If you’ve been reading popular science websites or magazines lately, then you may have heard the news: we don’t understand how airplanes work.

For example:

This fact may surprise you, given that humans have been successfully designing, building, and flying airplanes for over a century now. But I’m afraid that the articles are pretty clear:

In this post, I will consider the question of *why* we don’t understand heavier-than-air flight.

Science is hard, of course, and not understanding things is generally the default. There are also a number of things about flight that make it complicated; any of these things would be a natural reason for why we, as a species, have failed to understand flight. But after reviewing these possible reasons, I think you will agree with me that the *true* reason we don’t understand heavier-than-air flight is something deeper and more shocking than we usually acknowledge.

### What we *do* understand: lighter-than-air flight

The first human flights were really just human floats. Hot air balloons first carried humans through the sky in 1783, but the basic principle behind the hot air balloon goes back to at least the third century in China, when it was realized that a container of heated air floats above the colder air surrounding it.

Flotation of this kind is natural and not really puzzling to anyone. Hot air is less dense than cold air, so riding in a hot air balloon is the same principle as holding onto a life raft to keep you afloat in the water: gravity pulls heavier things downward, leaving the lighter ones to be displaced upward.

In technical terms, the relevant concept is Archimedes’ principle: the upward buoyant force on an object is equal to the weight of the fluid it displaces. This, in turn, is an immediate consequence of the deeper principle that is a bedrock of physics: the universe is a giant energy minimization machine. If the universe can lower its potential energy by displacing your hot air balloon upward to make room for heavier (colder) air, then it will do so. All you have to do is hold on.

In this way, “flight by flotation” is a simple, natural, and almost timeless idea. It feels clear and intuitive at every conceptual level.

We just found it impractical for some reason.

**Do we not understand air?**

By contrast, heavier-than-air flight is an unsettling business (which, as emphasized above, we do not understand).

One natural culprit for our failure to understanding flying through the air is that maybe air itself is too complicated to understand. After all, air contains 3×10¹⁹ molecules in every cubic centimeter, with each molecule flying around at about 1000 miles per hour and frequently running into other molecules. So if you wanted to *really* describe a single cubic centimeter of air, you would need to simultaneously solve 9×10¹⁹ coupled, second-order differential equations (*F = ma*, but for each direction of motion for every molecule).

Given this situation, if you were a high school physics student who had just learned Newton’s laws, you might well decide that there is no point in ever trying to apply them to something like air. It has simply too many *degrees of freedom* (a technical phase that means “things to keep track of”).

But it turns out that you would be wrong to despair, because a beautiful thing happens — a thing that is one of the key enablers of modern physical science. In situations like this, all the many quantities that don’t really matter (the individual positions and velocities of the air molecules) get rapidly mixed up by the fast motion and all the collisions. Then, if you allow yourself to not care about those things, then the only things left for you to keep track of are the ones that are explicitly *conserved*: energy, mass, and momentum. Air (and other fluids) turn out to be relatively simple precisely because nature gives us so few conserved quantities. So we end up writing down just a couple equations to describe those conserved things, and we allow ourselves to not care about all the others that are rapidly scrambled.

This approach (often called *hydrodynamics, *even when it has nothing to do with water) is essentially the path of maximum ignorance. We allow ourselves to remain ignorant about absolutely every quantity except for the ones that nature has mandated as special: energy, mass, and momentum. But in that ignorance is a kind of freedom, and it gives us science that works.

**Do we not understand the principles of flight?**

Perhaps, you might think, we lack an understanding of the basic forces that enable flight. For example, Aristotle taught that all objects are endowed with either “gravity” or “levity” as a fundamental property of their being, and that only the ones with “levity” tend to rise above the ground. This description might seem silly now (does it?), but one might naturally wonder whether we have yet to hit on the right way to think philosophically about what it is in the nature of matter and its interaction with other matter that enables flight.

A natural worry — but, it turns out, a completely unfounded one. The basic principle that enables flight is no more complicated than Newton’s laws of motion, known since the late 1600s.

The standard textbook introduction to flight will tell you that there are four relevant forces: *weight*, *thrust*, *lift*, and *drag*. But in Newton’s language, there are really only three: the force of the earth on the plane (*weight*), the force of the engine exhaust on the plane (*thrust*), and the force of the air on the plane (which we arbitrarily decompose into its vertical and horizontal components, *lift* and *drag*).

If you want to find out whether your plane is going to fly or fall, you just need to draw the free body diagram, sum up the forces, and see whether the resulting arrow is pointing net up or net down.

I’m not saying (yet) that solving for all the forces in this free body diagram is easy. But I am saying that basic principles of flight — the corresponding force balance — is not more complicated conceptually than Newton’s laws.

Physics has undergone many revolutions since Isaac Newton, but it turns out that none of them are necessary (or especially relevant) for understanding how an airplane can stay in the air.

**Do we not know how to solve the equations?**

Sometimes the process of solving equations is hard (for example: I once encountered an algebraic equation that was so tricky that I only managed to solve it by finding the smartest high school kid around and giving it to him). The basic equations governing fluid flow can be very difficult indeed: in fact, that there is a million dollar prize to be had for the first person who can prove that a general solution even exists.

So maybe, you might think, the problem with understanding flight is not that we don’t know what forces are involved, or what equations to write down, but only that no human has yet been born who is clever enough to solve them.

This may be true, but only in a sense that also turns out to pose no impediment to modern man. Because it turns out that even equations that you cannot (or don’t want to) solve exactly can be solved *approximately*, so long as you are willing to put up with a sufficiently tedious process of approximating the correct equations with “almost correct” equations. For example, it may be too hard to solve the equations that describe a continuous flow of fluid, but not too hard to solve equations for a bunch of discrete “parcels” of fluid that have their properties (energy, mass, momentum) updated at discrete points in time. This is called a “finite element method,” and if you tried to implement it by hand it would be so mind-numbingly repetitive that you would quickly decide that no degree of understanding was worth it.

Luckily for us in the 21st century, we have computers to do such things for us.

[I think it is generally underappreciated that the great miracle of modern computers is not that they can do things that humans cannot. It’s that they can do things that are easy for humans and then do them 100 trillion more times without getting bored.]

Those un-bore-able computers allow us to solve the relevant air flow equations with a large amount of quantitative accuracy, so that we can predict the lift and drag forces produced by nearly any geometric shape.

So the situation is looking up (no pun intended). But I’m afraid that our basic puzzle remains:

**Is it chaos theory?**

Anyone who grew up in the 90s knows the dangers of ignoring chaos theory.

The idea of (mathematical) chaos is that, in a chaotic system, infinitesimally small variations in the initial condition give drastically different outcomes after a long enough time. In technical language, mathematical systems have a *Lyapunov time*, *τ*, such that small differences in the initial conditions turn into exponentially large differences in the final state after a time *t ≫ τ*.

Predicting the state of any system at times much longer than its Lyapunov time is not literally impossible, but it is effectively impossible. The accuracy with which you must know the state of your system and the level of precision with which you must perform your calculations become so high that no amount of computing power will save you.

Fluid flow can easily become chaotic, which often means that it becomes mathematically intractable and the best-laid equations of mice and men are not useful. The propensity for a fluid to become turbulent is characterized by the *Reynolds number*, *R*, which is a dimensionless number that tells you how important the fluid’s inertia is. When *R* is small, the flow of a fluid at any given point depends only on the environmental conditions of that point (the pressure gradients and geometrical boundaries), and not on whatever happened upstream. When *R* is large, on the other hand, the flow at a given point in space and time depends very much on the past trajectory of the fluid flow, and how much momentum the fluid has built up.

[Side rant: the usual definition of Reynold’s number is that *R* is the “ratio of inertial forces to viscous forces”. I have always hated this definition because what the hell is an “inertial force”?? I think *R* is much better viewed as a ratio between two time scales: the time *T _{stop}* that it takes for an

*L*-sized region of fluid to come to rest through braking by viscous friction, and the time

*T*that it takes for the fluid to move a distance comparable to

_{move}*L*. Thinking of

*R*=

*T*makes it clear that when you are at low Reynolds number you are in a world without inertia, where any fluid motion is brought to a rest immediately. High Reynolds number is the world we usually think of and have intuition for, where moving things tends to remain in motion.]

_{stop}/T_{move}The flow of air over the wings of an airliner corresponds to very large Reynolds number: 200,000 or so. This is a big enough number that any normal person would anticipate chaotic (i.e., turbulent) air flow.

But it turns out that for situations like air flow over a wing, the Reynolds number that corresponds to chaotic/turbulent air flow is very high: *R* > 400,000, rather than the naive guess that it should just be *R* > 1.

As I see it, this is just one of those mysterious numerical coincidences in nature. There is no *a priori* reason to expect that air flow across an airplane wing is smooth instead of turbulent, and therefore that flight is possible and predictable rather than impossible and unpredictable.

But it happens that we are lucky. And since we are lucky, the flow of air across a wing is not especially difficult to describe, mathematically speaking.

**Is it… evolution??**

We have now arrived at a funny situation. As emphasized by this post and many articles before it, we don’t understand flight. But it’s not because air is difficult, or because the relevant laws of motion are difficult, or because we can’t write down the relevant equations, or because we can’t solve them. So what can possibly be the reason that we don’t understand heavier-than-air flight?

Having ruled out the obvious candidates, allow me to suggest a more radical possibility. Maybe the problem isn’t with nature, maybe the problem is with *us*.

Consider: we are products of evolution. The only goal of our species’ evolution is to keep us alive and reproducing. Nature has no generic mandate to make us capable of understanding; such a mandate exists only when understanding is in the benefit of our survival.

So maybe there’s a very good reason why our evolutionary history would not select for a naked, clumsy ape that feels comfortable saying he “understands” heavier-than-air flight.

🤔 🤔 🤔

This might be the first time I have had to dig up rusty aerospace engineering fluid dynamics knowledge to grok a post and comment. I actually TA-ed for undergrad fluid mechanics and fluid mechanics lab for 3 semesters, grading Navier-Stokes homework problems. Those office hours triggered the worst case of imposter syndrome I’ve experienced in my life (in my defense I was a control theory guy and shouldn’t have been assigned to TA fluid mechanics which I have zero intuition for).

I’m curious about how come you gesture at Navier-Stokes and CFD, but don’t comment on the simplified Bernoulli equations which are the workhorse of homework problems? My first “satisfying but wrong” understanding of the idea was that in a conservative incompressible flow with no gravity potential changes, P1+1/2 rho v_1^2 = P2 + 1/2 rho v_2^2, which means that if P1 >P2 then v1< v2, and then you make a geometric argument that the top of the aerofoil is a longer distance so the air is flowing faster, so the pressure is lower. In fact, I put that kind of explanation in my high-school science fair project poster in like 1989, which was a ping pong ball floating stably on a jet of water. The other popular one illustrating pressure differentials and lift is the kissing eggshells one...and those anecdotes about ships running into each other while passing in the fog because of the sideways net force pulling them together. Another popular class of familiar phenomena is spinning balls... ping-pong is best, but baseball and cricket too. And the effects of roughing up one side (in cricket) to change boundary layer separation points and generate "swing." I still don't quite get why my mental models of those situations is wrong, but I at least know they are kinda wrong. I used to be able to chug through the math and get to the right answers anyway once, now I can't even do that. Truth be told, though I've taught the subject, I never quite got comfortable with any of this shit and it is downright embarrassing. Oddly enough, things get weirder but less falsely comprehensible once you get to separated boundary layers (pimply golf balls...), vortex shedding, shock waves etc. There at least you basically have no tempting intuitions leading you astray into an illusion of understanding. True ignorance over false consciousness. We just make up these phenomenological abstractions and do math with them, but it looks more like sociology models of how rumors spread than like self-respecting physics. Anyhow, I still don't understand lift, so thanks for that.

Phew – I was expecting Bernoulli, too…

It can be an evolution problem – some facility big enough that automates the experiment-learn process, might just be made to convert a understanding problem into a cultural existence 🦥

I think of flight as like sailing with the forces happening backwards.

So, instead of lateral force transformed into pushing you forward, pushing you forward is transformed into lateral force. The air is shoved downwards, pushing you up. Aerofoils, wings, flaps, etc. Are just like jibs, mainsails and so on, creating a stable transformation of the force.

I’m might be wrong, but I’ve been sailing and gliding, so who knows. It’s just my intuition.

Isn’t this a perspective that falls flat? The 5th grade answer is that anything with enough horizontal force and an incline angle will “fly”, right? Baseballs and gymnasts and the beer you toss to a buddy fly, with none of the sophisticated engineering of an airplane. I’m not convinced that flight is no more than a function of the vector of energy (fossil fuels transformed into propulsion). If I take a wing and put it at a 45 degree decline but apply enough propulsive force to it the wing will fly, and if I take an airplane and reduce the horizontal speed below a certain threshold, the airplane will fall. What we really don’t understand is heavier-than-air-flight-with-an-insufficient-energy-vector.

>*What we really don’t understand is heavier-than-air-flight-with-an-insufficient-energy-vector.*

Yes, if we catapult an airplane with a huge capapult, it will fly.

What we don’t understand is regular flight, based on lift due to airflow.

My argument wasn’t that catapulting equates to flight, but rather that regular flight always requires regular energy expenditure. Toss a beer can and it will momentarily fly; attach an engine (or other propulsive energy source) to a beer can with the proper angle to the ground and it will fly until that energy source runs out, aerodynamics-be-damned. In other words, I don’t need to know much about lift or drag if I have sufficient ongoing thrust. Everything else is optimization.

After getting my undergrad in aerospace engineering, the best I could come up with is “air is sticky”. I can’t completely explain what it means, but that property is what makes it all work.

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It seems like there could be several fundamental mechanisms at work or a mechanism that functions quite differently depending on the sitation. Flight by large planes that are not articulated is different from the flight of tiny insects which is in turn quite different from the inexplicably efficient flight of a spinning, dimpled golf ball. Yet all of these situations come about as material objects interact with the medium in which the are suspended.

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I’m so glad to see this. I took flying lessons back in 2010, and I could not get my head around what caused lift. Many people tried explaining it, but it never made any sense to me.