Hello and welcome to lecture number 4 of this lecture series on turbo machinery aerodynamics. We have been talking about the fundamentals of axial flow compressors and in the last couple of lectures, we have had some discussion on the various aspects of the fundamental aerodynamics involving axial flow compressors. We have also had lot of discussion on the various nomenclature and terminologies that are used in axial flow compressor analysis and subsequently design of axial flow compressors. So, we have been discussing about the various aspects that are involved in design as well as the nomenclature associated with the fundamental axial flow compressors. In the last lecture for example, we were talking about what are known as cascades and the aerodynamics associated with cascades. Cascades if you recall are very simplified versions of axial flow compressor geometries wherein one can get very lot more details in terms of the measurements that can be carried out on an axial flow compressor blade and it also gives us the fundamentals of understanding of the aerodynamics of flow in axial flow compressors. So, cascade analysis plays a very significant role in terms of the understanding of the two dimensional geometry of an axial flow compressor blade. So, before one undertakes a very detailed design and development of the complex 3D geometries which are involved in modern day axial compressors, a cascade analysis and cascade experiments would give us a great insight into the performance as well as the various characteristics associated and the flow physics associated with the flow in an axial flow compressor. Although in a very simplified format because the actual flow in an axial flow compressor is highly three dimensional which is something which is not captured in a cascade analysis. Nevertheless, a cascade analysis does give us a lot of insight into what is going to happen in design of a particular geometry of an axial compressor blade. So, what we are going to do today is to continue discussion on cascades for some time and subsequently discuss about losses which are involved on a two dimensional scale. So, the 2D losses associated with a compressor is what we will discuss in lot detail in today's lecture. So, today's lecture is going to be about the following topics. We will start with performance parameters which are associated with cascades and subsequently we will spend lot of time discussing about the two dimensional losses associated with an axial compressor stage. So, before we take our discussion to the loss different loss parameters involved, let us discuss about the performance parameters which one can expect from a cascade analysis. So, basically there are two distinct performance parameters that one can derive from a cascade analysis. Although one may take a variety of measurements in a cascade there are basically two parameters that one would be interested in that is something that the designer would be very much interested in knowing as to what these two parameters are. And these parameters are what is the total pressure loss that is involved in a particular cascade geometry and the second is the static pressure coefficient that is the pressure rise on the blade surface. So, loss the first parameter being the loss it gives us an indication of what are the kind of losses that one can expect from a particular cascade geometry and which obviously also will give us some hint towards what the efficiency of such a compressor would is likely to be. The second parameter being static pressure coefficient static pressure coefficient is measured on the blade surface and this therefore, is an indication of the blade loading. So, the amount of load that the blade is designed to be is something that can be derived from this static pressure coefficient which basically involves measurement of static pressure on the blade surface. And so, these are the two parameters that we are going to discuss in detail for some time today subsequently we will of course, be discussing about the various losses involved. So, let us first talk about the total pressure loss coefficient. Now, in a cascade of course, various measurements that one can carry out in a cascade can vary from velocities the different components of velocities then the total pressures the static pressures as well as the flow angles. So, from these measurements that one carries out in a cascade one of the parameters that we can derive is the total pressure loss coefficient. This basically is an indication of the loss in total pressure that one is likely to encounter in a particular cascade. And so, total pressure loss coefficient is defined as p 0 1 minus p 0 2 divided by half rho b 1 square. So, here p 0 1 refers to the total pressure at the inlet of the cascade p 0 2 is the total pressure loss at the exit or at the trailing edge. And the denominator refers to the dynamic pressure at the inlet where rho is the density and v 1 refers to the velocity at the inlet of the cascade. And total pressure loss is or in fact in general losses are usually expressed and in expressed in the form of the symbol that is omega. Omega with a bar at the top means an average total pressure loss. So, omega bar refers to the average total pressure loss at the trailing edge of a particular cascade and p l c the subscript refers to pressure loss coefficient. Now, as you can see here pressure loss basically depends upon of course, the difference in total pressure between the inlet of the cascade and the exit of the cascade. And with reference to the dynamic pressure that is available for the cascade. Now, in an axial compressor rotor for example, you would not encounter a total pressure loss in the absolute frame of reference because there is energy added in the compressor. And therefore, that manifests itself in the form of a rise in total pressure, but there is a loss in total pressure in the relative frame of reference which of course, we will discuss in lot more detail in later lectures. So, the total pressure loss coefficient that you have just seen here is the difference in the total pressure loss between the inlet and the exit of the cascade with reference to the dynamic pressure. Now, this total pressure loss is very sensitive to the angle at which the flow enters the cascade or the incidence angle as we have discussed in the last class. So, this total pressure loss is highly sensitive to the incidence angle and we will see how it varies as the incidence angle changes. So, that is something we will discuss in little more detail later that as the incidence angle increases there is also a tremendous increase in the total pressure loss. And beyond a certain point the total pressure loss increases substantially to such an extent that one can qualify the cascade to have stalled. So, at very high angles of incidence for example, if I am assuming that you have seen C L versus alpha curve for an airfoil that is as you keep increasing the angle of attack of an airfoil the lift coefficient increases, but that increases only up to a certain point after which the lift coefficient drops drastically and this is the point which is referred to as a stall of an airfoil. So, very similar thing happens in the case of a cascade also because the compressor blades are basically aerodynamic bodies and are airfoil shapes, which means in a two dimensional cells they behave like an airfoil. So, there is a certain stalling angle which in this case the angle of attack refers to the incidence angle. So, at a certain angle of incidence the blade can stall leading to substantial increase in total pressure. So, total pressure loss is one of the parameters that one would be interested and what is the other parameter? The other parameter that we can infer from the cascade study is the static pressure coefficient which is denoted by C P C subscript P and C P refers to the static pressure coefficient of a cascade and C P is measured or calculated as P local minus P reference divided by half rho v 1 square, where P local is the blade surface static pressure that is at each individual point on the blade surface we take a static pressure there that refers to that is basically the local static pressure and P reference is the reference static pressure usually this is measured at the cascade inlet. From a cascade study one would normally express the C P distribution in the form of C P versus non dimensional axial length which is x by c that is where c is the chord. So, C P versus x by c is the C P distribution and C P distribution in some sense integration of the C P distribution is basically the loading of the blade. So, C P distribution can give us an indication of the loading of a compressor blade which is why C P also plays a very significant role for a designer because C P is indicating the load that is the loading associated with a compressor blade and therefore, a designer has a lot to learn from the C P distribution that is obtained from cascade studies. So, from a typical cascade analysis these are the two fundamental parameters that one can expect one is the total pressure loss coefficient and the other is the static pressure on the blade surface or the C P distribution. And so how we say that one can express these performance parameters and what is it that one can gain from looking at the magnitude of these parameters. Let us take a look at the total pressure distribution on a cascade. So, if you look at the total pressure distribution so I have what I have plotted here are two different graphs. One is the total pressure loss coefficient or omega and the second is the deflection angle that is at the trailing edge and so these are the two different parameters that we are looking at and on the x axis we have position along the cascade that is along the cascade axis and there are two distinct positions which have been marked here one both of these correspond to location of the blade trailing edge. So, as you traverse or move a probe from one end of a cascade to the other end the probe would basically see different trailing edges of different blades. If you are measuring total pressure from the total pressure probe there are certain points where there is a very drastic rise in the total pressure rise total pressure loss coefficient you can see here that there are two distinct peaks which you can see which correspond to very substantial increase in total pressure loss. Elsewhere the losses are close to 0. So, these two points correspond to the trailing edge of two different blades. So, there are two blades here trailing edge of these blades have those regions where there is a substantial amount of viscous dissipation and losses and therefore, there is a total pressure loss occurring in occurring at these locations and the total pressure loss peak at these locations and that is what is indicated here in terms of increased total pressure loss at two distinct trailing edge locations. Now, the other parameter that I have plotted here is the deflection angle. Deflection angle is basically the angle at the trailing edge as you measure at the trailing edge very similar to the incidence angle which is measured at the leaning edge. We also have a deflection angle which is measured at the trailing edge. So, as you approach the trailing edge one can obviously see a substantial increase in the deflection. So, what you see here is that other than the trailing edge of the blades at mid span locations or mid passage locations the losses are very low. There is hardly any total pressure loss at those locations which is expected because in the mid span mid passage location the flow is close to potential flow. There is the no viscous losses taking place there and. So, there is no reason for any total pressure loss occurring in these regions whereas, close to the trailing edge is where the viscous effects dominate and one would see certain amount of total pressure loss in the trailing edge regions of different blades and which is what is indicated by this increased total pressure loss just around the trailing edge of these blades. Now, I mentioned if you recall a few minutes ago that as you increase the incidence angle beyond a certain incidence the blade tends to stall and what is meant by stalling is I assume that you have already seen airfoil stall. That is at certain angle of attacks airfoil the flow separation begins at the on the suction surface of an airfoil for positive incidences. And as you keep increasing the incidence angle the flow region or the extent of flow separation increases to such a point that beyond a certain point the flow separation engulfs the entire suction surface leading to what is known as stalling of the blade. We are likely to see a very similar thing happening even in a cascade as you change the incidence angle. As the incidence angle increases beyond certain level the cascade also experiences what are known what is known as blade stall. So, let us take a look at one example of a cascade which is operating under two different modes or conditions. The first one is the normal mode of operation where there is hardly any total pressure loss taking place and the flow is very much aligned to the blade surface. As you can see here the velocity vector is in the direction of the blade angle at the leading edge. And therefore, the flow is well behaved there is no flow separation taking place, but as you increase the angle of attack or the incidence angle from the suction surface since it is a positive incidence that is taking place here. There is flow separation taking place from the suction surface of the airfoil. And this is what is referred to as a stalled or separated operation of the compressor of the cascade. So, for as you keep increasing the angle of incidence one is likely to encounter separated flow from the suction surface for positive incidence and for negative incidence one might see separation from the pressure surface of the cascade. Now, I also mentioned that total pressure loss is extremely sensitive to the incidence angle. So, I have now plotted total pressure loss versus incidence angle that is how total pressure loss is a function or how does it depend upon the incidence angle. So, as you change incidence angle one can see that total pressure loss changes drastically. And that is more true for negative for positive incidence angle that is total pressure loss increases drastically for increased positive incidence angle around 0 degrees and a few negative incidence angles total pressure loss is more or less constant beyond which again total pressure loss increases. Now, as a thumb rule what people have kind of observed is that as you keep increasing the incidence angle. And if you notice that the total pressure loss has doubled then the angle at which the total pressure loss has doubled is kind of taken as the point at which the cascade has stalled. So, double the total pressure loss as compared to the lowest total pressure loss indicates the angle of incidence for which the blade has stalled. Of course, this is a an empirical kind of a thumb rule which people have observed by carrying out experiments over several cascade blades that this kind of a thumb rule has been arrived at. So, let me quickly recap what I have been talking about in the last few minutes. We are discussing about the performance parameters associated with cascades and how is it that we can use cascade data in terms of the data that is obtained from a cascade, how is it that that we can make use of the cascade data. Now, in the next part of this lecture that I am that we shall be discussing we will be talking about losses in a compressor blade we will be focusing our discussion on the 2 D losses associated with the compressor blade. And we will of course, be discussing more details about the 3 D losses in a separate lecture in today's lecture we will basically be talking about the 2 dimensional losses in a compressor. So, before I discuss about the 2 D losses let me also discuss about losses in general in a compressor blade. Now, there are different ways of classifying losses in a compressor in an axial compressor one set of losses is referred to as viscous losses like for example, the total pressure loss which I discussed with reference to a cascade is a kind of a viscous loss. There are whole lot of losses associated with the 3 dimensional flow these are known as 3 D losses or 3 D effect losses like tip leakage flows and secondary flows etcetera. In a transonic compressor one may also encounter shock losses and there are also losses associated with mixing taking place of the mixing of the shear layer taking place at the trailing edge of the blade and suffered to as the mixing losses. So, it is very important for a designer to understand the various types of losses and the origin of these losses because it is basically an indication of the performance of the blade itself and estimating the amount of losses that particular blade is likely to incur is a very important aspect of the whole design process itself. So, we will today be discussing about the losses which are associated with the 2 dimensional flow in a 2 D sense basically that we can classify losses also in terms of 2 D losses as well as 3 D losses like what we have discussed now and we will see that there are certain types of losses which we can is which we can empirically calculate by assuming certain simplifications on the geometry as well as the flow and arrive at certain empirical correlations for estimating these losses. So, we will be discussing about some of these losses in today's lecture and we will also discuss detailed loss estimation associated with the 3 dimensional flows like the tip leakage flows and secondary flows in a separate lecture. So, estimating these losses as I mentioned is very crucial basically it is primarily in the sense that if we have to design a blade which has lower losses or design loss control mechanism estimate estimation of these losses is very crucial. And the main difficulty associated with estimation of these losses is the fact that from the measurements that one carries out in experiments or even in computations isolating these losses into different components it is often a very difficult task as to how we can distinguish between these different losses. And many many are times when we take measurements or even from computational data what we get is an estimate of a combination of different losses and not individual losses itself. So, for a compressor if you were to talk about the total losses taking place across a compressor blade then it is some total of all these different forms of losses like the viscous loss or 3D losses or shock losses or the mixing losses some total of all these losses put together gives us an indication of the total losses taking place in a compressor. So, let us talk about these losses in little more detail we will first talk about viscous losses. Now, viscous losses can be attributed to at least 3 different sources one is the profile loss which is on account of the profile of the nature of the airfoil itself that is basically the viscous effect on the blade surface. Annulus loss is basically attributed to the growth of boundary layer along the axis of the compressor especially for a multi stage axial compressor it can be quite significant. End wall losses basically refer to the boundary layer effects in the corner or the junction between the blade surface and the casing of the hub. The other set of losses the 3D effects as I mentioned one could have secondary flows which we will discuss in detail later on flow through secondary flow basically occurs as flow passes through a curved blade passage. Tip leakage flow is basically referring to the flow from the pressure surface to the suction surface at the blade tip basically true for rotors or stators which are hung from the casing. So, these are two distinct forms of losses besides of course, there are shock losses and mach and the mixing losses which we will discuss towards the end of this lecture. And so viscous loss is what we will be discussing in terms of the 2D losses that are associated with compressor blades 3D losses will be discussed in detail in a separate lecture. So, before I discuss about 2D losses in general what we will do now is to relate the losses in a thermodynamic sense to the entropy rise across a rotor because if you recall from fundamental thermodynamics any loss generating mechanism would obviously, lead to an increase in entropy. So, we can in principle relate entropy rise across a compressor to the losses that are taking place across the compressor itself. So, that is what we are going to do and try and relate the entropy to the losses across a compressor. So, the loss that we see toward from our measurements or computational analysis basically manifests itself in the form of stagnation pressure loss or entropy increase. So, as we know entropy change or entropy increase is related to the ratio of total pressures. So, delta S by gas constant R is related to the total pressure ratio which is minus log P 0 2 by P 0 1 which is also expressed in terms of the total pressure loss that is minus log of 1 minus delta P naught that is stagnation pressure loss divided by P 0 1. So, this right hand side let us expand that in an infinite series. So, delta S by R is also equal to delta P naught by P 0 1 plus 1 by 2 delta P naught by P 0 1 whole square plus the infinite series. Now, if you neglect the higher order terms because the square and higher order terms of delta P 0 is negligible what we see is that the entropy change delta S by R can be related to the loss delta P naught divided by P 0 1. Now, we have already defined total pressure loss coefficient in our cascade analysis that was defined as delta P naught divided by half rho V 1 square. Therefore, the total pressure loss omega is delta S by R into P 0 1 by half rho V 1 square or delta S by R is omega rho V 1 square by 2 into P 0 1. So, what we can see is that there is a direct correlation between the entropy rise and the total pressure loss and they are directly proportional as we had expected that entropy rise across the or the total pressure loss that is incurred in an axial compressor or in any turbo machine for that matter is related to the change in entropy across that particular compressor blade. So, this is trying to relate the losses across a compressor in a thermodynamic sense to the entropy rise across this compressor blade. And so, in general if we were to isolate all these different components of losses the profile loss, the 3 D losses, the shock losses and the mixing losses and so on. The net loss across a compressor or a turbo machine is a sum total of all these different individual components of losses. So, the overall loss in a turbo machine can be summarized as the sum of all these different components of losses the profile loss, the shock losses which is true only for a transonic machine if the flow is supersonic. And the secondary flow loss or and the tip leakage flow loss these are of course, 3 D losses and the end wall losses. Of course, this is the mixing loss has been added in the profile loss as we will see little later mixing loss is usually added to the profile loss itself. And we are going to estimate mixing losses with sum total of mixing and profile losses empirically by calculating or by estimating the mixing and the profile loss together. So, that is what we are going to do next in terms of estimating the losses across a compressor in a 2 D sense by adding up the profile loss and the mixing loss. So, we will take a look at the 2 D losses for the moment and 3 D losses in a separate lecture all together. Now, 2 D loss is of relevance or significance only in the case of an axial flow turbo machine it is not really true for does not hold much significance for other forms of turbo machines like the radial or the centrifugal kind of turbo machines. And the main significance of the 2 D losses is that some of these can be estimated from a cascade analysis. We have seen that cascade analysis is a simplified form of analysis of an actual turbo machine where the flow is assumed and ensured to be 2 dimensional. And so, some of these losses can be estimated from the measurements that are carried out in a cascade. So, what are these different forms of losses? So, 2 D losses are mainly associated with the blade boundary layers and the shock boundary layer interaction in the case of transonic machine and separated flows and wigs. And there is an additional component of loss which is incurred which is basically because of mixing of the wick with the free shear layer downstream of the blade. And that produces an additional component of loss which is called the mixing loss. Now, it is observed that the maximum losses obviously occur near the blade surface because that is where the viscous effects are maximum. And so, one can and if other 3 D effects are neglected then the maximum losses obviously occur near the blade surface because of the effect of the viscous viscosity on the flow itself. And the minimum losses are expected or found to be further away from the blade surface towards the edge of the boundary layer itself. Now, let us now classify the 2 D losses themselves. We have classified losses in general, but now let us take a look at what are the different forms of 2 D losses that we would be interested in. Now, 2 D losses can be classified in terms of these different parameters. One is the profile loss which is basically because of the boundary layer which also includes flow separation which is either laminar or turbulent separation. One may have wick mixing losses as we have just now discussed. One may also have shock losses and one might encounter trailing edge loss just because of the blade itself. So, these are the different forms of losses that one can expect in a 2 dimensional sense. We will first take a look at the profile loss and then we will now we will derive an expression for empirically calculating profile loss some total of profile plus the mixing loss. Of course, we will not do the detail derivation here as it is already given in some of these text books which I will explain little later. Now, the profile loss depends upon 2 different sets of parameters. One set of parameter is related to the flow itself and another set of parameter is to do with the blade itself or the blade geometry. Now, the flow parameter the different flow parameters which influence the profile loss are the Reynolds number, the Mach number, the curvature of the blade, the inlet turbulence, the free stream unsteadiness and the resulting unsteady boundary layers, the pressure gradient and the shock strength. So, these are the different flow parameters which are likely to affect the profile loss and in terms of the blade parameters, the thickness of the blade, the camber, solidity sweep, skewness of the blade, stagger and the blade roughness these are the different parameters which are associated with the blade itself that can influence the profile loss. So, these are 2 sets of parameters which can contribute to profile losses. Now, the mixing loss that we have discussed is basically associated with mixing of the wake with the free stream flow downstream of the blade trailing edge and in addition to the fact that it depends upon just the wake of the blade, it also depends upon the distance at which the measurement is taken. That is if one were to measure the wake right downstream of the trailing edge of the blade, one might see a very increased level of interaction between the exchange of momentum and energy between the wake and the free stream whereas, if the measurement is taken far downstream of the blade, the exchange of momentum and energy between the wake and the free stream is minimal. So, basically the mixing losses are associated with exchange of momentum and energy between the wake and the free stream and the transfer of this energy results in a decay of the free shear layer and increased center line velocity and increased wake width. So, these are all effects of increased mixing losses that are likely to be encountered in let us say a cascade experiments that one might carry out. Now, as you proceed downstream further downstream, the effect of the wake mixing diminishes and further downstream one would see a uniform relatively uniform flow, because the effect of wake mixing would have substantially diminished by then. So, which means that at a substantially downstream distance the difference between stagnation pressure further downstream or far downstream and the trailing edge will basically represent the mixing loss. That is if we can estimate the stagnation pressure at a region which is far downstream where the effects of mixing is negligible and also the stagnation pressure right at the trailing edge of the blade, the difference between these two should theoretically give us the mixing loss. But what we will see very soon is that most of these correlations which we are going to encounter are based on measurements which are downstream of the trailing edge by half the chord length or one chord length or some distance in between these two, which means that we may be missing certain amount of mixing losses. We may not be including all the mixing losses which one should have, which will occur only if the measurement is taken right at the downstream right at the trailing edge and at a distance which is far downstream of the trailing edge. And if there is flow separation taking place which may occur at increased incidence angles then the loss also will include some amount of losses on account of this additional wake which is generated because of the flow separation itself. So, mixing losses as I just mentioned is theoretically estimated if we have the measurements right at the trailing edge of the blade and the difference between this and the stagnation pressure far downstream where the effects of mixing is negligible. So, let us now try and estimate and try to derive some expressions for estimating some of these losses and we will basically be estimating the profile loss and the mixing loss together which is a normal practice because it is not really possible to segregate these two components of losses even though theoretically as I mentioned it should be possible for us to segregate the mixing loss if we have of course, the measurements at the exact trailing edge and far downstream. So, the sum total of the profile and the mixing losses along a streamline we can estimate as omega bar p plus m refers to profile plus mixing is equal to the difference between the total pressure for upstream or in the free stream p 0 t minus p 0 2 which is the downstream total pressure divided by the dynamic pressure in the inlet. Now, this is the general expression for the losses itself, but we would like to have an expression which is related to the measurements in little more detail rather than a generic expression for losses. Now, if we need to determine what we have just now discussed to return down here it is necessary that we also relate the static pressure difference and velocities to the displacement and momentum thickness on the blade boundary layer at the trailing edge because the profile loss is basically referring to the boundary layer effect and the losses on account of the viscous nature of the flow itself. So, it will be necessary for us to estimate the boundary layer thickness in terms of the displacement and momentum thickness at the trailing edge to be able to estimate the profile and mixing losses together. So, there is a detailed derivation for what I am going to discuss now given in the book by Lakshminarana. So, I would suggest that it is a straight forward derivation which is why I am not discussing that detail here. So, I suggest you can go through the derivation given in chapter 6 of Lakshminarana's book where it is shown that the losses total pressure losses which I have in general are related to 2 into p 0 t minus p 0 2 by rho v 1 square can be related to the sum of static pressure difference and the velocity square. So, this is equal to 2 into p t which is the static pressure at the free stream minus p 2 which is static pressure at the trailing edge divided by rho v 1 square plus v t square which is the free stream velocity minus v 2 square by v 1 square. Now, from this to the next expression which I have written here is what is given in the book by Lakshminarana.  Now, here delta and theta refer to the displacement and momentum thickness delta refers to the displacement thickness which is also an indication of the blockage theta is referring to the momentum thickness which is directly related to the total pressure loss. So, if we expand this in series and neglect higher order terms then the loss expression can be simplified as omega bar p plus m sequence square alpha 1 is equal to 2 into theta plus theta tan square alpha 2. So, here what we see is that if we neglect the higher order terms the loss is directly proportional to the momentum thickness. So, there is no displacement thickness term coming here if of course, we were neglecting the higher order terms and expanding in that way the displacement thickness terms actually cancel out which is expected, because if you look at total pressure loss the main contributed to that is basically the momentum thickness, because momentum thickness is directly an indication of the total pressure loss taking place in a in a boundary layer. Displacement thickness tells us what is the kind of blockage associated with this flow. So, if we were to neglect the higher order terms one can relate the profile and mixing loss together to the momentum thickness by the correlation that I have just now discussed. So, if we look at the significance of this loss correlation that I have just discussed we can see that the profile loss can be estimated simply based on the momentum thickness and the above loss correlation obviously includes both the profile loss as well as the wake mixing loss. And then in the event there is flow separation obviously, there will be additional losses which are incurred this is because in the presence of flow separation alters the pressure distribution drastically which is basically true beyond the separation point. And therefore, there is an increase in momentum thickness and displacement thickness with flow separation with coming into picture and obviously, that leads to an increased overall loss of profile plus mixing losses. So, presence of flow separation will only add up to the losses which are over and above what we have just now discussed for a normal boundary layer. Now, in addition to what we have just now discussed one may also have certain amount of deviation taking place at the trailing edge of the blade and we can estimate that based on the momentum and displacement thickness itself on how the deviation can be estimated based on the change in momentum and displacement thickness. So, boundary layer growth and subsequent decay of the wake as we have seen will also cause deviation at the outlet of the blade which again we can estimate we have seen that there is component of the outlet angle coming in the loss term. So, that can be related to the displacement thickness that is delta and the momentum thickness and of course, the inlet angle. So, 1 minus theta minus delta and tan alpha is basically giving us some indication of the deviation taking place at the exit which is of course, an estimate because there are other terms involved here which have been neglected because they are higher order terms and the losses in terms of higher orders can be neglected. Now, what is an implication of this estimate of the deviation is that viscous effect in a turbo machine always leads to change in the turning angle and it generally tends to decrease in the turning angle itself. Now, there are different parameters which can influence the displacement and momentum thickness some of them we have already discussed earlier on like variation of free stream velocity the Mach number the skin friction on the blade surface the pressure gradient which is also a function of the turning the turbulence intensity and the Reynolds number. So, these are different parameters which can influence the displacement and momentum thickness and therefore, in general they can also influence the profile and mixing loss together. So, if we summarize the discussion on the profile loss one can estimate the loss that is profile plus mixing loss by two different ways. One is of course, based on measurements which we carry from our cascade analysis and the other is of course, to carry out computation and computational analysis which of course, we will discuss in lot more detail towards the end of the course. We have a few lectures dedicated towards discussion on how we can compute flow through turbo machine and losses etcetera. So, there are two distinct ways of estimating this one is by calculating the inviscid potential flow and also the displacement and momentum thickness and subsequently of course, use the data which we have got from cascade analysis towards estimating these losses. The other method of course, is to use a Navier stokes based computational code and here of course, one can get the local and integrated losses directly without having to of course, segregate the losses. It is still possible to segregate the losses, but there are I mean there are uncertainties in terms of how well these computational codes can estimate and successfully segregate different components of losses. Some of these of course, issues we will discuss in detail in later lectures. So, so far we have discussed we have been focusing our discussion on two distinct components of losses the profile and the mixing loss. There are two other forms of losses which I mentioned the shock losses and the 3D losses. 3D losses we will discuss in detail separately. Let us now take a look at the shock losses and the effect of Mach number on the shock loss itself. Now, there is a direct correlation between Mach number and the shock losses. The static pressure rise obviously increases a function of the Mach number itself it increases with Mach number which means that as the static pressure rises there is an increase in pressure gradient with Mach number which means as the pressure gradient increases the momentum thickness will also increase, because with increasing pressure gradient the boundary layer is drastically affected as a result of that and therefore, it affects the momentum thickness as well as the displacement thickness. And when momentum thickness increases it also leads to an increase in losses as the Mach number increases. On the other hand increase in Mach number also leads to increase in shock losses, because shock loss is directly a function of the shock strength and shock strength is directly a function of the Mach number. So, as Mach number increases shock strength increases and therefore, shock losses also will increase. And of course, at transonic speeds the shock losses are also besides the Mach number they are also very sensitive to the leading and trailing at geometries and therefore, which is one of the reasons why transonic compressor blade geometry is substantially different from a normal subsonic blade geometry. And one of the reasons being that it the losses are directly functions of the blade geometry as well as the leading and trailing edge geometries. So, if we were to estimate the 2 D shock losses in a compressor what would need to besides knowing the geometry itself need to know the following that is basically we need to know the Mach number and its associated parameters. The an estimate for the 2 D losses in compressor will need to include at least 3 of these following parameters one is the loss due to the leading edge bluntness with the supersonic upstream Mach number. The second parameter being location of the passage shock which of course, can be determined from inviscid theories because shock and its location can easily be estimated from the gas dynamics that we have that you probably have learned. And once the shock strength is known the shock losses can be estimated because shock loss is directly a function of the shock strength. The third parameter which is probably the most significant and the trickiest of them is the shock boundary layer interaction that is the losses because of boundary layer growth. And interaction between the shock and the boundary layer is something that is quite difficult to estimate. And there are no empirical or theoretical correlations as such which can estimate shock and boundary layer interaction. And this happens to be continues to be an area of research for many people who work in the area of shock boundary layer interaction on how one can accurately estimate and predict shock boundary layer interaction. And of course, needless to say for weak shocks the interaction between the shock and the boundary layer is minimal. And therefore, the loss contribution is also minimal. However, as the shock strength increases the corresponding shock boundary layer interaction becomes quite significant and estimating that becomes even more critical and tricky. So, these are three different parameters that one we need to take into account when one is trying to estimate shock losses in at least a 2 dimensional sense. The strength of the amount of loss of course depends upon the shock strength which again is a function of the Mach number and the geometry. Now, let us now take a look at one of the empirical correlations amongst many others which have been kind of being proposed over many years now. And is being widely used to estimate the shock of the losses total pressure losses associated with the shock in a transonic compressor. And this is an empirical correlation which has been successfully used and validated with experimental data. Of course, this was proposed long back by Freeman and Comste in 1989. And the shock loss is basically a function of the total pressure loss which they have correlated is to a normal shock plus an empirical term here. So, delta P naught loss the stagnation pressure loss across the shock divided by P 0 1 minus P 1 which is a static pressure. This is basically for a normal shock plus an empirical term here 2.6 plus 0.18 into alpha 1 prime minus 65 degrees and multiplied by 10 power minus 2 into alpha 1 minus alpha 1 prime where alpha 1 prime is the blade inlet angle. So, here this difference between these two tells us the incidence. So, this is basically valid for very low incidence angles and it has been seen that it is reasonably accurate up to about an angle of 5 degrees. And as incidence exceeds 5 degrees then the prediction of shock losses by using this correlation is not very accurate. Of course, these are correlations which have been derived using the two dimensional assumption and as we know that actual flows are seldom two dimensional in nature which is also true for the other loss components like profile loss and mixing loss. Most of these which have been estimated here are with the assumption that the flow is necessarily two dimensional. So, the presence of three dimensionality or three dimensional flows will only complicate the losses and which is why we have devoted a separate lecture all together which discusses purely the 3 D flows, the duplicates flows and the secondary flows. So, the shock loss correlation that I have shown here is just one of the many correlations which are available. This is one which has been validated and widely used for estimating shock losses in a transonic axial compressor. So, let me now quickly recap our discussion in today's lecture where we discussed two distinct aspects one was to do with the cascade analysis and performance parameters which we can derive from a cascade analysis. We discussed about two distinct parameters one is the total pressure loss coefficient and the other is the static pressure rise or C p. Total pressure loss is an indication of the losses taking place across the cascade and the static pressure rise or C p distribution gives us some indication of the loading of the blade and these are two common parameters which one can derive from a cascade a simple cascade analysis. The second part of the lecture was discussion on losses we classified losses in general as profile loss, mixing loss, 3 D losses and also the shock losses associated with a transonic axial compressor. In out of all these we have devoted today's lecture towards discussion on 2 D losses and I discussed two distinct components of these losses the profile plus mixing loss and the shock losses in general. So, it is possible for us to empirically estimate these losses based on certain correlations which I have discussed today. These are empirical correlations which can make use of data which one obtains from a cascade analysis and one can estimate losses taking place in a particular cascade geometry using the data that is obtained from the cascade study and also try to segregate losses in terms of profile plus mixing as one component and shock losses as a separate component. So, these are the different aspects that we had discussed in today's lecture performance parameter from the cascade analysis 2 D losses in a compressor stage basically focusing on the primary losses. And in the next lecture what we will do is to take up some problems for solving we will basically have a tutorial section in the next lecture. We will solve a few problems and I will also leave a few problems for you to solve as tutorial exercise problems. So, this is what we will be taking up for discussion in the next lecture.


Hi everyone, welcome to Figure Fantasy. So today we'll be previewing a couple of figures that will be dropping in the future. This is actually based on the Japanese version of Figure Fantasy. This actually will just inform you of what is in store for us in the future, not really on when they are dropping. At least you have a preview of what to expect in the couple of months down the line. Okay, so in this video, I'll just be covering defenders, vanguard, and helper. This is actually just a part one. So let's start with our first defender. Okay guys, our first defender is Okada Ayane. So she is with Galatea. Much needed help in the defender category. So she is actually S tier. Her attack deals additional damage if the target is frozen. Then for ultimate, it's a damage over time. Every 0.5 seconds for three seconds randomly has a 70% chance of freezing for one second and also has the ability to have an additional shield and increase defense. Okay, passive is if the figure of an ally HP becomes 50% or less, she gets a shield and also gets 10 energy. Actually the ally gets the shield. Okay, and also for her special, it's going to be 25% chance damage that will be reduced. 25% chance that there will be damage reduction by 50%. Okay, just wanna fix actually what I'm actually reading. So moving forward, she's going to be a good addition to Galatea. Again, much needed firepower. Galatea right now is pretty much very underpowered and she could really turn the tide for Galatea. And take a look at her 2D art guys. What a beauty to have for your team. She actually uses her own gaming or small gaming console or mobile console. So she's probably a good gamer. So let's move on to our next defender. Okay guys, so for our next defender, it's going to be Izumi in a bike, which has only one wheel. So what do we call this bike? It's a mono bike, mono pad, mono, I don't know, but it's a single tire bike. But anyway, she's with Snow A and she is with, she's S tier by the way. Her attack guys is damage buffs if she damages her buffs. Sorry, buffs her damage if she is shielded. Her ultimate is going to be, she has 2 forms by the way. She has a defender form and a vanguard form. For her defender form, she has 30% of HP converted to shield and vanguard form after attacking four times, gets a shield and receives 30 energy. Allies in the same room receive a shield as well. Okay, so for her passive, crit rate decreases while shield is active and her special defensive form, 25% chance that damage will be reduced by 50% and at vanguard form, recovery rate HP is restored after a dodge. Okay, so again another impressive addition to our list of figures. Hopefully we could, you know, we could get these in the near future and let's take a look at her 2D art. Again, very, very good art and very high in detail as well. So let's proceed to a couple of vangards that we have. Okay, so first on our vanguard list is going to be Kizuna, so she is Galitea and she is going to be S tier. For her attack guys, it's an increase in normal attack, nothing too special, but for ultimate damages a target three times and also adds damage over time every 0.5 seconds and also provides damage negation. Her passive is going to be ally attacks in the same row and has a 20% chance to attack the target as well. So any ally that is in her, you know, in her row will have, you know, an opportunity to attack your target at 20% and her special is going to be evasion and HP recovery. So this is again another welcome to Galitea for S tier and by this time I think Galitea will have enough reinforcements that she will be, Galitea brand will be stronger and can compete with other brands. So let's take a look at her 2D art again, mostly similar to what she has with her 3D art, but again, nice art to have. Okay, next on our Vanguard, we will proceed. Okay, so what we have here is Rin. So Rin is Tenma, so she is classified as A tier attack, so increases normal damage, so very simple, but her ultimate is multiple, she deals multiple damage for five seconds, freezes the target and damage increases as well. Okay, so for her passive is a 50% chance for continuous attack, ignores defense and HP absorption. I'm sure she's going to be dealing a lot of damage to the target right in front of her and her special guys is going to be evasion and HP recovery. So again for Tenma, my thoughts on her is she's going to be a welcome addition, especially if she's going to be Vanguard. The current Vanguard, which is Albert, I don't think he's going to be better than her. I think she's going to be overpowered in terms of the damage that she will be dealing and this is a great help to Tenma moving forward. Okay, let's take a look at her 2D art and look at that pose and really love to get her for my Tenma team. So let's proceed to the last Vanguard in this list. Okay, so the last Vanguard in this list is going to be Date Masamune, so he is Galitea A tier. So attack, his basic attack has a chance to stun. Ultimate also has a stun damage increase of about 10% to 12% energy recovery and reduction of enemy energy. So he can reduce enemy energy. Take note of that. That's important because energy is used to put out your ultimate. Next is going to be his passive, deals additional damage that's just his passive and for his special is also evasion and HP recovery. This guy is going to be very good in terms of his ultimate in terms of the reduction of enemy energy and again, I welcome a lot of support for Galitea at this point because they need really a lot of new heroes. Hopefully that we get them as soon as possible. So we're going to be proceeding to our helpers. Okay, so the only helper that we have is Zong Liang. So Snow A in terms of brand, S tier in terms of rating and Zong Liang attack increases normal damage so very basic but his ultimate is healing effect of enemies is reduced by 40% so he takes out healing of your enemies and reduces it and also does burning damage. Okay, for his passive buffs adjacent allies as well grants a lantern that guards against lethal damage and turns it into a shield. Okay, a lot of protection there for his own allies and for his special 24% crit damage increase to adjacent allies and 10 energy after getting a lantern. So again, he's a good support damage or healing reduction to your enemies and also guards against lethal damage for your own allies and also adds energy to them. So he is with Snow A tier. Again, we at this point I see only a few healers and only I think two stand out really in the bunch but again for support we need different kinds of variety at this point. So this is going to be good at reduced healing for enemies and I'm sure he's going to be a crucial part of your lineup because he prevents lethal damage as well. So let's take a look at his 2D art. Okay, 2D art special and very detailed. Just like the 3D art that he has, lot of lanterns in that 3D art and 2D art. So that is it guys for the part 1 for the future figures. So for those of you guys who have stalked this far, please subscribe. This really helps my channel a lot and also guys don't forget to click that bell icon so you don't miss any figure fantasy videos in the future. Also don't forget to put a like to this video. Thank you guys, take care, stay safe, this is The Warden and I'm out of here.


It's great to be here, I can't believe it was five years since last had an opportunity to be here and talk to you all. I only have 15 minutes and I've been told stay on time because I guess the other gym typically runs way, way, way over. I used to run an airline, so I'm going to keep this thing relatively on time. So given that I have 15 minutes, I would love to talk to you about what Red Hat's doing and some of the extraordinary things we're doing around Linux and then things that we're doing on top of Linux. But I'd rather use this time to really reflect on the impact over the last 25 years that Linux has had, not just on technology, but more broadly what I see happening in business. So if you'll bear with me, I'd like to just spend a few minutes talking about at least what I see every day dealing with large enterprises and the impact that it both has already had, but I think also the impact that we'll be able to see going forward. So I'm going to use a S-curve analogy around it. And I want to talk about, from a technology perspective, the adoption that we've seen and the impact that's had. I then want to talk about more broadly the impact that I think it's having on business itself and the way organizations are operating, getting things done. And then I think what we can do going forward. Now, I'm going to go through this next page very, very quickly. This page was originally to basically talk about history of Linux, but if somebody on there is going to do what I saw last night and see, kernel 2.6, that's not really history. That was just a few years ago. So we often think about with technologies when we think about an S-curve is a level of adoption where something starts and it works and slowly kind of builds and over time it gets better and Clayton Christensen innovators dilemma as a model that innovation happens. But the more is I started to reflect on the history of Red Hat and it's really hard to talk about the history of Red Hat without talking about Linux. And it's hard to talk about Linux without talking about Red Hat because the two were so long, so closely of work together and continue to work well together. But when I really think about Red Hat as a business and enterprise adoption around Linux, there was certainly a lot of adoption happening in the 90s as what I'll call really bleeding edge people started to use Linux and think about it. And Red Hat was a big part of that passion about open source and what open source is and could mean. But frankly as a business model, we didn't have one. It was selling, literally selling t-shirts and coffee mugs and that's not necessarily really good business. And if I had to say beyond the technology and Red Hat's historically been either the largest or second largest contributor to Linux over two decades now. But I'd say aside from our technology contributions, one of the major contributions that Red Hat has made broadly to Linux and to open source is business model innovation, right? We ultimately decided that having a business model that involved selling t-shirts and coffee mugs probably wouldn't work and try to box product at CompUSA. That didn't work really well either. And so a group of our folks got together and said well wait a minute to really make this phenomenal technology available for enterprises. We need to create a level of stability. So how do we kind of take the power of this open innovation and kind of create kind of long term binary compatibility and certified ecosystems and all the kind of downstream things that enterprises need. And so we kind of created this business model around, we call it enterprise open source where we take all open source software and then offer it with a defined life, etc, etc. I'm not here to talk about our business model but more to say that's kind of was around 2001, Red Hat's kind of major innovation around business model. I think this is important to talk about for a minute because the amazing thing to me about Linux isn't that it started bottom up. It's that in the enterprise it started top down, right? So where Red Hat got its first major wins, we're with investment banks running massively mission critical exchanges, right? Trading platforms, right? It wasn't working its way up from file servers and kind of from bottom up. It really started our big initial wins on the enterprise side were with the large investment banks running trading platforms. Now you pause and think about that. I don't want to draw a caricature of investment bankers but many of you who've been involved with startups will recognize this. These typically aren't people who care a lot about freedom. These aren't people who kind of even care about cost, right? These are people who care about best possible feature functionality at any cost and they adopted Linux. Very simply, Linux on x86 was faster than Unix on risk. And within just a couple of years, all of Wall Street moved. Now I joined Red Hat in 2008 and my mental model of Linux was, I think of many was, it's kind of a thing that's working its way from the sysadmin up. And I started looking at Red Hat's list of customers. And I started scratching my head saying, these are amazing customers running their most mission critical applications. Why? And the answer is because it's better. And you kind of scratch your end and say, okay, it's better and it's also cheaper. So it's better and it's cheaper. So why isn't everybody running this? And I think what we're seeing over time is we continue to make progress. And I'll come back to that comment. But just again to talk about the power of sharing. The reason that Linux runs these large trading platforms has a lot to do with the fact that the Navy had a problem of wanting to build new anti-missile defense systems post Falkland Islands and some of the issues happening there with anti-ship missiles. And so they needed a real time operating system and we worked with them. We meaning the community, not just Red Hat worked with them to build features into the operating system to allow for real time performance. Those were the same features that investment banks needed to run some of the most highest performing mission critical financial transaction systems in the world. That's one example of just myriad numbers of very odd kind of combinations and collaborations that happened to create. What was even in the early 2000s, really an extraordinary powerful system. And we've now spent the last kind of since 2001 with the initial Red Hat Enterprise Linux, really driving from the top down, right? It's been more, not how do we add feature and functionality, it's been more driving against frankly inertia and lack of knowledge about the power of this. But to have a technology that's both better and cheaper is an extraordinary place to be. And so I think Red Hat's been kind of at a forefront of kind of watching that adoption. Last year at Red Hat Summit, which is an annual event we have around Red Hat's kind of product stack. Paul Cormier, who's been, had a products and technology at Red Hat for a lot of years, kind of made a proclamation and I should back up on that. We have this on our walls all over, right? This is a famous Gandhi quote and I'll have to say for the real historians in the room, there's still some controversy over whether Gandhi really said this or not. So for those of you who wanna research that, you can kind of see all the back and forth. But regardless, I think it is a really powerful kind of statement of how disruptive change happens.  And so last year at Summit, Paul puts this up and basically for the first time says, you know what? At least at the level of Linux, we've won. Now let me pause on that for a second, but I mean by we've won. Red Hat spent from its early days through until recently, I would argue fighting a battle to say open source is a viable alternative to traditional software stacks. But I think we've done that and I think we demonstrated that over a long period of time. Our next aspiration for open source is to not be a viable alternative, but to be the default choice for next generation software stacks. And I would argue we are there. If you look at that new workloads, whether those are things that are being built on the cloud, whether those are big data, analytics, machine learning. The default choice for those net new workloads is running on Linux. It's not a viable alternative, it is the default choice, which is an extraordinary thing. Because not only is it extraordinary obviously that this thing that started off as this very, very small movement has grown to be the default choice for next generation of technology. It also now serves as the platform on which the most net new innovation happens, right? Big data, Hadoop, Spark, all the things happening there, happening on Linux. SDN, all the innovation that's happening there, happening on Linux. Containers happening on Linux. It has become the platform on which the whole next generation of technology is developed, so it really is the platform for innovation. I'm actually thrilled to see if you ever need validation of that, that Microsoft is starting to embrace Linux. Thrilled to see that. I was actually, I'll say a few weeks ago, I was with Satya at his office and I like stole four Microsoft pins as souvenirs. Because when's the Red Hat CEO actually at Microsoft? But it's amazing to see the collaboration with a whole set of technologies, both in Linux and driving things into Linux. But importantly, driving the software stack around that. So I'm very interested to hear what Wim has to say. And we've had a phenomenal working relationship now over the last couple of years, which is just amazing, right? We really have two operating systems now in the data center, Linux and Windows. And again, I would argue most net new workloads are actually innovation is happening around Linux. That's at a technology level. I want to spend a few minutes I have left talking about what to me as a traditional executive, if you want to say before I joined Red Hat, has also seen in terms of the power of what Linux has done. And I would argue long term might be even a more impactful on the world than the technology itself. And that's the social DNA that is developed around how to coordinate thousands of volunteers to get things done. I talked to CEOs around the world. This is not one. I just thought this was a good picture, but that's not a CEO that I talked to. We live in a world that is dramatically accelerating and the problems that businesses face now are no longer how do I coordinate thousands of people doing road tasks in a static environment, right? The problems most business, because those problems have been solved, right? They've been automated with either robotics or information technology or additive manufacturing, right? The jobs that are left are people who have to apply either initiative or judgment or creativity to their jobs. And this is all happening in a rapidly moving context for how fast business moves. And so what I hear over and over and over again from executives is my structure, my management structure is not helping me innovate. It doesn't help me attract the best people. It doesn't get the best ideas out of my people. And it doesn't allow me to be agile and able to change, and that's a problem. I actually did write a book that Jim mentioned, and I want to be clear what the book is. It's not anything that I've done. It's what I have learned from Red Hat, which Red Hat learned from the open source community. And that again is, if you think about what management is and traditional management hierarchies bureaucracies, all that stuff. Management is coordinating people's behavior to get common things done. And Linux really developed a whole different way to coordinate behavior by breaking problems into small pieces, how interactions happen. And it's amazing to look at that, and Red Hat's applied that to how we run the business, and so me coming in from the outside kind of saw, wow, this isn't just a great way to develop software. It's a great way to coordinate bright people who have to apply judgment and creativity to their jobs, but still able to coordinate to get things done. In the case of Linux, something that's of such high quality that it runs nuclear submarines and stock exchanges. In traditional business, how do you actually coordinate people to get other things done and accomplish tasks when, again, you're not dealing with people doing rote tasks in the static environment. But you're trying to get best and brightest to work on difficult problems. That, I think, can be another massive legacy of Linux in open source. Is literally a different way to organize people to get things done in the world. And in the interest of time, I'll go through this quickly. You can look at that in a lot of regards. It's not just within a company. Many of the problems that we need to solve cross companies, right? So if you think about something, and I get asked this a lot by executives and pharmaceutical in automakers, it's, wow, the problems I have to solve today span my corporate boundaries. So how do I actually coordinate, collaborate, share across corporate boundaries, right? In this example, no single technology company I don't think is gonna fully solve the autonomous driving problem. No single automaker is gonna solve the autonomous driving problem. The need for coordination across technologies, state and local governments, the makers of the vehicles, etc, etc, etc. Require a whole different way of coordination. And again, that's something that we've learned so much from Linux about a different way to coordinate across corporate boundaries, no matter where people work. And these are three names I could have put up 100 names of companies that, these is literally in the last couple of weeks I've pulled these, of companies that talk about open source not from a technology perspective, but how they're running their business. Nike has specifically said we want to leverage the principles of open source and sharing to help us deal with industry transitions. IKEA talks about open source mindset and how it runs. Toyota just open sourced all of its hydrogen patents. It's becoming much more involved in thinking about how to drive innovation broadly, and I could put quotes from Ford and GM. The term open source and thinking about working cross-corporate boundaries is becoming the norm for leading edge companies. I would argue it's gonna become more and more the norm broadly if companies wanna survive. So we can certainly talk about Linux and what the 25 years have done for a technology stack. And beyond the technology stack is the platform that allows a new set of innovation. But I would actually argue the social DNA that was developed to coordinate a different set of volunteers to work together is already massively impactful on business and will be going forward. So quickly on what's next, I think the same social DNA as well as the technology stack can really help us solve the mega world problems. So if you start thinking about things like global hunger, curing diseases like cancer, global warming, these are massive, massive, massive daunting problems, right? There is no institution, there is no country, there's certainly no individuals that are gonna solve this. This is actually requires, again, a different way of people thinking and working together. And I do think that Linux as a technology, but more broadly as a social DNA about how people can work together has the potential to dramatically impact human outcomes and get some of the most daunting challenges. I used this statement at Red Hat Summit a few weeks ago that our ability to harness and distill the best ideas will determine the progress of human progress for the next century. And when I say the ability to harness and distill the best ideas, that is how you get people to work together, right? And what we know is the traditional way we've done that, proscribing, hierarchies, that does not work. That does not get best innovation out of people. And I think what Linux has done and broadly what we're seeing continue to innovate around open source is really develop a better way to harness and distill the best ideas. So again, we can talk about Linux for what it's done in technology. But I think its impact so far on the world has been so much broader. And where I see the possibility for it to go is truly extraordinary. So I'm just thrilled to be able to be a part of that. And I'm thrilled that Red Hat's been a part of that. So happy birthday, happy 25th. Thank you all and thank you all for your work to make it all possible. Thanks.