Complex arithmetic and airplane wings

April 23, 2012
By

(This article was first published on sieste » R, and kindly contributed to R-bloggers)

I was once told that the reason that such a shape was so commonly used for aeroplane wings was merely that then one could study it mathemtically by just employing the Zhoukowski transformation. I hope that this is not true!

(R. Penrose, “The Road to Reality”, p.150)

Penrose here talks about a complex holomorphic mapping also known as the aerofoil transformation.

What you need is a cirlce in the complex plane, described by the complex function z, that passes through the point -1+0i. Then the transformation

w = \frac{1}{2}\left(z+\frac{1}{z}\right)

transforms this circle into the cross section of an airplane wing.

Let’s look at this in R. Fortunately, R has some complex number arithmetic already built in. A suitable circle is constructed by

z <- complex(mod=2,argument=seq(0,2*pi,len=100))+(sqrt(0.5)+1i)

and the plot below (Fig. 8.15 in “Road to Reality”) is produced by

par(mfrow=c(1,2),mar=rep(3,4))
plot(z,type="l",asp=1,main="z-plane")
plot((z+1/z)/2,type="l",main="w-plane",asp=0.5)

Zhoukowski airfoil transform

Maybe at some point I also understand what Penrose means by

… the (idealized) airflow around [the wing] can be directly obtained from that around a ‘wing’ of circular cross-section

Is it true that a vector field on the surface of the circle directly transforms into the wind field around the wing? How do you transform a vector field by the above equation? Does the transformed wind field explain why the plane flies? Would be nice indeed, but this is another post.


To leave a comment for the author, please follow the link and comment on his blog: sieste » R.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.