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Circulation & Kutta–Joukowski

Bernoulli tells us that faster air means lower pressure. To explain why the air is faster over the top, and to get a single clean formula for lift, we need the idea of circulation and the theorem that turns it into force.

Circulation

Circulation $\Gamma$ measures the net “swirl” of the flow around a closed loop $C$ . It is the line integral of velocity around that loop:

\Gamma = \oint_{C} \vec{V}\cdot d\vec{l}

(1)

$\Gamma$: circulation [m²/s]

Draw a loop around a wing and you find a non-zero $\Gamma$ : the flow genuinely circulates around the section — faster over the top (with the motion) and slower underneath (against it). Superimpose this circulation on the oncoming stream and you get exactly the speed difference Bernoulli needs. The circulation is the lift mechanism.

The Kutta–Joukowski theorem

For an ideal flow, the lift per unit span is exactly proportional to the circulation:

L' = \rho_\infty\, V_\infty\, \Gamma

(2)

$L'$: lift per unit span [N/m]
$\rho_\infty$: freestream density [kg/m³]
$V_\infty$: freestream speed [m/s]

What fixes the circulation? The Kutta condition

An ideal fluid admits infinitely many flows around a wing, each with a different $\Gamma$ — most of them whipping impossibly fast around the sharp trailing edge. Real air cannot do that: viscosity forbids the infinite speed. The flow instead leaves the trailing edge smoothly, with the rear stagnation point sitting right at the tip. This is the Kutta condition, and it selects one specific value of $\Gamma$ — the one nature actually picks. Our panel solver imposes exactly this condition to close its equations.

Fig. 1. The wing carries a bound circulation

\Gamma

. When the flow starts, an equal and opposite ‘starting vortex’ is shed — total circulation stays zero (Kelvin's theorem).

Linking back to the coefficient

For a thin section, theory (next article) gives the circulation as

\Gamma = \pi\, c\, V_\infty\,(\alpha - \alpha_{L=0})

(3)

where $c$ is the chord and $\alpha_{L=0}$ the zero-lift angle. Substitute into (2) and non-dimensionalise by $q\,c = \tfrac12\rho V_\infty^2 c$ :

c_l = \frac{L'}{\tfrac12\rho V_\infty^2 c} = \frac{\rho V_\infty\cdot \pi c V_\infty(\alpha-\alpha_{L=0})}{\tfrac12\rho V_\infty^2 c} = 2\pi\,(\alpha - \alpha_{L=0})

(4)

Out drops the famous lift-curve slope of $2\pi$ per radian — derived properly next. Notice the chain: circulation → Kutta–Joukowski → lift coefficient. Every link is exact for ideal flow.