FLOW AND PRESSURE FIELD OVER A WING SECTION

(Really, really simplified)

The Simple Problem

We would like to (easily) calculate and visualize the steady flow and pressure fields around a section of a reasonably long airplane wing, such that we can consider this to be a two dimensional problem.

In order to simplify a very difficult problem, we will first assume that the air flowing around the wing is incompressible (a reasonably good assumption for subsonic flight). Our continuity equation is therefore:

where u and v are the components of velocity in the x and y directions. We can define a stream function such that:
This stream function has the significance that lines of constant psi are streamlines, that is, lines parallel to the flow velocity.

Now we're going to make some really big leaps in simplifying the problem. We will assume that the flow is irrotational and inviscid. This means that the air just flows smoothly without swirling around, and there is no boundary layer around the surface of the wing due to friction. This problem essentially has now lost just about all of its practical aspects. Nevertheless, the stream function now satisfies the Laplace equation everywhere:

How I solved it

In order that this code could be optimized to execute on the Cray YMP, I solved this problem using the red-black Gauss-Seidel iterative method with successive overrelaxation. I used a NACA 4412 airfoil, which was mapped onto a rectangular grid of nodes. The nodes representing the wing were blocked out on each iteration so that the wing's constant streamline boundary condition was met. The left and right boundary conditions consisted of a linear variation in the stream function, and the top and bottom boundary conditions were, of course, constant stream functions.

Output plots were created directly from within the program using the HDF library to create hdf rastermaps, which were later converted to gif files. I calculated a normal rainbow pallet to use on the stream function plots, and made a slight modification to this pallet for the pressure plots in an attempt to make them more useful.

Click here to see the big ugly FORTRAN code used to calculate all this. Click here if you have any interest in seeing the code for creating the NACA airfoil data file.

Now for the really cool stuff!

If you have an MPEG movie viewer configured, click on the buttons below for MOVIES showing the stream function and pressure plots using varying angles of attack.

Streamline Function Pressure Distribution
The pressure distribution plots look pretty strange and may not seem to make much sense. I believe this is simply because I didn't come up with a good color palette to illustrate the data. The important thing, though, is that they look cool, and when you come right down to it, that's the real purpose of this project!

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