 So, first of all, thank you to organizers for inviting me. And I'm here to present also my collaborator, Vladimir Saveliev, who had talked here the last time for the renormalization of Navier-Stock's equation. I spoke also about the atomization process. And here, today, I'm coming also to the very old problem, which is turbine cascade of energy in Lyon, in our laboratory in the central of Lyon. In December, we organized the conference, European conference, which is turbine cascade 2, again coming back to the very old problem of turbine cascade. And here, I will speak about the phenomenological model of a stationary turbine cascade, but in the framework of scaling symmetry approach. Ah, yeah, it's not working. So no need to state it obvious that in the turbines, in a very high Reynolds number, the turbine cascade idea is the key idea. And the turbine cascade is that in 3D incompressible flow, the energy is carried from large to smaller scales. And done at smaller scales, it is dissipated due to the viscosity. And there is a zone of intermediate scales in which the viscosity is supposed to be negligible. And the only physical parameter in the classical turbine cascade is the energy flux. If the energy flux don't work to smaller length scales. Since the viscosity is discarded, you can write the continuity equation in the space of turbine length scale R, and where here is the energy density, and here is the flux of energy. And V is a typical velocity increment at the scale R. So done as a classical cascade, classical theory supposes the small scales statistically stationary and with the local equilibrium hypothesis. You can find that the flux energy is the constant. It is usually denoted as epsilon. And from the simple dimensional reasoning, you can come up to the famous minus 1 third low, minus 3 fifths lows in the space of k of the wave numbers, which is called k o 41 low. And the constant here is principally unknown. And it is impossible to find from dimensional arguments. So measurement gives this constant approximately 2. There is another way which could be proposed. It is to formulate the energy flux as a functional of the energy density. Here is the energy decay rate. And here is the energy density. And the kernel is here. And you see in such a formulation, all scales participating to contribute to the flux at a certain scale R. And this kernel is unknown. But imagine that the cascade contributes to so multiple events that we can assume that this dimensionless kernel depends on the dimensionless variable R to R prime only. That means imagine that we could introduce the scaling symmetry in this collective contribution of all scales. And then the continuity equation is this. And so we will work with this equation. But the question is how to derive such kind of equation? How to derive it by a phenomenologically way? That means from the scheme of Richardson and not from the Navier-Stokes equation. But if we will have such kind of equation, if we will have the explicit expression of the flux of the energy, then we could find the auto-similar solution. And we can define the beer flux or integral of motion which will drive this collective distribution to the given scale R in the flux of the energy. And then we will look at a very interesting question. What is the behavior of the Turbine system before the Kalmogorov-Ovokov scaling law? So-called what are the intermediate asymptotics as precursor of the Kalmogorov scaling law? And this is the scaling symmetry approach. So it was applied in somehow in many areas of physics, for example, in the physics of high-energy elementary particles in phase transition in fluids, in magnets, in polymer solutions. So there is a system, a highly disorder system, and going towards smaller scales, this system exhibits some universality. And this universality is insensitive to the details of micro-dynamics. And the micro-dynamics is absorbed by the integral of motion. And the question is how to find these intermediate asymptotics. So we will speak, we will go through the mechanism of fragmentation. So the fragmentation is just a production of random fragments or particles by the continuous breakup of clusters. So you see there is a breakup, and each step of breakup gives a new population. And this time, we have the kinetic evolution towards smaller scales, where alpha is in between 0 and 1. And within this fragmentation process, there is a special fragmentation process proposed by Kolmogorov also in 1941, in the same year when he proposed his famous turbine theory. And Kolmogorov said that let's consider parent particles, which splits into dot particles with the distribution of alpha, independent of the parent particle size. And he described this process at the constant fragmentation frequency. We will consider the breakup process with the frequency, which is the power function of the size of parent fragments. And here, new 0 is the dimensional constant. So now, first of all, let me formulate the energy cascade in the framework of fragmentation, but also in the framework of scaling symmetry. So the kinetic energy is handed down to the smaller scales at random. And this population of smaller scales, the density of torbent energy is evolving in time. So here, you have the density of specific energy in the element dr on time t. And the energy flux is going towards smaller scales. And r is going to alpha r, alpha in between 0 and 1. Alpha is random with a certain partition function. And the energy energy is represented as a scaling transformation. So we compressed r, and we compressed the function of r. And you see in such a formulation, this energy is just a random part of energy carried from parent eddy as a scaling transformation. So the frequency of fragmentation mimics the torbent energy decay rate. And there is an accumulation of 0 size eddies, which mimics the effect of viscous dissipation. And epsilon is the energy flux at r goes to 0. So there is a energy flux, which is a function of time, which is function of r. But at the r goes to 0, you have the viscous dissipation. Now, we can construct the population-like equation just here. Here you have the energy coming from parent particles transmitted per unit time in the element alpha r, alpha plus d alpha r. And then you integrate over all alpha with the partition function here. And this term is the production term, or the gain. And this is the loss term. And the balance of two gives the evolution of the energy in dr, energy density in dr. And the partition function, or the spectrum of fragmentation, is normalized. And so then you get rid from dr. And you come into the integral fragmentation equation, or evolution kinetic equation, for the specific energy. And this is, in fact, a well-known equation done by Philip in 1961 for the population balance equation. We just have rewritten it for the energy density. But if you integrate this equation over r, it will be the constant. d over dt will be 0 equal to 0. And you can interpret it as a fact that in viscous flows, the mechanical energy of liquid is conserved. And we are not quite happy with this equation, because we have no energy flux here. And let me spend one minute to express the motivation to rewrite this equation, or speaking scientifically to renormalize this equation. And the motivation is its follows. So the fragmentation equation in integral form, it is the ensemble of jumping particles. And each particle jumps, and it has no velocity in r space. There is no explicit expression for the energy flux. And the difference between two terms, between gain and loss terms, may be very big, according to this partition function. And we don't know how to extract one term from another in exact way. And so let us assume that we somehow can rewrite this equation by the continuity type equation. And done, done. This is the evolution of finite number of quasi-particles breaking in r space, details irrelevant, because this is a smooth. Solution will be the smooth function. And there is a explicit expression. There will be the explicit expression. And then we can get self-similar solution. And each particle, each quasi-particle will move in r space smoothly at the effective velocity like that. And you see it depends on all other particles, on the position in r space, all other particles. And that means that we introduce the non-locality. And when we have r equal to 0, it calls in fragmentation theory, condensation of 0 quasi-particles. It will mimic viscous dissipation. And the problem is to find such an equation which will give the equivalent statistic from both approaches. And in order not to be boring, I will just show few steps how to get such an equation without any details. But I'm open to speak about details, mathematical details. So the first step, it is to reproduce the scaling transformation by the development of the operator acting on the function this. And it is nothing than the Taylor expansion of exponential function. So that's why we rewrite the scaling transformation by the scaling transformation operator here. And then the production term in the integral fragmentation equation may be rewritten by the exponential scaling transformation operator. It gives nothing. The second step is to use the work of Savilev and Nambu, which is devoted to redorabilization of the Boltzmann equation, and which we used for renormalization of the Navier-Stokes equation. In this work, they was shown the exact form of the scaling transformation operator, exact form. And then we introduced this exact form in the production term in the integral fragmentation equation. It is the production form. Here the exact form here with the integration with the partition function here. And then you see the flux of energy is appearing here. And Q1 is the secondary partition function, which is expressed by the primary repartition function. Here you have the first moment of the logarithm of alpha. And the secondary partition function is renormalized. And we can come in to the renormalization of the fragmentation equation. This is the continuity form of the fragmentation equation. And you have the expression for the flux of the energy. But it doesn't help to show this because you have unknown partition function, unknown secondary partition function. And you should still presume the partition function. And we presumed here the partition function as a power function in order to get a similar solution. So if you presumed such a repartition function, then the first moment of logarithm alpha is calculated exactly. And if you put here alpha power gamma, beta power gamma, and you compute this integral, you will obtain that the secondary partition function will be equal to the primary partition function. So finally, you come to the fragmentation equation in renormalized form, or in the form of the continuity equation in the space of r with the flux of the energy. Here we used the alpha gamma is exponential logarithm alpha gamma logarithm alpha. And in order to put the power here, here. So we must solve this equation to find analytical solution of this. And there is useful mathematical identity with such a presumed repartition function. For example, put here the q alpha, and you have the useful mathematical identity is here. And it will help us to obtain the autosimilar solution of the continuity equation for the energy density. For that, we propose the self-similar transformation since the frequency of fragmentation is proportional to r power mu. Then this time power 1 over mu is the characteristic scale for self-similar transformation. We propose this self-similar transformation. And we apply this self-similar transformation to the fragmentation equation. Like this example, the energy density is this. Then you calculate the derivatives of the product, and you come to two terms. And finally, you have the equation for the autosimilar solution. And we are interested in the fixed-point solution. So we are interested in how the turbulence is coming to Kolmogorov-Obekov scaling law. This is a fixed-point solution. So that's why in order to have the equation for the fixed-point solution, we assumed here the stationarity. And we integrate this equation. So we have the integrated equation here equal to constant. And you see we obtained the new parameter. It's the beer flux or integral of motion, which is denoted here like J0. And we will, after, call it epsilon 0. And this parameter may appear during the evolution when the system exhibits autosimilar behavior. And then using our mathematical identity here, just multiplying this expression on the left and on the right, you will have a very simple equation and with the known solution. And if you put here the power function of r, you obtain this solution in the class of a confluent hypergeometrical function. So at Dan, you use inverse self-similar transformation and you obtain finally the solution expressed in the class of hypergeometrical confluent function. And the question, how to use this solution? If it is known that confluent hypergeometrical function can be expressed by elementary function, if the parameter a is integer. And if it is integer, then you can rewrite this solution in the class of elementary function and to analyze what's going on. So in this a, we will choose from the physics of turbine cascade. And so up to now, it was found in our article in 2012. And now, how to find from the physics of turbine cascade the value of a? Another message here is that the solution is characterized by two-dimensional constant beer flux and the constant of nu 0 in the low fragmentation frequency. So from dimensional arguments, the energy density is v square over r. The flux of energy is r square over t cubed. Then let me propose that the constant nu 0 is also can be expressed in terms of beer flux. And then from the dimensional analysis, we can find mu and x. x is 1 over 3 and mu is minus 2 third. And a is equal from the solution 1 plus gamma over mu. Then if gamma is 1, a is 3 in order to be integer. And we can rewrite the final solution in the class of elementary function for the energy density and for the flux of the energy density and here in the framework. And if we use the translation towards very large time, then we will come exactly to the stationary solution, which is the Kalmogorov Obokov law. But with the constant coming from this solution, you put here this transition and you will get 2 as a constant. And now we compare with the Kalmogorov Obokov 41. This standard definition of the second order velocity structure function, we put in this definition of the second order of the velocity structure function, our long time exact solution with the constant 2. We obtain this, then we integrate to obtain the structure function. We integrate and we obtain exactly C2 equal to. And I'm finishing with the interpretation of the result. So here you see the graphics of our solutions. You see that energy density involves continuously from minus infinity to plus infinity towards ko 41. And the energy flux evolves to its equilibrium value on very large time. And it is interesting to see in logarithmic coordinates. So on negative times, the most energy is accumulated at small scales. That means that the turbulence, and this is provocation, we state that the following those results, that the onset of turbulence is on the small scales. The gradient of the velocity is small scales. Acceleration is small scales. And here you see small scales. The energy density is infinity. Done, done. And the energy flux is also infinity. And then at time 0, there is a second order of phase transition. The energy density exhibits 1 over r. The flux of energy jumps from infinity to the final value. And on positive times, we go to Kolmogorov over ko 41 low. Analytical solution for the structure function. And this time, we go to r power minus 2 third. We also have done some direct numerical simulation in the inertial zone. And direct numerical simulation gives very, very similar result with the theory for the energy density function for the flux. For the flux. And for the decay, it is not very good because there is no viscosity in our theory. And I conclude that, in fact, what we did, we applied the renormalization group analysis, which was applied in many areas of physics. We applied to the turbulence, not on the basis of the Navier-Stokes equation, but like a scheme, larger scales goes to the smaller and smaller. But requesting that there is an energy transfer and the energy conservation. And we proposed the continuity equation for the specific energy in the space of turbulence scales. We obtained the exact or the auto-similar simulation in the case of decaying turbulence. And we analyzed the behavior of the energy density and of the flux of energy. As the intermediate asymptotics, as the universal behavior in the precursor state and the final state described by the Kolmogorov-Obakov power law. And the formal scenario is what I said, that at negative times there is the accumulation of turbulence on small scales. At time 0, there is like a second-order phase transition. And energy flux jumps discontinuity from infinity to finite value. And on positive time, the distribution approach to the long-time limit distribution, and which is Kolmogorov law, but in our solution, the constant appears from the analytical solution as the limit of large time. So thank you.