 After seeing this video, you will be able to then attempt the tutorial that we have given you. So, even if you know how to do it, please pay attention to the video, do not do the tutorial till I give you the go ahead. So, a funny shaped airfoil, in fact this airfoil shape is called a reflexed airfoil and it does have some particular uses to help change the pitching moment of a wing. This amount of reflex is probably a little bit more than you would see in practice. I put this much in just to make it really clear the way the curvature was behaving, but this is quite a bit of reflex here. So, it's probably a little exaggerated for a common situation, but again, we're just doing that for the purposes of showing you how this type of airfoil behaves. Okay, so let's start out by putting up some radius of curvature vectors in here. So, here's on the upper surface the radius of curvature on the front half of the airfoil, but on the back half the radius of curvature flips and comes from above. So, like we've done now a few times, that means in this region here we're going to expect the pressure to be lower than P infinity, right, so that there's a net force inwards or the other way of saying that is the pressure increases as they move out along the radius. So, if it increases P infinity is out here, so P here has got to be lower. So, if P is lower than P infinity, then the C P in that region should be less than zero. And the opposite happens here, the pressure will increase towards the airfoil along the radius direction, always along the radius direction, so here the C P is going to be greater than zero since the pressure will be greater than P infinity there. Okay, on the lower surface we have a radius of curvature in the front half that looks like that, and on the back half it looks like that. So, doing the same things here, this means that the pressure in this region has to be greater than P infinity, right, P increases along that direction, so C P is positive here and over here we're going to have the opposite, right, C P will be negative here because the pressure has to increase in this direction, so the high pressure is P infinity, low pressure will be on the surface. Okay, so that's the basic behavior, now let's try and do some sketches of the C P here, so I've put an axis together here with our usual negative C P's up on top. One is the stagnation point C P, and that's the highest C P that we'll ever have, so in terms of the positive value. So we'll have a stagnation point here at the leading edge, and in general for an inviscid flow we'll have a stagnation point here. We'll talk more about some details of the stagnation point, we're not at the trailing edge as we go along, but for now we're expecting a stagnation point on an inviscid flow back here. Okay, so those two points I've drawn in, now somewhere in the middle of the airfoil here we're going to have all of this sun flip occur where the C P's go from negative to positive on the upper surface and the lower surface vice versa. We'll know exactly where that's going to be, expected to be where that radius of curvature flips roughly. They don't have to be switch signs in exactly the same spot in the upper and lower surface. That's not true, but in the general vicinity we would expect that to be the case. So I've drawn this green sort of fuzzy area to say I don't know exactly where it's going to be, but somewhere in this vicinity in terms of X the C P's are going to cross through zero on both sides. So I'm just going to pick a spot here and say that's where the C P's are going to cross through zero and I'm going to use that to help connect my dots. Okay, so now next what am I going to do? Well, now it gets a little bit more hand wavy. Beyond knowing that they're positive and negative values, I'm going to have to pick some values here. So I'm going to start by picking the value on the lower surface on the back half here. Why do I pick that one? Well, no particular reason really. One is that these lower, the higher pressures are a little bit more bounded in that we know they can't get above one, whereas the negative C P values can go on for any value really. Of course the incompressible assumption will break down probably at some point, but at any rate, so I just have less to pick from here. So I'm just going to pick this. This is completely arbitrary. I know it has to be in this vicinity somewhere because the pressure in this region should go positive, C P should go positive. So I'm picking a dot. Don't make anything out of it beyond picking a dot. So nothing more than that and just saying that it's positive. Okay, and I'm going to now also just pick a dot for the upper, the lower surface C P, which will be negative in this region. Again, don't know what it will be. I'm just picking something at this point. So this is even more arbitrary than this. Beyond it being above this line, that's the only thing I really know for sure, and it's the same thing here. It's below but less than the C P of one. Okay, so with those two dots picked, I can draw the C P distribution for the lower surface and then the upper surface will go through the stagnation point as well since that will be at this spot. And I'm going to draw C P crossing through zero also for the up, the lower surface through the same spot. So again, an arbitrary choice on that one. It could cross through zero somewhere else, but should be somewhere in this vicinity. Okay, now I'm going to put in the lower surface C P the rest of it. Okay, and sketch in a couple of dots here. So the C P in this region I'm picking again similar to this. It's got to be somewhere between zero and one. I really don't know what it is. So I'm just picking this value. So just to summarize, very arbitrary beyond being positive, very arbitrary beyond being positive, very arbitrary beyond being negative. Okay, the location here is arbitrary. These are not really arbitrary. These are going to be stagnation points. Okay, now the last part is less arbitrary because once I've lowered this down, I need to produce the lift coefficient that I've given you. So I said the CL of 0.4. The CL will be the area under the curve, but I have to be a little careful because this part here is actually going to be a negative lift if you will. This is the first that's going to act downwards because the pressure now on the lower surface is less than the pressure on the upper surface. So this actually subtracts off lift. So whatever I have over here is going to have to supplement and go beyond what it might otherwise do because it's got to produce, it's got to overcome this negative contribution to the lift. So I need to calculate some areas here. So this area here, this is our rough numbers here. So this is roughly about from the half chord. So let's call that 0.5. And then this height here, let's call that a height of about 1. So the area here is something like about 0.5. So that means the area over here is going to have to be 0.9 so that this area subtracted from that produces an area of 0.4 in total, which is the lift coefficient that I need. So I need the area here to be about 0.9. And then the x distance here is about half a chord again. And so that means that in order to produce an area of 0.9 I need to go up from roughly about here since this is about the average Cp on the lower surface that I've picked. So I need to go up a distance of 1.8 so that 1.8 times this 0.5 equals my 0.9. So that blue dot there is about 1.8. So now just connect the dots again like that and that's what we think the Cp looks like. So what are the things again just emphasizing what I pretty much know for sure, or what I don't know for sure. I know there's a stagnation point at the leading edge. There's one at the trailing edge. Somewhere they're going to cross through 0 as we're expecting. The upper surface is going to have a negative Cp on the front half and a positive Cp on the back half. And we're expecting the opposite on the lower surface. I've said the Cl is 0.4 so that means I have to I have the sum of the areas here making sure that this one I realize is a negative really. So the difference really between these two areas has to be my Cl of 0.4. Okay, so I actually simulated this using the X-ray software in the incompressible potential mode and when you simulate that for a Cl of 0.4 here's what the Cp distribution looks like. So actually pretty similar to what I just sketched. I overestimated the Cp value here. Mine's more negative than this. But if you flip back here, let me do that. In fact, it's not bad. It's certainly the right shape qualitatively. And in fact, this airfoil does cross through zero fairly close to each other, not exactly, but very close. So generally the trends, clearly the trends are right on. And with a little bit of guess work actually I did pretty well on the Cps, but the trends right on. So again, important part here is just how do you apply streamline curvature? How does this work? How does it relate the Cp distribution to the geometry and therefore the geometry to the lift? Okay, so before we go ahead, let us take any doubts if anybody has about the arguments that were presented by this author in arriving at this distribution. And you notice that the estimated curve just by reasoning and simple arguments is not very bad as compared to the value predicted by a software called Xfoil. So we have to apply a similar reasoning today in trying to arrive at the Cp distribution. Now all of you do not have the same airfoil. Just to inform you, those of you who are experts in looking left and right and trying to copy from your friends. You may have the same one, but actually there are five airfoils in the whole class. So there are five different airfoils and these five airfoils we are now going to apply the same logic to arrive at the pressure distribution. So first of all, are there any questions or doubts? Hello. Yes. Why do we have stagnation point at trading edge? Why not? What is the stagnation point? The velocity becomes zero and the pressure is at its maximum. Is it P infinity equal to P? So what will be stagnation point when the value of u local velocity is equal to zero? Yes. So this is a inviscid flow assumption, number one. So in the inviscid flow assumption we need to have a full pressure recovery. We are not looking at losses due to viscous effects. So in case of unseparated flow there has to be full pressure recovery at the trailing edge. So in the case of airfoils, especially when we assume inviscid flow and unseparated flow you will get Cp, you will get a full pressure recovery and therefore you will have a stagnation point at the leading edge and at the trailing edge. It's an assumption. So when you actually look at the aircraft wing you may not get complete recovery. There will be little bit of loss. Any other question? Yes. My name is Gautil. So my question is that like we have seen here that on the back side of the airfoil the net force is downward due to the pressure and on the front side of the airfoil the net force is upward. So it will create a pitching moment and it will increase angle of attack. What will that create the stall in the airfoil? See, it will create a pitching moment no doubt about it. Now whether that pitching moment will be large enough to overcome the inertia of the airfoil to actually create an alpha increase we don't know, right? Moreover, if you look at an airfoil analysis it's a 2D flow. So in a wind tunnel for example we assume that the airfoil section goes across the wall from this wall to this wall so it is constrained not allowed to rotate. So yes, there can be pitching. There will be in every airfoil in general there will be some kind of a pitching moment because in this case of course there will be a huge pitching moment because clearly front part and the rear part are having loads in the different direction. But the pitching moment may not lead to increase in the angle as you are expecting. It may not if the whole wing is constrained it may not but it also can be. So in a freely moving airfoil in an airfoil in free space what will happen is the pitching moment will make it attain a position where the net moment is 0 and that could be beyond alpha stall and hence the aircraft can stall. And sir, what happened? Yes. At the trailing edge we are not getting Cp equals to 1 the reason is we have viscous flow or something else. No, if you look at this particular CFD analysis you are not getting it recovered. But in the theoretical calculation which was shown by the author he assumed a fully full pressure recovery. It is an assumption. So it is an assumption that there will be full pressure recovery but there is no way of knowing how much it will be. When you reason out by just physics it is very difficult to know how much it will be. So he assumed that there will be a full recovery. Yes, any more questions? Yes. Sir, the maxima or minima in this Cp graph does it correspond to the camber, maximum camber in the airfoil or Cp? I think it does. If you look at the airfoil it does. The peaks in the front portion and the rear portion are appearing at the places where the camber is maximum. So it will be. By simple argument, camber leads to acceleration of flow and therefore there will be more suction at maximum camber location. Yes. But the thickness distribution can also alter that slightly. So you cannot say it is only camber. It is camber plus thickness. Both of them play a role.