 Thank you so much for inviting me here to give an opportunity to give this talk. So I will be discussing today the work which we did essentially with my team when I worked in the U.S. We started essentially this work with Bob Stelengwerth and Mila Stanić when I've been at the University of Chicago. Then we continued with Aklan Bomek, Zakari Adela, Arun Pandyan, Norah Svisra at the Carnegie Mellon University and I will be essentially presenting this work with the cumulative contribution of our group. We had discussed scale coupling in strontial driven Rihmaar Meshkov flows and this work had been published partially, some of the results are also in preparation and the work was supported by the U.S. National Science Foundation and also by the U.S. Department of Energy. So Rihmaar Meshkov instability controls many processes in nature and technology in high energy density regimes as well as low energy density regimes as supernova infusion, scram jets and combustors as well as impact dynamics of liquids and soils. In this environment the flow dynamics is often characterized by strong shocks, constant densities, large disturbances and which essentially means that the flow feels changed sharply and rapidly. On the one hand side, on the other hand side there is a very small contribution of dissipation and diffusion. As a result, this dynamics is characterized by appearance of interfaces and extensive interfacial Rihmaar Meshkov mixing and even though the title of our conference turbulent mixing, in fact, the more we study these problems, the more myself I am coming to a conclusion that we should talk primarily about interfacial mixing rather than turbulent mixing. So what we did in this work, we performed, we conducted the systematic study of Rihmaar Meshkov instability induced by strong shocks. It means that we considered Mach numbers up to 10. We applied theoretical analysis, which included zero order theory developed by Rihmaar, linear theory developed by Rihmaar Nishihara and Bovchuk as well as group theory analysis to study highland linear regimes and order and disorder in Rihmaar Meshkov flows as well as we developed an oval empirical model to describe the numerical simulation data. We also applied smooth particle hydrodynamic simulations which was implemented in the SPHC code which was developed by Bob Stellingwer for many years. So what we have found that instruction driven Rihmaar Meshkov instability is initial conditions are the key factors of RMI evolution that importance in that they exist. We kind of identified a characteristic amplitude scale that enables maximum initial growth rate of Rihmaar Meshkov instability. We also found that order and disorder in Rihmaar Meshkov flows are defined to large extent by the interference of the initial perturbation waves. We also found that there is a coupling, essentially strong coupling of the interfacial and bulk dynamics specifically significant amount of shock energy goes into the material compression and background motion. Only small portion of shock energy is available for interfacial mixing. At the same time the small scale dynamics is non-uniform and heterogeneous both at the interface which is essentially a known fact as well as in the bulk which was not so known before. So I will describe briefly what is the Rihmaar Meshkov instability and Rihmaar Meshkov mixing is like you know what is a phenomenon application outline of the dynamics as well as the challenging of modeling and quantification why this problem is essentially so challenging to study. Rihmaar Meshkov instability was discovered theoretically by Rihmaar and experimentally by Evgeny Meshkov. It occurs whenever a shock wave refracts an interface with influence with different values of the acoustic impedance. Shock can be planar or perturbed, it can be steady or unsteady. Shock may propagate from the light fluid to heavy fluid and vice versa. The interface can be planar and perturbed meaning that essentially Rihmaar may develop when the shock is perturbed and the interface is planar or when the shock is planar and the interface is perturbed. Rihmaar plays an important role in a broad range of processes and nature and technology at various scales from Supernova to the characteristic scales at gigameters up to the initial confinement fusion where the characteristic scales are microns. The pulse shock dynamics of Rihmaar Meshkov instability essentially after the shock passes the interface is a superposition of essentially two motions. One motion is such that the fluids and the interface between the fluids move as a whole unit on the one hand side and on the other hand side there is an interface perturbation that starts to grow because of impulsive acceleration induced by the shock. If we will be looking at this interfacial dynamics which is actually well illustrated by this figure by Katie Pressrich in the dynamic experiments we see that there are the large scale essentially coherent structures which we might be seeing here and here and they are induced essentially by initial perturbations and there are the small scale structures which are essentially irregular even though it's hard to tell how turbulent these structures are. If we will be talking about how to quantify this type of dynamics reliably we will see that in principle there are a lot of challenges. Especially this challenge is significant when we are talking about extreme parameter regime characterized by strong shocks, cotton resistances and large scale perturbations. Only limited information is available about RMI evolution under these conditions. On the one, because first of all experiments are rare because the flow implementation diagnostics control are very challenging. Eulerian simulations are also a challenge because on the one hand side they should capture shocks track interfaces and on the other hand side they should also account for very accurate dissipation processes. But most importantly if we will be talking about this I would say technological challenge there is a very specific challenge which is coming with the Richtmeier-Mischkopf type of nature flow because if we will be looking at the instability evolution from a linear regime after the shock passes the interface up to the interfacial mixing regime what we would know and what we know already that this evolution is characterized by several power-low dependencies. Say we have a linear growth with time in the linear regime there is a sum between the non-linear regime then there is a decay for the velocity in the non-linear regime and then there is an interfacial mixing and because at each of these substages the dynamics are essentially power-low substantial dynamic range and high precision and accuracy of the observation are required to quantify this power-low accurately and usually this information is really hard to obtain in experiments and in the simulations including those simulations that we conducted at the high class power supercomputers. So what we did we conducted a systematic study of strong shock driven Richtmeier-Mischkopf instability our methods included rigorous theoretical analysis we developed also an empirical model because there might be the situation when the rigorous theoretical analysis are limited in their power and they applied Lagrange simulations as PHC for Mach number from one to essentially two-six. We achieved good agreement between the simulations and the serine experiments in our verification and validation studies and we identified new properties of the evolution. So I will discuss briefly the methods of study so we include theoretical analysis, diagnostics and benchmarks as well as the simulations and parameter regimes. So for the theoretical analysis which is common for all these actually methods of analysis is that they usually do not apply adjustable parameters they provide sufficient number of benchmarks to compare with the experiments and simulations. One of them is the zero-order theory which developed originally by Richtmeier however we always refer in our simulations to the work of Nishihara and Wolfchuk they provide us with powerful formulas in the whatever extreme parameter regimes we need to obtain and this zero-order theory describes the interaction of a planar shock with a planar interface and derives the characteristics of the post-shock background motion of the fluids. Linear theory describes the interaction of a planar shock with a slightly perturbed interface and when I mean slightly perturbed it means that the ratio of the amplitude to wavelength cannot exceed it stays within 1% or so because it should be very, very small initial perturbation and linear theory derives the linear growth rate of aromai and a broad range of the machinout numbers and we are referring usually with the theory of Nishihara and Wolfchuk and Wolfchuk 2001 and this is the review paper by these others. We compare also with the Wichlin and Linear models and there are all sorts of Wichlin and Linear models that try to account for because measuring aromai growth rate when the initial perturbation is 1% it's a very challenging task so people usually try to make it a little bit larger but when they make it a little bit larger the amplitude starts to influence the growth rate and the question is what the influence is so far there was a more formal approach developed by Velikovic with which we conducted a comparison of the empirical models that were developed by Schwartz and Kallix, Butler et al. In order for us to study nonlinear dynamics we proceed to the group series analysis that was developed by me and Kallix and we analyzed the nonlinear dynamics of Riechmeier-Meshkov instability and also described the order and disorder in Riechmeier-Meshkov flows On the top of this we actually developed another model to describe the simulation data with a statistical confidence The data and diagnostic benchmarks includes the vector and scalar field in the bulk and at the interface such as the amplitude growth growth rate and interface morphology From the zero order we derive the control parameter and we compare it with very cautiously for the planar interface velocity and we achieve agreement of our simulations within 1% accuracy For the linear series we conduct special high-resolution runs to evaluate the value of the growth rate within 4% to 5% accuracy with the linear series simulations We achieve reasonable agreement very similar to what is an experiment with the weekly nonlinear models and for the group series we reproduce very well the flow fields and the morphology of the interface and the novel empirical model it actually identifies, it definitely agrees with the available data So the smooth particle hydrodynamic simulation that we apply is a Lagrange mesh free numerical method that allows for relatively easy treatment of the PDEs governing the fluid motion so in a sense it's not a pure patical method it's not a continuous method this is a method that is in between because it essentially partitions the flow fields and reproduces the motion of the fluid as a motion of the particles The method applies the particle and kernel approximation to reduce the PDEs to a set of ordinary differential equations which is much easier to integrate The method is in principle suitable for high performance simulations and is diverse, proven and robust This is an outline of the SPH principle in which I will not be going in much details In fact, actually the idea here that in a particle approximation any function can be represented with this equation replacing delta function with a certain kernel operator we may actually apply this kernel approximation like this and then calculate the derivatives using this expression and this way reduce the PDEs governing the fluid motion to the ordinary differential equations The parameter regime in which we have been working were all instruction driven regimes including the Mach number 3510 output number from 0,3 to 0.95 which means that the density ratio in our cases was varied from essentially 1 to 3 to 1 to 40 The initial perturbation was varied from a planar interface up to 100% of the wavelengths for the initial amplitude We made additional rounds for the planar interface for very small amplitude interfaces and conducted extensive validation and verification studies including standard shock and knock problems we compared with experiments and achieved pretty good agreement Typical number of particles in the simulations that was 10 to the power of 5 we noted that when it was 10 to the power of 4 actually agreement with the problem such as a knock problem it was still within 10% accuracy so 10 to the power of 5 it was sufficient number of number of particles for us The parameters of these dynamics that was essentially ideal metatomic gases with a standard gamma 5-sort and the parameters if you look cautiously at these values such as the density of the light and heavy fluid or typical temperatures it would be easy for us to note that such gases don't exist in reality and it's essentially a mathematical model so essentially this is a fluid which is a rarefied rather stiff gases and the reason why we selected this parameter is because we wanted to ensure that there is a high energy density per proton and it was really a large increase of the energy density before and after the shock passage At the same time when we were comparing these experiments we needed to scale these values several orders of magnitude and we still had a reasonable agreement with what we had So it's an RMI evolution and many people who sit in this audience probably are well familiar with this picture and this actually evolution shows us the post shock dynamics of Riecht-Meyer-Meschkow instability and we do observe that there is a superposition of the two motions. First of all after the shock passage from the light heavy fluid to heavy fluid in our case both fluids and interface between them start to move and they move as a whole unit so we see this displacement this time. On the other hand there is an interface growth because of the impulsive acceleration induced by the shock and we also observe these interface growth as you may see from here. If we will look so there is a background motion and there is an interfacial growth If we will look at the span of scales essentially this kind of estimate tells us what the challenges of this problem are and why probably the problem cannot be solved for so many years or even decades maybe essentially like you know 60 or 40 years when the shock goes from the light to heavy the largest velocity in the fluid system is the shock velocity If we will be looking at the velocity of this background motion with which both fluids and interface between them move after the shock passage it depends on the mach number, output number, gammas on variety of factors but to large extent it is essentially about 10% of the shock velocity so we have one order of magnitude down. If you would like to look at the initial growth rate then the initial growth rate it also depends on various output number, mach number typical gammas etc but on average it is about 10% of the background motion of velocity so from the shock velocity we need to go two orders of magnitude down. If we will be looking at the nonlinear regime in order to quantify nonlinear dynamics accurately we need also to go about 10% of the initial growth rate which give us already 10 to the minus 3 of the shock velocity and if for whatever reasons we would like to proceed to the mixing regime we need to go one order of magnitude one more which give us essentially 10 to the minus 4 of the shock velocity. So in a sense we have a phenomena where we need to go from a scale up to one to the scale to minus 4 and we need to track and accurately measure each and every number which makes the problem really challenging to study. So the background motion develops whenever the shock is planar and interface is planar so we do not need to have any perturbation for the background motion. Whereas the interfacial growth which develops because of impulsive acceleration due to the shock, if the shock is ideal and planar and steady we need to have a perturbed interface for the instability to start to grow. So our benchmarks as I mentioned for these simulations was to look for the zero order linear and haline and linear series as well as the gas dynamic experiments. So magnitude of the background motion velocity so this picture shows us essentially the color with which we color the velocity component as the direction of shock propagation in case when there is a planar interface and there is an interface between the two fluids which is somewhere around 10 in this picture but we see that actually the light fluid the heavy fluid and the interface between them they are essentially invisible because they move with the same velocity. Whereas there is a perturbed interface we do see that there is a velocity growth on the background of this background motion and the ratio of typical magnitudes of values of here background motion is essentially about 10% or so. Agreement that we achieved for the value of V infinity and we call it V infinity because it means that we have an infinitely planar interface within 1% of it. Clearly we conducted the comparison with the linear growth rate with the series that was developed by Volchuk and we conducted separate resolution runs with a very small less than 1% initial perturbation amplitude that we achieved very good agreement 4-5%. In principle this agreement remains reasonable when the amplitude perturbation becomes 6% of the amplitude wavelength. So if I will be looking for the late-time evolution of the flow fields essentially there are two criteria because we have a power-low dynamics and we essentially don't have a substantial dynamic range to quantify velocity very accurately we better look at some qualitative features and one of them is the velocity field typical velocity field a frame of reference that moves with the velocity of the background motion and what essentially this field tells us that while there are very small scales component structures we don't have essentially zero velocity away from the interface and we have intense motion in the vicinity of the interface. The other actually important quantitative parameter is the flattening of the bubble front and we do observe the majority of our simulations when we propagate far enough to the non-linear regime that is called bubble flattening and we celebrate. We would like also to look what would be the how our data would fit compared to experiments and we compare with the gas dynamic experiments which is essentially experimental cases of 5, 10, 8, even and these are the experimental data versus numerical simulation data and we see the agreement as pretty good. So quite often especially in the funding agencies communities people say that simulation is a discovery tool and I would say that you know this is why I place it like in the waters so it's a discovery tool but in fact actually sometimes what is really quite important is to formulate the problem cautiously. So in our case the series essentially was working very closely with the simulations and we tried to formulate the problem properly such that our simulations would have actually really good meaning and we may obtain the meaningful results. So one of the the key results that we have obtained as far as that first of all we identified to the first time by our knowledge that there is a characteristic amplitude scale that defines the maximum initial growth rate and it did it very cautiously. We identified we found that there are non-linear small scale dynamics at the interface and in the bulk implying that there is also the scale coupling between the interfacial waves plus we found also that there is a strong effect of wave interference of the initial perturbation waves on the odor, disodorant, rihmar and mescal flows. So the first thing it's about RMI initial growth rate because this growth rate initial because essentially we tried to measure it at some initial stages and we would like to separate between initial growth rate and linear growth rate. When we would be looking at this picture we see essentially what happens to the amplitude after the shock passes the interface the interface is first compressed and after the shock passes the interface the interface perturbation starts to grow. We, I mean depending on the so of course when the amplitude is very small the linear theory can provide us with certain values but these values are because the growth rate is proportional to the amplitude and the amplitude is small the growth rate is small it means that it's really hard to measure. We try to make it larger but then various effects should start to play a role and it's hard to say what the result will be. So what we did we conducted systematically in a broad range of the mach and output numbers and we measured the initial growth rate at the very initial moment of time and this time scale was essentially defined not by the V0 not by the value of the initial growth rate but rather by the typical time scale which is set by the background motion velocity because this time scale is really a robust parameter it can you know it has a clear physics meaning and it can be easy controlled in experiments so we don't actually don't put additional error bars into the scaling. This is what linear and linear theory predicts how the amplitude should depend how the initial growth rate should depend on the amplitude and it's natural that it's a linear growth and linear theory works only when the initial amplitude is very small. There were a number of weakening linear analysis that tried to calculate the correction factor to this linear theory one of them it was the series and it also predicts that there would be some growth of the initial growth rate with the initial amplitude and while this growth would be smaller than linear when the amplitude is pretty large it still would be someone at own function of the initial amplitude. This is the case I believe it's a mach3 at 0.6 and what we see that in fact actually initially in a good agreement with the linear series between linear analysis we have a growth of the initial growth rate with an initial amplitude however when the sum maximum value is achieved our numerical simulation result shows that there is a non-monotone dependence of the initial growth rate and there is a decay of the initial perturbation of the initial growth. We repeated this exercise for the bunch of the mach and output number and what we have noticed is that there is indeed the non-monotone dependence of the initial growth rate with an initial amplitude and there are some interesting features here because while the maximum value of the initial growth rate does depend on the mach and output number at least for ideal fluids the position of the characteristic amplitude scale at which this maximum growth rate is achieved is insensitive is relatively insensitive to the characteristic mach and output number. Furthermore if you will be measuring for instance the position of the middle line velocity which is easily to obtain in experiments we also see that the position of the maximum of which we are here is very correlated with the position of the minimum of the middle line velocity means that we don't have essentially a numerical artifact and if we will be comparing our maximum and minimum value we see that there is a really good correlation for the maximum of the initial growth rate with the minimum value of the initial of the middle line velocity even though the middle line itself doesn't have a it's not a kind of really special position to measure now we try to scale to play with data a little bit more and we scale this dependence which is the ratio of the initial growth rate to the background velocity and this parameter essentially has a physics meaning of a fraction of energy that is available for interfacial mixing and its value is pretty small and we see that if we have a scale with an output number we see essentially quite good universal behavior so dependence of the initial of this parameter with the characteristic amplitude usually I am very against empirical models because I do prefer to develop rigorous series but they abide by the situation when we do need an empirical model at least try to understand what our data are because we might develop a good series in this regime we might develop a good series at a very very long time but when we have an initial time and finite amplitude we do need to understand what our data tells us actually what is the situation because in this case we have a characteristic amplitude scale we try to fit our data with the following dependencies which account for the two cases first of all we have a linear when the amplitude is very small we have a linear growth according to the linear series second when the amplitude becomes very large we have a decay and this is decaying function and we apply exponent rather than any other power law or a rational function because we see in the simulation that there is a characteristic amplitude scale so the parameters C1 and C2 are free adjustable parameters and we try to identify this parameters value from our data sets essentially we consider over 20 curves we do it curvewise then we do it through the data points considering the entire data set through the 300 data points and as well as about 20,000 pairs of data points so it's really well designed statistics and if we have this dependence for the model we can easily transform it into the linear fit which is essentially presented here and was a value Z versus X which is an initial amplitude according to this expression we should have a linear dependence and we do observe the good collapse of data along this curve so essentially we see that when our amplitude is large enough after we are speaking greater than 0.1 and up we do have a very nice exponential decay this picture shows our result it's a ratio between the initial growth rate and linear growth rate as well as the experimental data points linear growth rate actually the difference in the intercept appears because the linear growth rate is not really accurate it's really a challenge to identify accurately in the experiments and simulations however the thing which we are looking here is essentially the slope of this line and the slope of this line in our model in the simulations as well as in experimental data including the experimental data of Aleutian I would say in 1990 something they agree up to the third significant digit as soon as we know these parameter values for our C1 and C2 what we see is that in our data set is defined very accurately within few percent of accuracy essentially 6 percent here and 7 percent here as soon as we have our expression for the initial growth rate we can evaluate the maximum value of the growth rate the characteristic amplitude scale as well as the maximum fraction of energy available for interfacial mixing and I would like to point your attention that this would be true for any Mach an output number and even though we consider that essentially only ideal gases with a gamma equal to 5 sort we believe that the sensitivity of these results to the gamma would not be very large because in fact in this expression there is no any gamma dependence and this is one more time like you know these are experimental data coming from these are data from our results our model this is experimental data by Aleutian and these are experiments of which have been taken essentially referring to the experiments of Dimonti and Remington where they evaluated Mach number as a large or Mach number as a greater than 20 various models such as the model of Schwartz and I think also Glendigind model to provide some evaluation this is why we have such a significant scatter evaluating Mach number as a 15 or 15.3 but what we see here that essentially this decay should be defined by the initial amplitude only and should not be independent of the Mach and output number flows so it's like another comparison in fact the question is why this characteristic universal scale appears because what does it say us we make our initial amplitude larger we have more energy put into the interface we make initial amplitude even bigger then there is even more energy sitting at the interface now we make it infinitely large and we see that essentially there is no any growth rate of the instability it's in a sense it's natural because there is a strong dependence on the verticity left by the shock and when the amplitude becomes very large then the verticity effect essentially will be zero on the one hand when the amplitude is very large we have just a smooth transition from one fluid to another fluid and no instability might occur what is quite important that there is indeed a characteristic scale because we do absorb the exponential dependence rather than a power law or a rational function of the initial perturbation amplitude so the question is when we start to increase the amplitude what happens to the energy because we are trying to put more energy to the interface and instead we having a larger growth rate we have a growth rate decay well it appears that in fact the large perturbation amplitude influences not only the dynamics at the interface but also dynamics in the bulk specifically if you will compare for instance the position of the transmitted shock here in case of planar interface and in case of perturbed interface we see that when the amplitude becomes very large the shock starts to propagate transmitted shock starts to propagate quicker and in principle this means that this type of dynamics can be used also as an additional diagnostics for the flow we observe a lot of small scale structures in our simulations at the interface which is kind of natural for us to observe such as development of the Kerring-Gelgoltz instability but we observe also some other structures which are small scale structures in the bulk and I can point your attention for instance to this picture which shows the typical temperature field what we have here we have observation of cumulative jets we call them reverse jets because they propagate to the the direction reverse to the propagation of the spike and they appear they appear because two flows collide at a small angle of attack it's a standard cumulative jet it's very energetic it's short-lived but it creates hard spots inside the bulk whose temperature can be essentially two-fold of that of the ambient which is kind of a case here and definitely similar structures are observed also in experiments which might be kind of demonstrated here and here now one more thing to which I probably will tell just in passing it would be a question how the order and disorder in Riechmeier-Muschko flows might appear and traditionally we do know that initial conditions and flow dynamics of RMI currently research is focused primarily on the effect of the wavelengths and amplitude of the interface perturbation even though what we have found it has not been discussed before such as exponential decay of the initial growth rate with an initial amplitude but what we would like to do also we would like to study the co-influence of effects of the relative phase and the amplitude of the perturbation waves on the large scale structure of the bubbles and spikes in Riechmeier-Muschko of theory and Riechmeier-Muschko flow and we applied the group theory analysis keeping going so what we have found that in fact the phase of waves and the interference of waves constituting the initial perturbation have a dramatic effect on the order and disorder in Riechmeier-Muschko flow I will be skipping probably the group theory considerations just would like to show you two nice pictures so what is this it's a flow evolution in one case and the flow evolution in the other case at certain moments of time output number is the same Mach number is the same it was 5 the moments of time at which the snapshots have been taken given here there are two waves they have in both cases the amplitude of the waves is the same and the wavelengths of the waves are the same the only difference that in this case the waves are anti-phase meaning that the relative phase between these waves is pi whereas in this case the relative phase between these waves is pi over 2 and if I would like to present it like here so essentially we have two waves and different phase is presented by this parameter so what we see here is that this silent parameter that present in every essentially initial conditions in every experiments and simulations appears to be to play a very definitive role because if we have an anti-phase configuration we have a well designed order quite non-linear stages of instability varies when the the phase is random such as a pi over 2 we have appearance of disorder people might call it turbulent I would still call it chaotic or maybe rather than stochastic because in this case we do know that there is a parameter such as the relative phase of waves which we may control and which we may vary creating typical order or disorder what is the one I am almost done so in fact actually this parameter plays important role not only quantitatively creating order or disorder but it also start to influence the growth rate of the instability for instance when we have the same ratio of the first and second harmonics here we might have we might vary the initial growth rate substantially essentially two-fold it is pretty large when the phase is say in-phase, anti-phase and random phase and these lines which are given here they are just given for a convenience they are not functions they are just lines and I am concluding so the conclusion besides that we performed the it's not the first actually but we conducted the systematic study of the Riechmeier-Meschkow instability induced by strong shocks we applied the theory and the simulations and we collaborated really strongly together to obtain some new results so what we found we found that the velocity of the background motion it's a kind of polish and a secret in our Riechmeier-Meschkow community because everyone knows that it exists everyone observes it however people really hardly measure and monitor it what we say that this parameter is important it's robust, it's reliable and it should be applied not only to say to quantify the flow but it also should be applied for the scaling we found also that a significant part of the shock energy goes into compression and background motion of the fluids only small fraction of the shock energy remains for interfacial mixing there is a strong dependence of this shock energy interfacial shock energy on the interface perturbations when the amplitude of this interface perturbation is very small the growth rate grows linearly this time however when the amplitude becomes larger there is a non-monotone dependence of the initial growth rate there is a characteristic variable which appears to be a universal quantity of pharomide dynamics at which this growth rate is achieved and which is not dependent on the mach and output number and as a result we can calculate a lot of useful values such as the maximum initial growth rate maximum fraction of energy available of interfacial mixing etc and the decay of the growth rate with the initial amplitude is exponential rather than a parallel so at late time bubbles flatten decelerates and the fluid interface can always keep some order whereas the dynamics in the bulk at small scales is non-uniform and heterogeneous and I think that would be it