Fundamentals7 min read1 / 9

Lift is pressure

A wing does not “grab” the air or ride on it like a boat. The entire lift force is delivered by one thing: air pressure acting on the skin. If we can describe the pressure everywhere on the surface, we can compute the lift exactly.

Pressure is a force per unit area

Static pressure $p$ is the normal force the air exerts per unit area, measured in pascals ( $1\,\text{Pa} = 1\,\text{N/m}^2$ ). Crucially, pressure always pushes inward, perpendicular to the surface. On a small patch of area $dA$ with outward unit normal $\hat{n}$ , the force the air applies to the wing is

d\vec{F} = -\,p\,\hat{n}\,dA

(1)

The minus sign encodes “pushes inward.” To get the total aerodynamic force we integrate over the whole surface:

\vec{F} = -\oint p\,\hat{n}\,dA

(2)

Lift $L$ is the component of $\vec{F}$ perpendicular to the oncoming flow; drag $D$ is the component along it. Skin friction also acts, but it points along the surface and contributes almost nothing to lift — we meet it later under drag.

Fig. 1. High pressure under the wing pushes up; suction on top pulls up. Summed over the surface, the leftover is lift.

The pressure coefficient

Raw pressures depend on altitude, speed and weather. We strip all of that out with the dimensionless pressure coefficient, comparing the local pressure to the freestream and scaling by the dynamic pressure $q_\infty = \tfrac12\rho_\infty V_\infty^2$ :

C_p = \frac{p - p_\infty}{\tfrac12 \rho_\infty V_\infty^{2}}

(3)

$p_\infty$: undisturbed (freestream) static pressure [Pa]
$\rho_\infty$: air density [kg/m³]
$V_\infty$: freestream airspeed [m/s]

Using Bernoulli’s principle (next article) the local pressure is tied to the local speed $V$ , and equation (3) collapses to a remarkably clean result for low-speed flow:

C_p = 1 - \left(\frac{V}{V_\infty}\right)^{2}

(4)

Local speed

V / V_\infty

1.40

C_p = 1 - (1.40)^2

C_p = -0.96

Faster than the freestream → suction ( $C_p < 0$ ). This is what holds a wing up.

Reading a real pressure plot

Figure 2 is the pressure coefficient our solver computes around a NACA 2412 at $6^\circ$ , plotted the way aerodynamicists always draw it — with suction pointing up. The upper surface sits deep in suction (large negative $C_p$ , especially near the nose), while the lower surface is close to neutral or mildly positive. The gap between the two curves is the lift.

Fig. 2. Solved

C_p

over a NACA 2412 at

\alpha = 6^\circ

. The area between the upper (suction) and lower curves is proportional to the lift coefficient.

From pressure to the lift coefficient

Projecting (2) perpendicular to the chord and non-dimensionalising gives the section normal-force coefficient as the area between the lower- and upper-surface pressure curves:

c_n = \frac{1}{c}\int_0^{c}\left(C_{p,\text{lower}} - C_{p,\text{upper}}\right)dx

(5)

For the small angles of normal flight the lift coefficient is essentially this normal-force coefficient ( $c_l \approx c_n\cos\alpha$ ). So “more suction on top” and “higher $C_L$ ” are literally the same statement. In the next article we explain why the air speeds up over the top in the first place.