How do aeroplane wings work? [duplicate]

Question

I wonder why this question has not been posed on this forum. Is it such a dumb question that nobody needs to ask because the answer is obvious?

A search on the internet for "How do wings work" yields millions of hits, each one bafflingly different from the next.

I would have thought that any answer would include a working example, a simple back-of-the-envelope set of calculations that show that applying real life numbers to the offered theory would show how it works. After all, that's how most science works. You have an idea, you put numbers to it and then see if it works out in practice.

• That's how Kepler demonstrated his theory of planetary orbits.
• That's how Boyle demonstrated the relationship between the pressure and volume of a gas.
• That's how modern physicists used Einstein's theory to explain the variation in the orbit of Mercury.
• Whether or not Galileo actually used different sized canon balls to demonstrate his theory is open to debate.
• That's how the Michelson-Morley experiment disproved the theory of Ether

The list goes on and on. The scientific process follows a well-worn path:

1. A theory is developed to explain a physical phenomenon
2. The theory is developed into a mathematical equation
3. Numbers are applied to the theory,
4. Experimentation demonstrates the validity of the theory.

Further experimentation maybe supports the theory, disproves the theory or demonstrates the limitations of the applicability (Newton remained unchallenged until modern physics started to investigate domains beyond human experience, of the very fast, the very heavy or the very tiny.)

Amongst the myriads of explanations for how wings work there appear to be two major competing theories: one invoking the Bernoulli principle to explain the pressure variations around the wing which generate the force and one relying on the momentum change of the disturbed air to account for the aerodynamic force, the vertical component of which balances the weight of the aircraft.

In my search for an answer, I have been unable to find any even "back-of-the-envelope" calculations, using the mathematical equations derived from the theory, real-life numbers and experimental data to support any of these theories. Is there any paper, journal of textbook which offers a theory and supports it with worked example using real-life data?
For example, one which could demonstrate how a 1-tonne Cessna 172 remains in straight and level flight at 5,000 ft above the ground at 100 kts? (And I mean a theory that does not rely on a look-up table of Cl/Cd values)

[EDIT] Since posting this question It has been pointed out that two similar questions have already been posted: How do wings generate lift? (12 answers) How complete is our understanding of lift? (9 answers)

Both of these posts generated much discussion but contained no worked examples to support the various claims. As noted in my question, Physics is an experimental science where the validity of a theory is supported by experimental evidence that real-world numbers applied to the theory correspond to the experimental results.

My question, whilst on the same subject, presents a specific request for a worked example.

Answer 1

I think your fundamental premise is unfortunately false.

Kepler's laws were purely empirical. He had no actual theoretical basis for them. He used a tremendous amount of data (thanks to Tycho Brahe) and essentially observed the trends. He just happened to be correct.

Somewhat amusingly (at least to me), beyond Kepler's three laws that we celebrate today, Kepler believed in a fourth law that was absolutely not correct. He was equally confident in the fourth law, but we virtually erase it from history because it wasn't correct.

There is no simple back of the envelope calculation for a modern physics based deviation in Mercury's orbit.

Many physical phenemona do not reduce down to an elementary algebraic equation.

Aerodynamics -- including the generation of lift -- is very well understood. It has also been validated through experiment (many wind tunnel tests and flight tests) where the theory and experiment are in very close agreement. In fact, we have several approaches to the development of the aerodynamic equations -- depending on the assumptions and simplifications we wish to make.

Engineers have a very good understanding of how these levels of theory compare and which are better / worse at predicting reality in different situations.

The issue at hand is that the fundamental theory of aerodynamics does not boil down to a simple equation with immediately obvious physical interpretation.

Everything in aerodynamics can be explained in terms of (and rigorously derived from):

1. Conservation of mass
2. Conservation of momentum
3. Conservation of energy
4. Gas equation of state
5. Some other gas relations like the dependence of viscosity on temperature

These five things are intuitive, observable, testable, etc. There are multiple ways to get from these five things to aerodynamics....

Unfortunately, they are not simple algebraic equations. Instead, they are a coupled system of nonlinear partial differential equations (1-3) (with some algebraic equations thrown in (4,5)).

The 'modern' way uses the least mathematical abstraction. It is called CFD (computational fluid dynamics). In CFD, we divide the domain of interest into a large number of tiny elements. We initialize all of the elements to an initial guess (that is wrong). We then solve the above equations on every element, applying appropriate boundary conditions. This makes a small correction to the properties at every element. We then do it again, and again, and again....

CFD is a very brute force approach with very little mathematical abstraction. After millions and millions of calculations, if we're lucky, the properties in all of the elements stops changing (there is theory on how this should happen too). We then say that the solution is converged. We can then sum pressures and shear to get lift and drag, we can produce beautiful images of the flow, etc. We have 'solved' the flow.

Although CFD is very powerful, it is not very satisfying in terms of intuition (which is what you're after).

There are other ways to attack aerodynamics (and the calculation of lift). You mentioned Bernoulli's equation -- Bernoulli's equation is most frequently derived from the conservation of energy, but it also can be derived from the conservation of momentum.

Potential flow theory (which most pre-CFD low-speed aerodynamics theory was based on) is usually derived from the conservation of mass -- but can also be derived from the conservation of momentum.

Potential flow theory allows aerodynamic calculation with much less brute force than CFD -- but instead, it adopts a great deal of mathematical abstraction. Complex variables (imaginary numbers), a potential function, sources, sinks, and vortices (mathematical singularities), superposition of elementary solutions, etc.

All of this can be rigorously derived from the fundamental equations -- and all of it has been validated through comparison to experiment. And more than 100 years of successful application to the design of aircraft. Engineers have deep understanding of their applicability and limits.

Unfortunately, although all of this is true, it does not mean that it can be explained in one or two sentences without heaps of math.

Any simple explanation of lift is wrong.

Any rigorous and correct explanation of lift is going to get messy.