How Do Planes Actually Generate Lift?

This is part five of Lunch Break Physics, the series that has argued a suction cup isn’t sucking, a fridge doesn’t make cold, the sky isn’t blue because it reflects the sea, and ice floats because water freezes into a roomy cage. Back in part one I dodged the wing and promised that “lift deserves its own lunch break, and it’ll get one.” Here it is. The physics below is standard, and the sources are at the bottom. If I got something wrong, tell me in the comments.

The story you were told

Here’s the explanation in almost every textbook, museum placard and pub argument. A wing is curved on top and flatter underneath, so the path over the top is longer. Two little parcels of air split at the front of the wing, one going over and one going under. They have to meet up again at the trailing edge, so the one on the longer route over the top must go faster. Faster air has lower pressure (Bernoulli), the pressure underneath is now higher, and that difference pushes the wing up.

It sounds airtight, and there’s even a real equation in it, but NASA has a whole page on why it’s wrong [1].

The fatal assumption is the innocent-looking phrase “they have to meet up again at the trailing edge.” There is no rule that two parcels of air which happened to be neighbours at the front must be neighbours again at the back. When you actually measure it on a wing that is making lift, they are nowhere near each other. The air over the top arrives at the trailing edge well before the air underneath does [1]. The whole chain of reasoning is built on a coincidence that does not happen.

The demo

Same wing the whole time, with the air flowing in from the left. Drag the angle-of-attack slider up and watch what happens to the streamlines behind the wing, then release the tracer pair and watch where the two dots actually end up.

As you raise the angle, the streamlines leaving the back of the wing bend downward. The wing grabs a wide sheet of air and flings it down, and the air behind it doesn’t close back up. The two tracer dots start together at the leading edge, one over the top and one underneath. Whenever the wing lifts, the top one reaches the trailing edge first. They never meet up. (A symmetric wing at zero tilt makes no lift, and there they do tie. That’s the special case, not the rule.)

The wing throws air down

Forget paths and meeting points and look at what the wing does to the air as a whole. Air reaches the wing moving roughly level, and a large mass of it leaves heading downward.

Changing the motion of that much air takes a force, so the wing must be pushing down on it. By Newton’s third law the air pushes back up just as hard, and that upward push is lift, which you can see in the river of air heading downward behind the wing.

The shape of the top surface is a detail here, and it isn’t the cause. What matters is that the wing turns the flow. A curved, cambered top helps it turn a lot of air smoothly, which is why efficient wings look the way they do. But a flat plate tilted into the wind turns air too, which is why a paper plane flies and why you can feel lift on your flat hand out of a car window. Tilt matters far more than curve, and that tilt is the angle of attack, the slider in the demo, and the single biggest lever a pilot has [1].

And the pressure difference? It’s real. In normal flight the air over most of the top moves faster and sits at lower pressure than the air underneath, and if you add up that pressure over the wing you get the lift [1]. Pressure and flow-turning are not competing explanations, and neither comes first. Air only follows a curve if the pressure is lower on the inside of the curve, so the low pressure above the wing bends the air down, and the bending keeps that low pressure in place. Bernoulli was never the problem.

A little bit of math

You can size lift without any of the hard aerodynamics. Newton’s second law says force equals the rate at which you change momentum, so the lift is just how much downward momentum the wing hands to the air every second:

Lift  ≈  (mass of air turned per second)  ×  (downward speed it picks up)
      ≈  ṁ · Δv_down

It’s only approximate, because a real wake has no single tidy speed, so read Δv as an average.

That gives you two knobs, how much air you push down and how hard you push it. The demo shows the second knob: raise the angle of attack and the air is turned more steeply, so the lift arrow grows. (The demo only models smooth flow, so its arrow keeps growing all the way to 18 degrees. A real wing stalls before that, see below.) Speed turns both knobs at once. Keep the same wing at the same angle of attack in the same air, and more air passes each second while each kilo gets thrown down harder, so lift grows with the square of airspeed [3]. That’s why a fast plane can carry its weight on a smaller wing, and why it lowers flaps on approach: flaps add curve, and on some designs area, so the wing still turns enough air at landing speed [4].

Circulation

So the top really does move faster. But if it isn’t because of a longer path, then why? This is where the elegant bookkeeping lives, and it’s what actually sets the amount of lift.

Aerodynamicists describe the flow around a wing as two things added together: the straight oncoming wind, plus a faint extra flow wrapped around the wing, forward over the top and backward underneath. The second piece is called circulation, written Γ. Draw an imaginary loop around the wing and add up how much the air flows along it as you go round. With no lift that total is zero. Around a lifting wing the faster flow over the top wins [2]. (No air actually goes round in a circle here, and nothing needs to spin. Circulation is just the total you get by adding up the flow along the loop.) Add that share to the oncoming wind and it speeds the top up and slows the bottom down. That is the faster-top, lower-pressure picture, with no mention of path length.

What fixes how much circulation? The sharp trailing edge. Real air is slightly sticky, and it can’t whip around a knife-edge and back up the underside, so it peels off cleanly at the point. Only one amount of circulation lets it do that, and a wing in smooth, unstalled flight settles into exactly that amount. This is the Kutta condition. In the textbook model (steady, smooth flow, no wingtips) it pins down the circulation, and with it the lift, as long as the wing isn’t stalled [2].

Once you know the circulation, the lift falls out of one compact line, the Kutta–Joukowski theorem:

Lift per metre of wingspan  =  air density × flight speed × circulation
L′  =  ρ · V · Γ

Denser air, faster flight, or more circulation (more camber, more angle of attack) each turns the lift up [2]. Speed counts twice, because for the same wing at the same angle of attack the circulation itself grows with speed, which is the same square-of-speed rule as before. The circulation that speeds up the top is the same flow-turning that flings the wake down, so counting the wake’s downward momentum and counting the circulation per metre of wing, under the same assumptions, give the same force. So you work out the lift one way or the other, never both added together.

The same trick, elsewhere

Once you see lift as “turn a fluid, get pushed the other way,” it shows up all over the place, and always for the same Newtonian reason.

A helicopter rotor blade is a wing going in circles, and a propeller is the same thing turned to pull. Both throw air, a rotor straight down and a prop backward, and ride the reaction. A hovering helicopter holds itself up by throwing a column of air downward, with no ground to push against.

A sail set at an angle to the wind is a soft vertical wing. It turns the wind and gets pushed sideways-and-forward, which is how a boat can sail at an angle into the wind (though never straight into it). The keel underwater is another wing. It turns water to stop the boat sliding sideways, and when you add the two pushes together, what’s left over points forward.

A race car’s rear wing is exactly a wing, mounted upside down. It turns air upward, so the reaction presses the car down onto the track for grip.

A curving football, or a table-tennis ball with spin on it (the Magnus effect), works on the same reaction. The spin shifts where the air peels off the ball’s surface, so the wake gets thrown off to one side and the ball is pushed the other way and curves.

Other ways to stay up

Plenty of things hold something up or push it along and look like they belong on that list, but they run on different physics.

A hovercraft, or an air-hockey puck, rides on a cushion of higher-pressure air underneath. A fan keeps pumping air in as it leaks out under the skirt, and the pressure on the underside carries the weight. Airflow is essential, but there’s no wing in a passing wind. It’s closer to the suction cup run in reverse.

A hot-air balloon is pure buoyancy. The warm air inside is less dense than the cool air outside, so the balloon floats for the same Archimedes reason that ice floats on water. It would rise just as well sitting still in a sealed hangar.

A rocket doesn’t need air at all. It throws its own exhaust downward and rides the recoil, so it works in the vacuum of space where there’s nothing to turn. Newton’s third law again, like a wing, but with its own propellant doing the job of the surrounding air.

Then there’s the “blow over a strip of paper” party trick. Hold a strip of paper to your lips and blow across the top, and it rises. People trot it out constantly as proof of the equal-transit story. But there’s no second parcel of air taking a shorter path underneath, and the paper is limp, not a wing. Your breath leaves your lips at roughly room pressure. The jet then follows the curve of the drooping strip, and following a curve needs lower pressure on the inside, so the pressure on top of the paper drops and the paper lifts.

Newton’s third law runs through all of them, so that is not the tell. The tell is where the push comes from: a wing turning passing air, a fan-fed cushion, buoyancy, or a rocket’s own exhaust. Only the first is lift in the wing sense.

Consequences you can see

So the next time someone tells you the air on top has farther to travel, you can nod at the part that’s true (the top really is faster, and the pressure really is lower), and then point out that the wing earns none of it by making air race along a longer path. It earns it by throwing a river of air downward and letting that air push the plane back up. No ground required.

Lunch Break Physics runs Tuesdays at noon. Last week: why ice floats. Next Tuesday: why the shower curtain attacks you. Got an everyday-physics puzzle you’d like poked at? The comments are open.

Sources

• [1] Equal Transit Theory, NASA Glenn Beginners Guide to Aeronautics: the “longer path / equal transit time” theory is one of the most widely circulated and incorrect lift explanations. A symmetric airfoil and a flat plate both lift, planes fly upside down, and particles over the top reach the trailing edge before those underneath, so the pressure difference is real but the equal-transit reasoning behind it isn’t. The flow turning is what matters, and the distance travelled plays no part.
• [2] Kutta–Joukowski theorem, Wikipedia: the flow around an airfoil is the superposition of the free stream and a circulation Γ set by camber, angle of attack and the sharp trailing edge (the Kutta condition, which uniquely determines Γ and therefore the lift). The lift per unit span is L′ = ρ·V·Γ.
• [3] Lift Equation, NASA Glenn Beginners Guide to Aeronautics: lift grows with air density and the square of speed, L = Cl·(ρV²/2)·A.
• [4] Flaps and Slats, NASA Glenn Beginners Guide to Aeronautics: flaps and slats add wing area and effective camber for low-speed takeoff and landing.
• [5] Ground effect (aerodynamics), Wikipedia: the ground obstructs wingtip vortices and reduces downwash, cutting induced drag and raising the lift coefficient.