 So, good morning everybody, I am Manjul and Samin is my thesis advisor and this work we are doing in collaboration with Manikandan Mathur. So, the topic is Lagrangian Coherent Structures resulting from 3 dimensional axial water stress axial vortex breakdown. So, first I will define not define basically I will see the examples of axial vortices. So, they are very common feature of and have a wide range of applications in fluid flows as you see is one of the very common example bath of vortices and when it comes to aerospace applications you see these axial vortices resulting in delta wing, flow pass delta wing and as well as in swirl combustion chamber you see that there will be a axial vortex. So, they have a wide applications. So, I will define what is breakdown in axial vortices. So, it is defined as an abrupt change in the in abrupt change in this core of the axial vortex when we go beyond certain Reynolds numbers which we will define later on. So, based on these structure that they result in they are classified as bubble type breakdown you see a bubble here and a spiral type and as well as the double helix type. So, what happens is stagnation point appears and after that a recirculatory region is there. So, here this recirculatory region is resulting in a bubble shape, here it is a spiral shape and here is a double helix type. So, we study this problem in a closed domain. So, this is the two dimensional schematic of the problem that we are studying. So, this is a closed circular cylinder in which the top lid is being rotated with a rotational speed omega the dimensions are as shown and the relevant parameters of this study are Reynolds number and aspect ratio which is the height to the radius. So, as you see when this omega is beyond certain values you will see this first I will talk about the how the dynamics is here when this plate is rotated what happens is that this fluid at the top which is pushed to the side walls and then it is bent towards the non-rotating stationary plate when it reaches here it is again bent upward at the axis and then you will have a recirculatory region. In three dimensional cases you will have actually a spiraling down flow so this has simplified 2D schematic. So, when we increase the omega beyond certain speed this nice axial vortex actually breaks down something like this which we call at the as the bubble type vortex breakdown. So, you have a stagnation point here you have a stagnation point here actually four stagnation points are here and you say that these are the two vortex breakdown bubbles. So, based on the combinations of Reynolds number and gamma you can have different numbers of breakdown bubbles as we will see along. So, to simulate this problem in three dimensional we are solving the incompressible Navier-Stokes equations in cylindrical polar coordinates in this form. So, you can notice here that for the radial momentum equation the qr is not the conventional radial velocity ur instead we are using a variable r times ur. So, this is basically done to achieve the to avoid the singularity at the axis and the grid that we are using to solve this problem is staggered grid. So, you do not have to solve q theta and qz at the axis so that is how we avoid the singularity at the axis and again the Reynolds number is omega r square by nu. So, these equations they are solved using the fractional step algorithm based on the Orlandian, Versailles and Camus's works. So, this is the main part of the algorithm where you discretize the Navier-Stokes equation and then you solve for some approximate velocity which is with the wrong pressure and then there is a Poisson equation which you solve with that approximate velocity and then you enforce the continuity equation to get actually this equation and then after you solve for the correct velocity and then correct pressure using these relations. So, I am not going to detail of this fractional step algorithm. This is actually the Scudius map which he obtained in 1984 when he conducted extensive experiments on this problem and he presented that the number of breakdowns they lie in this map of Reynolds number versus aspect ratio as such the solid lines are obtained by him. This dashed line it separates the region of steady and unsteady flow regimes. The dots are the cases that we have simulated numerically in this presentation I am talking about the aspect ratio 2.5 and more or like precisely I will confine myself for this Reynolds number. So, you can see in this map that for some aspect ratios and Reynolds number you have only a single breakdown bubble for some you have 2 and for a very small region you have here 3 breakdown bubbles. So, first we look at the Eulerian picture of the flow how the flow looks like. So, when you have aspect ratio 1.75 if you look in this picture 1.75 somewhere lie here 1850. So, you have a single breakdown bubble here which is shown here these are the iso surfaces of 0 axial velocity and these are the contours of axial velocity in a plane and this stream lines actually look like this this is a recirculatory region. Similarly for this combination of parameters you have 2 distinct breakdown bubbles and again the corresponding stream lines are like this. So, then I will talk about this aspect ratio so how the transition happens in aspect ratio as we are increasing the Reynolds number. So, as you have seen previously Reynolds number 2200 it is steady you have 2 distinct bubbles and this is a horizontal slice which shows the contours of axial velocity you see it is perfectly axis symmetric case. As you are increasing the Reynolds number further you see both the breakdown bubbles merge and instead of 4 stagnation points at the axis you have actually 2 stagnation points now when you still the flow is axis symmetric then again you increase the Reynolds number and you see that they merge as in this shape and start moving along the axis still the flow is unsteady, but it is perfectly axis symmetric. When you increase the Reynolds number further 23500 rotating azimuthal wave appears which in the flow which is rotating near the side wall and the wave number we have determined of this wave is to be 5 which we will see in the next slide. So, as we are actually increasing the Reynolds number you will have a non axis symmetry and unsteadiness. So, sorry yeah so that is how these both the cases look like we will not play. So if you see that this breakdown region is actually moving then it disappears then appear again, but the flow is axis symmetric, but in this case you can clearly see an azimuthal wave which is rotating and you can see clearly the wave number is 5. Now we but not for the much detail you can get from this flow so I go to the Lagrangian picture of the flow. So for the Lagrangian picture first I will talk about the motivation for this study. The first motivation comes from a paper Stevens et al. in 1999. They have done the dive visualization of this flow in the unsteady regime for aspect ratio 2.5 and their dive visualization clearly shows some so many rib like structures at the axis. The physical significance of these structures is still not understood very well and not much have been talked about these structures. This is the first motivation that we have and the second motivation come from the Sotheopolis paper in 2001 where he has shown that if you have a small perturbation to the breakdown bubble then actually a fluid can exchange across the breakdown bubble with the surrounding flow. You can see here he has distributed some particles in the breakdown region and then after a long time you will see the busting events and particles will escape the bubble this called the emptying and filling mechanism of the bubble which I am not going in the detail. So to study these two problems this is the topology of the breakdown which we have recreated from the Wekin's book. So in the two dimensional you can see the bubble axisymmetric bubble is actually an invariant surface. So this line actually will not allow you to penetrate the bubbles across the surface. So whatever is inside the bubble will be inside and whatever is outside that cannot go inside it will be just moving around the bubble. So bubble actually act as a bluff body to the flow. In the three dimension also it is a sphere, it is an invariant surface. So just to visualize this thing what we have done we have actually advected some particles inside the breakdown region, advected some particles outside the breakdown region and see what happens to the flow whether they really get exchanged or not. So this is the movie, so if you can see here the green particles which are inside the breakdown bubble they are actually just moving up and down and rotating but they do not cross and similarly the particles which are outside the breakdown bubble they do not penetrate inside the breakdown bubble. So they just move around with the flow no particle actually crosses and then how the topology will be in the perturbed case. So you can see here this is plane one, so this point is actually these are two saddle points and this is the unsteady manifold, this is the unstable manifold and this is the stable manifold. So you can see when the perturbation is present there are some trajectories which might actually escape the bubble through this unsteady manifold of this point which we will show with these two movies, so sorry this should be 3400. So 3400 we have seen this is the axisymmetric case and unsteady so these bubble do not actually cross the surface while in this case they will eventually actually go out of the breakdown bubble and mixing will be there because of the non axisymmetry that is present in the flow. So now we move on to the first motivation of the Lagrangian study that we had. So we want to compute the Lagrangian coherent structures of the flow. So Lagrangian coherent structures as defined by Haller they are the shearing material surfaces which are independent of the reference frame. These are some of the few examples that you find that are given in this paper. Then how do we calculate LCS is first we compute the finite time Lyapunov exponent of the flow. So how it is computed? We have a parcel of fluid which is advecting in time at the later time the fluid parcel becomes something like in this shape. So we construct a matrix which shows the relative separation between two adjacent particles with respect to the at some time t0 plus t with respect to the initial time. So we construct a matrix like this then a Cauchy green matrix is defined something like this here t is the transpose of matrix m. Then we find the Eigen values of this Cauchy green matrix and then the FTLE field FTLE field is defined as the maximum using this maximum Eigen value of this matrix is defined given by this formula. So the FTLE is classified as forward FTLE and the backward FTLE. So forward FTLE is obtained when the particles are advected forward in time and similarly when the particles are advected backward in time it is called backward FTLE. So this is the comparison of this FTLE field with the Eulerian picture that we have. So this is the Reynolds number 3500 you see this is the this plane shows the iso contours of axial velocity and this is the FTLE field with different integration time with respect to the first initial time when the particles are advected. You can see that the FTLE field show a rib like structures which is populated in this axial area and it is the whole region actually is populated which is analogous to the to those pictures that we have seen in the dive visualizations of Stevens et al. And similarly so what is the meaning of these rib like structures so forward FTLE is basically gives you the manifolds which are most unstable that is where actually the maximum stretching of the material is happening. So these are the candidates for the unstable maximum stretching manifolds when we actually can compare the two for the FTLE field for two different Reynolds numbers which we have seen 3400. So this one is without the non rotating azimuthal wave this is with the rotating azimuthal wave but in both the cases we can actually clearly see both the cases they are populated with these rib like structures. So what is the significance of these structure we have to further study this but what we can say now is these are the candidates for the where maximum stretching is actually happening for the material lines these red structures. So with this I want to conclude so we have seen that axisymmetric bubbles are actually invariant surfaces but material cannot exchange through these surfaces and unsteadiness actually breaks that symmetry. So symmetry is actually barriers to the mixing and the finite time Lyapunov exponent actually captures the flow characteristics which are absent in the Eulerian field and we can actually conclude those rib like structures they are quite meaningful where the stretching is happening we further have to conduct the study for that and yeah that is the questions.