution of mixing layers. This work is run by myself and my advisor, Samin, in collaboration with Process Balai Srinivasa. The brief outline is I'll first give a brief introduction on mixing layers and what the objective of our study is brought by the brief details on our numerical scheme which we are using and the computational parameters, the domain, et cetera, et cetera, and some preliminary research and the discussions followed. Mixing layers are shared with, characterized by the velocity difference, the prevalent in nature, like when two streams merge together, as well as in engineering flows, like what we have inside an aircraft combustion chamber or jet exhaust. So there is this characterized by a velocity u1 and u2 with a difference. It can be continuous or non-continuous, but the viscous effects will eventually lead to a continuous interface. When the velocities are higher, or rather the velocity differences are higher, the compressibility of X starts to kick in and the compressibility of X are quantified by convective Mach number which is defined in this way, where C1 and C2 are the speed of sound in either streams. Most notable effects of compressibility on the mixing layer, the first one is that the instability is three-dimensional when the flow is compressible, or the mixing layer is compressible. Whereas in any compressible flow it is two-dimensional with characterized by the roller structures. And compressibility reduces the growth of the shear layer, as well as turbulence, statistics are concerned, the production is reduced by the effects of compressibility. The motivation behind this work is that most of the analysis till now have been either concentrated on the linear regime or on the self-similar regime during the late times. We are studying the effects, compressibility effects during the transient evolution using our numerical simulations. In particular, we are studying the evolution of turbulence, statistics, and trying to deduce something from that. The numerical method which we are using is called the gas kinetic scheme. It is the Boltzmann equation solver which solves the velocity distribution function f with a classical BGK approximation for the collision term. The advantage of using a Boltzmann solver is that once we solve for one particular variable f, the distribution function can obtain the flow variables, mass, momentum, or energy, or the primitive variables, density, velocity, pressure, or temperature from f itself using the moments of this distribution function. So the scheme is a finite volume scheme where we discretize the domain into finite volume cells like in any other conventional scheme. But the difference is when we compute the flexors, we use an integral solution of this equation f in terms of an initial distribution f0 and an equilibrium distribution towards which it is driving. We use this f to calculate the flexors. So this is a typical cylinder phase. What we are using is a second order BG scheme which means that we are linearly reconstructing the distribution functions in either in each all cells. So here we have a... This represents the initial distribution f0, which is discontinuous, and this is the equilibrium towards which it is driving, G. So this is the non-equilibrium initial conditions which enables the scheme to capture any non-physical phenomena in the flow. However, in this whatever we are doing, these simulations are in the equilibrium regime, so we are not taking advantage of this particular nature of the scheme. So the scheme works like this. So we start with... At any given time step, we start with the cell average density, energy from which we can reconstruct to the cell interface values and using that to describe the initial and the equilibrium distribution functions. Once we get that, we use the integral solution to get the f, the actual distribution function, which is the function of time. And the flexors are just like the primitive variables, the corresponding flexors are also moments of this f. Just only that they are one order higher. And once we have the flexors, in any other conventional finite-volume scheme, we do a timing integration and update the cell average values. So the domain which we are using is a cuboid domain of these dimensions, and the flow is in the x-directions, and the velocity varies in the y-direction. So we have an imposing p-ro-d boundary conditions in the x-direction, which is a stream-wise direction. So we are actually doing a temporal evolution and not a spatial evolution on the mixer, which would require a longer domain and a larger grid size. So in the span-wise direction also, we have periodic boundary conditions and it is stress-free in the y-direction. The domain is discretized into phytol by 2, 3, 6, and 1, 2, 8 grid points, or rather cells. And we initialize the mean flow field with a tan hyperbolic profile. We do the simulations for two values of compressible Mach numbers, convective Mach numbers, which is 0.2, which is in the incompressible regime, and 0.7, where the compressibility effects are significant. The initial Reynolds number, which is based on the moment of thickness, I'll define the moment of thickness in the later stages, is 160, which is same for both the cases. And the medium is higher, with the characteristic flow properties as 1.4 for the specific ratio and 0.7 for the random number. We do the simulations up to non-dimensional time of 800 for the incompressible case and 1200 for the compressible case, which are well into the cell-similar regime. So the density is initialized uniformly and pressure is also uniform, and for the temperature, we have the Croco-Bismann relations. And to accelerate transition to turbulence, we give some initial perturbation field. These initial conditions, perturbations are generated using a digital filter method which was originally developed for generating inflow conditions, but later adopted for temporary whirling flows like in incompressible mixing layers. This method generates a 3D perturbation field based on a prescribed length scale and the Reynolds stress profile. So in our simulations, we give a Reynolds stress profile, which goes in in Y, and the peak values of these Reynolds stresses are specified based on what Pandano and Sarkar obtained in our DNS studies in 2002. The length scale which we use is the vorticity thickness, which is roughly four times the momentum thickness, initial momentum thickness. So this is how the momentum thickness evolves with time. This is the momentum thickness scaled by the initial momentum thickness, and this is the non-dimensional time defined this way. As you can see, there is a significant reduction of growth rate, which is already well known in these cases. And the momentum thickness is defined in this way. So we have a bar quantity, which corresponds to the Reynolds average quantity, and the tilde represents the density weighted average. So our flow variables, not flow variables, the velocities are density weighted average, and the fluctuations from that are used to calculate the turbulent statistics. However, what we are interested is we are studying this region, not the self-similar region. Nevertheless, we compare our self-similar quantities with existing literature. This is the normalized growth rate, self-similar growth rate, normalized with incompressible value against Mach number. So we found this is an agreement with the existing empirical curves. These barons and Langley's are, these curves are empirically obtained. So what we observe at a non-dimensional time of 200. This is for the incompressible case and this is the compressible case. So here we can see the span-wave structures or the two-dimensional roller structures still prominent in the flow, which is actually remnants of the linear instability region. But such a pattern may be exiting here, but it's still, it's not as clear as what is there in the incompressible case. So how to study these turbulent statistics which are varying in time? Apart from that, at any given point of time, the turbulent statistics are a function of y or they vary in the transverse direction. So to eliminate the dependence on y, we define an average quantity, capital F, in this way. This, where f can be any turbulent statistics like the Reynolds stress or the kinetic energy or any of the budget quantities. The budget quantities are normalized with delta u cubed by the instantaneous moment in thickness and the Reynolds stress and Tk are known with the square of the velocity difference. The first quantity which we studied was the turbulent kinetic energy and this line is for the incompressible case and it is significantly lower than that in the incompressible case. So what is this reduction in turbulent kinetic energy? So we look at the turbulent kinetic energy budget. We have the transport terms, the pressure, sorry, the production, dissipation and this is the pressure dilatation term and these are the mass flex fluctuations terms. These two terms actually go to zero in an incompressible case. Even in our simulations for 0.7, these two terms are found to be negligible. So here we report only the reduction and dissipation. So we can see that it is actually the reduction in production of kinetic energy which is responsible for the lower Tk levels. What causes this reduction in protection? We look at the production term. This is how, this is the production term. We do not have any other mean flow gradients. So this is the only term which will be there and we observed that in our simulations, the mean flow gradients are roughly comparable for incompressible cases and incompressible cases. So it is actually the Reynolds transform R12 which is contributing to the reduction. We look at the budget of R12. Similar to the Tk budget, we have production, dissipation, then the pressure dilatation term and the max flow stems. Again, these two are negligible. We investigate only the production and dissipation terms. Once again, we observe that it is a reduction in production which is responsible for lower R12 values. So what is the production term? The production is given by this relation. So this is the transverse fluctuations. It might be, it has to be by going with the certain arguments, it has to be the reduction in or the lower levels of R22 which is responsible for a reduction in production here. We, coming to the budget of transverse fluctuations, here also we have the same terms. And when we take all three components of this, the diagonal components of the Reynolds tensor, we get the Tk budget and where we have pi representing the pressure dilatation which is which we observe to be 0 and even in the commercial case. However, here we have the production also is 0 because we do not have any flow gradients responsible for the production in this case. Mass flow stems are negligible. So what is this pressure strength them doing? So these three terms, the corresponding stream wise and span wise fluctuations, these three components together becomes the pressure dilatation terms which is almost 0. So we can write it in this way. So what this represent is that there is production in U square and pi11 is kind of extracting the energy from this component and redistributing it to the other two terms. So we have to investigate how this redistribution works in the compressible and incompressible case. So here we have this is the dashed lines are for the incompressible case and solid lines for the compressible case. So during the early stages there is a clear difference here. Which means that the redistribution is less efficient when we go to higher Mach numbers. The redistribution term is here. Or this is the pressure definition of the pressure strain rate term. This is the correlation of the pressure fluctuations and the velocity gradients of the velocity fluctuations. What we observed is that the velocity gradient fluctuations remain largely unchanged by the compressibility effects. They are at the lower scales. It is a reduction in p dash which is responsible for a less effective redistribution here. So it is a kind of a cycle. We have the higher compressibility means a lower pressure fluctuations which results in a less efficient redistribution from u square to v square as well as to w square. But it is a v square which is responsible for the stress production u v term. So a lower v square means a lower stress protection which means a lower Reynolds stress u v. And a lower u v means again a lower total kinetic energy protection because all of the TK production happens in this component which means a lower u square. And again the redistribution means it is being less effective. So to conclude, summarize what we observed is that the transient evolution during the interim regime is more or less similar to what we observe in the self-similar regime. It is actually the reduced production which is responsible for the lower TK levels of equivalence and eventually the slower growth rate of the mixing layer. And it is ultimately due to the less efficient redistribution at higher Mach numbers caused by the lower RMS fluctuations of pressure. Thank you. Any questions? Slide 11. Yeah. Yeah, I did only for possible Mach numbers. Even these curves are obtained from previous. You mean these two points? Yeah. For that I have to do simulations. I have to do simulations for the entire range of Mach numbers. So I have not drawn a curve. It is just points on this. So this solid curve is from Barron's empirical relations. It is based on experiment. This Langley's curve is largely experiments because this is from the 1970s. Whereas 2006 Barron's empirical curve. So in 2006 they have both experiments and simulations taken into account. So actually we developed this for some future purpose. In this particular study we have, that is what I told we have exploited the what is the non-inquilibrium effects are there. Thank you.