How a wing lifts

Engineers write the lift of a wing as one short equation:

L = ½ × ρ × V² × S × C_L

• ρ (the Greek letter rho) is the air density. At sea level in the standard atmosphere it is 1.225 kg/m³. It falls with altitude.
• V is the airspeed, the speed of the aircraft relative to the air, in m/s. A plane flying into a 10 m/s headwind at 50 m/s over the ground has an airspeed of 60 m/s.
• S is the wing area in m². The toy plane has 16 m².
• C_L is the lift coefficient, a number with no unit. It packs in the wing's shape and its angle to the airflow.

The part ½ρV² is called the dynamic pressure q, in pascals. At 50 m/s at sea level, q = 0.5 × 1.225 × 50² ≈ 1,531 Pa. Then lift is simply q × S × C_L. With C_L = 0.4: L ≈ 1,531 × 16 × 0.4 ≈ 9,800 N, which is the weight of the toy plane. So at 50 m/s it can fly level with C_L = 0.4.

What each term does.

• Speed is squared. Double the speed and the lift is 4 times as big. Triple it and you get 9 times as much. That is why a plane needs a long run on the runway.
• Density counts directly. Half the density gives half the lift at the same speed and angle. High up, a wing must go faster, or use a larger C_L, to carry the same weight.
• Area counts directly. A bigger wing lifts more at the same speed. Flaps increase both the effective area and C_L for landing.

Angle of attack. The angle of attack α measures how steeply the wing meets the oncoming air: it is taken from the chord line (an imaginary line from the front to the back edge) to the direction the air comes from. It is not the angle of the plane to the horizon. In the linear range the lift coefficient grows steadily with α. In the toy model used here, C_L = 0.08 × (α + 5°), so the wing still lifts a little at α = 0° and makes no lift at α = −5°. Beyond about 15° the wing stalls and C_L stops growing (lesson 4).

Because lift must equal weight, a slower plane needs a bigger C_L, and so a bigger α. For the toy plane (9,810 N, 16 m²) in level flight: 80 m/s needs α ≈ −3°, 50 m/s needs 0°, 40 m/s needs 2.8°, 30 m/s needs 8.9° and 25 m/s needs 15°, right at the stall.

Why does a wing lift at all? There are two descriptions, and they are the same physics seen from two sides.

1. The wing turns the air downwards (Newton). To push the plane up, the wing must push air down. Lift is the rate at which the wing gives air downward momentum. A rough estimate for the toy plane at 50 m/s: its wing (span about 11 m) affects a tube of air about that wide, roughly 5,800 kg of air every second. To produce 9,810 N the average downwash is about 9,810 / 5,800 ≈ 1.7 m/s. That is gentle, but it is a lot of air. (This is an order-of-magnitude estimate from momentum, not an exact figure.)

2. The pressure is lower above and higher below (Bernoulli). The air flowing over the curved upper surface speeds up and its pressure drops. The pressure difference between the two sides, added up over the whole wing, is the lift force.

These are not rival explanations. The pressure difference is how the wing pushes on the air, and the downward turning of the air is the result. The Flow Lab shows the streamlines and the pressure on a wing in detail, so this course does not repeat it.