 Apologies for this. So, I will talk about thermal convection, but it is also connected with the stability stratified flows. So, we will take buoyancy driven flows as a theme because they are connected versions. So, I will basically focus on what is the spectrum for thermal convection. Is it Kolmogorov or non-Kolmogorov? So, this is everybody knows about this equation, but we need to just focus on Kolmogorov theory, since it is a very broad conference. So, I thought I will put this slide. So, for fluid incompressible fluids, we have a spectrum which is Kolmogorov k minus 5 third and constant flux. The flux is constant throughout, but this will this could change with buoyancy because buoyancy works at all scales. Here we assume that forcing it is at large scale, but in buoyancy we are forcing at all scales thermal forcing. So, that is the key and we need to address that issue as we go along. So, the two systems which we will study in this talk is RBC R11 and when heating is at the bottom and cooling at the top and stability stratified flows which is reversed like earth atmosphere, dense fluid at the bottom or cold fluid at the bottom and hot at the top. Now, so this is the equation which we will deal with is a business fluid. So, we have buoyancy term which is here theta is the temperature difference from the conduction temperature. So, it is a fluctuation over the conducting profile and this is an equation for the temperature which includes that temperature gradient. So, for stability stratified flows this will be negative dT by dz. So, the whole thing becomes negative and for RBC it is positive. So, there are few parameters which again for people who are not familiar with this field the final number is nu by kappa which is viscosity by thermal diffusivity and Rayleigh number which is buoyancy by non-linear buoyancy by viscosity sorry buoyancy by viscous effects. Now, I think we will ignore this I will just explain what they mean at later time, but we no need to get into that those parameters. So, stability stratified flows there are several regimes. So, we focus on the third regime which is isotropic regime because we are looking for Kolmogorov like spectrum or deviations. So, we need to assume isotropy which is not the case when gravity is too strong. So, most of the simulation done for stability stratified flows were focused on this quasi 2D regime like Lindbergh there was a talk in the morning. Strong gravity will make it quasi 2D and what we are going to discuss in this talk will not work. So, we are working on isotropic regime approximately isotropic regime that requires that Richardson number must be order 1 we need to be careful about this. So, this one slide I would like to flash because this is the important slide. So, let us look at the energetics of Kolmogorov flux. So, we are in the Fourier space. So, these are two spheres one black sphere and a blue sphere and this pi k is the flux coming out of the black sphere and pi k plus d k is the flux coming from blue sphere. Now, Kolmogorov assumes this flux to be constant equal the reason being there is no forcing in this region and there is no dissipation in fact, related to your talk yesterday. So, but what happens if there is a forcing if there is forcing then the flux can change also if there is a dissipation is strong then the flux can change and we can very simple energetics argument that d pi by d k will be supply by the external forcing f k minus d k. Now, these are relevant for buoyancy driven flows f k because buoyancy will act at all scales and we like to see what are the effects of this. Now, just a simple analysis that if f k is less than 0 and ignore d k let us ignore this. So, basically I take the energy out the forcing is such a way that it takes the kinetic energy out. So, flux will decrease with k. So, this will decrease with k it will not be Kolmogorov like it will decrease with k. What if the forcing feeds energy then flux should increase with k. So, these are in fact, these will be the key for these two systems stability stratified flows and relevant. So, in fact, let me just tell you before and this will be state stability stratified flows flux will decrease with k and this will be relevant where flux will increase with k. So, this one important point that this is independent of isotropy assumption I did not assume any isotropy because this is energetic argument and we work with Fourier space which basically has multi scale built in. So, this is I had interesting conversation with Shelly and so this is connected with one paper he has written in 1999. So, this is a famous theory by Bolgionov-Nubakov in 1959 for stability stratified flows it was not for relevant it was for stability stratified flows. So, this is a flux plot. So, as I said I will focus on flux. So, what happens to the flux kinetic energy flux it will decrease with k why because kinetic energy gets converted to potential energy and then to dissipation because it is a stable system kinetic energy is lost to potential and the flux will decrease with k. So, this is decreasing with k and this is the loss of kinetic energy. So, the example which I like to give is if there is energy supply from the federal government and money supply not energy supply and keeps going down to a smaller and smaller scale to grass root level if there is no corruption then the money should be same, but if there is a corruption then money coming at the bottom will be less and this happens for stability stratified flows. So, as a result in fact, this is a crux of the Bolgionov-Nubakov argument the flux is decreasing with k this is non-linear with equating the terms and its simple analysis flux is decreasing with k minus 4 to 5 and spectrum is minus 11 by 5 it is not 5 third. And so, let us focus only on this regime I will just I will not discuss about some other circle points, but we will ignore that ok. Now, some earlier work I will again skip, but as I said most of the work focused on strong gravity which was not looking at this regime we had to be careful in looking for isotropic regime. So, there is lot of work well involved which was discussed today morning talk. So, let us see whether this Bolgionov-Nubakov theory is correct or not it is a scaling argument may work may not work. So, we looked at this by spectral simulation. So, these are the plots for kinetic energy normalized. So, the two normalization is above is 11 by 5 and bottom is 5 third. So, 11 by 5 is better not very great, but it is better is 1000 cube simulation not the largest one, but it is that is what we got and these are the theta basically temperature or density spectrum we have this is flatter 7 by 5. So, is also prediction of Bolgionov-Nubakov that it will be 7 by 5 k is minus 7 by 5 ok. So, let us look at this one these are dissipation and that is the forcing sorry. So, f k is sorry f k is the forcing supply by the force. So, f k is negative that means kinetic energy is getting converted to potential this loss of kinetic energy ok. So, they are all consistent and Bolgionov-Nubakov from various angles we would say that it works it is satisfactory. Now, let us look at. So, I will. So, now let us look at RBC that is the main topic. So, there is lots of work RBC is a different problem which everybody would say that yes what is the spectrum. So, I will not again discuss this, but one major point is that experiments are difficult because Taylor hypothesis does not work. So, how do I deduce the spectrum from the experiment because there is no mean flow I cannot resort to Taylor hypothesis. So, there has been lot of confusion what is the spectrum. So, let us look at the energetic energy text argument again and see whether we can resort we can get some solution we can get some answer. Now, there is a work by field theory work by Prakashia, Zatak, Lavov and they kind of claimed that if the same equation actually equation is the same for Ralebo-Nard as well as for stable stratified flows then Bolgionov-Nubakov theory should work the spectrum must be minus 11 by 5, but there is a problem in the argument that this energetic argument is not correct there is one place where the f k the sign of f k is different. So, let us look at in detail. So, this is what I showed this picture before what should be the f k for Ralebo-Nard remember it was negative for stable stratified flows because there was a conversion from kinetic energy to potential stable system kinetic energy is lost to potential, but in Ralebo-Nard quite clearly the plumes feed energy to kinetic. So, the f k must be positive f k cannot be negative for Ralebo-Nard we know the thermal temperature feeds kinetic energy we start with just start take a normal fluid start heating it convection starts. So, f k has a different sign for Ralebo-Nard then stable stratified flows. So, the whole phenomenology is not going to work Bolgionov-Nubakov does not work for Ralebo-Nard that is a simple argument from energetic. So, this is the picture earlier we all that the flux was decreasing with k, but here the flux is not decreasing, but it should increase with k because thermal plumes are feeding energy to to kinetic. And then it should kind of flatten because maybe the buoyancy becomes weak at small scales and it should flatten. So, we are looking for this kind of flux profile and what happens to the spectrum? Spectrum should become shallower because the flux is increasing. So, spectrum should become not 5 will basically minus 1.5 or so it should not be steeper it should be shallower than 5.3 in this regime, but 5.3 here. So, we so this started doing the simulation. In fact, we did a very big simulation this 4000 cube simulation with our code Tarang with it is a machine in caust, Sahin 2. We reach Raleigh number quite large in spectral this is the highest 10 per 11. So, I will show you what we get it is there is bit of surprise. So, we are I am expecting something shallower than 5.3 not Bolgionov-Nubakov. So, we find this is a spectrum for 4000 cube it is kind of 2 decayed 5.3 this is 5.3 scale is normalized to 5.3 and the flux is very flat constant. So, I am not getting increase of the flux I was expecting the flux to increase because buoyancy is feeding energy to kinetic and it is quite quite clean 5.3 there is a I mean we have some other model, but I will skip that part. So, this is details are there in a 2014 paper of ours in 2016 this review article new journal of physics which you will you will get more details, but now the question comes why is flux not increasing? Now, we are in a kind of dilemma the flux does not increase for another subtle reason that in. So, we looked at all the terms of momentum equation or the velocity equation we find that buoyancy is this is a buoyancy term. So, these are magnitudes the arrows are magnitudes of various terms of momentum equation it turns out the pressure gradient is much bigger than the buoyancy term several times bigger. So, buoyancy even though they are active they are thermal plumes they are not very large. So, it turns it is feeding energy, but it is not the most significant term it is a pressure gradient which is driving the flow and that is why Raleigh-Bernard has very strong similarity with hydro turbulence 3 D. So, that is why we are getting 5.3. In fact, the pulmograph constant for Raleigh-Bernard is very close to hydro 1.8 or so we got in simulation that is what we get. So, this was kind of work by my another student Ambrish Pandey and we could show this a connection that pressure gradient is the most dominant. In fact, it was very very surprising for us I mean why should pressure gradient be thus most dominant term, but it is for Raleigh-Bernard it is buoyancy is weak compared to pressure gradient. So, it is basically hydro now I will show you some more evidences why we are getting that. Now, before that I like to say about temperature distribution temperature spectrum in Raleigh-Bernard. Now, we do not get a single spectrum, but we get dual lines there are two lines. So, this is E theta temperature fluctuation the top line and bottom line. Now, this is again I mean why why should we get these lines and they are not numerical artifacts. It turns out the top line is 1 over k squared and this line is fluctuation. Now, the reason why 1 over k squared is coming is because of this walls. So, there is a boundary layer. So, the temperature drops flat then drops again and if I look at theta which is the fluctuation it has a linear growth ok. Now, these are bit of mathematics it is easy to get. So, these are mean to total temperature, but theta has this this growth and if I do the Fourier transform I get sorry if I do the Fourier transform I get this and. So, the Fourier transform of the large scale average temperature is this this is and if I square this I get 1 over k squared. Now, I believe it should be also show up in pipe flow or so. The mean flow has a signature in the fluctuation. Now, I do not know about how, but here we really get this feature in all our simulations. Now, let us look at why Kolmogorov spectrum we revisit. So, one argument I said is the pressure gradient is the most dominant one even for a Lebanon. Let us look at anisotropy is the flow strongly anisotropic. Now, these are the numbers e per by 2 e parallel. So, we look at e per by 2 e parallel this anisotropic parameter for isotropic it is 1, but these four different parallel numbers these are runs for different parallel numbers we get numbers from 0.63 to 0.3 and for parallel number 1 it is the least anisotropic or most isotropic 0.73 is close to 1. So, the flow is not strongly anisotropic even though they are plumes, but they basically if I look at statistically they are close to isotropic that is one signature which comes from here. These are a free slip, but for no slip as well we get very similar results. Now, these are called ring spectrum which is a good measure of anisotropy. So, these are angular. So, we just do for different angles we look at the e k and it looks except this region where there is some small stretching up it is isotropic. So, these four relevant are relative for it from number 1. So, it is I mean if I did fluid turbulence other if I did not look at this region I would say this like hydro no issues. Let us look at cell to cell spectrum sorry not cell to cell energy transfers. It is supposed to be local for hydro now let us look at what happens for RBC is it local local and forward. Now, the picture is this is a picture for. So, now if you are not used to this figure. So, giver shells are here and receiver shells are below. So, shell 10 the diagonal will be here 10 to 10 is 0 10 cannot give to itself, but 10 to 11 is positive red is positive is positive and 10 to 9 is negative. So, it is and it is basically banded that means the most energy goes from n to n plus 1 and it is very very similar to hydrodynamic turbulence. So, relevant and hydro are very very close they are not very different. Now, quickly how much time I got at the 10 minutes only. So, real space probes now since there is no mean flow especially if you do experiment in cylinder then how do I apply how do I get spectrum. Now, most experiments there are few PIVs, but mostly it is temperature measurements near the edge. So, these are we are doing for a cube some simulations. So, there we probes at various phases and we look at velocity or temperature mostly temperature and from that we extrapolate to velocity. Now, so we look at the frequency spectrum and related to E k Fourier spectrum. Now, if there is no Taylor hypothesis because there is no mean flow like in that one direction like a wind tunnel your E k suspect because E f we cannot extrapolate to E k. So, we try to see this in a in a cube not in a cylinder actually cylinder of course, we our code does not work for cylinder, but cylinder Srinni has done lot of experiments and also there are other people Wunder Eilerts. So, it turns out the mean flow rotates in cylinder easily, but in in a cube for our Rayleigh number the mean flow does not reverse it just goes on and on along a diagonal. So, that can be treated as a mean flow for applying Taylor hypothesis and then look for spectrum. So, that is what my student Abhishek again he did lot of this simulations with these are no slip boundary condition with open form temperature a velocity vertical velocity feel at one of the probes it is positive fluctuating and opposite diagonal it will be negative. So, this in fact, there is a roll going around in fact, you can see here it is going down like this. So, the spectrum it is highly fluctuating there is not a very high resolved simulation high grade resolved simulation. So, there is a 5 third here you may not believe me, but ok I mean we are claiming it is still 5 third we are trying to do bigger simulation, but possibly E f and E k are 5 third from this simulation. Now, there are works where you try to get spectrum in the middle put the probe in the middle of the box and the spectrum is minus 2 it is not minus 5 third and f minus 2 is purely because there is a mean no mean flow this in Landau's book those no mean flow gives you f minus 2 spectrum. Now, can we relate it to other systems now this argument energetics argument I really like it it is very general. So, it should also work for Rayleigh Taylor because it is a gravity driven flow Rayleigh Taylor is unstable. So, the same argument should work why should I mean it is quite obvious that it is unstable and similar energetics. So, gravity is feeding energy to kinetic energy. So, 5 third should work and there are peoples I mean the lot of work by Buffett's work Stefano. So, in 3 D there are claims that it is minus 5 third there are some peoples of course, there will be peoples which will say minus 11 or 5. So, but I think we should also kind of trust some energetics or some theoretical arguments and I am strong believer that RTI in 3 D should give minus 5 third. Bubbly turbulence it is again buoyancy driven bubbles are shaking the fluid up it should again be minus 5 third. These are some conductures from this theory and I hope it works unstable stratification like salt water above pure water this should again be minus 5 third. So, stable stratification is 11 by 5, but Rayleigh Bernard is or unstable stratification should be minus 5 third. Now, what about 2 D RBC? Our boundary layer. So, this part is not clear to us right now 2 D RBC I can make some again similar energetic arguments. So, 2 dimensions RBC there is inverse cascade. I think lot of people see that there is inverse cascade and it is natural there will be inverse cascade. So, if I look at these 2 regions A and B in the case again this is the way number. So, we should be bigger A or B. So, if I feed energy then energy is coming to large scale. So, small k. So, pi A must be bigger than pi B. So, because I am feeding energy in the middle. So, it will be unlike. So, this 3 D and 2 D will have different RBC behavior. So, pi k bigger and the flux decreases with k modulus mod flux. So, we expect 2 D could have some signature of Bozino Bokov in RBC. And I kind of conducted that may be boundary layer if I take a section that could turbulent boundary layer could show minus 11 by 5 5. So, there could be difference between bulk and the boundary layer spectrum. We do not know I mean that part is not very clear at the moment. So, I conclude. So, stabilized flows for number of Richardson order 1 we get Bozino Bokov. So, Bozino Bokov is the argument is quite nice correct it works we could verify its simulation. So, this part I will skip I did not really say this I did not show it. So, in fact, if our gravity is weak then we get Kolmogorov. So, this now for relevanard Kolmogorov scaling in 3 dimensions and the pressure gradient is much bigger than Bozino Bokov. So, there are lot of signatures why hydrodynamic turbulence and relevanard have lot of similarities. And boundary layers is a different physics which we do not we are not very confident what would happen there and I stop here ok. Thank you.