Camber Line equation and Thickness Distribution equation

Question

I'm a high school student, not majoring in aerodynamics, so I'm struggling to understand the distinction between the Camber Line and Thickness Distribution equations. As I understand it, the Camber Line equations describe the curved line representing the average contour or curvature of both the upper and lower surfaces, while the Thickness Distribution equation provides the thickness of the airfoil at a specific point.

When I input these equations into Desmos, it generates the following image (for NACA 2412). My aim is to calculate the cross-sectional area. However, I'm concerned that integrating the Thickness Distribution equation might yield the cross-sectional area of a symmetric airfoil. The second image shows the upper part of the airfoil as curved and the lower part as flat. Since NACA 2412 isn't a symmetrical airfoil, how can I find its cross-sectional area. Could you explain me this difference.

Answer 1

It seems that you generally have the right idea. You're doing great. One comprehensive and relatively modern reference on this subject is NASA TM 4741. You should check it out if you don't already have it.

The details of the NACA airfoil definition say that the thickness is measured perpendicular to the camber line. So we can't just add and subtract half of the thickness to the camber line equation in order to get the upper and lower surfaces. This will be close, but not quite correct.

Instead, we have to calculate the slope of the camber line and use that to direct the thickness offset. This will require taking the derivative of the camber line.

You said you're in high school, so you may or may not be in the middle of Calculus right now. If you haven't gotten there, you should be able to go talk with a Calculus teacher and get some help. If you're in Calculus now, you should be able to handle these derivatives. Note that the camber line equations are often piecewise -- one equation for the front of the airfoil and another equation for the back.

Unfortunately, the equations in the paper aren't numbered, so referring to them is a little cumbersome. The equations to reconstruct the airfoil are the first two equations on page 11 of the paper.

$\left(\frac{x}{c}\right)_u = \left(\frac{x}{c}\right)-\left(\frac{y}{c}\right)_t\ sin\left( \delta \right)$

$\left(\frac{y}{c}\right)_u = \left(\frac{y}{c}\right)_{cam}+\left(\frac{y}{c}\right)_t\ cos\left( \delta \right)$

Where you flip the sign on the RHS to get the lower surfaces.

They also appear on page 113 of Theory of Wing Sections by Abbott and von Doenhoff in slightly different notation which I find a bit more compact.

$x_u = x-y_t\ sin\left( \theta \right)$

$y_u = y_c+y_t\ cos\left( \theta \right)$

$x_l = x+y_t\ sin\left( \theta \right)$

$y_l = y_c-y_t\ cos\left( \theta \right)$

where

$\theta = atan \left(\frac{dy_c}{dx_c} \right)$

Despite the notation differences, $\delta = \theta$.

This should get you moving in the right direction, let us know how it turns out.

Part of me thinks the area of the cambered airfoil should be exactly the same as the un-cambered one (even after all this work) -- but part of me leaves room for them to be slightly different.