 Thank you very much for your patience, all of you that are here. I will only take a few minutes, so I'll skip a lot of slides and I'll go directly to the message. So, this is a work performed with my postdoc now, Mikhail Gouding, who was a former PhD student of Norbert Peters, and it is about internal intermittency and finite Reynolds number effect for the passive scalar mixing. So, I'll skip all the importance of turbulence for mixing, the presence of a large number of scales, but we are definitely interested in performing here numerical simulations in a forced homogeneous isotropic turbulence in which the scalar will be injected through a large-scale scalar gradient, and in that we look at the statistical description of turbulent flows and turbulent scalar mixing and turbulent scalar mixing by structure functions. So, I'll skip all the details about the code. The code is incompressible, it solves incompressible Navier-Stokes equations by a pseudo-spectral method. The scalar is transported by this transport equation in which we consider that the Schmitt number is equal to unity, and we have an imposed mean gradient in one direction. Let's say y, so gamma is the mean value of the scalar gradient. This is the range of the Reynolds number we saw for typically six situations of a Reynolds number, so ranging R lambda based on the Taylor microscale ranging between 88 up to 754, and these are all details of the simulations. These are images of the scalar, so in which we see that the large-scale coherent motions we may have here with a strong gradient at the edges, and this is an image of a scalar dissipation in which we see a strong intermittency, so alternates between regions in which things are smoother, and regions in which the dissipation is much more important. The signals show up, such zones in which turbulence relaxes, and another zones in which we have a stronger activity. Of course, this is better illustrated when we look at the scalar gradient here represented, so this is known as the phenomenon of internal intermittency. To describe that, we usually work with structure functions that can be for the velocity field, any order structure functions for the velocity field, so increments between two points of the space separated of a scale are, and of course we have transport equations for these increments and moments, and we'll be particularly interested in increments for the scalar for which we have under conditions of asymptotic results. We have transport equations which relate a second order moment to the mixed third order moment, and of course the mean value of chi, which is the dissipation of the scalar variance. Now when we look at the even order structure functions, and here we look at the n equal to two, four, and six, so second order, fourth order, and sixth order, so this look like that as a function of the normalized scale, but small scales they are analytical, so for the second order moment we perform a Taylor series expansion, so the second order will be proportional to r square, fourth order r4, etc., in the inertia range, so for scale larges, if that is allowed to exist by the Reynolds number, so we have what we call anomalous exponents, these are the values we obtain for this case which is an Erlanda of 500 and something. Now, because we lose all that in the transport equation obtained from the first principles, one of the questions that I started to address two days ago is that of self-similarity or the way we can normalize all the quantities that are involved in the transport equation and in the structure functions themselves, so which is the good normalization of all these things. And usually, as I have shown in my talk on Monday, the Kolmogorov and Kolmogorov or Book of Corsin because we speak about the scalar are the right quantities to normalize the second order structure functions, because remember I started from the transport equation for the second order moment and we have shown for the velocity field that the mean value of epsilon, so the kinetic energy dissipation rate and the viscosity were the good parameters to build the Kolmogorov scale that were adequate. Now for the scalar, for the second order moment, they perfectly normalize these are the compensated second order structure functions for the scalar then they perfectly, they are perfectly normalized with respect to Kolmogorov or Book of Corsin variables, so all these variables. But if we normalize the fourth order structure functions here that we see that are standard arrangement, here this is the result for the velocity field and here for the scalar, so both the velocity and the scalar, the higher order moments do not normalize with respect to the classical Kolmogorov or Book of Corsin variables. So we need to understand that and to look for adequate similarity scales for fourth order, sixth order, etc. In order to explain intermittency and go back to transport equations and adequately normalize that in order to be able to solve that and obtain eventually what happens for the scaling range in the inertial range for instance, it is not the only question of course. So how do we do that, there is that in the dissipative range as I have mentioned there analytically, so we develop in a Taylor series for end order and we write that like that as the end order moments of the scalar gradient will be written as n divided by two moments of the scalar dissipation. So this n divided by two moments of chi will come into play and I'll skip other details, we show that when we normalize by the classical Kolmogorov or Book of Corsin variables then what it comes into play is this yellow rectangle here which involves n divided by two power of chi and mean value of that and renormalize by the mean value of chi to the power n divided by two. Of course this is dimensionless but the ratio as we see will evolve as a function of the Reynolds number. Of course for n equal to two this is equal to one so it simplifies and we go back at the second order moment to the classical result of Kolmogorov or Book of Corsin but at higher order moment we'll still have for instance for four chi square divided by the mean value of chi to the power of two and we show with our numerical simulations that of course at left you have normalization with respect to the classical key OC Kolmogorov or Book of Corsin so we do not have normalization, the curves do not collapse whereas when we use our modified scaling in which we use here the square power of chi instead of the square of the mean value of chi of course we have a much better arrangement and a much better superposition and collapse of the curves. The same results holds for higher order moments and I will finish by going back to the transport equations that can be obtained here we are for the scalar so from the advection diffusion equation this is a divergent term so the transport through the velocity field this is a production term and at the right hand side when we write the transport equation for higher order moments of two n we will obtain a term in which we obtain the classical diffusive transport which was present of course at the level of second order moments as well but this is a dissipative source term in which we have the direct connection correlation between the two point dissipation and delta phi to the power two n minus two so the next order lower of the scalar increments of course if n equal to one then here this is zero and we go back to the classical result in which that we only depend on the mean value of chi but if it is not the case where we have delta phi square for instance for n equal here to two so fourth order moments multiplied by chi this will be a function of r so it will not be a constant and we of course we close our budget for the fourth order moments over the whole range of scale with a non-linear curve here so the transfer of the energy for the fourth order moment which will correspond actually to a transfer of the variance of the variance or energy of the energy and will be non-uniform through the scales however we close our budget and this is the last slide with dissipative source term also collapses when we normalize with our modified scaling function that we have proposed. These are our conclusion we proposed the modified KOC scaling in which the higher order moments of chi will come into play. The result is backed up by the self similarity analysis that I do not develop here because of the limited time but it is exactly the calculations I have developed this Monday for it was for the variable viscosity that can be applied for the passive scalar here and it just has to be developed and the question is now are the two of them the modified and the classical KOC scaling are equivalent for infinitely large Reynolds numbers then it is for now an open question that probably further numerical simulations will help us to solve that as far as we have for now this normalized functions here still increase with the Reynolds number almost at a lambda of 750 that we have so if somewhere there will be a plateau that for now we do not know. Thank you very much. So the message is once again that things are not clear when you look at higher order moments that are really relevant for internal intermittency and understanding how smooth regions in the mixing do cohabitate or live together with other regions in which mixing is much more important it is another way in looking at how mixing is done and how good is the mixing so the second order moment is only the energy average of the scale it is a first order approach thank you very much.