I have a doubt about the derivation of the Kutta-Joukowski theorem for a Joukowski airfoil. I know the results, but my main objective is to know how get these ones.

Consider for the initial plane a cylinder centered on $\zeta_0$, with a circulation -$\Gamma$, in an uniform flow with an atack angle $\alpha$:

where $a$ is the circumference radius, and $b$ the intersection of the circumference with the real positive axis, $\xi$. The parameter $\beta$ is the angle between the horizontal line and the line that links $\zeta_0$ to $b$. The center of the circumference is:


enter image description here

For this flow we have the following complex potential:


The Joukowski transform is:


To determine the complex potential on the $z$ plane we need to find the relation between $\zeta$ and $z$. And we get:


That is an awful relation. We not only have a square root, but also an $\pm$ symbol. Substituting this on the potential complex and then find the residues of $\left(\frac{dW}{dz}\right)^2$ and $\left(\frac{dW}{dz}\right)^2 z$ would be pratically impossible. What should be the next steps?

[Note: I didn't find any paper or book that shows the derivation of this theorem on the $z$ plane, but only on the $\zeta$ plane, which I already know. Do you know anything that can help me?]


you have transformed circle into ellipse by, $z = \zeta + \frac{b^2}{\zeta}$ ,

Now you have to do the inverse by tranforming ellipse into circle,

So the standard text books (Milne Thompson, etc) use,

$\zeta = \frac{1}{2}z + \frac{1}{2}{\sqrt{z^2-4b^2 }}$ .

Myself tried to transform ellipse into circle with the aove and suceeded.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.