Fluid dynamics · circulation and lift

Airfoil Circulation Studio

A Joukowski wing section spans a transparent wind tunnel. Change angle of attack, camber, thickness, and free-stream speed while spatial streamlines, surface pressure taps, and the pressure-coefficient trace update together; disable the Kutta condition to compare the same geometry without circulation.

Angle of attack

\alpha

$5.0^\circ$

Circulation

\Gamma

$9.32\,\mathrm{m^2\,s^{-1}}$

Lift coefficient

C_L

$0.770$

Lift per span

L'

$274\,\mathrm{N\,m^{-1}}$

Mapped thickness

t/c

$11.6\%$

Ideal drag

C_D

$0.000$

Wind-tunnel controls

Design one lifting section

Apply Kutta conditionChoose the circulation that leaves the sharp trailing edge smoothly.Spatial streamlinesAnimate material markers along mapped potential-flow streamlines.Surface pressure $C_p$ Teal marks suction; amber marks positive pressure.Pressure traceCompare upper and lower surface pressure on the tunnel backboard.Joukowski source circleReveal the offset circle that maps into the wing section.Ideal force balanceShow lift and the deliberately zero potential-flow drag.Angle of attack

\alpha

5.0^\circ

Circle camber offset

m/c

2.0\%

Thickness design

t/c

12.0\%

Free-stream speed

U_\infty

24\,\mathrm{m\,s^{-1}}

Orbit the transparent tunnel; compare Kutta on and off before pushing toward high angle of attack.

This Lab solves two-dimensional, steady, incompressible, inviscid potential flow around a Joukowski airfoil. The Kutta condition makes flow leave the sharp trailing edge smoothly, and $L'=\rho U_\infty\Gamma$ gives lift per span. The model excludes boundary layers, stall, viscous drag, finite-span effects, and compressibility; $C_D=0$ is D’Alembert’s paradox, not a claim that real aircraft have no drag.

Start with the essentials

Focus question: Why does the same airfoil admit many circulation solutions mathematically, while attached physical flow selects one?
One-sentence intuition: The Kutta condition requires finite velocity and smooth departure at the trailing edge, selecting one circulation $\Gamma$ ; the resulting pressure difference supplies lift.

Core mathematical model

Joukowski conformal map

z=\zeta+\frac{b^2}{\zeta}

An offset circle in the preimage plane maps into an airfoil with a rounded leading edge and sharp trailing edge.

Complex potential outside the circle

W(\zeta)=U_\infty\left(e^{-i\alpha}\eta+e^{i\alpha}\frac{R^2}{\eta}\right)+\frac{i\Gamma}{2\pi}\log\eta,\qquad \eta=\zeta-\zeta_c

Uniform flow, the cylinder image term, and a point-vortex circulation form the analytic circle-plane solution.

Pressure and lift

C_p=1-\left\lvert\frac{V}{U_\infty}\right\rvert^2,\qquad L'=\rho U_\infty\Gamma,\qquad C_D=0

Bernoulli converts surface speed into pressure, and Kutta–Joukowski gives lift per span. Zero potential-flow drag exposes the missing viscous physics.

Common difficulties

The Kutta condition is not hidden thrust

Typical misconceptionTurning on the Kutta condition adds an artificial source of lift.

Better mental modelIt adds no external force; it selects, among mathematically allowed circulation values, the one with finite speed and smooth departure at the trailing edge.

Zero drag is not a real-aircraft conclusion

Typical misconceptionIf the pressure field produces lift but no drag, a real airfoil should also fly without loss.

Better mental modelInviscid potential flow omits boundary layers, wakes, and skin friction, so it necessarily misses drag and stall. That contradiction is D’Alembert’s paradox.

Run the experiment

01
Establish the zero-lift baseline
Choose Symmetric · zero lift while leaving the Kutta condition enabled.
What to observe: Upper and lower streamlines and pressures are nearly symmetric, with circulation and lift close to zero.
02
Increase only angle of attack
Choose Cruise section and watch the upper-surface pressure taps and lift vector.
What to observe: The upper flow accelerates, its pressure coefficient falls, and the selected circulation and lift per span rise together.
03
Remove the trailing-edge selection
Choose Kutta off and compare the flow around the unchanged geometry.
What to observe: Circulation and lift vanish, but the trailing-edge flow no longer represents the selected attached-flow solution.