Explained: Airfoil Cosine Spacing

[@williamsworkshop1389]

Awesome! I didn't expect a video of this. It helped a ton and I finally got it working in my program (C#, making a program that outputs an entire wing instead of just an airfoil). Been watching all your videos and I'm impressed, definitely got yourself a new subscriber. Can't wait to see more!

[@JoshTheEngineer]

I had wanted to make this video anyway, and your question gave me a little motivation to do it, so thank you! And thanks for subscribing, it means a lot!

[@emilyy6816]

I have struggled with this concept for the last week and I'm so grateful that I found this video. Thank you SO MUCH! You explained it perfectly.

[@JoshTheEngineer]

Awesome, you're welcome!

[@tuzobarca]

Great vid. I just graduated with my degree in mechanical engineering but i still find myself watching these videos. Thanks.

[@JoshTheEngineer]

Thanks, and congrats! It doesn't hurt to keep on learning.

[@Sonhadoresdc]

Josh, thank you very much. I was thinking in a way to solve this spacing problem at my code (I was constantly trying linspace, sine and cosine spacing, but not at the same way you did).

[@JoshTheEngineer]

You're welcome! Glad it was helpful!

[@zhenhaojing3384]

Thank you!

[@JoshTheEngineer]

My pleasure!

[@treadless_ca]

What is the reasoning for using 0.5* [1-cos(beta)]? In the end, doesn't it result in the same grid spacing as simply using 0.5* [cos(beta)]?

[@JoshTheEngineer]

In the most general sense, the overall goal of this non-uniform spacing is to get a different distribution of data points that go from the leading edge (x = 0) to the trailing edge (x = 1). The reason we need the data points to go from 0 to 1 is that the chord length is actually a normalized chord length. This means that you can take our generated airfoil shape, multiply it by whatever chord length you want, and all the X and Y data points will scale accordingly. So based on that need to go from 0 to 1, we need the function to increase from 0 to 1 as we go from beta = 0 to beta = pi. If you look at the plot that I have on the whiteboard, you can see that the cos(beta) plot (green) goes from 1 to --1. If we just multiplied that by 0.5, it would go from 0.5 to -0.5, so that won't work. Then you might say, well we can just shift it up by 0.5 and that would be fine. That's actually what the 0.5*[1-cos(beta)] is doing, and you can see that if you distribute the 0.5 term through to get 0.5-0.5*cos(beta). At the leading edge (beta= 0), we get 0.5-0.5 = 0. Then at the trailing edge (beta = pi), we get 0.5-(-0.5) = 1. To sum it up, the main reason is so that we get a distribution (on the vertical axis of the plot on the whiteboard) that goes from 0 to 1, and simply using 0.5*cos(beta) would go from -0.5 to +0.5.

[@treadless_ca]

excellent explanation, thanks!

[@AndrewBaglini]

thanks!

[@JoshTheEngineer]

You're welcome, friend!