Wind tunnel
Ideal flow around a Joukowski airfoil. Change its shape or angle of attack and the streamlines, surface pressure and lift are recomputed as you drag.
78 kt
Height of the mean line above the chord
Presets
- Lift coefficient
- 0.87
- Lift per meter of span
- 854N/m
- Zero-lift angle
- −2.3°
Surface pressure
Upper surfaceLower surface
Cₚ
x/c
The upper surface is at lower pressure than the lower one. The shaded area between the curves, summed along the chord, is the force normal to the chord; in ideal flow that is Cₗ cos α, 0.87 here.
Airspeed changes neither this plot nor Cₗ, only the force: lift per meter of span grows with the square of the speed.
With 2.0% camber the section still lifts at 0°; it stops lifting at −2.3°.
The model
Ideal flow around a circle with circulation has an exact solution. The Joukowski transformation, z = ζ + 1/ζ, maps that circle onto an airfoil, and the Kutta condition sets the circulation so the flow leaves the sharp trailing edge smoothly.
Bernoulli’s equation turns local speed into surface pressure, and the Kutta–Joukowski theorem turns circulation into lift:
Cp = 1 − (V / V∞)²
L′ = ρ V∞ Γ = ½ ρ V∞² c Cl
The flow is steady, two-dimensional, inviscid and incompressible. With no boundary layer there is no drag, separation or stall, and a two-dimensional section has no wingtips. The chord is 1 m and the air density 1.225 kg/m³, sea level on a standard day.
The method follows NASA Glenn’s explanation of conformal mapping and MIT’s Kutta–Joukowsky lecture.
Also in the lab: Why airplanes bank, Flying in moving air and Density altitude at KLAF.