Positioning and size for a Joukowsky Airfoil

Question

Joukowsky transformations are absolutely eluding me because I can't find numerical examples (I have a learning disability and formulas have always made absolutely no sense to me). I'm trying to get an understanding of how to set up a Joukowsky tranformation for an airfoil and it's positioning.

Given a thickness/chord ratio of 12%, Camber of 4%, and angle of attack of 3 degrees.

So far what I've done is trying to get the radius from the thickness/chord ratio. From diagrams it appears that to do that $$ \frac{Thickness}{Chord} = \frac{b}{a} $$

Considering that the chord is entire length making the assumption then that $ a = 100$ $$ .12*100 = b $$

This gives the chord of b as $12$

to get radius from chord $$ \frac{12^2}{12*4} $$

Giving a radius of $ 3 $

I've also seen that $$ Camber = \frac{\epsilon_i}{2a} $$

Taking again $a=100$ and $camber = 4%$

$$ .04*200 = \epsilon_i = 8$$

This is where I don't know what to do next.

Answer 1

Joukowski airfoils use a transformation to turn a circle into an airfoil, essentially. This is why the mathematics are so complicated. There are a lot of resources out there on making Joukowski plotters. Here is an automatic one you can use to check your work (http://airfoiltools.com/plotter/index?airfoil=joukowsk-il). Basically, you define a range of points from 0 to 2 pi (circumference of a circle) and apply the transformation equations. The transformation translates the original circle center to a new location, then applies the airfoil transformation. This is a pretty good resource (first few pages) on how to implement it: http://www.jimhawley.ca/downloads/Joukowski_airfoil_in_potential_flow_without_complex_numbers.pdf

Personally, I think the author in the last link describes the transformations fairly clearly and "demystifies" the equations. He even attempts to recreate a NACA 2412 airfoil using the method. Hopefully this is at least a little helpful.