 Alright, the last topic we are going to look at is going to be lift. So we've spent some time looking at drag on bodies. Now we are going to look at lift and lifting bodies. Okay, so here we have a lifting body. This could be a finite wing. And what I've drawn in the image, our number of things, the width of a wing section is referred to as being the span, and we use B for that. The cord length is from the leading edge. The leading edge is up here to the trailing edge. The trailing edge is down there. So the cord length is C. And the thickness would be the maximum thickness along the cord length, and so that is denoted by T. And we calculate the area, usually the plan form area, that would be B times C. And finally, I've drawn a line, although fairly poorly. This is supposed to be the cord line. And so what the cord line does is it goes from the trailing edge up to the leading edge and then it extends. And the angle of attack of the airfoil or the wing section with respect to the free stream is alpha. And that's denoted there. And V infinity obviously is the free stream velocity. So the net result of this, remember we said lift was always perpendicular to V infinity and drag was parallel with it. So what you're going to get out of this is you will have a lift force on this object and you will also have a drag force. Okay. Now the way that we characterize lift characteristics on any kind of body, be it an aerodynamic body or not, we use the lift coefficient, very much like what we saw for the drag coefficient. And the nature of that is going to depend upon whether it's a finite body like we just saw or if it's a two-dimensional section of an airfoil. And if it's a finite body, we give it capital L to denote that this is 3D lift. L would be the lift force and then we divide by our dynamic pressure and some characteristic area in this case, plan form. So that would be 3D lift. And if you are only looking at a section, a section of a cross section of an airfoil, not a three-dimensional section, then we would use L prime to denote lift per unit length and dividing by, again, the dynamic pressure and we would use the cord for the length. And so this is referred to as being 2D lift and L prime is lift per unit span. So there's a designation a little L and a big L and it's similar for drag coefficient. For any kind of lifting body, we always do characterize the drag through the drag coefficient just like we saw earlier for cylinders and spheres. Okay, now lift and drag, both of these, C L or C D, these are going to be functions of a number of variables. One obviously is going to be the angle of attack of our body with respect to the free stream. And the other one is the Reynolds number and that will be based on, actually I shouldn't put D there because the characteristic length here is going to be the cord length. So when we're looking at lifting bodies at an airfoil, we will compute Reynolds number based on the cord length. And what people have done is they've collected enormous amounts of data characterizing all different types of airfoil cross sections. Those are plotted up in plots with the lift curve or drag or moment characteristics. So what we're going to do now, we're just going to quickly take a look at a NACA airfoil. And what we're going to be looking at is the NACA 2412. And so we'll begin with the video clip. And what this NACA 2412 is doing is it is undergoing a sweep process where the angle of attack is changing. So let's take a look at that. So here we have the NACA 2412. I draw a cord line in there and from that I'm able to calculate 1.7 degrees. Lift coefficient from experimental data is 0.45, not from the data here but that's from other curves that what other people have collected. Then figuring out the angle of attack. Here we have it at 3.3 degrees. Lift coefficient would be 0.55. You can see the flow is going smoothly over the airfoil. Put the angle of attack a little higher. Here we go up to 6.7 degrees. The lift coefficient is now around 0.9. And then the final angle of attack that we go to, draw on the cord line here, 10 degrees. You can see some separation. Lift coefficient is now 1.2. So it's getting quite high and there's onset of separation, what appears to be separation here, the trailing edge. If we were to go a little bit higher, 12, 13 degrees, that's when you start getting separation and you would see the consequences of that with separated flow on the top of the airfoil. Now, in terms of data what we normally will do is we will take the lift curve or the lift coefficient and we plot it as a function of alpha. And so for an airfoil like we just looked at, the one with camber, what happens is the lift curve slope at alpha equals 0, or not the slope, the lift curve, the lift coefficient is going to be above 0. So if we were dealing with like an ACA-0012 or an ACA-0015, a symmetric airfoil, the first two numbers denote that it's symmetric. We would have the lift coefficient that would be at 0 at alpha equals 0. So this is alpha equals 0 degrees. But what they would do is they would go in and they would do this alpha sweep and they would, using a force balance, would measure the lift coefficient as a function of angle. And then eventually what's going to happen is you're going to get up in here where the lift drops all of a sudden. And this is where you're getting the onset of separated flow. And this is what they refer to as being stall. If you're in an aircraft, that's the last thing you want to have happen because you will quickly start dropping or the pilot will lose control. Either one of those is very, very bad. So anyways, this would be the characteristic for an example of a NAC. We looked at a 2412. The first two numbers here denote camber and I think it denotes the thickness at a given location. Actually no, this denotes thickness. I think it's how far off of the chord line it is and where it would be. I apologize, I can't remember what the first two numbers were, but I know they denote the fact that it is not a symmetric airfoil. And then it would go down. And if you're dealing with something like a 0012 or 0015, then we would pass through zeros. So that would be a symmetric airfoil. So that is how the data is plotted and then there will also be moments. Sometimes they'll have moments about the quarter chord and there will also be drag. And so you'll get the drag characteristics for the airfoil. So with that and looking at what we know in this course through Bernoulli and looking at our airfoil section, what we saw in the video, we saw the flow coming along, stagnation stream line coming up over the top and then there's a stream line coming off the trailing edge. But what's happening here is the upper region has a higher velocity and there are a number of different ways that you can explain that. Some people say that the fluid has to take a longer path on the top. That's kind of a junior type of approximation. Really what's happening is you get circulation that is built up due to the flow around the airfoil and that gives you a net lift as a result of the circulation. And that would be something that you'd study in a more advanced course in fluid mechanics. That V upper is a higher velocity and consequently, and VL down here, this is a lower velocity. So assuming that we have your rotational flow, we can apply Bernoulli's equation. So if we have V low lower and V upper higher and the left and the right have to equate to one another, the consequence of that is that up here we have a lower pressure and I'm denoting the region above the airfoil. So we would be looking up in this region here and then down here with respect to that, this is a higher pressure. And consequently, what happens is we have a pressure differential and that pressure differential is what results in the lift on the airfoil. So P upper is lower than P lower and then we get net force up and that results in the lift and that's what we saw in the lift curve. So the lift coefficient plotted as a function of alpha. So that is lift. We've looked at drag, external viscous flows and that concludes the course. So I'd like to thank you for your attention. I hope you enjoyed it. I hope you learned something about fluid mechanics and introductory course. If we were to go on and do more advanced things in fluid mechanics, we would look closer at things such as the Navier-Stokes equations, we would look at solutions there. We would look at potential flow modeling. So the situation where we have sources, things, vortices, and you could compute flows around an airfoil like this, we could look at things like compressible flow, we could look at turbo machinery, open channel flow. We haven't looked at that, that would be of interest to civil engineers. So there are many, many different areas in fluid mechanics that could still be explored. This one just scratches the surface, a little bit of an overview. We went fairly quickly through a lot of the different aspects of fluid mechanics but hopefully it has you interested in fluid mechanics and willing to pursue further studies. So thank you for your attention. Goodbye.