Joukowski Airfoil Flow

What is modelled

A circle of radius R, centred slightly left of and above the origin, is placed in a uniform stream at angle alpha. The Joukowski map w = z + a²/z turns this circle into an airfoil with a sharp tail at z = a. The flow past a circle is known exactly, so the flow past the airfoil follows by conformal mapping.

Equations

• Complex potential in the circle plane W = U (ζ e-iα + R² eiα / ζ) + (i Γ / 2π) ln ζ, with ζ measured from the circle centre.
• Kutta condition (finite velocity at the tail) Γ = 4 π U R sin(α + β), where sinβ = camber rise / R.
• Physical velocity u − i v = (dW/dζ) / (dw/dz), pressure coefficient Cp = 1 − |V|² / U².
• Lift per span L' = ρ U Γ (Kutta-Joukowski), CL = 2 Γ / (U c).

Checks run in the page

• Velocity at the trailing-edge point of the circle is zero.
• The numerical line integral of velocity around a loop equals the Kutta circulation.
• The lift from integrating C_p around 720 surface panels equals the circulation lift, and the pressure drag is zero.
• The highest surface pressure coefficient is exactly 1 at the stagnation point.

Honest simplifications

The flow is inviscid, steady, incompressible and two-dimensional. There is no boundary layer, no stall, no drag and no wake, and the lift keeps growing with angle where a real wing would stall. The chord is taken as 1 m only to give the lift a number in newtons per metre of span. Thickness and camber are the Joukowski parameters, not a standard wing section.

• Choose Symmetric, zero lift: the lift vanishes at zero angle and the surface pressure curves of the two sides coincide.
• Choose High lift: the suction peak near the leading edge deepens as the angle and camber rise.
• Raise the speed and watch the lift grow with its square in the third plot.