 So there are some coefficients, now the lift that you generate on a body it is not just a function of velocity and curvature, there are many many more things, okay. For example it depends upon the shape because different shapes are going to create different streamline curvature, it depends upon angle of attack because the angle of attack also will create a difference in the curvature, it depends upon viscosity because viscosity is the property of the fluid that will we affected and that will affect the friction acting on the body compressibility because that will affect on the transfer of momentum between the fluid, okay. Then area, free stream velocity and density all these things and many more will affect the amount of lift generated, so therefore what we do is to take care of all the important things we what we want is a simple equation that will capture the essence and to do that we basically have this equation which says that lift is a function of the density of the fluid, the velocity, the area and there is a coefficient called Cl or the lift coefficient which simply relates the numerical value of lift with the parameters like area, velocity and density that directly affect the amount of lift created. So let us say there are two fluids, one with density rho 0, one with density rho 1, everything else remaining same, the lift generated will be more if rho 1 is more than rho 0, okay. So that is why similarly velocity, same body at different velocity will generate a different amount of lift, so that is the reason why, so intuitively we know that the force will be proportional to rho infinity, rho infinity to V infinity square n area, now area in this case is the area on which the flow is actually attached or working, okay. So if there is a situation where the area is such that some area is not playing any part in the lift generation, then on that area we do not have to, we cannot count that area because that area is not exposed or not being affected, it is not transferring its momentum, it is not playing a role. So the lift coefficient is non-dimensional, it is a coefficient, it captures almost not all but almost all dependencies and you determine that experimentally, okay. Now there are two more coefficients which are important, the first one is drag coefficient and this drag coefficient is very similar to lift coefficient, only thing is we call it, we use the coefficient cd instead of cl, now this depends on many, many things, okay. There is something called as a form drag, skin friction drag, other drag components, we will discuss this in detail in the next chapter when we look at the drag estimation. Drag estimation is a very important part and by the way after we do the next exercise then you will be now in a position to attempt the assignment number 2 which is the Midsum Assignment about which I will talk after I finish the next lecture. So there is also a moment coefficient because the forces acting on the body are not going to act at only one specific point, they are acting all over the body because the entire body is going to be immersed, the whole body is immersed in the fluid, okay. So whenever you have a body which is exposed to fluid, apart from the forces of lift and drag which act on it and the weight because of its own mass, there is also a moment which is acting on this body and this moment is called as a pitching moment because this is the one that makes the nose go up or down. So the pitching moment is also defined by a coefficient called Cm or the pitching moment coefficient and it relates the moment with half rho v square s but now the problem is that there is something called C there and that C is introduced because moment is a quantity which is forced into length not just force, lift is a force, drag is a force but moment is force, moment is force into length, so you need a length dimension. So we choose some characteristic length and that characteristic length C is normally the chord length for an aerofoil, okay. So chord length is added otherwise the units will become dimensional. So to make a moment coefficient non-dimensional, we all this is convention, okay. This is not physics. For example, how did this half come into picture? Why should d be equal to half rho v square sc, scd? Why not rho v square scd? Can anybody answer this question? Why do we have this half there? Nobody knows this? What do you think? Even if we do not have half it will be fine because there is a change in cd. Exactly. That is one thing. You could define a new cd called cd dash which is equal to twice cd and then you can say d is equal to cd dash rho v square s. But there must be a reason why people have put this half, why not one-third, why not one-fourth? Yes? Maybe because half rho v square is a dynamic pressure. That is right. That is exactly right, that is very right because half rho v square is dynamic pressure and this relates to dynamic pressure, it is a function of dynamic pressure. So people have said, let us put half rho v square so that we can call it as drag is equal to cd into dynamic pressure into area. That is the reason. It is just for convenience. But as long as you are clear and I am clear that we are using cd of the formula cd half rho v square s, we are both on the same page. Tell me, rho infinity, any doubt what it is? No, it is the density of the surrounding fluid. V infinity, any doubt? No, it is the velocity of the body but what, which velocity? It is the relative velocity. It could be ambient velocity if the body is stationary. It could be velocity of the body if the area is stationary or it could be a relative velocity. Correct? What about s? What is the area s? Which area? Please answer this question. Which area is this s? Yeah. I am Vinay. It is a, any reference area, it can be the wing area, it can be any reference quantity used by the OEM or I mean. So when you say wing area, is it the top and bottom surface? No, it is the projected area. So is it projected area and top view or bottom view? It is the same. If you look at only the wing, I can understand but if you are looking at the aircraft now there could be some differences. So what I am saying is, you are right. It is a reference area. So this reference area, one has to be clear. So when I say CD is so much, you should be clear that this half has to be used. You should be clear that the density is, this density, velocity is this velocity and you should be clear that I have given you CD assuming the area to be this area. In normal, unless otherwise stated, this area s is considered to be the projected area in the top view. Not the area along the curvature, not the area of top plus bottom, area in the top view, projected top view. Yes, Farah, you have a question. So with the change in angle of attack, the projected area would change. So do we, do we consider that? That is the thing. That is also an important thing. With the change in the angle of attack, the projected area may change. So we do not. So we do not consider, when you define this as, we do not consider the change in the projected area that can happen because of change in angle of attack. So as long as all of us are clear, so I can say it is the projected area in the top view at zero angle of attack, whatever it is, in the top view of the aircraft or the airfoil as shown in the paper. So as long as you and me are on the same page, we will not make any mistake. This is the area where there are, this is the interesting point. This is the area where there are maximum confusion and maximum errors in calculations. I will explain to you when I come to one particular example. Pressure coefficient basically quantifies the difference in pressure. Pressure coefficient is what is important to us. So what is pressure coefficient? It is basically the local pressure at any point in the flow field minus the pressure of the free stream far away from the body that is P infinity divided by Q infinity which is dynamic pressure half rho V infinity square. In fact, this value Cp is more useful than the absolute difference because you might say what we need is the absolute difference in the pressure acting at every part of the body. Above and below, to take any part of the body, above and below there is a pressure difference. Give me that. I will multiply by the local area and give you the net force. It turns out that the value of pressure coefficient at any point is more important for us. I will show you why and the dependencies are very similar. So this is one example. This is a NACA force. I hope all of you must have read about NACA airfoil by now because I mentioned last time that NACA family is a known family, it is there in all textbooks, so many sources. So I hope you know that 4502 means something. This is the 2% thick airfoil and it is a very, you can see it is a very thin airfoil. 2% thickness is very less and it also has some kind of a camber. So if you plot Cp versus X by C versus the ratio of location upon chord, you get a distribution as shown here. So one of these, now notice interesting observation is that the scale is inverted. Normally above 0 we put negative value and below 0 we put positive values. This is the convention again. You need not follow this convention but when you look at the Cp curve and make some conclusions, my request to you will be look at the scale first and if the scale is different from what you are used to, you have to invert the image in your mind because many conclusions you will draw looking at the shape of the Cp curve. So now tell me what happens if the value of Cp is equal to 1 plus 1. What does it mean? Let us start with 0 first. So when Cp is 0, it means that P is equal to P infinity. So that means on this airfoil somewhere at this portion ahead of the tip, you notice that Cp is 0. There are actually two curves here. There is one curve here which is like this and then like this and the other one is I think like this. There are two curves here. This is the pressure Cp plot for the upper surface and the lower surface. So we have to draw some conclusions. Let us see why. How can you obtain the Cl from Cp and also Cd from Cp? So basically what you have plotted here is the pressure distribution. This is the pressure below free stream on top and pressure above free stream on the bottom. So we have plotted a pressure around the airfoil. So here pressure is, so if you want to calculate the lift, what you do is from the leading edge to the trailing edge, you have to just integrate the difference in the pressure between the lower and the upper surface and this cos theta ds is basically to take care of the shape as you go around the body. So this is the method to do it. So I can replace Pl by Pl minus P infinity and Pu by Pu plus P infinity. So I can get Pl minus P infinity and Pu minus P infinity. If I divide by Q infinity s and I take s as equal to the chord length into unit dimension because notice that lift acts on an area not on a line. So we are assuming unit span. So the depth of this airfoil is one unity. The length is C. So the area is C into 1. So with that interestingly you will get Cl will be integration from 0 to C of Cp lower minus Cp upper into dx by C where x by C is the fractional location of the point along the chord. x by C equal to 0 at leading edge, x by C equal to half at midpoint, x by C equal to 1 at the trailing edge. So basically this is what you do, you plot. So this is the pressure coefficient at the, so from here it goes down to 1 and then it goes like this. This is Cp of lower surface. You go from here right up to this value and then come down. This is Cp at the upper surface. So if you just calculate the area between these two, that area will be the lift coefficient. So basically larger the gap between the pressure distribution on the upper and lower surface, larger will be the Cl. That is why this particular graph is very interesting. It just shows you if there is some net area between the two that means there is some lift. As you can see the Cl value is 0.5. So if you plot, can it be possible that there is no lift here, so we can plot. And I am just saying that Anderson chapter 5 has got more details. You are supposed to refer to it about the Cp methodology. Last thing we will see is effect of airfoil camber and effect of thickness. So you can see here, there is some net area, although very small. The value of Cl is non-zero, 0.0001. So this is a symmetric airfoil at zero angle of attack. Actually speaking if it is a zero lift, but due to some numerical error or a very small value, there is some very small value of, if this is a cambered airfoil such as we saw last time. So I will show you this to you in a better way. So you can see that the airfoil is symmetric and it is supposed to be zero angle of attack. So it should have the same on both sides. So therefore the Cp will be the same, delta p will be the same and negative and the lift will be zero. But numerical calculations indicate that there is some value, this is because of numerical error. And if you look at, if you look at now a symmetric, a cambered airfoil, you will see that the values are, there is some finite area and therefore there is some finite lift coefficient. What about this airfoil? Naga 4202, this will have some net Cl because there is some non-zero area. But you can see that the Cp is shooting up very high and then coming down and here also it is coming down and going up and then coming down. Now we see the effect of thickness. So because of thickness, the Cp on both sides will be reduced. So you can notice again, same formula is applied once again and you can see that there is going to be a difference in the pressure. So these kind of airfoils are better and if you plot with angle of attack, the variation of the Cp, you will see that the top surface undergoes continuous increase in the pressure coefficient. It can go up to minus 2 and up to 1. So can you go more than 1 in Cp? Can you have Cp more than plus 1? Think about it. So the material for this presentation has come from these sources. I have already mentioned them inside. There is a very interesting edX course by MIT called as Introduction to Aerodynamics that explains the various pressure coefficient diagrams and in the next class, it will be a tutorial followed by a quiz. I am going to run a software and show you the various pressure distribution. So on that note, we will stop for the day.