Kutta-Joukowski theorem

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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}\,.

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