Lift Equation

The lift equation is three easy terms and one hard one. Dynamic pressure you can compute from a density and a speed. Wing area you can measure with a tape. The lift coefficient is where the entire subject of aerodynamics has been quietly folded up and put away: it carries the aerofoil section, the angle of attack, the planform, the twist, the flaps, the Reynolds number and the Mach number, all in a single dimensionless number.

That is not a swindle, it is the point of the equation. By pulling the speed, size and density out into terms you can vary at will, what is left is a number that belongs to the SHAPE and the ATTITUDE alone. A wind-tunnel model tested at one-tenth scale and one-third speed gives a $C_L$ that applies unchanged to the full-size aircraft, provided the Reynolds and Mach numbers are close enough. Every wind tunnel ever built runs on that promise.

The trap in this equation is the reference area $S$, and it catches people because it is a convention rather than a physical fact. $S$ is the full planform area of the wing including the notional portion carried through the fuselage — the part that is not there. It does not change when the flaps come out, even though the real wetted area does. And because $C_L$ is DEFINED by this equation, a lift coefficient quoted without saying which area it was divided by is a number with no meaning. Mixing the exposed wing area with the full planform is good for a silent error of 10 to 15%, always in the direction of a flatteringly large coefficient.

Where does the lift come from? Not, please, from the equal-transit-time story about air rushing over a longer top surface to meet its partner at the trailing edge. Air over the top does arrive first, and by a wide margin, and the explanation predicts far too little lift besides. The honest short answer is that the wing turns a large mass of air downward, and Newton's third law does the rest; the honest long answer is circulation, which is the same statement in the language that lets you compute it. The pressure distribution and the downwash are two descriptions of one event, not competing theories.

A last practical note. $C_L$ is very nearly proportional to angle of attack over the working range — about 0.1 per degree for a thin two-dimensional section, less for a real finite wing — right up until it is not. At the stall the curve rolls over and comes down, and every relation on this page that treats $C_L$ as freely available stops being true at that point.