 So we see if it works. Thank you. Okay. I'll set. Okay. So I'll introduce our next speaker. So it's Dr. Manita Denaliuk. And she's going to talk on similarity of anisotropic variable viscosity close. Thank you very much for the introduction. I thank you very much the organizers for allowing me to present this work. So this work is being performed in collaboration with my postdoc now Mikhail Gouding. He's a former PhD student of Professor Norbert Peters who was present at the last edition of the meeting here. And we started to collaborate on this subject and we established an European proposal. And suddenly he passed away one year after that. So two years ago. So I dedicate a part of this presentation and part of this work to the memory of Professor Norbert Peters. The other people who contributed are my students. So this is a part of a larger question that is an appraisal and prediction of variable viscosity flows. Which theory you will see that if things look relatively simple for classical flows, where things are much more complicated for variable viscosity flows. This is the outline of the presentation. So I'll start with the importance of variable viscosity flows with only two short examples. I'll continue with the morphology and phenomenology. Two examples from our previous studies performed in my group. And especially I'll look at the round jet, specially evolving round jet experimentally studied and a temporal mixing layer that is numerically studied. And then I'll go to the core of the subject, which is a self similarity. And I'll mainly address here the question with the tool of the two point statistics. And for the pedagogical reasons I will first present some results in a classical flow. So with the constant viscosity and then I'll go to the variable viscosity flows. And some results in this partially evolving variable viscosity jet. And then I will conclude. The importance of the flow stands in the fact that many practical applications are connected with variable viscosity. And I should say that a lot of attention is being according to variable density flows. But much less importance is according to variable viscosity flows. So to understand which is the effect of variations of the viscosity will concentrate on flows in which the density is matched. And only viscosity gradients are present or viscosity variations are present. So here you have three examples. This is a fluid that comes in an ambient fluid at the same viscosity. So this is a classical jet you see with the classical structures that are developed. The second example is the same fluid that issues in an environment which is 20 times more viscous. So both of them are density matched. So all this structure modification of the structures are due to variations of the viscosity. So you see that the morphology of the flow is very, very different with this mushroom like structures. And the last image is the same fluid that comes in the environment which is 40 times more viscous. And the morphology of the fluid is even more different with kind of pancakes structures and even regions of the center of the jet which are completely unmixed. Another example that I chose for today comes from geophysical applications and these are fountains of magma on the bottom of the ocean and which are the questions that are interesting in this and the shape of the mixing of the edges of that and the distance at which this impinging jet will be able to go. I'll go to some examples that were studied in my group for a few years now. So these are images obtained with a planar laser induced fluorescence and at the left you have a constant viscosity jet in which we mix here nitrogen with nitrogen. And at right you have a propane which issues in an environment which is of nitrogen. So I should state that the nitrogen is 3.5 times more viscous than the propane. So it is the example here is a variable viscosity jet with the ratio of the viscosity of an ambient fluid through the viscosity of the core jet equal to 3.5. So it is not very large. Of course earlier we had ratio of the viscosity that are much more important. However we will see here differences. The initial condition is the same. Here we see the classical Kelvin Helmholtz vortices and at right we have a hint of this large scale that will appear much earlier. But especially we see a mixing of scales that are distributed over a much wider range and the transition towards turbulence is much enhanced and it appears much earlier. The physical interpretation that we offer of that is that because of three kinds of instabilities that are born at the edges that are instabilities that are born because of the viscosity variation stratification instabilities that are associated to the jet. So Kelvin Helmholtz and sometimes depending on the shape of the injector we have instabilities at the wake that is created here. So because of that we have islands of viscous fluid that are enticed into the jet core and this recreates stagnation point that will stop the mean velocity. So the decay of the mean velocity will be much faster. The fluid will stop because it goes into something on an environment which is much more viscous and lateral fluctuations were born. So the transition towards turbulence will be accelerated will appear earlier and the mixing will increase. These are images also obtained in our specially evolving jet with planar laser induced fluorescence. At the left we have the mean scalar field in a constant viscosity flow. So it's nitrogen with nitrogen and in a variable viscosity flow propane with nitrogen. And at right we have the RMS root mean square so typical fluctuations of the scalar field constant viscosity and variable viscosity. So what we see for both of them at a larger potential core in a constant viscosity field so the flow will go much further and here it is a potential core that is much more reduced. And also the maximum of the fluctuations that appear here that appear much earlier very early in the very near field then here in the constant viscosity that will appear later. So once again we have turbulence that will be born much earlier. Another example is a temporarily mixing layer that is studied using DNS. So I chose not to give a lot of details here but you have two streams one rapid and one slower with a variable viscosity. So and the rapid stream is less viscous than that which is slower. So what we see that evolution through the time of different quantities here is the evolution of the mixing layer thickness. Defined like that when the ratio of the viscosity increases so between the two streams this mixing layer thickness evolves much rapidly. So the mixing layer opens much more and mixing is enhanced and also all fluctuations are born earlier. This is the profile of the mean velocity in our temporal evolving jets. So this is the rapid stream at the low viscosity stream. This is the slow stream at a higher viscosity stream. And we see that when the ratio of the viscosity here results go up to a ratio of viscosity of 9 but we had the simulations at a higher ratio of the viscosity. So we see that of course the rapid stream is slowed down because of the world viscous world fluid world that is situated in this region. Whereas the slow one is of course accelerated and much more differences we see when we represent the derivative of the mean velocity that is represented here. We looked in order to quantify which are the terms responsible of that. We started from an aviastox equation and we derived the transport equation for the mean velocity field in here in a temporal mixing layer. And of course these are the classical terms and all these four terms are specific to the variable viscosity because we have viscosity derivatives, viscosity fluctuations, viscosity fluctuations and derivatives of that etc. The most important terms that appear of course is specific to the variable viscosity term is this in red. You see that it is the most important and this is down to the fact that the rapid stream is at a lower viscosity so we are in a counter gradient configuration. So this will contribute to the fact that the absolute value of the mean velocity will evolve much faster in time so the evolution will be enhanced. If we go now we look also for the temporal mixing layer to the turbulent kinetic energy and here these are the squares of the three velocity components and again we represent that as a function of time. This curve corresponds to the ratio of viscosity of one and it increases up to I have here up to nine and we see that the turbulent kinetic energy increases so this validates and corresponds perfectly to what we have seen in the jet earlier that turbulence is enhanced and it appears much earlier. In order to correlate that to the transport the first principles and the transport equation that we derived that from the Navier-Stokes equation and we derived the transport equation for the total kinetic energy. We see classical terms that appear. This is the production term that's still the most important as a magnitude here but in which the derivative of the mean velocity is very enhanced because of the reason I have presented earlier. The profile of the mean velocity is modified because of the viscose wall that is situated nearby. And we have a supplementary terms of course the second and the third line that correspond and are specific to the variable viscosity terms. Here we have a production term is positive here that corresponds to the fact that blobs of fluids which are viscose that will correspond to a lower kinetic energy. The production of that will be a result of the viscosity gradients and this is an extra dissipation so it's a negative term here that will be due also to the viscosity gradients. So we have shown once again that turbulence fluctuations are born faster, mixing is increased and the next question we are interested in in order to solve all these equations and to have a prediction of how mixing is evolving is do we have self-similarity? As you know if I come back to a round jet it is one of the flows that are self-similar. The Reynolds number whatever is calculated are lambda or etcetera stays constant along the axis of the jet. Is that the case again in a variable viscosity jet or variable viscosity flows in a general and we'll do that by looking at the two point statistics. And for pedagogical reasons I'll just introduce some things for a constant viscosity and of course I'll go immediately to the variable viscosity. This is the image of the cascade as introduced by Richard so on etcetera and Kolmogorov said that if the Reynolds number is sufficiently high then all these small scales should be locally isotropic and universal. And one way to understand what happens at the scale is to consider increments of differences of here u but it can be any other turbulent flow a turbulent field between two points of the space separated by a distance r so this is an increment or a scale. And we can calculate moments of second order moments, third order moments etcetera. The second order moment correspond to the energy at the scale r and all smaller scales and of course they are correlated they correspond to the spectra. The third order moment correspond to the energy flux at the scale r. And now Kolmogorov said enough he said several things in K41 that for K in 1941 says the theory known as K41 that for sufficiently high Reynolds numbers he introduced the similarity hypothesis and this is the first similarity hypothesis that said that at the end order moment of velocity increments when they are appropriately normalized that should be universal functions of a scale r appropriately normalized with respect to the scale that is known as Kolmogorov scale and defined by that by this expression. Based he derived that based on a phenomenological basis and there is also the second similarity hypothesis that I will not take care in this presentation that deals with the scales much larger or that's situated in the inertial range if the Reynolds number allows that to exist. So I concentrate the following of my presentation on two issues related to the first similarity hypothesis which are the functions f can that be invariant in when different parameters of the flow varies space time Reynolds etcetera. So when I change one parameter of the flow if we normalize appropriately the functions do they stay similar so do we have self similarity. I recall that that is very important because when we go back to the equations having solutions self similar solution and self similar transport equation it is much easier to be tackled analytically and or numerically etcetera. And which is the expression of the similarity scale because Kolmogorov provided the expression but that was based on a phenomenological basis. Another thing that Kolmogorov did in 41 in another paper is he derived his equation so from Nagia stock so from the first principles this is known as the fourth fifth low so the third order moment are related to the second order moment and this is the energy transferred at any scale. If the Reynolds number is sufficiently high and if we placed ourselves in the inertial range then this reduces to the third order moment equal to four fifth mean energy dissipation rate time the scale or what we have seen for instance with this result published in 2006 is that if we consider the maximum of these functions and we normalize appropriately so by epsilon r so we compare that with fourth fifth. We need Reynolds number as high as 1000s here for fourth turbulence and as high as 10000s here for decaying turbulence so very high Reynolds numbers in order to have this constant for fifth to be to be to be valid. So for most of the flows these are experimental points that we had a bad time for most of the flows in laboratory or real flows etcetera the atmospheric boundary layer should be somewhere somewhere here are affected by the finite Reynolds number effects. And in order to solve this we went back to the first principles and we really arrived with Kolmogorov equation or for the transport for the scholar etcetera etcetera but by consider what we call the large scale terms or forcing terms here. That if we only represent these terms as a function of the scale r these terms are not important at small scales but they are more and more important at intermediate scales and larger scales. So large scales these are these forcing term that is associated actually to the shear or to decay or if we are look at that for the scholar for the mean temperature gradient etcetera. So it's everything that is flow specific that will appear and will be represented in this forcing term. So now we have the good tools part of K41 was really writes and we have the good tools in order to apply that to real flows and the finite Reynolds number flows. And if I advance now to go to the similarity the first similarity hypothesis that concerns scales in the dissipative range and inertia range if the Reynolds number allows for that to exist. So by working on this transport equation but completed by forcing at a large scale we may apply the self similarity analysis. The self similarity say once again if we appropriately normalize all that that these terms should stay similar so should be the same when completely normalized. So one form of that is the equilibrium similarity as proposed by George 92 for spectra and by our group later on for structure functions etcetera. And mathematically that said that for instance for the second order structure functions that should be written as a shape function that depends on the scale normalized with respect to a characteristic length scale. And a coefficient in front of that that will depend on any parameter P that I put it in in red that can be the space the time but it can be also the energy injected in the flow of for instance the Reynolds number. So when the Reynolds number increases that this term may stay self similar if we normalize with respect to adequate scales. Same thing for the third order moments and etcetera all terms that will appear in the equations we are interested in. Now this is a self similarity that comes in to play self preservation as was introduced by Townsend in 56. And he understood by that a particular case of self similarity when this parameter is only the space for instance X for specially evolving jet or it can be the time. So it's when the when the flow stays self preserving that following an evolution in space or time. Now I will substitute this functional forms in the scale by scale transport so this transport equations here we choose as a normalization basis the dissipative term here. So we multiply by whatever will appear here the scale L divided by the square of the characteristic velocity and divided by the viscosity so this will become a constant. And we impose that each term this is the equilibrium similarity that each terms should stay constant because one of the terms is constant. And all the other coefficient should should stay constant during this transformation. Convangore was speaking about a fine transformation of the flow. So independently on the language we use the state that when we apply modification of the flow that all these things should stay self similar. So we should have real real coefficient in front of that. So this third order time that's that represents the energy transfer. It will lead to the Reynolds number built with the characteristic length scale and with the characteristic velocity scale and with the of course with the viscosity. Now I'm in the case with the constant viscosity that should be constant during the transformation. And this term will lead to a relationship between epsilon that is here. It is the energy transfer at the level of the smallest scales divided by the viscosity and multiplied by the square of the scale and divided again by the square of the velocity that should be constant. This is a system these two terms a systems of two equations with two unknowns the velocity and the scale. If we solve that then the solution is the Kolmogorov scale. So this comes from the first principles and from the transport equation here Navier's talks. Moreover if we impose that the characteristic velocity should be the variance for instance. So that is characteristic of the larger scale. So characteristic length scales can be identified with the Taylor micro scale. And you know the definition so it can also be a solution. But the natural solution that emerges is the Kolmogorov solution. So if we speak about a complete self-similarity or complete self-preservation that signifies that all scales will stay self-similar or self-preserving during the transformation that all these scales are proportional. So eta Kolmogorov scale and lambda the Taylor micro scale will be proportional. All the other terms the forcing term will lead to other constraints that will lead us to the conclusion on how the characteristic scale evolves through the scales. And of course this was validated by many studies but maybe the most representative is that of the spectra normalized with respect to the Kolmogorov variables that do collapse over the dissipative range and scales or wave numbers here smaller and smaller. In conformity with the first similarity hypothesis. Now this was for constant viscosity flows but our issues is for variable viscosity flows and is self-similarity valid or not. So we derived the details appear in all these papers and two of these papers. We derived the transport equation for the total kinetic energy. This is a summation over i so this is the total kinetic energy at a scale r. So this is the time derivative of the total derivative. This is the diffusive term. This is the energy transfer to the scale and note that this form of the equation takes into account anisotropy because this is a divergence term and here we'll have the Laplacian term here. We are not yet developed as a function as in a spherical coordinates for instance which is the case when anisotropy is valid. So this is a very general form. We have divergence and Laplacian and etc. And anisotropic flows can be tackled with these tools. So this is a production term etc. So all these blue terms are classical terms but in which we have considered viscosity variations. So for instance here note that for the diffusive term we have the viscosity that is coupled with the total kinetic energy at the scale, the bed scale. And all these red terms are specific to the variable viscosity flows because we have gradients of the viscosity and gradients for instance of the viscosity increments. So as soon as our flow is the constant viscosity all these terms go to zero and all these terms reduce finally to the form that we had for classical flows. One of the aims of us is to close this term because we have many, this is the classical term that is unclosed but this is another term that is unclosed because we have the correlation between the viscosity and the total kinetic energy. And the higher order terms will have other terms that will be, will have to be closed. So if we succeed in closing all that then we will be able to predict the kinetic energy so how mixing is performed at each stage of the flow. Now if I simplify a little bit the things and I consider the central region of a round jet in which local homogeneity I'm still in an anisotropic context is considered. So I neglect some of the terms so this is a little simplified form of the equation. We have here a transport, a turbulent diffusion, a production, production or destruction by viscosity gradients and the dissipation. Note that the dissipation is defined by mean values of the viscosity times the square of the velocity gradients. So for that we apply our equilibrium similarity or self similarity analysis and we impose that all these functions should be written as a product between shape functions. Of psi which is the scale normalized with respect to a characteristic length scale and coefficients in front of that and we reapply the exercise. Then we have supplementary difficulties because we have this new term that is the total kinetic energy coupled with the viscosity of the two points. And this can be written as a reference viscosity at that position times a function that we can consider. And the new term that is very specific to our variable viscosity flow is that the decay over x of the viscosity. Because we go in an environment which is more and more viscous but it can be less and less viscous but at variable viscosity. And of course the decay over x of the total kinetic energy because in our jet that the kinetic energy decays. So that will be written as the derivative over x of the reference viscosity times the derivative of the kinetic energy and of the function times of the shape function here. And of course we do the same thing for the viscosity. So a little bit of attention to this mixed term viscosity total kinetic energy that can be decomposed in a mean value of a reference viscosity. And in the following at least for now we only consider this mean value but we can also have a contributions from the fluctuating value of the reference viscosity. So if we replace all that and we do the same kind of exercises previously for the constant viscosity flow. Terms are the same except the viscosity was replaced by the reference viscosity because the viscosity varies. And we have this new term which is specific to our flow because we have the decay of the reference viscosity and the x derivative of the kinetic energy. So it can be shown that again if we look specifically at the very small scales the Kolmogorov scale naturally occurs. But the Kolmogorov scale that will be defined now with the variable viscosity epsilon defined in a variable viscosity flow. And the viscosity that is a reference viscosity. So for now I have only considered that that position in the flow the mean value of the viscosity. It can be shown of course that if I look at larger scales at the Taylor micro scale can be also derived as a characteristic scale. In which again I should take into account the reference viscosity in our flow and variable viscosity dissipation as a reference that was considered. If the flow is completely self-preserving then all scales are proportional of course but for now we do not have a clear evidence for that. But we have a better insight on these slides and if we suppose power loss now for the mean velocity for the characteristic velocity. Typical fluctuation of the velocity and for the reference viscosity. And we consider that the mean velocity will decay at x to the minus nu. The typical velocity will decay with x to the minus nu. So this is a capital U and the reference viscosity will vary at x to n nu. Let's say n nu positive if we go in a more viscous environment. So if we replace all that in our systems of equations and again with a particular attention to this term here. Which depends on the derivative of the viscosity. Then and we combine all these we obtain different constraints. And if we work on them that this is a very important result that we can cover different scenario. The first of one is with if n nu is equal to zero. So this corresponds to the fact that the viscosity does not vary. We are in the constant viscosity case that of course and U equal to one. So the mean velocity will decay as x to the minus one which is a classical result in a constant viscosity jet. But if n nu different from zero. So if we have any variations non-negligible variation of the reference viscosity which was for us the mean value of the viscosity. Then if this coefficient this exponent is different from zero then and you cannot be equal to zero. So we do not have a conservation of the jet momentum. So all the classical things that are known for the jet are modified. Moreover and I did not I do not develop here but if we work also apply the same analysis to the transport equation of the scalar. So the mixing fraction that will produce the mixing in our in our jet. And if we come back to our result in the Navier stocks that we can show that the Reynolds number calculated the characteristic velocity characteristic scale. And the reference viscosity cannot be constant during the decay. So as soon as we have the viscosity variations then the product between the velocity and the characteristic scale can be kept constant. And if the viscosity is constant which is the case of the constant viscosity jet then the Reynolds number will be constant will be kept constant. But if the viscosity varies this Reynolds number cannot be kept constant. So we conclude at this stage that self similarity or self preservation cannot be achieved in a variable viscosity jet even if the viscosity increases or decreases. Now because we pointed out this pessimistic conclusion so the variable viscosity jet are not easy to be tackled the difficulties are much more important that in the classical jets. So life was not meant to be easy as one Australian prime minister used to say. Now we try to test a little bit and to and to look at the different functions. So this is our correlation between the viscosity and the second order moment here. And if we test that at the different diameters downstream these are results from a PIV performed in our specially evolving jet. But we see that self similarity at least for these things holds on the axis of the jet in which the mixing is relatively well performed. But if we go on the shear layer and the borders of the jet then these functions are not similar they do not have the same shape. Now an important point was the closure of this term because remember I used the reference viscosity that came from how we wrote this term here. And I decomposed in a mean value times the correlation between the fluctuating viscosity and the kinetic energy. And we tried to see if the mean viscosity is sufficient to close all that and so we looked at the shape of this mixed structure function and the second order structure functions of the shapes are different at least at large scales here both in the central region of the jet or the axis of the jet and also in the shear layer where the difference is more important. So the mean viscosity is not sufficient to reliably close this term. So we went more we looked at fluctuations of the viscosity and here we consider the RMS of the viscosity times the second order structure functions as a closure. And this closure is much more so the ratio of these two functions is nearly constant over a range of scales that is more important. So this closure which is not perfect but is much better than the mean value of the viscosity is supported by experimental data and the reference viscosity is even more complex than the mean viscosity. So the analysis can be performed again if we went back to our analysis but now the reference viscosity will be this fluctuation of RMS of the viscosity. So this signifies even more that self similarity is not likely to occur in variable viscosity flows with large scale viscosity gradients because it is required that the mean value of the viscosity should be locally constant over the scales that are relevant for the flow of that position but also that the RMS so next order fluctuations of the viscosity should be locally constant that for the similarity to be valid over that region of the space. So as a conclusion of that is that self similarity is much harder to be achieved in variable viscosity flows. Now if I look at just a rapid look at higher order moments transport. So this is the transport equation for the fourth order moments in which we have this we have the now the divergence of the term but we have the fourth order moments here production term that will be specific of the flow. Again the decay of the viscosity times the decay of the fourth order moments here that can be decomposed etc. But the dissipation of the mean energy dissipation rate does not appear as such. We have the correlation between the second order structure functions and the dissipation. So things are much more complicated. We need a closure here for this term and for this case I will have a talk on Wednesday afternoon and I will develop that for a passive scalar and a little bit for an active scalar. K41 so even the Kolmogorov scale is not does not have the classical expression because it cannot be written as a function of epsilon. It can it should be written as a function of epsilon square here because this term will be developed at the square of epsilon divided by the adequate scales. So things are even much more complex in in variable viscosity. The last result that we performed and this is a work in progress or everything is working progress but we perform a DNS for homogeneous isotropic turbulence with variable viscosity. So now we went we want to go to a little bit simpler flows. So the flow is homogeneous isotropic turbulence and we introduce the viscosity variations of the scales that are much smaller and we look at the correlation between the dissipation velocity gradients and the viscosity. And when I skip the details of the simulations I'll just these are images. These are low viscosity in blue regions and high viscosity here and during the transition the mixture the mixture field will be wrinkled and it will become convoluted by vorticity. So we have a mixing during the time in flow which is homogeneous isotropic turbulence. And in this flow I recall that homogeneous as decaying homogeneous isotropic turbulence is not self-preserving because the Reynolds number decays simply. So very small scales that can be locally et cetera but the Reynolds number decays. So this is basically a flow that is not self-preserving and we show that of course variable viscosity will not be self-preserving. So this when we have results of all that I will stop here. These are the structure functions we have obtained during the decay. Well we show that in some way Taylor's postulate is confirmed because the ratio of the viscosity stays that does not influence on the shape of the functions when we appropriately normalize all that. But during the decay the shape of that evolves so we do not have a self similarity we do not have self-preservation in a variable viscosity flow. So as a conclusion I have shown that in flows with viscosity gradients or viscosity gradients we go faster towards turbulence we mix faster. So things look to be simpler and is better for applications like combustion and the questions from combustion people actually were the origin of this flow of this study. But on the other way for things that are more fundamental et cetera classical theories are little useful. So we have to redefine and to go back to the first principles and to redefine epsilon to redefine the production correlation between the viscosity et cetera and especially if initial flow of the constant viscosity field is self-preserving as is the case for a round jet especially volume round jet then the variable viscosity flow cannot be self-preserving or not completely self-preserving. It can be over distances over the space but not over large distances because air lambda will evolve will go in some environment which is more and more viscous. And other flows in which the basic flow is not self-preserving but invariable viscosity will not be self-preserving again. Thank you very much.