 Hi, very good morning to all, Myself Vishnu, I am from India Institute of Technology Madras. It is my pleasure to introduce to you my research topic on direct numerical simulation of lead rotating Rayleigh-Bernard convection. To be much more clear, I will be studying, I will be investigating the effect of rotation shear by rotating the top lid of a Rayleigh-Bernard convection system, basically a thermal convection system. So the motivation for such a study is coming from technological field, one of such field is the combustion in which there is a premixed soil combustor which has a solar which imparts the azimuthal velocity to the air fuel mixture and under certain air flow conditions, there is a vortex breakdown bubble which is formed and such vortex breakdown bubbles are used for aerodynamic stabilization of the flame. So we need to study the dynamics of such vortex breakdown bubbles to understand the stability of such flames, where to stabilize the flame as well as to understand flame flashback because the dynamics of the vortex breakdown bubble can be such that it may move inside and it can cause flame flashback. So such a study is important, so the dynamics of vortex breakdown bubble need to be studied. So we simplified such a problem to a thermal convection or basically a Rayleigh-Bernard convection system in which we have a cylinder in which we heat the bottom and cool the top and we start rotating the top lid. So by doing this so we have the top lid being rotated and the fluid particles which are adjacent to the top lid, they spiral out because of the centrifugal force which is imparted by the top rotating lid and this causes the spiraling of the fluid particles from the center to outside near the side wall and due to the mass conservation flow will happen from the center of the cylinder towards the top lid and thus there is a circulation of the flow and this closes this circle. That means there is a spiraling of fluid from the top to the outside and it goes down and moves towards the inner region and moves towards the center upwards. So the basic thing to be noted is that how big, what's the size of the cylinder that is given by the aspect ratio and how much are we heating that's given by the Rayleigh number, how much are we rotating the top lid that's given by the Reynolds number and what's the fluid which we are concerned with that's given by the Prandtl number. So to introduce my equation we have the continuity in the energy moment and the energy equation. So one of the difference which I want to note is that we use a characteristic velocity because here the velocity is imparted both by the rotation of the lid as well as due to the free fall velocity due to thermal convection. So you see that this is quite different from the other equation because it has the effect of Reynolds number as well as the Rayleigh number part even in the energy equation this is so. So this is a characteristic velocity which I am concerned with. It has both rotation r omega part as well as the free fall velocity due to thermal convection. So the Reynolds number is defined as omega r square by mu, Rayleigh number is the usual definition of Rayleigh number. The way we have this B-star parameter but it's not an independent parameter it can be found out of the other independent parameters such as Rayleigh number, Reynolds number, gamma and Prandtl number. One thing to notice is that the gamma which is the aspect ratio of the cylinder is defined as H by r which is quite different from aspect ratio which is defined for Rayleigh brand connection systems and the Prandtl number which I mentioned earlier. So the numerical scheme which we are talking about is a central second order finite difference scheme in the space. It's in a cylindrical coordinates and time step is by RK3, Rengar-Kutta third order. So you may be thinking what's advantage of such a characteristic length. So if you do such a non-dimensionalization which includes both the free fall velocity as well as the rotation of the cylinder, the advantage is that we can study a range of flow regimes in which on one extreme we have if you put Reynolds number equals 0, we have the Rayleigh brand connection flow and the equations concerned are with the Rayleigh brand connection system. So in the other extreme if you put Rayleigh number equals 0 that means if we shut off the heating and that means there is no buoyancy force, we will obtain an equation for just a lead rotation cylinder. So this such a study, such a flow has been well studied and in 1984 a SCUDI has done experiments on such a flow for different parameters and in this, in this presentation I will call such a flow as a SCUDI flow. So we validated our code, one of the validation which I use is like without any buoyancy force because we find, we didn't find any experiments on lead rotation or rotating RBC. So I shut off the buoyancy force and I could validate with the SCUDI's experiments the stagnation points and our results will match the stagnation points of SCUDI's experiments without any buoyancy force. And I couldn't find any of the literature for lead rotation RBC. So I have to be satisfied with validating the code with numerical results which are done by Lugd and Abode in 1987. And however you must note that this, this is an, this the computation they did is for an axiometric case but our code is a non-axiometric code. So while comparing our personal computation with the Lugd case we find that the bottom stagnation point even though it matches the overall structure of the vortex breakdown bubble is not matching. So with this limited literature I'll proceed to the investigation of the flow. The main, main thing which are concerned in such a flow is how is the heat transferred? So we use a Nusselt number to quantify such a heat transfer. So since I mentioned earlier we use a different scaling and due to this our Nusselt number is being modified as Nusselt number LRRBC that is lead rotation RBC which includes the usual Nusselt number definition for a RBC as well as we have this force convection part. So one thing to note is that if you put Reynolds sample equal to 0 we'll obtain the Nusselt number definition for a Rayleigh-Bernard convection system. So with this knowledge we'll proceed to investigation of the flow and the heat transfer yeah. So what we did is that for a particular Rayleigh number so whatever study I'm going to say is about for a fixed gamma that is a fixed aspect ratio of 2.5 and a fixed Prandtl number of 0.677. So in the top figure we can see that Rayleigh number is fixed as 2 to the power 5 and we start rotating the top plate. So this zero means it's a purely Rayleigh-Bernard convection system and we see that as you start increasing the rotation of the top plate the heating or the heat transfer which is quantified by the Nusselt number or the modified Nusselt number starts increasing. So second by second graph it shows if you fix a Reynolds sample that is if you start rotating without any heating and if you go on heating it is quite obvious that heat transfer increases because we are going on increase in the Rayleigh number. So this thing I will come to the end of my presentation this is what we are mainly concerned with. So we looked into how we say how is the hot fluid being convected due to this slit rotation. So this is the corner of the temperature that actually this slice of the temperature at a vertical plane and we find that these are RBC and if you start increasing the Reynolds number we find that the Nusselt number increases as well as the hot fluid get confined to the top lower part of the domain. So we thought of understanding okay how is a heat mechanism being possible in such a system because the Nusselt number is keeping on increasing as we start rotating. So we thought of identifying plume structures in such a flow to understand what is the mechanism of heat transfer. So what we quantify identify as a plume is be taken from a paper by P. R. Y. Tal in 2016 this criteria these two criteria is based on the fact that a plume is a coherent structure which has a strong correlation between the upward W velocity and the temperature fluctuation. So by using these two criteria we identified points in our domain and we called such points as a plume. So in this figure we can see that like this structures are identified in the flow domain which satisfy these two conditions. So what we can observe is that in a Reynolds number 1000 these are for Raleigh number fixed Raleigh number 2 into 10 power 5 and gamma could 2.5 and Prandtl number 0.677. So what we find that as you keep on increasing the Reynolds number that is the as you keep on increasing the rotation speed of the top plate you find that the plumes size is been decreased. However one thing to remember is that I will remind you is that Nusselt number keeps on increasing as the Reynolds number is increasing that means as you keep on increasing the rotation speed we earlier found that the Nusselt number increases but the plume size is decreasing. So this we can infer that the heat transport by the plumes as you go on increasing the rotation is being reduced. So there is something with transport heat. So we thought of investigating what is that with transport heat at higher rotation. So maybe the flow field might give a clue to that. So we looked into the flow field these are basically the streamlines and the W velocity slices in the vertical slices. So what we found is that it's quite clear that in a Raleigh-Bernard connection system where in we have Reynolds number equals 0 that is we are not rotating the top plate we just have the last scale circulation that means the fluid goes ascended from one side and descent to the other side. So we keep on increasing the rotation of the top plate. So what we found that this last scale circulation has been broken we can see in this Reynolds number equals 100 ks in which the last scale circulation is broken and as you keep on increasing the Reynolds number there is a sole flow which is established in the center core. You can see the sole flow established in the center core. So if you further keep on rotation we find we found that at Reynolds number equals 2200 there is this axial core or axial vortex core is breaks and the vortex breakdown bubble is formed. So here in this figure we identify such a vortex breakdown bubble as a contour of W equals 0 contours. So here we can see that vortex breakdown bubble is formed by breaking of this axial vortex. So one thing is that the last scale circulation has been broken up by rotation. Moreover the flow has been shifted from a convection dominated flow to a rotation dominated flow. So I will introduce some of the concepts for a some of the flow patterns for a rotation system. So basically we have the top plate rotation as I said earlier mentioned due to this top plate rotation there is a azimuthal velocity which has been imparted and such azimuthal velocity causes a azimuthal flow and there is a prime that we call as a primary flow. So however there is a secondary flow also and because of the secondary flow there is a meridional flow which is set up in the system. So basically we have the primary flow and secondary meridional flow. So and there is one more thing you will notice is that as I earlier mentioned we have that Rayleigh-Bernard system with the top plate rotation. So as you keep on rotating top plate the fluid that there is a boundary layer actually in the top top and such a boundary layer is been twisted due to the top plate rotation. So due to this top plate rotation and twisting of the boundary layer there is a Ekman layer which is formed and such a Ekman layer actually due to rotation of the top plate the fluid particles gains the centrifugal force and be like a centrifugal pump they are spiral out from the top plate. So there is a flow set up. So basically there is a Ekman layer over here and we have the thickness of Ekman layer defined root of Ekman number in this flow. Basically the fluid is deflected ready from the top plate to the side walls. Further the Ekman layer feeds something like a vertical layer which is called the Stewardson layer. Actually Stewardson layer is a sandwich structure between these two thicknesses. So we can identify Stewardson layer and the fluid which moves which gets twisted from the top plate get deflected along the side walls towards towards the bottom in the Ekman layer and get deflected in the bottom and moves towards the central core. So this is a whole loop and the Ekman layer and the Stewardson layer forms this complete this total loop. So we can identify this Stewardson layer from this definition which is a sandwich structure between these two thicknesses. So we thought investigating such layers existence of such layers in our flow field and whether it says about how complex is the flow field. So what we did here is that we compared a no buoyancy case that is we don't have the heating with a buoyancy case that is we have a heating as well as a rotation. So this this figure is a no buoyancy case and this is a buoyancy case of vertical velocities and this is a no buoyancy case and this is a literotaching buoyancy case with literotaching literotaching the lid and this is temperature conduce that is the size of temperature. So what is interestingly note here is that we have no fluctuations in these are these dashed lines are the Stewardson layer or here also we can see that Stewardson layer and what you notice that the temperature conduce show fluctuations in near to the Stewardson layer if you start rotating and heating but such fluctuations are not found in near to the Stewardson layer in no buoyancy case. Moreover we can see certain fluctuations which are growing which are clearly seen over here which are not found. So this we also looked into a top view that is we took a section of the cylinder near to the bottom plate around 0.2 if you normalize by 1 it's about 0.216 and we found that with just the rotating the lid that is without any heating we found the flow to be steady and axisymmetric but such a axisymmetric flow turns to non-axisymmetric once you start heating that means once you start heating the top bottom and cooling the top. So moreover we see that there are waves which are generated near to the these are like azimuthal velocities conduce and we can see that the waves are generated near to the side wall. So we care from this we can infer that the side wave side wall waves will break the axisymmetry of the system. So with such a complex flow dynamics we also thought of looking into how is the evolution of the Nusselt number or the modified Nusselt number in time. So what you find is that for a just rotating the top plate without any heating or without any buoyancy force we found that it's a steady value of Nusselt number basically it's a force convection without any thermal natural convection effect. However without any rotation that is the classical Rayleigh Binance system for this Rayleigh number 2 into 10 power 5 shows that the heat transfer oscillates it has a periodic oscillation. However if you start heating and rotating that means for a Rayleigh number equals 2 into 10 power 5 and Reynolds number equals 2200 what you find that there is a turbulent fluctuations which are happening in the bulk Nusselt number. So from this we infer that the complex the advection in such complex flow field causes this turbulent fluctuations in the modified Nusselt number as well as this causes the increased heat transfer and obviously the increased modified Nusselt number. Yes coming to the later part of my presentation it's about parametric study so we thought of looking into what will happen for a litrotaging case just the litrotaging case what is happening if you start heating. So it is quite obvious that we start heating Nusselt number increases. So for Rayleigh number equals to that is only just the forcing part if you go force convection part if you go on increasing the Rayleigh number the Nusselt number increases. So with this whole data we have we tabulated and we found that for a fixed Reynolds number Nusselt number increases and fixed Rayleigh number increasing Reynolds number the Nusselt number increases. This is normalized Nusselt number with respect to Rayleigh equals to 0. So because it's one sorry with this idea we defined a we defined a Rayleigh number effective which has contribution of Reynolds number and Prandtl number and we found that the Nusselt number modified Nusselt number scales with a Rayleigh number expanded near to one third. So with this I conclude that there are instabilities in the severson layer which breaks the axis symmetry of system the plumes the heat transfer by the plumes is reducing as you start rotating. So and the advection in such a complex flow field is responsible for the enhanced heat transfer if you start rotating and finally we found a new scaling low for the litrotating RBC and with an exponent near to one third. Yeah so with this I summarize and set a study. We are yet waiting for experiments to validate our results. Thank you. Thank you for your attention. You mean over here right? Yeah you mean the difference between this and this? Yes. Yes. So it's actually a vortex you can see the vortex core over here. So if you start increasing for the Reynolds number this vortex core actually breaks down to a the vortex the vortex core break downs break down. Yes yes that's what first is it expands further it expands and then it compresses again it expands. So there's a region of W equals 0 foaming or basically if you draw a streamline over here we can see there's a close bubble. So such bubbles are called vortex break down bubbles. No no it's not a green one is a W contour. It's a velocity contour vertical velocity contour. It's not a temperature I'm sorry it's a vertical velocity contour. So I did I just want to show the vortex break down bubble which is generated due to higher rotation. Actually I didn't think about it maybe I think yeah I think the same thing yes yes yes thanks.