I wrote this article a long time ago about how to use the class-shape-transformation (CST) parameters to create an airfoil:
I think some people wish to find the CST parameters for an existing airfoil, which can be used later for an optimization purpose (for example). If you want to do this, basically you have to minimize the error between your CST-generated airfoil and the target airfoil. In this post, I will give you some simple examples in MATLAB about how to do this. First, let us define the following error measure in a matlab-like language: error = mean(abs(yt-yp)), which means that we want to compute the mean absolute error between the y-coordinate of the target airfoil (yt) and the CST-generated airfoil (yp). We can easily see that the ‘error’ is the y-coordinate deviation of your CST airfoil to the target airfoil, measured at every point in the x-coordinate including the upper and lower surface. In order to do this, you have to create a computer code that calculate this error and then set the error as the output of the code. The input of the code itself is exactly the CST weights themselves. Please note that the principle is similar regardless the parameterization method that you used (e.g., CST, PARSEC, or anything else).
Let me you give an example. Here we want to fit the RAE 2822 airfoil with CST parameters of 4 weights each at the upper and the lower surface (so in total there are 8 parameters to be determined). The main program is the ‘airfoilfit.m’ with error term as the output. The ‘airfoilfit.m’ code calls the ‘CST_airfoil_fit.m code’, which is the CST generation code that I modified specifically for this purpose. I also include the sample of the RAE 2822 airfoil for this demonstration.
In order to find the CST parameters that fit your airfoil, we have to do the optimization procedure with the objective is to minimize the error as mentioned before. This can be done simply by using fmincon (or you can use the unconstrained fminunc) as follows:
[Wopt]=fmincon(@(W) airfoilfit(W,yt,XL,XU,0),[-1 -1 -1 -1 1 1 1 1],,,,,ones(1,8)*-1,ones(1,8),);
where yt is the y-coordinate of the target airfoil (lower+upper surface), while the XL and XU are the x-coordinate of the lower and upper surface, respectively. I also set the trailing edge thickness to zero to match the RAE 2822 data. Here, what we want to find is the W (CST weights) that returns the lowest error, where the initial solution is [-1 -1 -1 -1 1 1 1 1] and we set the minimum and maximum search bounds as -1 and 1 for all weights. Note that the first four weights are for the lower surface , while the rest are for the upper surface. The output of this optimization procedure is the ‘Wopt’ that hopefully could match the RAE 2822 airfoil.
Just run the script ‘demo.m’, and.. voila! you will have the CST airfoil that closely matches the RAE 2822 airfoil. The Wopt values that I found is [-0.1235 -0.1735 -0.1787 0.0583 0.1262 0.1516 0.2070 0.1957], with the error value of 0.0002. The shape is not perfectly similar especially in the lower surface, since it is more difficult to be approximated. This can be further refined by adding the amount of the CST weights, or by setting a different initial solution. I suggest to try it with different initial solution in order to find the optimum CST parameters that return the lowest error. Following is the final plot from the demo.m code:
Please download the demonstration and code in the following dropbox link:
The code is quite a draft and perhaps is quite ugly, so please modify it by yourself if you want a better-looking code.