Modelling airplane wing

Question

In the X-Z plane I have defined an airfoil shape using the function z = f(x). See figure 1. However, since the chord length of real airplane wings increases along the wingspan, I thought I needed to improve this function and found the following formula to describe the relationship between the tip chord, root chord, and wingspan:

$$x=\frac{\left(c_{root}-c_{tip}\right)}{\frac{b}{2}}y+c_{tip}$$

where $$c_{tip}=1\ \ \ c_{root}=3\ \ \ \ \ \ \ \frac{b}{2}=20$$

(The formulas shown are not important in figure 2. I just wanted to show the function f(x/(1+y/10)) but I can provide it to you if you want)

However, I first tried replacing x with 1+y/10 in my z = f(x) function, then I tried multiplying x with 1+y/10, but neither provided me with an accurate model. Later, instead of multiplying with x, I wrote it as $$f\left(\frac{x}{1+y/10}\right)$$ and it gave me figure 2. What I actually want to understand is the mathematical basis behind this phenomenon, that is, how could I express this situation in mathematical terms if I needed to?

I still don't know if this model is correct, because in figure 2 it seems to me that only the chord length has increased, but shouldn't the area of ​​the airfoil increase? For example, when c = 1, Area = 2, and when c = 3, shouldn't Area = 6?

Answer 1

Is your goal to generate 3D points that lie on the surface of a wing, or to have a closed form analytical equation for a wing surface?

The first is perhaps a little easier. When using this approach, I generate a "block" of points that I then apply transformations to. Think of the wing like a chunk of clay -- start with a block, then stretch it, twist it, shear it, etc. until it is in the shape of the wing.

The points are initialized in a simple array (creating the block) and then the transformations are applied in sequence.

To generate an analytical form in geometry, we usually use parametric functions instead of explicit (or implicit).

Explicit functions are ones like $y=f(x)$. If you wanted to define a circle explicitly, you would write $y=\sqrt{r^2-x^2}$ and $y=-\sqrt{r^2-x^2}$. We need two parts because a circle fails the vertical line test when written as $y=f(x)$.

Implicit functions are ones like $f(x,y)-C=0$. To define a circle implicitly, you would write $x^2+y^2-r^2=0$. This is sometimes easy to write and work with, but a pain to actually evaluate the points on the circle -- you must perform an elaborate solution procedure for every point.

Parametric functions are ones like $x(t),y(t)$. To define a circle parametrically, you would write $x=r\cos(\theta),\ y=r\sin(\theta)$.

If you wanted to stretch your circle, you could just apply a transformation

$x'=r\cos(\theta)$

$x=2\,x'$

$y=r\sin(\theta)$

Here, I use $x'$ to denote "x-prime", an alternate variable similar to x. At other times in life, you will see a prime used to denote a derivative -- I'm not doing that here.

I think that starting with airfoil equations written parametrically (even if you substitute $t=x$ to have $z(t)$ and $x(t)$) will make your life easier. You can then think about each of the transformations you want to apply as a single operation that can be built up one on top of another.

What software are you using to do this in? It looks like Mathematica or MathCad or something. I can help with the Math and Aerospace parts of this, but I don't have access to that software, so I can't try it myself.