Why do we need the Kutta condition?

Pure inviscid potential flow around a sharp-edged airfoil is non-unique — any value of circulation Γ gives a valid solution. The Kutta condition picks the physically realised one by requiring the flow to leave the trailing edge smoothly, which corresponds to the value of Γ that real viscous flow selects in the limit of vanishing viscosity.

How does this connect to lift?

The Kutta–Joukowski theorem says the lift per unit span is L′ = ρ U Γ. Tilting the airfoil increases the angle of attack, which forces a larger Γ to satisfy the Kutta condition, which increases L′ — exactly what you see when the streamlines bunch up more on the upper surface.

Is this a real CFD solver?

No. Joukowski is an exact analytic solution of incompressible inviscid flow with an enforced circulation. It is fast and beautiful for visualisation, but it ignores viscosity, separation, compressibility, and three-dimensional effects.

Airfoil Streamlines (Joukowski)

Potential flow over a Joukowski airfoil: stream-function bands with the Kutta condition fixing circulation.

About this model

The Joukowski conformal map z = ζ + b²/ζ takes a circle in the ζ-plane to an airfoil-shaped curve in the z-plane. Around the circle we can write the exact potential flow of a free stream plus a doublet plus a vortex with circulation Γ, and the Kutta condition fixes Γ so that the rear stagnation point sits on the trailing edge. This simulator paints the resulting stream function ψ on a grid; equally spaced ψ contours are the streamlines, and their local spacing is inversely proportional to the speed — tightly packed lines on the upper surface visualise the high-speed/low-pressure region that creates lift.

Who it's for: Intro aerodynamics, complex analysis, and conformal mapping; complements the lift, Bernoulli, and vortex-ring simulators.

Key terms

Joukowski airfoil
conformal mapping
potential flow
Kutta condition
circulation
stream function
lift

How it works

A Joukowski conformal map turns potential flow around a circle into potential flow around an airfoil. The Kutta condition fixes the circulation so the rear stagnation point sits at the trailing edge — that circulation is the lift. Streamlines bunch tightly above the upper surface (high speed → low pressure → lift).

Key equations

z = ζ + a² / ζ (Joukowski map)

Γ_Kutta = 4π U R sin(α − θ_TE)

L′ = ρ U Γ (Kutta–Joukowski lift per span)

Frequently asked questions

Why do we need the Kutta condition?: Pure inviscid potential flow around a sharp-edged airfoil is non-unique — any value of circulation Γ gives a valid solution. The Kutta condition picks the physically realised one by requiring the flow to leave the trailing edge smoothly, which corresponds to the value of Γ that real viscous flow selects in the limit of vanishing viscosity.
How does this connect to lift?: The Kutta–Joukowski theorem says the lift per unit span is L′ = ρ U Γ. Tilting the airfoil increases the angle of attack, which forces a larger Γ to satisfy the Kutta condition, which increases L′ — exactly what you see when the streamlines bunch up more on the upper surface.
Is this a real CFD solver?: No. Joukowski is an exact analytic solution of incompressible inviscid flow with an enforced circulation. It is fast and beautiful for visualisation, but it ignores viscosity, separation, compressibility, and three-dimensional effects.