The four-digit NACA airfoil is perfectly decent, which is a bit surprising considering it was first published way back in 1933. Since it was designed before computers, I’m sure that it could be improved.
The general form of the thickness equation is:
y = \sqrt{x}+ a x + b x^2 + c x^3 -(1+a+b+c) x^4
This does not provide the proper thickness, but ensures a rounded leading edge and a zero-thickness trailing edge. The original NACA parameters (adjusted to have a zero-thickness trailing edge) are [a, b, c]=[-0.4244, -1.1842, 0.9575]. These were selected by the aerodynamicists to fit their experience with airfoils: a maximum thickness at 30% of the chord, a trailing edge angle that looked OK, and a particular bulbousness of the leading edge. These sound a little arbitrary, but they did a great job considering the difficulty and expense of running physical experiments.
Now we have computers! We can test hundreds of silly ideas in minutes! For example, what are the best parameters for minimizing drag on symmetric airfoils? The original tests of the NACA 00xx airfoils were at a Reynolds number of 3 million, so we’ll match that in XFOIL.
How should we optimize these? Simple gradient descent does a great job of finding local extrema, but airfoils are very strange functions. Instead, I’ll use random search. That explores a wide range of candidates, and refines the best of them.
First, let’s try minimizing the drag for a 15% thickness airfoil (NACA 0015) at zero angle. After some minutes of random search, we find parameters [-0.0950, -0.9250, 0.9493], and the drag is down to 61% of the original! This result is shown in green on the plots below, and the reference NACA airfoil is in red. It looks more like a laminar flow airfoil, with the maximum thickness farther back. Unfortunately, this airfoil is over-optimized for level flight: it misbehaves at higher angles. It is extremely bad at generating lift, it has high pitching moments, and it only maintains the drag savings at low angles. Unless you can ensure that the airfoil will only operate at zero degrees, find a new airfoil.
Let’s try again, but maximize the average L/D over the range of one to five degrees. This gives [0.3295, -2.2740, 0.0746], yielding a 27% increase in the average L/D over those angles! This can be seen in blue above. Surprisingly, this one’s maximum thickness is at essentially the same location as that of the NACA airfoil. This one is slightly thinner toward the nose, and substantially thinner at the tail, leading to a near cusp! Where the NACA airfoil has a 10.3 degree tip angle, this one has a 0.15 degree tail.
Why should we be constrained to only three parameters? How about different numbers? Here are the results with between zero and six parameters:
Clearly, there are limited returns with more than 2 parameters. I think that the polynomial representation is somewhat limiting here, so a different scheme could be useful. Regardless, some simple optimization gives a nearly 30% increase in L/D!