Complex parametrization of Airplane wing?

Question

I read once about complex parametrization with fluid-dynamics objects such as airplane wings, something related Rieman Zeta function. What are the mathematical models this kind of things such as airplane wing?

Answer 1

The key term is Airfoil.

Joukowski mapping is used in aerodynamics because it is very easy to differentiate and it has good features such as plotting the unit circle on the [-1,1] (proof). You can build the wing just by random but it makes the investigation of different models pretty hard. For example, you can try the analysis of the following parametrization here, here and here for major parts of the airplane. Even though they look almost the same as the Joukowski aka Joukowsky transform, their analysis is far harder.

phi = linspace(0,2*pi,200); 
z = -0.1 + 0.4i + 0.8*exp(1i*phi); 
w = z + 1./z; 
plot(real(w),imag(w)); 

where we selected the $z$ so that $\left|z\right|\leq 2$: its maximum is 1.43 and minimum 0.65 by manual calculation. If you select too large values, you will get a circle so not a good airfoil. By the settings, I got the following complex plot

where real part is on the x-axis and imaginary part on the y-axis. You can make this into a wing by scaling it to the right direction. And here the code in Mathematica for completeness:

Later when you have found the model for your requirement, it is a time for visual-design with tools such as Rhinoceros. You can go a long way still with mathematical computation softwares such as R, Matlab and Mathematica.

Currently, the time of producing this kind of mold is 7 hours for a cockpit where you can have a microcontroller-chip such as Teensy/Arduino. Wings also take quite long time to produce.

When you try to make the model ready for manufacturing, make sure you use different color for inside layer than with outside layer because some 3D printers does not like much the positions where the layers go over one another like twisted ball. You can see below a good design and a bad design. You waste a lot of time with this kind of twisted designs because some printers think them as corrupted. The same problem is with Jukowski wings so you need to make sure your design does not have a twist!

If you really like to have twisted designs, you could apply smoothing functions such as convolution on the other side. You can find more about it here. The key is to make the cut point stronger so it is not brittle and possible to manufacture in reality.

Hope this gets your wings flying!

Further information

1. A Math function that draws water droplet shape? and more about Joukowski mapping

2. DP example with a glider with simplified forces such as gravity and drag: equations here, the physical model here and some flying profiles here with different speeds.