 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. So, this can be expressed further as the loss component omega p plus m sequence square alpha 1 this is equal to this is the alpha 1 is the inlet angle is equal to 2 into theta plus delta square by 1 minus delta the whole square plus tan square alpha 2 into 1 minus delta whole square by 1 minus theta minus delta the whole square minus 1. 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.