A Boeing 747 at 250 tonnes weighs 2.45 meganewtons, and at cruise there is nothing under it but air so thin you could not breathe it. The wing has to push that air down hard enough, and fast enough, to hold the whole aircraft up — continuously, for eleven hours. It does, and the equation that says how is short enough to fit on one line. What follows is that line, taken seriously: what each term does, what happens when you run out of one of them, and why the explanation most people were given at school cannot possibly be right.
Density of the air, speed squared, wing area, and one dimensionless number that carries everything else — the shape of the wing and the angle it is held at. Speed is the term that matters, because it is squared: fly half as fast and you have a quarter of the lift. The bar below shows lift against weight. Move the sliders until they balance, then climb: as the air thins you need either more speed or more angle, and there is a limit to both.
Tilt the wing and lift rises in a dead straight line — about 0.08 per degree for a wing of this shape — until suddenly it does not. Past about fifteen degrees the airflow can no longer follow the curve of the upper surface, it separates, and the lift collapses while the drag goes through the roof. That is a stall, and it has nothing to do with speed: a wing stalls at an angle, and you can stall at any speed you like. The flow lines are computed from a real vortex of circulation sitting at the quarter chord — note that the air ahead of the wing is already moving up before it arrives, and that everything behind it is heading down.
Climb, and the air thins, so the slowest you can fly gets faster. Climb, and the air gets colder, so the speed of sound drops and the fastest you can fly gets slower. Two limits moving toward each other. Where they meet there is exactly one speed the aircraft can hold, and above it there is none — pilots call it coffin corner, and it is the reason a heavy aircraft cannot simply climb above the weather. Add weight and watch the ceiling come down.
You were probably told this: the top of the wing is curved, so air going over it has further to travel, and since it must meet its partner at the back it has to go faster, and faster air has lower pressure. Every clause of that is either wrong or unjustified. There is no reason the two parcels must arrive together — measurements show the upper parcel arrives well early — and even granting the claim, the numbers do not come close. Drag the slider and see how much lift the story actually buys.
Reynolds number is the ratio of a fluid inertia to its stickiness — how much the air behaves like a thrown ball versus how much it behaves like honey. It spans eight orders of magnitude between a fruit fly and an airliner, and that is why you cannot scale a wing design up or down. At a 747 Reynolds number the air is all inertia and thin boundary layers. At a fruit fly Reynolds number the air is thick and reluctant, steady aerofoil theory stops working entirely, and insects fly by shedding and recapturing vortices instead — flapping is not a primitive version of a wing, it is a different solution to a different problem.
| flyer | chord | speed | air density | Reynolds number | regime |
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The lift equation, the International Standard Atmosphere and the Reynolds number are exact as implemented. ISA returns 1.22500 kg/m³ at sea level against the defined 1.225, and the density of 0.4135 used throughout occurs at 9,984 m — about 32,760 ft, a real cruise altitude. A 250-tonne 747 weighs 2.4517 MN, and holding that at 250 m/s on 541 m² of wing needs CL = 0.35070, which is the figure the page is built around. Viscosity comes from Sutherland formula, giving a cruise Reynolds number of 6.4 x 10⁷. The lift-curve slope is finite-wing theory, a₀/(1 + a₀/πeAR), which for this wing aspect ratio of 6.57 gives 0.0807 per degree and puts cruise at 4.3 degrees above the zero-lift angle. The myth calculation is the honest one: if the upper surface is x longer and the parcels really did arrive together, Bernoulli gives CL = (1+x)² - 1, so a realistic 2 per cent buys 0.040 — about a ninth of what is needed. To reach 0.351 the top of the wing would have to be 16.2 per cent longer than the bottom, which no aerofoil on any aircraft is.
What is simplified. The stall is a smooth curve fitted to the shape real wings have, not a computed separation — real stall depends on aerofoil section, surface finish, Reynolds number and how fast you pulled, and a swept wing can stall at the tips first and pitch up rather than down. CL,max and the flap increments are typical values, not Boeing data. The flow field in section 2 is a genuine potential-flow calculation — a uniform stream plus a point vortex of the circulation that Kutta–Joukowski demands for the lift shown — so the upwash, downwash and streamline curvature are real, but it is inviscid: there is no boundary layer, so it cannot itself predict the separation that section 2 is about, and past the stall the drawing keeps being tidy when the real flow has become a mess. Compressibility is ignored below the Mach limit, drag is induced drag only, and the whole aircraft is treated as a wing with a weight hung off it.