 I'm talking about understanding large-scale inhomogeneous turbulence from a kinetic theory perspective. Actually, this idea has been formed more than 10 years ago, and with my collaborators, Steve Ozak, Ilya Starosovsky and Victor, and I think the idea is pretty interesting, even though we have not made much of progress in the last 10 years, but still a few new results. So first, let me to give some introduction about why we are so much into the large-scale inhomogeneous turbulence problem, and why we need to go beyond the Navier-Stokes-based closed formulation. And I'll give some review of the kinetic theory basics, and then especially about kinetic theory that pertains to finite Knusson number. And then talk about some formulas that is applicable to finite Knusson number without perturbations expansion, and that gives some further insight about the fluid properties in finite Knusson. And then the core of my talk is how that is relevant for understanding homogeneous turbulence. And I'll give some arguments and some results, and then I'll conclude my presentation. So I have worked in industry for a long time, so our most important thing is to address real world engineering problems. And so this is one of the examples that we collaborated with development of the Tesla Model S. And as you know, there are many electric cars now coming in the market, and they don't really have much of the wind tunnel, the traditional ways to design and develop cars. So they rely on much more on simulation tools. So the top picture is the one that initially came up with the design model, and then they solely based on simulation. And eventually the model that came to the market is the one in the lower picture, that has the drag value, which is 25% lower, which is one of the lowest drag cars in the world. And that's very important to speak about the simulation of turbulence, because the flow around it is turbulent. And there's another project we did with Porsche. This is more of an aerodynamics, but this aerodynamic around the car is more of the airflow that goes into the air intake, and then penetrate into behind and try to air cool the brakes. Because this car is highly performing, so the brake when you hit on it has a high temperature. These trace particles are colored by temperature. And I mean, this is a colorful picture, but it is actually a lot of quantitative analysis with it. This is something we did with Jaguar. And that you see the pressure fluctuation with or without the wiper. And actually this actually has a sound, unfortunately I cannot play the sound here. So another interesting project is a NASA ERA. And this is one of the airplane designs they are working on that are both of the most quiet noise. And the two design concepts, the baseline and the modified, you see the pressure fluctuation from the flaps going through the gap. And the modified one has much lower fluctuation just showing the movie. And also they put various brackets to reduce the sound. And unfortunately I can play the sound later. I cannot show the true geometry because it is confidential, but it is covered by another irrelevant picture. But this actually has a sound. You can actually, when you do the simulation, you generate a frequency. Then you can actually play on the loudspeaker and see how that sounds like what you actually hear from the real situation. So what we are talking about, airflow is fully described by Navier-Stokes equation. There's no argument here. However, due to the limitation of many things, so we are only able to simulate large-scale part or the average part of the flow. So we mostly, I mean actually we are always dealing with the Reynolds average, the form of the Navier-Stokes. So when you take the average, everybody knows because of nonlinearity, you get an additional term. That term is called Reynolds stress. And it's defined in such a way that it is actually the second order fluctuation velocity which is measured from the mean. So in order to have a close form of this equation, you need to express that Reynolds stress in terms of mean quantities. And for a long time, over 100 years ago, that Reynolds stress has been approximated by the Boussinesq approximation, which says that Reynolds stress is proportional to the mean rate of strain with a scalar coefficient in front. And why this is conjectured in this way is because it assumes that the eddies, the small eddies, act like molecules. They interact and they do many things so that as a result of the effective microscopic effect is that of eddy viscosity times the gradient of velocity. And eddy viscosity you can also analyze and it is scaled by some representative length scale and time scales. And when you do this quick analysis, it doesn't have to be specific, you will find this effective interaction length scale. And over the larger scale representative length scale, then this shows the ratio of these two lenses is in the order one range, which makes a question, this assumption is valid. I will talk a little further about it. So now an airflow can also be described, as we know, by kinetic equation, Boussinesq equation. And so see how that can connect it back to the Navier-Stokes. So when you look at the Boussinesq equation and the right hand side is representing a collision process. And no matter how detailed the collision process might be, but it must satisfy the fundamental conservation laws. And so when you take the velocity moments of the right hand side, because of no net increase or decrease of mass momentum energy, the right hand side vanishes. So you get this formally without an approximation, a continuity equation. And I'll write once again the continuity equation. And the first moment, because that's a general form, but when you take a specific moment, the first one is nothing but the mass continuity equation. And the velocity is in the continuity equation, represent a fluid velocity, it's nothing but average of the microscopic or the particle velocities. And when you take the velocity moment, then you get the non-continuity equation for the momentum. But then you left with a momentum flux tensor, which is represented by the pi, which is similar in a similar form as the velocity deviation from the mean and the second order average of that. And I want to point out this momentum stress tensor is fully determined when you solve the Boltzmann equation. There's nothing unknown about it. The only thing unknown is at the macroscopic level, this pi needs to be closed in order for a closed macroscopic representation. Now let's also review a little bit about the collision process. A collision process has a basic property that due to the Boltzmann H theorem, it drives the distribution function, whatever kind, towards a more universal equilibrium distribution function with some sort of effective time scale, which relaxes the function to equilibrium. And the equilibrium form has a specific, equilibrium distribution has a specific Gaussian form, which is, with this particular form is called the Maxwell Boltzmann distribution. That is a result of the detailed balance of the collision. And also one interesting thing is it is fully determined by the conserved fluid properties such as density, temperature, and the fluid velocity. So when you have the three quantity associated with the conservation laws, then this equilibrium distribution is fully determined. And anything that measures the full distribution and the difference to the equilibrium is a measure of a deviation from such an equilibrium. And that, or discuss, is related to the Kuhn's number. And if in a condition such that the microscopic interaction lens scale, or the, usually we know as the mean free path, divided by the characteristic or the typical fluid flow in homogeneous lens scale, is a small quantity, then we can introduce a small value on the right hand side representing the collision process is a much faster process. Then we can do a standard perturbation analysis by expanding F around the equilibrium by a small deviation from equilibrium F1 or some sort of a higher order deviation. And that is described systematically by the so called the Chapman-Escok expansion. In this expansion, you can relate the higher order deviation to the lower one via a hierarchy of relations so that in the end, the Nth order deviation from equilibrium is all the way related to the equilibrium distribution function as a high order time and spatial derivatives in this way. And now look back to the momentum stress tensor. Then the average of the stress tensor can be expressed into average based on the equilibrium distribution function or the deviation from the equilibrium distribution function in many terms. And the average is defined, average over N means is that quantity average over the F of N. Once again, we have this hierarchical relationship and then you can have these guys all expressible consecutively into the average with the equilibrium distribution function as the probability weight and in this way. So it's the Nth second order moment involves the equilibrium moments up to N plus 2. Okay. Now let's look at some specific examples. If you assume the distribution function is equilibrium, then this second moment give rise to just a delta function with a proportionality scalar as known as a pressure. And but if you are interested into some deviation from such a fully, fully equilibrium distribution function, but the deviation is small, then you'll find a correction to that in this form. There's nothing but the Navier-Stokes the right hand side. And this form is the typical Newtonian form because the stress is proportional to the rate of strength with the proportionality coefficient as the viscosity. Now if you are not happy with this, you can do a little bit further to look at the second order correction to this Navier-Stokes. You find that there is a well-defined form is this related to so-called the Bernetti effect that the second order deviation from equilibrium is a time, a total time derivative of the rate of, I mean the rate of strain plus all these tensorial kind of a combination where S is the rate of strain and omega is the anti-symmetric combination of velocity gradient, the vorticity tensor. And so later I will come back, this term represent the memory effect and these terms are responsible for secondary flows. And interestingly enough for kinetic theory, if we are using say BGK approximation then if the collision relaxation time is a variation can be ignored, you actually you can write the solution of the Boltzmann equation without expansion. So you will find the solution of the full distribution function is related to a path integral through the path of the equivalent distribution function. It means that the full distribution function, if you have the relaxation time is 0 then immediately assume a full equivalent distribution function. But if tau is not 0 but finite it actually is a integration over the path of the equivalent distribution function with a decaying factor. So this means that if the spatial variation or time variation scale is small or even as small as comparable to tau then you are more deviated from equilibrium. Another interesting thing is that equation actually have exact closed form solution in the microscopic way. Usually microscopic you cannot very rare to find exact closed form from the microscopic representation to microscopic representation. But in a special flow situation in a unidirectional flow for any finite nuisance number you can actually find a closed form equation for that Boltzmann equation. And once again if the mean free path lambda is a mean free path and is 0 or the collision time is 0 that the whole integral over the exponential form actually become unity. So you reduce to the well-known diffusion equation that governs the flow of channel flow and the quad flow and any flow is nothing but just this. And but if you do a little bit analysis say okay I look at a small deviation from the say small lambda then you find that this top one reduced to a telegraph equation and that actually you can take two limits one limit is tau equals 0 that give rise to the standard diffusion equation but if you think this term is actually more important than this first time derivative then it become like a wave equation. And it has a very interesting dispersion property and also interesting diffusion property that in the short time it goes like a ballistic and the long time is a diffusive. And this actually this whole equation this equation can be solved for any time so you actually you find that the random walk like process is exactly has the same solution as a Langevian equation. Now look at that equation for our familiar channel or quad flow situation. And the solution before I show the solution I want to talk about the first of all the solution actually reduced to the known Navier-Stokes solution at a Nusselt number goes to zero but when I had a finite Nusselt number actually solution resulting some finite slip loss on the wall. When you look at the momentum flux tensor it has this form compared to our familiar momentum flux in the usual Navier-Stokes in the unidirectional flow is just a viscosity times the gradient of velocity. And for this form for quad flow because the momentum flux at any location is the same for steady state quad flow that correspond to du dy is a constant. But this one actually shows even for momentum flux is a constant across different location in the quad flow it does not result in a constant velocity gradient. These are the exact analytical solution that a quad flow is constrained in a location plate at minus 1 and plus 1 and velocity at minus 1 and at this location velocity is plus 1. So the dash line is the usual quad solution which is very trivial and but when you I mean actually dash line is a small Nusselt number almost like a quad flow. But when you gradually look at a higher Nusselt number you find this curve actually bends and also it does not end at minus 1 on the wall it actually has a gap and that gap is the slip velocity because the fluid velocity is not there is a jump to the wall velocity and as the Nusselt number goes to infinity actually you can find analytical solution a simple tow to a given slip velocity. Similar analytical solution can be found for channel flow and this is at a small Nusselt number which is very close to a parabolic form. And as you increase Nusselt number you find that there is appearance of slip velocity and also deformation of this parabolic. And later on it becomes like a flat in the middle and there is a huge jump. Why there is a big jump also in the quad flow because there is some artifacts because the solution is assuming a constant mean free path usually the wall has the equilibrium mechanism. So the actually mean free path is never constant when you look at a distance from the wall getting closer. So near the wall the actually mean free path should be smaller so instead of a slip velocity you should see a sharp rise of velocity and then become more of a flat. And similarly it's like this that reminds us a lot about a fully turbulent channel flow profile so flat in the middle but then there is a sharp rise near the boundary. So that's so much about the review of the basic kinetic theory. Now let's look at what do we think about turbulence model with this picture that we can assume that now we select the number of fluid elements that have a specific velocity value but this velocity value is a real fluid velocity value. And now we can introduce a distribution function that measures how many, what is the fraction of fluid with this particular value located in the neighborhood of this location at a time t. And this black arrow represent an average, you know like a large scale average velocity as we know as the like average velocity. So because say if we are dealing with incompressible flow then when you integrate over all possible velocity value obviously the total amount of particles should be one. And now if you take the average of this velocity you get this bigger velocity as the like a Reynolds average velocity. And also you can measure the velocity difference between the velocity and the average squared that give you the RMS velocity. And with this process you actually you can write down the effective kinetic equation. And because the choice of the action term is exactly expressed. And we can also introduce a forcing term why we needed this force term like a non-local interaction because we want to make sure that the fluid flow will recover the desired pressure. And for incompressible flow situation we can determine this body force related to the pressure gradient through this very well known relation. And that's, so these two terms are not mixed serious. Now this force also will constrain the fluid flow is always flow in the incompressible manifold. Let me show it again. This is the kinetic equation describing the large scale motion of the fluid. All right now what is the collision process now? Now the collision process can be studied if we want in the future to really find out what is exactly. But now for our main current purpose is we can assume a BGK model. BGK model means the collision is described as the true distribution function to deviation from equilibrium with a characteristic relaxation time. And so all in here is how to determine this relaxation time and as well as what is the right equilibrium distribution function for turbulent flow, large scale turbulent flow. But then we also know there is a well known observation that for homogeneous turbulence the one point PDF of homogeneous turbulence is Gaussian. So you can find in many lecture notes. So the distribution function is e to the u squared over the RMS velocity squared. And if we translate that into a Galilean transformation to a reference frame of a mean flow then this immediately turning into this form and this is just a normalization factor. All right and this form is actually strikingly similar to the Balsme distribution function form and but there are some very important difference. First of all this was inside the exponential function this URMS squared is not the usual thermal temperature. All right and in practice this URMS squared is the fluctuation of the turbulent flow. Usually it is smaller than the mean flow squared and of course the mean flow squared in our low Mach number simulation is much much less than the thermal temperature. So if you look at this relationship the mean flow if this is an effective turbulent temperature that corresponds to this flow is like a particles but at a very very low sound speed or very very low temperature. However because we have a force that constraints it to incompressible flow manifold but it still have a very low temperature behavior or low sound speed behavior. Now look at further this collision operator if we think of this collision time which is once again to be further studied there are some relevant time skills associated with this large-scale turbulent time. One is kinetic energy over epsilon the inverse of rate strain inverse of vorticity and so on and that there are combinations. The turbulent kinetic energy is nothing but the URMS squared average and then if you plug these quantities into the estimation for the effective turbulent mean free path indeed you will find it is an order unity. And so what the kinetic process tells us is that the kinetic formulation represent any interaction as a relaxation to local Gaussian and the average turbulent flow is a flow of a large nozzle number and how many minutes sorry. So we can do some further about the constraints of the collision operator but when so once you have that if you have the only thing that you need to determine is the collision time when you have the collision time is determined everything is solved and as a consequence even though this is not need to be explicitly solved is you will have a resulting mean flow equation and the turbulent kinetic equation equation for turbulent kinetic energy but they are not the same as our typical known you know the Reynolds closer closure model but with these many many finite nuisance order terms and one interesting thing is I want to if we just assume the relaxation time has this is related to k over epsilon then the in the long wavelength limit the indeed the the turbulent Reynolds stress is related to the mean rate of strain and with the added viscosity defining exactly in this way is a capsule model but but the further more if you look at us the next order it actually also fully determine the form with these fully determine the coefficient of theta is actually just the k the turbulent kinetic energy with a constant of like 3 over 2 or something and you can show that it actually will predict this secondary flow where the standard one doesn't and then you can compare to some well-known closure models that included the the second order corrections and you can compare the coefficients and these these are naturally obtained but with the symmetry you have a zero for one particular streamwise or spanwise directions another interesting thing is and this is the rapid distortion that if you have a homogeneous flow but a suddenly have a shear in post and that the experimental show is actually is a continued decay before it arises and that the standard capsule model is given by this purple line and that the blue dots is the RMG based capsule model and if you use the equivalent kinetic expression that you'll find because I said the memory effect it the the red line is the is the model based on standard capsule and the blue line is the RMG one so I have this summary so the conventional turbulence closure is based on added viscosity and the effective turbulent connoisseur number is order unity and the high order correction in turbulence model closure have some issues and they is more desirable from a kinetic theoretical representation and once the Navier stokes is adequate for regular fluid flow but its average appear like a micro nano flow and that's basically my takeaway point thank you