# How to draw NACA 6-Series Airfoils?

In relation to the question on the NACA 64-2A015 airfoil I would like to know how to draw this airfoil.

At least these two reports [1,2] by NASA provide the equation for it.

However, I am not able to find the correct parameters to draw it, so that it would match up with the coordinates given in the table above.

Here is the Matlab code I used:

a = 0.4;
b = 1.0; % caution for NON-unity entries change the equation for h
% c = 1; to simplifiy the equation the chord is set to 1

cl = 1;

g = -1/(b-a) * (a^2 * (1/2  * log(a) -1/4) - b^2 * (1/2 * log(b) -1/4)); % g  = -1/(1-a) * (a^2 * (1/2 * log(a) -1/4) + 1/4)
h = 1/(1-a) * (1/2*(1-a)^2 * log(1-a) -1/4*(1-a)^2)+g; % simplified version for b = 1: h =  1/(b-a) * (1/2*(1-a)^2 * log(1-a) - 1/2 * (1-b)^2 * log(1-b) + 1/4*(1-b)^2 - 1/4*(1-a)^2) + g

x = (0:0.001:1);
y  = cl/(2*pi*(a+b)) * ( 1/(b-a) .* (1/2 * (a-x).^2 .* log(abs(a-x)) - 1/2 .* (b-x).^2 .* log(abs(b-x)) + 1/4 .* (b-x).^2 - 1/4 .* (a-x).^2) - x.*log(x) + g - h.*x); % y = cl/(2*pi*(a+1)) * ( 1/(1-a) .* (1/2 * (a-x).^2 .* log(abs(a-x)) - 1/2 .* (1-x).^2 .* log(abs(1-x)) + 1/4 .* (1-x).^2 + 1/4 .* (a-x).^2) - x.*log(x) + g - h.*x);

L6j01_x = [0, 0.005, 0.0075, 0.0125, 0.025, 0.05, 0.075, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1];
L6j01_y = [0, 0.01193, 0.01436, 0.01815, 0.02508, 0.03477, 0.04202, 0.04799, 0.05732, 0.06423, 0.06926, 0.0727, 0.07463, 0.07487, 0.07313, 0.06978, 0.06517, 0.05956, 0.05311, 0.046, 0.03847, 0.03084, 0.02321, 0.01558, 0.00795, 0.00032];

plot(x,y), axis equal, hold on
plot(L6j01_x, L6j01_y,'ro'), hold off


It seems my Leading Edge definition is not right. Any suggestions on how to adjust the front part of the airfoil-section?

• It's been a while, but I vaguely remember hearing during lectures that the leading edge of an airfoil is a circle. So at some point the line from the equation would transform into a circle with radius unknown, perhaps the NASA report server can help you further. – Koyovis Jul 31 '17 at 21:14
• Jup, you're right. If I see it correctly this is also stated in the figure/drawing from the NASA report above. But I was actually talking about the mismatch of my calculated line (blue) with the red circles. Especially the section 1-40% chord. – rul30 Aug 1 '17 at 6:22
• Oh right, I assumed the starting point needed to be offset to depart from the nose circle instead of from (0,0). – Koyovis Aug 1 '17 at 8:13
• If you're using Matlab, can it construct a polynom through your data points? – Koyovis Aug 1 '17 at 8:14

From this reference:

$$c_f=K_1\cdot \frac {t}{c} + K_2(\frac {t}{c})^2 + K_3(\frac {t}{c})^3 + K_4(\frac {t}{c})^4$$

For the 64A series:

• $K_1$ = 8.2125018
• $K_2$ = 0.7685596
• $K_3$ = 1.4922345
• $K_4$ = 3.6130133

$c_f$ is the particular scale factor for this profile. And then the reference continues with this head spinning further explanation:

Now, for a specified family and thickness, the thickness distribution may be determined without iteration. From the thickness, the scale factor is computed from the polynomial function shown above. Then, the scale factor is used to multiply the basic values of the psi and epsilon functions for this airfoil family. These scaled psi and epsilon functions are used in mapping the z-plane to the z'-plane shown in Figure 1. The Joukowski function zeta = z' + 1/z' then maps the z'-plane into the zeta-plane and these results are normalized so that the leading edge is at x=0 and the trailing edge is at x=1.