I'm going to write about things I find interesting. Because that is what I do, and because I have permission.

Everybody has heard about how a wing generates lift. It's simple. Airfoils have a funny shape, where the top longer than the bottom. Because the air on top has to 'catch up' to the air on the bottom, it has to travel faster, and by Bernoulli's Principle, it has a lower pressure. This difference in pressure pulls the plane up into the sky. Simple and bulletproof. Unless some smartass at the back of the class asks why Maverick and Goose can fly their F-14 upside down. The sad thing is, not only is this myth false, the truth is simple enough to explain to most adults; so it doesn't even fill that void. It will also usually vastly underestimate lift. Not only can the air on top 'catch up', it usually moves a substantial degree faster. There are formal methods to show the myth of equal transit time to be false. Any engineering or science student of 2 or 3 years experience could whisk the idea away with Reynold's Transport Theorem if they were so inclined. But you needn't have heard of Osborne Reynolds nor have the mathematical faculties to express his theorem to understand its implications. If you imagine a wing at zero angle-of-attack (that is, it's front-most edge and its back-most edge are at the same height), the idea of equal transit time would have you believe that the wind splits around the front, attaches at the rear, and continue on like nothing had happened. But Newton would say, if you want to impart an upward momentum on the wing (lift), you have to impart a downward momentum on the air. And I'm not prepared to call Newton a liar. The modern understanding was developed in the late 19th century by three men: Kutta, Joukowski, and Kelvin. Although with modern computers we can calculate better results than they could, as far as a qualitative overview is concerned, there is nothing better even today. The problem with predicting lift in the 1800s was viscosity. Viscosity was, then and today, a numerical troublemaker. But in those days, they couldn't just bully numbers into submission with calculators. They had to be clever. In his lab, Wilhelm Kutta was observing the lift produced by wings, and he observed something quite curious. It had been seen before many times, and had gone without explanation. It was the odd condition of a trailing streamline propagating out from the trailing edge of the wing. A streamline, in this case, is a path traced by a particle in time. There are two very important streamlines for a wing, one at the front and one at the rear. These two lines are unique in that they are the only two which actually touch the surface of the wing, at what are known as stagnation points (points where the fluid velocity is zero), and together the two represent the border between the air that moves 'over' and the air which moves 'under'. At the time it was common to completely neglect viscosity in calculations and analysis. And so the theory of the time predicted that if a wing had an angle relative to the air, the front stagnation point would be on its bottom side some ways from the leading edge, and the rear stagnation point would be on its top side some ways away from its trailing edge. While the front stagnation point was predicted correctly, that the rear stood right on the rear edge was surprising. Everyone suspected that it had something to do with neglecting viscosity, but nobody quite put their finger on how it worked. But Kutta made a second interesting observation. If a wing met the wind with some angle, it made lift, and if that angle was increased, the lift was increased. But as that angle increased, a vortex formed on its trailing edge and shed away. This may be innocuous enough to you and I, but Kutta was a contemporary of Lord Kelvin, the same man for whom the temperature scale is named. Kelvin had something else that was named after him though. Kelvin's Circulation Theorem. Without getting into the finer details of what circulation means, the idea is that it has to be conserved like mass or energy. If you create a vortex in a fluid, another spinning the opposite way must be produced. In reality it isn't always just a mirrored vortex next to the first; sometimes its a property of a propeller, for example. But it's always there. So if a vortex formed off the rear of a wing, the question asked must be 'where is its mate?'. The Kutta Condition, as it is known, elegantly combined these observations: a wing (or at least those with a sharp trailing edge) produces a vortex bound to its surface strong enough to maintain the rear stagnation point at the trailing edge. This isn't a vortex in the idea of a tornado spinning around the wing. Just a force in support of the air above the wing, and an antagonist to the wind below. Today, we know that this simple statement beautifully encapsulates a vast amount of viscous effects to a remarkable degree of accuracy. It is intuitively obvious as well; if the fluid has any viscosity it simply can't wrap around a sharp, knife-like rear edge of a wing. Especially if you imagine the particles closest to the wing violently changing their direction as they reach the rear edge. Bernoulli was never wrong. People just misused his work. The Bernoulli Equation can still be used to calculate the lift of a wing, if it's paired correctly with the Kutta Condition. But lukily, Kutta and Joukowski did the heavy lifting for us. The Kutta-Joukowski Theorem states simply that the lift of a wing (per unit span) is the product of the density of the fluid around it, the velocity of the fluid, and the strength of the bound vortex. Though not as simple as the Bernoulli Equation, the strength of the bound vortex can still be calculated with some care and college-level mathematics. This explanation is clearly not as short or simple or sweet as the myth of equal transit time, but looking back the Kutta Condition is no harder to imagine than the idea that the air on the top and bottom of a wing must reach the rear at the same time. Especially for the inquisitive minds that ponder why that must be, when there is no real explanation given.

I wrote that from memory, and kind of drunk, so there might be a mistake or two in there. Here's a picture that explains it really well. It's kind of interesting going back in time in fluid mechanics. The Navier-Stokes equations have been around for two hundred years, but because of their complexity to solve, they were always this sort of untouchable thing people knew was correct, but was always out of reach. And it was always viscosity. Viscosity always ruined everybody's party and people always looked for a way around it. The history of fluid mechanics is like a superhero film and viscosity is the villain. Prandtl, Munk, Betz, Kutta, Glauert, Von Karman: they're the heroes. But its never vanquished, it just doesn't die. Between the Kutta Condition and totally inviscid flow theory (known as Potential Flow), you could get a lot done, but there was also a lot of limitations. But as our mathematical prowess and tools increased, we could always strip away layers of abstraction. In the early 20th century Prandtl did his work on boundary layer theory, which opened up many new doors. 50 years later, two employees of the Douglas Aircraft Company invented the Panel Method, which allowed us to model large and complicated shapes. For the Douglas Company that meant they could compute loads on complete aircraft, rather than just wing sections or other small bits that Prandtl's work allowed. Since 1985 or so, numerically solving the Navier-Stokes equations is a possibility. Its commonly known as computational fluid dynamics (CFD). But guess what? Solving the equations directly is still nearly impossible. Not because we don't have the number-crunching power, but because it's not numerically stable. Its very difficult to trust any results you do get. So-called 'Direct Numeric Simulation', or DNS, is limited to the simplest problems - like flow around a cylinder - at incredibly small rates of flow. If you're familiar with Reynolds' Number, we're talking numbers smaller than 2000. And if you're not familiar with Reynolds' Number, an average wind turbine on calm day has a Reynolds' Number of 100,000,000. So we don't solve or calculate viscosity's effects anymore, we model it. Like how economists model people as completely self-interested, heartless assholes; because we're incalculable. We don't model viscosity directly; viscous instabilities are known as turbulence, and we model that. We use things like the 'Reynolds-Averaged' Navier-Stokes equations, which is basically a time-average. We use esoteric stuff to force the effects we like, and they have names like k-epsilon and shear-stress-transport, neither of which can I claim to understand. But they all have the same effect. Its like going over your data with a rolling pin. Flattening it out so that it doesn't shoot off to infinity when you're trying to solve it, but in the process destroying plenty of things that might have been really interesting.