Bernoulli’s Principle and the Equal Transit Myth
Like most kids that watched Top Gun, I thought airplanes were pretty cool and wanted to be a pilot. When I tried to look into the encyclopedia to understand how airplanes fly, I was given an explanation that depends on Bernoulli’s principle. But what is Bernoulli’s principle?
Bernoulli discovered in 1738 that as a fluid’s speed increases, its pressure decreases. Intuitively, this makes some sense. Imagine you are a particle of air approaching a high-pressure region. As you approach it, you start crashing into more and more other particles since they are packed more tightly together there. Crashing into particles is a drag and slows you down. If, on the other hand, you are approaching a lower pressure region, then there are fewer particles in front of you to counter the force of those crashing into your posterior. As a result, you accelerate into the low-pressure region. In practice this phenomenon works in both directions. If you cause an increase in pressure you will cause a decrease in speed. If you cause a degree in speed you will cause an increase in pressure.
That’s fine, but what does it have to do with airplanes? Well, the conventional explanation asks us to look at a cross-section of an airplane wing. Since the wing is shaped so that there is a longer curved surface on top and a shorter, straighter surface on the bottom the air across the top has to move faster in order to reach the trailing edge at the same time. Thanks to Bernoulli we know that if the air on top of the wing is moving faster than the air on the bottom, it will have a lower pressure. Since there’s more pressure below the wing than above it, the result will be to push the wing, and whatever’s attached to it, into the air. This explanation is sometimes called the “equal transit myth” and it is plain wrong.
Even as a child, I could never understand what principle compelled the air on top of the wing to reach the trailing edge at the same time as the air on the bottom. In fact, there is no such principle. For one thing, consider that unlike an airplane wing, a sail is nearly two-dimensional: the air flowing over the windward surface travels almost exactly the same distance as the air flowing over the leeward surface. If this conventional explanation for lift were correct, sails couldn’t produce lift, but we all know that isn’t true. Interestingly, experimental evidence has shown that not only do particles going over the two sides of a wing get to the trailing edge at different times, but in fact the particles that go over the top (or leeward side on a sail) reach the trailing edge long before their siblings on the other side. Bernoulli’s principle is still true, but the equal transit theory isn’t the reason that the particles on the leeward side are going faster.
Air Circulation and Starting Vortices
So let’s forget about that equal transit and start from the beginning. Imagine a sail that’s luffing in the wind. The air pressure is equal on the top and bottom of the sail. Now start sheeting the sail in slowly. All of a sudden, the sail is catching air on the windward side and deflecting it. This causes a higher pressure and hence lower speed region on the windward side. However, this isn’t lift. It’s drag. Right now all that pressure is in the same direction as the wind, which is of no use in going upwind. As those particles reach the trailing edge of the sail, the pressure difference between the windward side and the leeward side will be so great that the air coming off that edge will swirl into a tiny tornado, called the starting vortex. This vortex is absolutely essential to setting up the conditions for real lift to obtain.
Thanks to the Baron Kelvin, we know that angular momentum in a bounded fluid is conserved. In other words, when the starting vortex is created by our sail, there must be another vortex that spins in the opposite direction but with the same strength. This is called Kelvin’s circulation theorem. By some miracle of physics, the equal-and-opposite vortex is centered on the sail itself, and induces the air around the sail to go even faster over the leeward side and even slower over the windward side. This further enhances the pressure difference between the two sides of the sail and begins to generate lift proper. Now that it isn’t just the force of the wind hitting the sail but the force of the wind being bent around the sail that can push a boat in directions other than straight downwind.
The insight that there must be a circulation field to help drive the flow of air around the sail is called the Kutta-Joukowski theorem. Its major limitation is that it only explains the flow of a two-dimensional fluid. A lot of strange phenomena happen at the edges of airfoils that have to contend with three dimensions, but I’ll have to move on to those later.
Science in your Home
Most of this information is a rehashing of the explanations put forth by Arvel Gentry. He describes a simple experiment that you can perform in your bathtub in order to see the circulation fields around a sail-like airfoil. I really recommend trying this out at home.
Fill your bathtub with just a few inches of water. Not too much, just three inches or so should do. Then, sprinkle some kind of dark powder over the surface of the water. Gentry recommends pepper. I used ground nutmeg with success. Now, fashion an airfoil. A simple one can be made from a small piece of waxed cardboard like a milk carton. Start with a flat piece and bend it very slightly into a gentle curve. It shouldn’t curve as fully as a laser’s sail; Because water is so much thicker than air it doesn’t take as much curve in the airfoil to generate the same lift. Make sure the piece is tall enough to reach completely to the bottom of the tub and still be sticking up above the surface of the water. Place the cardboard airfoil into the water near one edge of the tub (although don’t let it touch the edge). Place it so the length of it is pointing to the other edge of the tub and parallel to the long axis. Now, move it in a steady line across the length of the tub and lift it out of the water when it’s still a decent distance from the other edge. Thanks to the spices in the water you should be able to see a vortex left behind where you first started the motion of the airfoil. That’s the starting vortex. When you lift it from the water you should see another swirl, opposite in direction and also larger and slower than the starting vortex. This is the circulation vortex that was following the airfoil along and is left behind when it is lifted from the water.
The whole operation should look a bit like this. Note the starting vortex that trail behind each time the wing is accelerated, and the effect of circulation around the particles just in front of the wing. Further experimentation can be had right in your browser thanks to some folks at NASA. I definitely recommend playing with that for a few minutes.
Let’s Be Honest, This Doesn’t Explain Everything
The biggest problem in this model of lift is that there is no explanation of why the circulation field that gets kicked off by the starting vortex should follow the airfoil as it moves through the airstream. It’s just as much a bit of voodoo as the old equal transit explanation’s assertion that two particles just had to get to the back of the wing at the same time. According to some new research, this circulation field doesn’t even exist, although after doing the bathtub experiment, I’m inclined to believe my eyes. As mentioned above, this description of things is inherently two-dimensional. The effects of a third dimension are harder to understand but also important to getting the most out of a sail, a topic that we’ll cover in future installments. Until then, stay cool.