 And it is part of the PhD work of my PhD student, Lea Rabnel, and the work done also with my colleague, Emilia Mbaria, and this is done in a context, a larger context of a big project in which also variable density jets are involved and it's a collaboration of his people, Alexandre Mout, Mirelle Amiel, and Fabien Selmet from Irfe, Marseille. So, I'm in one, and there in Marseille it is a collaboration. But now I mainly present mixing and entrainment in variable viscosity jets. As I started to present yesterday, a variable density and viscosity flows are a wide spread in many practical applications. And for instance, one of the application concerns combustion in which both reactants fuel and oxidizer air for the instance, they're mixed together and they create smaller and smaller scales. And at some point when the mixing will be performed then the flame will be stabilized. So it is very important for combustion application to understand how step by step mixing is performed and where mixing will be performed. But for all these things, and there are also other applications, we need for a theory and we should be able to predict what happens in what we call real flow in which viscosity varies, density varies, temperature varies, as is the case here. Some of the previous results we have obtained, for instance, in the free flows, here it was a jet, a specially evolving jet issuing from a pipe flow. And we have observed during the PhD thesis of Benoit Albault that finished in 2009 that the mean velocity field was stopped much earlier in a variable viscosity jet. So we compared an air jet, a classical jet, constant viscosity jet, air jet, with a variable viscosity jet in which we mixed propane issuing for a tube or a pipe with ambient air. And if the mean velocity field in a constant viscosity jet is a classical profile at one diameter, four diameters, and eight diameters, then in a variable viscosity jet at one diameter, this is the first profile, so the velocity is much smaller. At four diameters, this is the profile, so it becomes wider and the maximum is smaller. And at eight diameters is even wider and becomes smaller. So the fluid is simply stopped by the environment, which is, in this case, much more viscous. In the same case, for the same case, we obtained an increased mixing with the ambient fluid, and we also observed lateral fluctuations. There are V fluctuations, RMS of the V fluctuations that were increased in color for a variable viscosity jet with respect to the constant viscosity jet in which lateral fluctuations just start to be born at the first, over the first diameters of the flow. And we proposed this phenomenology, and viscous blobs are enticed into the core of the flow, and this will create obstacles and that will produce lateral fluctuations. So transition towards turbulence is much, appears much earlier in a variable viscosity jet. Other application concern, of course, combustion as I have already mentioned, and it is important to predict the lift of the turbulent jet flames. So for this, we need to understand mixing, and we'll use what we call a mixed infraction that usually called Z. Transported by this transport equation, we speak about an active scalar because this mixing fraction will give us locally the viscosity of the flow, and that will come into play into Navier-Stokes equation. So the two equations, Navier-Stokes and the transport of the scalar will be coupled, and we speak about an active scalar in this case. Here the example was for combustion. So fuel that will mix with an oxidant. Now the experiment that we performed in order to study this effect was to compare a classical jet in which we studied nitrogen mixed with nitrogen. Previously we had air, but for security reasons, they imposed us to put off the oxygen, so use nitrogen jet, so in which, of course, the ratio of the viscosity is equal to one. We have a propane jet, so this application came from combustion questions, and mixed with nitrogen, in which the ratio of the kinematic viscosity is of 3.5. So one, we have said that this low ratio of kinematic viscosity, we should not have maybe observed so much effects, but the effects are here, as you have already observed, and I will further show. The aim is, of course, to understand the model and to predict turbulent mixing in variable viscosity, and this is the outline. I have already done the introduction. I will rapidly present the experimental setup and the measurement we have performed with PIV for the velocity field and planar laser induced fluorescence for the scholar, and I'll go to conditional statistics, so interface determination, the dynamical field, some statistics, and just one statistic for the scholar field, and I'll go to entrainment and detrainment, so the opposite of entrainment for the variable viscosity flows. This is the experimental setup, so here we have a jet in which we inject the propane in the case of the variable viscosity. The diameter here is of three centimeters and it is placed into a coflow, a very low velocity here in which we inject the nitrogen, so it makes propane with the nitrogen. The diameter here is of 80 centimeters and it is an enclosure, and of course, over the first diameters we have windows in order to allow laser diagnostics to be performed. The initial velocity profile is a top hat and we wanted that, so the conversion was designed such as the top hat profile will be achieved here, so we did not want a lot of velocity gradients because we want to observe and to look directly on the effects or come from the viscosity ratio effects. The measurements as I had, we also performed numerical simulations as I rapidly mentioned yesterday, but now the focus is on experiments in order to trace the scalar. We used, as I mentioned, the nitrogen instead of air, but the tracer in this case, after a long list of choices, we used an anisol, so the seeding of the flow is using the molecules of anisol and we send a laser beam and then we visualize the flow as we see. And of course, we perform simultaneous measurements of velocity and concentration because as I have already mentioned, the two are coupled, so we need to understand both of them. This is a schematic of the experimental setup, so these are the windows, window, window, window, et cetera. And we have to measure the velocity field for the PAV, we have two cameras, so we have what we call 2D3C, so three velocity components in a plane. It can be over the axis of the jet here, is the diameter of the jet, over the axis of the jet, or we may go on the different distances if we want. And here is the camera, only one camera that will be used for the planar laser induced fluorescence. And of course, you may have details over that. This is an image of the experimental setup. These are the two cameras for the PAV and this is the one camera for the planar laser induced fluorescence. I'll go now to the conditional statistics and how to determine the interface and how you compare the two flows. So this is an image from the constant viscosity flow nitrogen with nitrogen, the classical Kelvin-Helmholtz vortices, and a right you have with the same initial conditions as I mentioned yesterday, variable viscosity, so it is a propane mixed with nitrogen and we see that the transition towards turbulence is accelerated, so it appears earlier. The viscosity gradients, of course, they are situated at turbulent, non-turbulant interface, so many things happen here, although somewhere at some downstream position we look at the statistics over the jet axis, et cetera, et cetera, but everything starts here over the first diameters where we have strong velocity gradients, and of course, in this case, scalar gradients or viscosity gradients. And many of the physical phenomena that are responsible for the mixing will happen here. So how we do now to determine where the interface is placed. So starting from images, we'll trace here as a function of the threshold, so this is the value of the local mixing fraction. We'll trace the local values of the concentration profile, but it's here, for instance, we make a cut. We have a maximum value of the concentration here and smaller and smaller, so we put the profile here, et cetera, so we make another cut here. Maximum value of the concentration will be our mixing fraction here. It will decrease. We'll put it here from right to left because larger values are here. And we put all of them back. We average, so this is the black curve. This is an average value over the interface, so where the concentration varies between more or less 01 and 0.6, so 0.6 corresponds here. It's almost the pure propane. 01 is almost pure nitrogen, so over the interface. And the point where the two straight lines will fit this profile, so this is a straight line, this is another straight line, and this point corresponds to a concentration more or less equal to 0.2, will correspond to the threshold value at which the interface takes place. This is one criterion that we chose. So once we decided which is the value of the mixing fraction at which the interface takes place, then we'll filter and we're trashing old images. So starting, for instance, from this image, we impose that the Z equal to 0.2, and then all which is larger will be like propane, so it will become white, or everything which is smaller will become nitrogen, let's say, so it will be black. So everything above the threshold and everything below the threshold will become like that. This is the interface, so of course, the next step, we select the gradients of that seven and we trace the interface exactly. And in the next and last step of this procedure, we'll eliminate everything which is placed into the core of the flow and we just keep the external part of the interface. And we look at exactly which is the position Y, at the jet at which this interface is placed. So now we know exactly where the interface is placed between it's like a cliff, but left you have the core jet, and then right you have the ambient or the core flow. Once we have that, we may are able to perform statistics and we perform statistics for the mean velocity, wait for many things, but I only present two statistics here for the mean velocity field, the longitudinal velocity field. But we trace that as a function of the position Y minus the position of the interface and normalize with respect to the mid-width of the jet. So here this point zero corresponds to the interface. So this is the profile for the constant viscosity flow, so nitrogen, nitrogen jet. The red one corresponds to the variable viscosity, so the propane into the nitrogen. And we observe that the interface, the mean velocity such as was calculated becomes negative. So for this Reynolds number, we speak in this case of the reversal flows and I will speak a little bit later about detrainment instead of entrainment. So at this low Reynolds flow, the flow, the Reynolds number, the flow goes in this direction in a variable viscosity in the core of the flow, but it changes the direction. It is a detrainment and the reversal flow into the ambient, the ambient, the two diameters downstream, we observe the same kind of phenomena. So again, here is the interface. The constant viscosity flow profile, which is normally observed, or jump here or change into the derivative of the flow that was already reported by other people. And again, it goes to values, the mean value of the velocity, it goes to negative values into the propane jet, so into the variable viscosity jet. Over at the distances, similar things at the five diameters and further downstream in which the jump over the, so the change in the slope of the velocity field here is even more enhanced and again, negative values in the variable viscosity jet. We also observe when we look at, and we perform a careful analysis of the interface in that in the constant viscosity jet, this is much smoother over the distances we have investigated, but in the variable viscosity, this is much more fractally because it is much more turbulent. And if we perform a mapping of the interface position, so with the sum of all of that, this is much wider or wider at least corresponding to a jet broadening in the case of the variable viscosity flow. So again, it is actually a sign of an improved mixing and an improved opening of the jet. Quite rapidly a result of the scalar field, so the mean value of the concentration of, and we observe that usually, that over other people reported that the interface, so you observe at least a small value of a platoon of the concentration. So you observe that in a constant viscosity field on the left, so on the side of the jet core, but in a variable viscosity we obtain small platoons both at the side of the jet and the other side. And this plateau is even wider when we go at the larger diameters down stream. But higher Reynolds numbers, this was at the 8,000th Reynolds number calculated with the jet diameter, but the higher Reynolds we inject the higher velocity with a higher energy. We see a less important disparity in the behavior of the two jets. So Kelvin-Helmholtz will appear earlier, but the mixing will be again improved. So we see a transition toward turbulence. There are still differences here. We still see in a variable viscosity that this is more turbulent. But, sorry, yeah, I'll finish. It's okay, it's almost done. And we see still differences between the two flows, but the differences will appear when the distance downstream will increase. And I'll go now to the last part with the physical processes, definitions and method, entrainment that is usually characterizing the jets and detrainment that is specifically slow. We also see that in the variable density flows. The entrainment is one of the physical processes by which fluid elements outside the interface, initially irrotational, may acquire the opacity. So they are brought into the movement. And we define usually, there are different methods, but one of the methods is to calculate, to quantify the entrainment is to calculate entrainment coefficient, C E. D is the normalized diameter of the jet in this way. And this is the derivative through the downstream position of the mass flux. So to calculate that, although we obtain the asymptotic values and they are reported values, for instance, from these people, 0.14, in other jets that have been reported. But nothing is, as far as we know, nothing is reported for the variable viscosity jets. There are different methods to calculate this, is to calculate this derivative. So to calculate the mass flux here and mass flux here. And there is also another method. I think there is a slide that is not okay. So it's either we calculate the derivative by calculating M here and here and taking the differences, or I calculated what happens through the interface. So it is important to have the position of the interface. So when we represent now C E, so the entrainment coefficient, as a function of the threshold concentration or threshold mixing factor that we have already selected, that should present a plateau. Again, this is our result for constant viscosity flow for different Reynolds numbers and should present a plateau. And then we obtain that the value that is in agreement with the literature was 0.14 or 18. But for a variable viscosity flow, it's really hard to have a plateau. We have somewhere, but especially at the lowest Reynolds number which was 8,000, this is this curve here. We have a plateau, but this coefficient is negative. So it is not positive as is the case in that. So we do not have an entrainment. We have a detrainment. This is a mass flux that goes away. So when the initial energy of the fluid is not sufficiently high to fight against an environment which is much more viscous or at least more viscous, then the fluid goes back at the edges and we have this phenomenon of detrainment. It was observed in other flows, especially in a variable density. And for instance, these people proposed the term of a banana peel model. So this year we have an injection here and the edges that it comes back, the edges it comes back. But higher Reynolds numbers, of course, it will not be the case because the fluid has enough energy to go further. And so what we see at lowest Reynolds number is a fountain regime, so the fluid goes back. But higher Reynolds number, so when the Reynolds number increases, we have this regime or prompt regime with a detrainment over the edges. And of course at higher Reynolds number, we have an entrainment and a mixing with the ambient. These are the conclusions. So we have reported how to, in a variable viscosity jet, how to determine the interface. We have once again, with respect to my presentation of yesterday, emphasized that entrainment and mixing is, are enhanced and are more important in a variable viscosity, provided the initial energy should be sufficiently high and otherwise we also observe the detrainment coefficient. We are going to perform calculations starting from the first principles for the entrainment coefficient and in order to determine analytically which is the initial energy that is necessary in order to really have a mixing up to or down to some downstream distance. Thank you very much. The confinement is much larger. It was designed to that is 80 centimeters through with respect to three centimeters. And no, I do not believe that. It does not, so we have checked that that's the, there is almost no influence of that. Sorry? What do you want to see? Bioncy, there is, oh, the ratio of the density is very small is 1.15. So that's not, but the same phenomenon if you think of that is was reported in a variable density flows in which the, we have a light fluid that is injected in a heavier fluid. And if the initial energy is not sufficiently high then it comes back. But now this, the viscosity is the problem. So the frictions that the initial fluid will. Just a minute, your attention. Yes. Should be small. Yes. Or very affixed. Yes. And move. Yes. So it is some instant. Yes. Yes. Thank you. Thank you. Thank you.