# The Mathematics of Airfoil Design

My question is kind of simple, does the calculus of variations find its way into the design of the shape of an airfoil?

I'm interested in what kind of mathematics gets used in basic airfoil design. I suspect the calculus of variations must be involved, but I know nothing about deriving the shape of a plane wing.

It depends on how you define "design" and where you're looking for the calculus. For example, a very rudimentary aspect of airfoil design, which is very limited but useful for a basic understanding of airfoil properties is Thin Airfoil Theory.

Also, very important to designing an airfoil or wing is determining the rate of change of various coefficients with respect to flow properties, say lift coefficient vs. aoa or moment coefficient vs. aoa:

$\frac{dC_l}{d\alpha} ; \frac{dC_m}{d\alpha}$

There's also the new theory of stall, which is useful in the same respect that T.A.T is.

In the more rigorous design of airfoils, the calculus is maybe more obscure. Airfoils are designed nearly exclusively via CFD (the exceptions being cases like an R/C maker). CFD involves the solution of PDE's such as the Navier-Stokes equation.

Alot of airfoil design uses conformal mapping. Zhukovski transforms are used to map the flow around a cylinder to a flow of an airfoil. The "Eppler" airfoil series also uses conformal mapping methods to create much more complicated airfoils; I believe the specific method that is used is a well kept proprietary secret.

I just use CFD and wind-tunnel tests. The most common CFD programs use Navier-Stokes equations with some additions to simulate turbulent affects. You'd be surprised how accurate they are when compared to the physical experiments. If you'd like to actually see the equations and the methods used to solve them. I can refer you to some nice PhD papers.