Airfoils, Lift & Pressure Distribution

The cross-section of a wing decides how it lifts. Learn airfoil geometry (chord, camber, thickness, leading and trailing edges), NACA 4-digit designations, angle of attack, lift, drag and moment coefficients, pressure coefficient and pressure distributions, the Kutta condition and circulation, thin-airfoil theory (C_l = 2πα), zero-lift angle, aerodynamic centre, stall and high-lift devices — with Python for thin-airfoil lift and NACA thickness.

An airfoil is the 2-D shape you see when you slice a wing front to back. Its job is to create a large pressure difference between the lower and upper surfaces with as little drag as possible, over a wide range of angles and speeds.

Two numbers dominate: the lift-curve slope (about 2π per radian, or 0.11 per degree, for any thin airfoil) and the maximum lift coefficient before stall (typically 1.2–1.8 without flaps). Camber shifts the curve up; thickness mostly affects stall and structure.

An analogy

An airfoil is like your hand out of a car window: tilt it slightly and it rises smoothly; tilt it too far and the flow breaks away and it just gets pushed back — that's stall.

Geometry

The chord line joins leading and trailing edges; the mean camber line lies halfway between upper and lower surfaces. Camber (curvature) lets the airfoil lift at zero angle of attack; thickness gives structural depth and gentler stall. Angle of attack α is measured between the chord line and the oncoming air — not the horizon.

Pressure distribution

Most lift comes from suction on the upper surface near the leading edge (C_p down to −1 to −3). Past the suction peak, pressure rises toward the trailing edge (an adverse gradient) that the boundary layer must climb; when it can't, the flow separates and the airfoil stalls. Integrating C_p around the surface gives C_l and C_m.

Circulation and the Kutta condition

Potential-flow theory explains lift as a circulation Γ around the airfoil, fixed by the Kutta condition: the flow must leave the sharp trailing edge smoothly. Kutta–Joukowski gives L' = ρVΓ. Thin-airfoil theory models the camber line as a vortex sheet and predicts C_l = 2π(α − α_L0) and an aerodynamic centre at quarter chord.

Stall, high lift and airfoil families

Practical airfoil design:

Formulas

Worked example: Section lift

Given: C_l = 0.669, ρ = 1.225, V = 60 m/s, c = 1.5 m

Find: Lift per metre.

Answer: ≈ 2213 N/m

Worked example: Decode NACA 4415

Given: NACA 4415

Find: Geometry.

Answer: 4% camber at 0.4c, 15% thickness

Worked example: Pressure coefficient

Given: Local velocity 1.4 × V∞ on the upper surface

Find: C_p.

Answer: C_p = −0.96

Thin-airfoil lift and NACA 4-digit thickness

import math
# Thin-airfoil theory and a NACA 2412 section
def cl_thin(alpha_deg, alpha_L0_deg=-2.1):
    return 2 * math.pi * math.radians(alpha_deg - alpha_L0_deg)
for a in (0, 4, 8):
    print(f"alpha {a:2d} deg: Cl = {cl_thin(a):.3f}")
# NACA 4-digit thickness at 30% chord for a 12% thick section
t, x = 0.12, 0.30
yt = 5 * t * (0.2969 * math.sqrt(x) - 0.1260 * x - 0.3516 * x**2 + 0.2843 * x**3 - 0.1015 * x**4)
print(f"NACA xx12 half-thickness at x/c=0.30: {yt:.4f} c (total {2*yt:.4f} c)")
# Lift of a wing section: L' = 0.5 rho V^2 c Cl
rho, V, c = 1.225, 60.0, 1.5
print(f"lift per metre at 4 deg, 60 m/s, c=1.5 m: {0.5*rho*V**2*c*cl_thin(4):.0f} N/m")
# alpha  0 deg: Cl = 0.230
# alpha  4 deg: Cl = 0.669
# alpha  8 deg: Cl = 1.108
# NACA xx12 half-thickness at x/c=0.30: 0.0600 c (total 0.1200 c)
# lift per metre at 4 deg, 60 m/s, c=1.5 m: 2213 N/m

A cambered airfoil (NACA 2412 has α_L0 ≈ −2.1°) lifts even at zero angle of attack. The NACA thickness polynomial reaches its maximum (exactly t/2) at 30% chord. Thin-airfoil theory over-predicts slightly (real slope ≈ 0.1/°) and says nothing about stall — that needs viscous analysis or wind-tunnel data.

Common mistakes

Applications

Key takeaways

Going further