# Kutta-Joukowski theorem

The Kutta-Joukowski theorem applies to 2-dimensional flow around an aerofoil, and gives the lift on the aerofoil as

$L = \rho v_\infty \Gamma \,$,

under the following assumptions:

## Derivation

The force per unit length on the aerofoil is

$\mathbf{F}=-\oint_C p \hat\mathbf{n}\, ds\,$

where $ds\,$ is the line element along the contour of the aerofoil and $\hat\mathbf{n}\,$ is the unit vector normal to it. Introduce the complex function $F = F_x + i F_y\,$. Then

$F = -\oint_C p( \sin\theta + i \cos\theta ) \, ds\,$,

where $\theta\,$ is the angle that $\hat\mathbf{n}\,$ makes with the $\hat\mathbf{y}\,$ direction. So

 $F\,$ $= -i \oint_C p( \cos\theta - i\sin\theta ) \, ds\,$, $= -i \oint_C p e^{-i\theta}\, ds\,$,

Note that if $z = x+i y\,$ then $dz = dx + i dy = ds ( -\cos\theta + i \sin\theta) = -e^{-i\theta} ds\,$ (we take the contour to be counterclockwise in the complex plane). For later use, note that $d\bar z = dx - i dy = -e^{+i\theta} ds = e^{2i\theta} dz\,$. Then

 $F\,$ $= i \oint_C p \, d z\,$,

Using Bernoulli's principle,

$p = p_0 - \frac{1}{2} \rho v^2\,$,

so that

$F = -i \frac{\rho}{2} \oint_C v^2 \, d z\,$,

Outside of the boundary layer, we can encode the velocity by introducing a velocity potential $w' = v_x - i y_y\,$. Immediately outside the boundary layer the flow is assumed to be tangential to the contour, although it might have opposite sense. In other words, $w' = | v |e^{-i \theta} \,$ or $w' = -| v |e^{-i \theta} \,$ along various segments of the contour. Thus $v^2 = (\bar w')^2 e^{2i\theta}\,$, and

$F = -i \frac{\rho}{2} \oint_C (\bar w')^2 e^{2 i \theta} d z\,$,

or

 $F = -i \frac{\rho}{2} \oint_C (\bar w')^2 d \bar z\,$,

which is the Blasius–Chaplygin formula. The function $w'(z)\,$ may be expanded as a Laurent series,

$w'(z) = a_0 + a_1 \frac{1}{z} + a_2 \frac{1}{z^2} + \dots\,$,

where we exclude positive powers since the velocity is to remain finite at infinity. Therefore

$a_0 = v_{x,\infty} - i v_{y,\infty}\,$,
 $a_1\,$ $=\frac{1}{2\pi i} \oint_C w'\, dz\,$, by the residue theorem, $=\frac{1}{2\pi i} \oint_C \mathbf{v} ds + i \oint_C ( v_x dy - v_y dx)\,$, $=\frac{1}{2\pi i} \oint_C \mathbf{v} ds + i \oint_C ( \frac{\partial \psi}{\partial y} dy + \frac{\partial \psi}{\partial x} dx)\,$, introducing the stream function $\psi = \operatorname{Im}\, w\,$, $=\frac{1}{2\pi i} \oint_C \mathbf{v} ds + i \oint_C d\psi\,$, $=\frac{1}{2\pi i} \oint_C \mathbf{v} ds\,$, using the assumption that $C\,$ is a stream line, $=\frac{\Gamma}{2\pi i}\,$, where $\Gamma\,$ is the vorticity.

Therefore,

$w'(z) = a_0 + \frac{\Gamma}{2\pi i}\frac{1}{z} + \dots\,$,

and

$(w'(z))^2 = (a_0)^2 + \frac{a_0 \Gamma}{\pi i z} + \dots\,$.

We can now evaluate $F\,$, or its complex conjugate, using the residue theorem:

$\bar F = \frac{i\rho}{2}\left[2\pi i \frac{a_0\Gamma}{\pi i}\right]= \rho \Gamma v_{y,\infty} + i \rho \Gamma v_{x,\infty}\,$.

Thus

 $F_x = \rho \Gamma v_{y,\infty}\,$, $F_y = -\rho \Gamma v_{x,\infty}\,$.