aerodynamics
I understand the common explanation of lift, which describes the airflow over the top of the wing as moving faster than the air below the wing. However, I don't quite understand why the air moves faster.
I've read some explanations citing a circulation effect among others, but haven't found a good, clear explanation for the exact cause. Can someone help me out?
My apologies, I won't be reading your entire question.
But still I will provide an answer. Why is that? Because flight does not require any of the things you talk about.
You could build an airplane that would fly with no "airfoil" shapes. You could build an airplane that would fly with completely flat rectangular wings made out of plywood. The important thing would be the angle of attack of the wings to the air. Consider a flat piece of wood, like plywood. Push it through the air in a direction exactly parallel to its flat dimensions and it develops no lift. Tilt the wood so the leading edge is "up" compared to the direction it is moving and you can feel the lift.
The lift can be thought of a few ways. Think of the air molecules hitting the surface of the wood. They bounce off, in a downward direction. Well if we are pushing air downward, we must have an equal and opposite force, which is the lift. Or another way: we are gathering air under the board, it gets a little pressurized. The pressure is pushing up on the wood. This is really the same picture as the first if you think about it.
All the rest with airfoils and so on, all this has to do with developing lift efficiently, developing lift while minimizing drag. An airplane with flat plywood wings would fly, but it would have a lot of drag and would therefore be very inefficient.
The first question you need to ask is: does an irrotational, inviscid, incompressible fluid really exist?
The answer is no (well, yes, sort of, if you consider super-fluids). The irrotational, inviscid, incompressible fluid is a mathematical creation to make the solution of the governing equations simpler.
Lift cannot exist without viscosity! That's is the most common misconception that comes from an undergraduate aerodynamics course. So it bears repeating. Lift cannot exist without viscosity.
When we look at potential flow though, we get pressure differences and these pressure differences result in lift, so what gives? First, the potential equations don't actually hold until the starting vortex is sufficiently far away. The discussion of sufficiently far away is, again, a vague concept. But it involves determining the velocity induced on the wing by the starting vortex using Biot-Savart law. Essentially it is "far enough" away when the induced velocity is small relative to the other velocity magnitudes in the problem. Viscosity causes this starting vortex to appear and this starting vortex is what causes pressure differences.
Additionally, in the absence of viscosity, circulation is conserved around a closed path. This is no problem if we make our domain large enough to include the starting vortex. However, we can't actually solve for the starting vortex with the assumptions made to get the potential equations, so we have to omit it from the domain. This means we need to have some sort of circulation within our domain and this is what becomes the bound vortex.
Here is an illustration (forgive me, I am most decidedly not an artist):
At start up, viscosity causes the starting vortex to be shed and it proceeds downstream. Potential equations cannot deal with this situation because they lack the viscous term. It's just not something they can predict. However, in the free-stream the flow behaves as if it were inviscid. So once the starting problem is overlooked, this vortex will persist forever because nothing will dissipate it. If we take that solid outer line as a control surface, we can integrate around it and find that there is no circulation. So Lord Kelvin can rest easy.
But, since this vortex lasts forever, it's not possible to track it forever or the solution to the problem becomes very expensive. And we are (usually) interested in the steady state solution (although unsteady potential solutions are also possible). So we make an artificial cut in our domain, that's the dashed line. When we make that cut, the integral of vorticity around the sum of the two smaller control surfaces must still be 0. This means that the vortex bound to the airfoil has a circulation equal in magnitude and opposite in direction to that of the starting vortex.
During this start up process, very large velocity gradients exist at the trailing edge. This is what causes that vortex to be shed. Once the vortex moves away, the velocity gradients become smaller and smaller, eventually reaching zero. This zero-gradient condition is handled automatically by viscosity, but it must be enforced in the potential equations through the Kutta Condition.
The reason we need the Kutta condition is purely mathematical. When the inviscid assumption is made, the order of the governing equations drops and we can no longer enforce two boundary conditions. If we look at the incompressible, viscous momentum equation:
$\frac{\partial u_i}{\partial t} + u_i\frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho}\frac{\partial P}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_j \partial x_i}$
we can enforce two boundary conditions because we have a second derivative in $u$. We typically set these to be $u_n = 0$ and $u_t = 0$, implying no flux through the surface and no velocity along the surface.
Dropping the viscous term results in only having the first derivative in $u$ and so we can only enforce one boundary condition. Since flow through the body is impossible, we drop the requirement that tangential velocity be zero -- this results in the slip boundary condition. However, it is not physically correct to let this slip line persist downstream of the trailing edge. So, the Kutta condition is needed to force the velocities to match at the trailing edge, eliminating the discontinuous velocity jump downstream.
John Anderson Jr explains in Fundamentals of Aerodynamics (emphasis in text):
... in real life, the way that nature insures the that the flow will leave smoothly at the trailing edge, that is, the mechanism that nature uses to choose the flow... is that the viscous boundary layer remains attached all the way to the trailing edge. Nature enforces the Kutta condition by means of friction. If there were no boundary layer (i.e. no friction), there would be no physical mechanism in the real world to achieve the Kutta condition.
He chooses to explain that nature found a way to enforce the Kutta condition. I prefer to think of it the other way around -- the Kutta condition is a mathematical construction we use to enforce nature in our mathematical approximation.
The explanation of flow over the top needing to go faster to keep up with flow on the bottom is called the Principle of Equal Transit and it really is not a great way to present the problem. It is counter-intuitive, has no experimental validation, and really just leads to more questions than answers in most of the classes it is discussed.
To sum all of this up and to directly answer your question: yes, wings do have lift in incompressible (and compressible), irrotational, inviscid flow. But only because the potential flow equations are a mathematical abstraction and the Kutta condition is a mathematical "trick" to recover a solution that generates lift under those conditions. Of course, not just any wing will have lift. A symmetrical wing at zero degrees angle of attack will not have lift.
Best Answer
Addon by Alex Qvist: When the air hits the front of the wing it flows in a steeper curve upward, than the bottom wing flow, This creates a vacuum on top of the wing, and this pulls more air towards the top of the wing, this air does the same thing but moves faster because of the vacuum pulling it in, and then the vacuum of course lifts the wing.
Main post:
The common explanation given is that it flows faster over the top of the wing because the top is more curved than the bottom of the wing. However, I understand why you would find this explanation unsatisfactory.
To start with, I think we need to identify the point at which the flow separates. Looking at Wikipedia, I'll post two images:
The argument that the wind flows faster over the top is mostly a consequence of geometry. First identify the point at which the flow separates, meaning the point above which the fluid goes over and below which the fluid goes under, this is slightly below the front-most point of the airfoil, due to the fact that it's angled slightly upward. If both paths take roughly the same time to pass over the wing, then the average velocity of the fluid from the point of separation to the tail where the flow rejoins will be roughly proportional to the distance from those two points.
Now, you may say, "but it will flow faster over the top even if the top isn't curved more!" You would be correct. A plane can function with no additional curve on the top of the wing, as the famous xkcd comic points out. Such a mode of flying, however, will still see the fluid passing over the top of the wing faster. A simple argument for this is that the point of separation is lower than the front of the wing, again, since the wing is angled up. A plane can fly upside down, but I don't know of a plane that can maintain altitude with the wings not angled up. The curved top, however, increases efficiency by intensifying that natural effect.
I hope that helps some, this is intentionally not a rigorous answer, and I want to recognize that I am not addressing the more hairy details of the actual fluid equations associated with this, which are required for a full explanation. In short, the fluid velocity over a surface isn't completely proportional to the distance traveled. Even without getting into that, however, I think your question is mostly answered.
Another Attempt
I realize that my answer up to this point may not only be incomplete, but might not answer the question. The question is why the flow on the top is moving faster than on the bottom. Let me post another image.
There are 2 things I want to note here.
Number 2 is particularly important because it is simply not correct to say that the speed is proportional to the distance between the separation and rejoin point. That way of looking at it may still have some usefulness. But I digress.
At this point I'm repeating Wikipedia's explanation, but refer to the 2nd image in this answer. Under some assumptions the fluid does not cross the blue streamlines. That means that when the channel size between 2 blue streamlines narrows or widens, the fluid speed changes correspondingly. I'm still hand waving away plenty of technical detail, but please let me offer this as basic level answer.
The fluid on top of the wing is accelerated and the fluid on the bottom of the wind is slowed down compared to velocity of the aircraft itself because the wing geometry and angle narrows the flow area above the wing and widens the flow area below the wing
This is the absolute best explanation I have. If you assume that the fluid is incompressible it works great, if not, it works less great but still works. There are also some other assumptions, I hope that the general point is still the same with all those included. The bold text is the best answer I have and I think it's a good one.