How complete is our understanding of lift?

Lift is generated because air molecules are bouncing into and rebounding off of the airfoil, on both top and bottom surfaces. It is the difference in the amount of momentum transferred in these collisions that creates lift. It is, (obviously), only the velocity of the air molecules that is normal (perpendicular) to the airfoil, that produces lift.

The Bernoulli principal is true, because the TOTAL average momentum of any air molecule in incompressible (subsonic) flow is a constant. Therefore, if the velocity of the air parallel with the airfoil increases, the normal component of the velocity must decrease to keep the total constant.

So, if the air is moving faster, the normal component must be slower, and it's pressure (against the airfoil) must be lower.

So, the longer distance to travel argument is only bogus if you try to assume that it can only be generated by a asymmetrical airfoil. Other things can change the travel distance (and resultant velocity) of the air across the airfoil as well. If a symmetrical airfoil is inclined to the relative wind, then as the air flows across the airfoil on the side where the airfoil bends away from the flow, the air must travel a longer distance (to fill in the gap created by the inclination) than air flowing across the surface on the other side, where the surface is inclined into the relative wind, and must either compress (supersonic flow) or move away from (change direction away) the airfoil.

This is because in subsonic, (incompressible), flow, the air cannot make an instantaneous change in direction when it gets to the leading edge of the airfoil. If the Angle of attack was 10 degrees, the air does not make an instantaneous 10 degree change ion direction. From the point of the leading edge away from the airfoil, the change in direction, and the resultant pressure, gradually changes as you move further away. the result ios that the flow of the air is following a curved path, and travelling a longer distance, on this side of the airfoil, than it is on the other side, even for a symmetrical airfoil.

Scientifically speaking, lift is perfectly understood. Lift is merely the vertical component of force generated by a body moving through a fluid. And we know perfectly well how to calculate forces on a body moving through a fluid since the Navier-Stokes equations was published in 1822. That is to say, we know the physics of it and it has to do with the viscosity of fluids (in the case of aircraft, air).

But using the Navier-Stokes equations to design a wing is like trying to use Quantum-Electrodynamics (QED) to cook the perfect steak. Since gravity isn't involved in the perfectionness of steak, all you need to formulate a perfect steak is QED.

The Navier-Stokes equations calculate forces on a single point on the wing. Therefore you have to repeat the calculations over the whole wing to calculate lift. Over the last 190+ years mathematicians and engineers have formulated simpler algorithms to calculate the result of the Navier-Stokes equations and over the last 30 or so years we've used computers to calculate lift. However, you can see how this doesn't tell you the ideal shape to generate the aerodynamic characteristics you want. You can also see how this doesn't explain "lift" in terms a human can understand. It's all just large arrays of numbers.

Is it possible to explain lift in terms a human can understand? Maybe. We've certainly given names to how certain shapes generate certain output when subjected to the Navier-Stokes equations. Names such as "Coanda effect" and "Bernoulli Principle" etc. In the end, nature/physics doesn't care what name we give to our interpretation of the result of the Navier-Stokes equations - if calculating the equations result in a vertical force vector upwards you have lift. Maybe, like quantum physics, we'll never get a complete intuitive understanding of what lift is. But we certainly have the complete theory to explain it.

Additional note: Apart from not being helpful in helping us formulate a theory of wing design, the Navier-Stokes equations are also problematic because it's computationally expensive. For example, it's often not practical to use the Navier-Stokes equations to simulate turbulence (even though it's possible in theory). So we often take shortcuts for certain forms of simulations using other simpler but less perfect equations.

TL;DR: we can very precisely model aerodynamic forces at the micro level; we can reasonably predict behavior at the macro level by aggregating micro-level models (CFD). We don't have a universally-applicable story for why the macro level behavior is what it is.

Fuller explanation:

At the risk of being a bit pedantic, I'm going to back up a couple of steps of abstraction in order to provide a more-complete picture.

The overall aerodynamic force on a body is decomposed into vectors normal to the direction of travel and parallel to the direction of travel, which are labeled 'lift' and 'drag' respectively; they are not distinct forces in & of themselves. Aerodynamic force itself is often decomposed at a different scale into pressure & friction; for the most part, friction only contributes to the 'drag' component while pressure contributes to both the 'lift' & 'drag' components.

Trying to tell a stylized story about why the integrated pressure & friction across the entire body result in a particular net force is challenging at best, since it is affected by the idiosyncrasies of each body; various models (such as Venturi, downwash, & circulation) really just provide designers & analysts with rough rules of thumb within particular flight regimes.

This last point is more important than it appears. As soon as you enter transonic flight (a mix of subsonic & supersonic flow at the surface of the body), drag increases precipitously (standing shocks creating adverse pressure gradients). Passing through to fully-supersonic flight you find yet another set of behaviors (because the leading shock radically alters the pressure distribution on the body). Don't even get me started on hypersonic flow (where the temperature change across the shocks is enough to decompose the N2 & O2 from the air itself).

Edit Peter Kampf's answer covered most of the same topics as mine, with pictures, so I'll just add this for completeness:

The principles of aerodynamics and fluid dynamics are what you would call "well understood."

The ambiguity is around what so-called "lift" is, which can be a nebulous concept. For example, if you drop a piece of paper it will drift slowly to the ground, essentially a form of gliding; this same air resistance is the basic force keeping a plane aloft. Would you consider this "lift"? Once you get into these arguments about semantics, things get vague.

Just as one example of the craziness, the FAA test, the same one you are taking, requires you to know the "four forces of flight" in which the so-called "lift" is the force that keeps the aircraft aloft. The only problem is that you can compute lift by equations that are in every book on aerodynamics and if you actually do this (like I did) you will find that the force generated is nowhere near enough to keep a plane in the sky. If "lift" were the force keeping a plane up, it would fall like a rock, so the FAA guidelines are simply completely wrong. It's just a huge semantic hairball that is not going to go away anytime soon.

The worst part of it is that EVERY pilot (or wannabe pilot) I have ever known thinks they know exactly what "lift" is and, even worse, their beliefs generally fall into one of 5 or 6 different categories with contradictory principles. This leads to huge arguments whenever the subject comes up. After 15 years of this, I just try to stay out of it, other than to tell the beginners not to make the same mistake (like I am telling you now).

How complete is our understanding?

• Complete enough to design and fly a number of complex aircraft of varying sizes, shapes and applications.
• Complete enough to extract power using it.

On a basic level, lift is the force generated as solid body 'turns' fluid while satisfying the conservation laws. The issue is not that we don't know what lift is, but there is no consensus about how to explain it. Most of the 'theories' of lift are just models try to explain the same thing based on the points of view of the people involved.

The way the pilot views the lift is different from that of an engineer or a mathematician. For the pilot, the lift is a force that keeps the aircraft in the air (and proportional to $rho V^{2} S$ and angle of attack, at-least till stall), while a mathematician can say that lift 'follows naturally' by solving Navier-Stokes Equation (whether it can realistically solved or not is another matter) for some conditions. Of course, this is of no practical use to either the engineer or the pilot. Both can claim (rightfully) that they are correct, while a physicist can object that NS assumes that the fluid is a continuum, while it is not the case in realty.

This is the reason for so many theories of lift. As the fluid flow is extremely complex, some simplification is done in each theory (like omitting viscosity in the Euler or potential flow theory). Based on the simplification, the theory is either useful in some (or most) of the situations or is outright wrong.

Which bits are still in dispute, and which bits are fully accepted?

Almost all 'theories' of lift accept that lift is a force and its requirements. As far as engineering goes, the issue is which bits of are necessary for the problem in hand.

For example, the potential flow theory can predict the lift as long as we are not approaching stall. After that all bets are off. There is no point in arguing about a result from a theory after using it in a situation for which it was not designed for in the first place.

This is the reason for arguments about lift. Some theories are developed to describe a particular situation (for example inviscid flow) and then applied in general, which obviously leads to confusion and dispute.

As far as engineering is considered we have enough understanding of lift to create the flying machines we need though not as much as to explain everything that happens with accuracy.

The problem here is that "correlation does not imply causation". Neither Bernouilli's principle nor Newton's laws of motion explain lift. Both of them give valid methods of calculating the lift force from the air flow pattern around the wing, but neither of them explain why the flow pattern is what it is.

Ideas like "equal transit time" at least try to give a reason "why," but experiments which visualize the flow pattern with smoke demonstrate that is just wrong.

The best "one-word explanation" of what causes lift is the viscosity of the air. Viscosity is the reason why there can't be any discontinuities in the overall flow pattern*. In particular, the air velocities on either side of the relatively sharp trailing edge of the wing have to be the same, otherwise the effect of viscosity at that point would propagate upstream through the air (at the speed of sound) and change the global flow pattern.

If there were no viscosity, no wing of any shape would produce any lift, or any drag force.

*Let's limit this discussion to subsonic flows. Introducing shock waves into the airflow makes a "hand-waving" non-mathematical discussion more complicated, but it doesn't invalidate the essential point I'm trying to make.

From this paper:

The principle of equal transit times holds only for a wing with zero lift. [!!]

[...]

The air passes over the wing and is bent down. Newton’s first law says that them [sic] must be a force on the air to bend it down (the action). Newton’s third law says that there must be an equal and opposite force (up) on the wing (the reaction). To generate lift a wing must divert lots of air down.

[...]

So how does a thin wing divert so much air? When the air is bent around the top of the wing, it pulls on the air above it accelerating that air downward. Otherwise there would be voids in the air above the wing. Air is pulled from above. This pulling causes the pressure to become lower above the wing. It is the acceleration of the air above the wing in the downward direction that gives lift.

We (those of us reading this) can conlude the following about our (humanity's in general) understanding of lift:

• We certainly understand it well enough to design aircraft, and there may be overlap with this knowledge in other areas, such as maybe wind-powered generator design.
• Many believe we have a fairly complete understanding of lift.

The second bullet is not at all in impune the excellent (and challenging!) work done through history in fluid dynamics, aeronautical physics, and aeronautical engineering. It is merely to allow for the possibility of future paradigm shifts in our understanding of those topics, even if those shifts do not affect common design practice or practical discussions of lift. A historical example of that last point would be General Relativity as a paradigm shift in our understanding of gravity, while Newtonian gravitational theory was still used for the moon program and is still widely taught and used for situations not requiring extreme precision.

In addition to links in Frederico's comment, see also: https://physics.stackexchange.com/questions/290/what-really-allows-airplanes-to-fly

Edit:

This NASA page discusses the controversy of "Bernoulli versus Newton" and concludes that both explanations of lift are "correct" and that there is even more to it. The Euler Equations and the Navier-Stokes Equations are mentioned. This page in the same series on NASA's site suggest that lift is fairly well understood by experts, but is poorly explained in the majority of popular sources:

There are many explanations for the generation of lift found in encyclopedias, in basic physics textbooks, and on Web sites. Unfortunately, many of the explanations are misleading and incorrect. Theories on the generation of lift have become a source of great controversy and a topic for heated arguments. To help you understand lift and its origins, a series of pages will describe the various theories and how some of the popular theories fail.

Lift occurs when a moving flow of gas is turned by a solid object. The flow is turned in one direction, and the lift is generated in the opposite direction, according to Newton's Third Law of action and reaction. Because air is a gas and the molecules are free to move about, any solid surface can deflect a flow. For an aircraft wing, both the upper and lower surfaces contribute to the flow turning. Neglecting the upper surface's part in turning the flow leads to an incorrect theory of lift.