Subsonic Airflow — Page 63, Lesson 70

Let’s pick this up right where the pressure distribution left off, because now we’re going to turn those pressures into moments — the aerodynamic pitching moments. Here’s the key idea: the distribution of pressure over a surface isn’t just the source of forces, it’s also the source of aerodynamic moments. A moment is a turning effect — a force acting at a distance from a pivot point. So when we change angle of attack, the pitching moment of the aerofoil changes, and there are two ways to think about that change. The first way: changes in the magnitude of lift acting through a moving centre of pressure — the CP. The second, simpler way: changes in the magnitude of lift always acting through a fixed point called the Aerodynamic Centre, or AC. Let me define the AC properly, because it’s one of the most important reference points in the whole subject. The AC is a fixed point on the chord line. It’s defined two ways. First: it’s the point where all changes in the magnitude of the lift force effectively take place. Second: it’s the point about which the pitching moment will remain constant at normal angles of attack. Now, here’s the subtlety. There is a nose-down pitching moment about the AC. That moment is the product of a force and an arm. The force is the lift acting at the CP. The arm is the distance from the CP to the AC. So the moment equals lift times that distance. Now watch what happens when you increase angle of attack. The lift force increases — that would tend to increase the moment. But at the same time, the CP moves towards the AC, which shortens the lever arm. These two effects work against each other. The lift goes up, the arm gets shorter, and the net result is that the moment about the AC stays the same at any angle of attack within the normal range. That’s the whole point of the AC — it’s the point where the pitching moment is constant. Let me give you the geometry. Imagine the chord line, leading edge on the left, trailing edge on the right. The CP sits somewhere along it, and the AC sits at a fixed spot. The moment arm is the distance between them. When the CP slides back towards the AC as angle of attack increases, that arm shrinks, and the moment stays locked. Now, a very practical fact for subsonic flow: for airspeeds of less than Mach 0.4, the AC is located at the 25% chord point — that’s one quarter of the way back from the leading edge — and this holds for any aerofoil, regardless of camber, thickness, or angle of attack. That’s a remarkable simplification. It means for most of the speed range you’ll fly in, you can treat the AC as sitting at the quarter-chord point, no matter what the wing looks like. And why do we care? Because the AC is an aerodynamic reference point, and its most direct application is to the longitudinal stability of an aircraft — that’s the stability about the lateral axis, the pitch axis. We’ll come back to that in Chapter 10. Now let’s look at a special case: the symmetrical aerofoil. Look at what happens to the pressure distribution as angle of attack changes. At zero angle of attack, the upper and lower surface forces are equal, and they’re located at the same point. So there’s no net pitching tendency from asymmetry. Now increase the angle of attack. The upper surface force increases, and the lower surface force decreases. So a change in the magnitude of lift has taken place — but here’s the crucial point — there’s no change in the CP position. That’s a characteristic of symmetrical aerofoils. The centre of pressure doesn’t move with angle of attack. And that gives us a beautiful result: the pitching moment about the AC for a symmetrical aerofoil is zero at normal angles of attack. That’s one of the big advantages of symmetrical aerofoils — no built-in pitching moment to trim out. So to tie it together: for a cambered aerofoil, the CP moves with angle of attack, and the moment about the AC stays constant because the lift increase is offset by the shortening arm. For a symmetrical aerofoil, the CP doesn’t move at all, and the moment about the AC is simply zero. Both cases, the AC sits at the quarter-chord point in subsonic flow below Mach 0.4.