Basic Aerodynamic Theory — Page 44, Lesson 54

Let’s start with the heart of this chapter — Bernoulli’s Theorem. I want you to hear the exact wording first, because it’s a definition you’ll carry for your whole career: “In the steady flow of an ideal fluid the sum of the pressure energy and the kinetic energy remains constant.” Now, that word “ideal” is doing real work. An ideal fluid is both incompressible and has no viscosity. Incompressible means its density doesn’t change as pressure changes. No viscosity means there’s no internal friction, no stickiness between the layers of the fluid. Real air has a little of both, but for the theory we treat it as ideal. The statement can be written as an equation: Pressure plus Kinetic energy equals a Constant. In symbols, that’s p plus one-half rho V squared equals a constant. Let me unpack each symbol. The letter p is static pressure. Rho, the Greek letter that looks like a curly p, is air density. V is the velocity of the airflow. So one-half rho V squared is the dynamic pressure — the pressure you get purely from the air’s motion. Let me give you a concrete worked example, because the numbers matter. Take a mass of air with a static pressure of 101,325 newtons per square metre, a density of 1.225 kilograms per cubic metre, and a velocity of 52 metres per second. Its dynamic pressure works out to 1,656 newtons per square metre. The calculation is one-half times 1.225 times 52 times 52. So now, pressure of 101,325 plus kinetic energy of 1,656 gives you a constant of 102,981 newtons per square metre. Notice every term is in the same units — newtons per square metre, which is a pressure. Now here’s the key insight from the figure. If you have a tube that narrows — a throat — and the velocity of the air at that throat doubles, then the dynamic pressure rises by a factor of four. Why four? Because velocity is squared in the dynamic pressure term. Double the velocity, square it, and you get four times the dynamic pressure. And because the total stays constant, the static pressure must decrease to compensate. So the significant point is this: Static Pressure plus Dynamic Pressure is a constant. And that constant has three names you must know — they all mean the same thing. It’s called TOTAL PRESSURE, or STAGNATION PRESSURE, or PITOT PRESSURE. You’ll meet the pitot pressure again when we talk about the airspeed indicator, because that instrument literally measures this total pressure. From this, you can see that flow velocity depends on the shape of the object the air flows over. And from Bernoulli’s theorem, an increase in velocity causes a decrease in static pressure, and vice versa. That “vice versa” is important — slow the air down and static pressure rises. Now, a caution. Those tubes I just described — the ones used to demonstrate the principle of continuity and Bernoulli’s theorem — are of no practical use in making an aeroplane fly. They’re just teaching tools. But we can generate an aerodynamic force to oppose the weight of an aircraft by using a specially shaped body called an aerofoil. Here’s how it works. Over the top surface of a lifting aerofoil, the airflow velocity is greater than the airflow beneath it. Because of Bernoulli, that higher velocity on top means lower static pressure on top. So you get a pressure differential — lower pressure above, higher pressure below — and that produces a force per unit area acting upwards. The larger the surface area, the bigger the force that can be generated. That’s the lift force, and it’s what opposes the weight. And here’s a nice connection: the flow over the top of the aerofoil looks very much like the tube on the opposite page — the one with the throat. The principle of continuity and Bernoulli’s theorem still apply to the airflow over the wing. Now let me introduce you to streamlines and the streamtube, because these are the tools we use to visualise that airflow. A streamline is the path traced by a particle of air in a steady airflow, and streamlines cannot cross. If two streamlines crossed, that would mean one particle of air was at two places at once — impossible. Here’s the reading rule. When streamlines are shown close together, that illustrates increased velocity. When they’re spread apart, decreased velocity. Diverging streamlines — spreading apart — illustrate a decelerating airflow and a resultant increasing pressure. Converging streamlines — coming together — illustrate an accelerating airflow with a resultant decreasing pressure. So the spacing of streamlines is your visual shorthand for both speed and pressure. Finally, a streamtube is an imaginary tube made of streamlines. There is no flow into or out of the streamtube through the “walls” — only a flow along the tube. Think of it as a pipe with invisible walls. With this concept, you can visualise the airflow around an aerofoil as being within a tube made up of streamlines. That’s the mental model we’ll build on next.