The Mathematics of Airfoil Design

[dydxforsn]

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.

 

[AIR&SPACE]

Answer: Sort of...

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.

 

[Aero51]

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.

 

[Vadar2012]

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.

 

[alphy]

Yes, the calculus of variations does play a significant role in the design of airfoils. Airfoils are designed to generate lift and minimize drag, and this involves finding the optimal shape that will achieve these goals. The calculus of variations allows engineers to optimize the shape of the airfoil by considering various parameters, such as the angle of attack, thickness, and curvature, and finding the combination that produces the desired lift and drag characteristics.

Specifically, the calculus of variations is used to solve the governing equations of fluid dynamics, such as the Navier-Stokes equations, which describe the flow of air over the airfoil. By varying the shape of the airfoil and solving these equations, engineers can determine the optimal shape that will produce the desired lift and drag forces.

In addition to the calculus of variations, other mathematical techniques, such as computational fluid dynamics and numerical optimization, are also used in airfoil design. These methods allow for more complex and accurate analysis of airfoil performance, taking into account factors such as turbulence and boundary layer effects.

In summary, the design of airfoils involves a combination of mathematics, including the calculus of variations, to optimize the shape for desired performance characteristics.