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# Airfoil

File:Airfoil.svg
Various components of the airfoil.

An airfoil (in American English, or aerofoil in British English) is the shape of a wing or blade (of a propeller, rotor or turbine) or sail as seen in cross-section.

An airfoil shaped body moved through a fluid produces a force perpendicular to the motion called lift. Subsonic flight airfoils have a characteristic shape with a rounded leading edge, followed by a sharp trailing edge, often with asymmetric camber. Airfoils designed with water as the working fluid are also called hydrofoils.

## Introduction

A fixed-wing aircraft's wings, horizontal, and vertical stabilizers are built with airfoil-shaped cross sections, as are helicopter rotor blades. Airfoils are also found in propellers, fans, compressors and turbines. Sails are also airfoils, and the underwater surfaces of sailboats, such as the centerboard and keel, are similar in cross-section and operate on the same principles as airfoils. Swimming and flying creatures and even many plants and sessile organisms employ airfoils; common examples being bird wings, the bodies of fishes, and the shape of sand dollars. An airfoil-shaped wing can create downforce on an automobile or other motor vehicle, improving traction.

While any object with an angle of attack in a moving fluid, such as a flat plate, a building, or the deck of a bridge, will generate an aerodynamic force perpendicular to the flow called lift, airfoils are more efficient lifting shapes, able to generate more lift (up to a point), and to generate lift with less drag.

File:Lift drag graph.JPG
Lift and Drag curves for a typical airfoil

A lift and drag curve obtained in wind tunnel testing is shown on the right. The curve represents an airfoil with a positive camber so some lift is produced at zero angle of attack. With increased angle of attack, lift increases in a roughly linear relation, called the slope of the lift curve. At about eighteen degrees this airfoil stalls and lift falls off quickly beyond that. Drag is least at a slight negative angle for this particular airfoil, and increases rapidly with higher angles.

Airfoil design is a major facet of aerodynamics. Various airfoils serve different flight regimes. Asymmetric airfoils can generate lift at zero angle of attack, while a symmetric airfoil may better suit frequent inverted flight as in an aerobatic airplane. Supersonic airfoils are much more angular in shape and can have a very sharp leading edge. A supercritical airfoil, with its low camber, reduces transonic drag divergence. Moveable high-lift devices, flaps and sometimes slats, are fitted to airfoils on almost every aircraft.

Schemes have been devised to describe airfoils — an example is the NACA system. Various ad-hoc naming systems are also used. An example of a general purpose airfoil that finds wide application, and predates the NACA system, is the Clark-Y. Today, airfoils are designed for specific functions using inverse design programs such as PROFIL and XFOIL. Modern aircraft wings may have different airfoil sections along the wing span, each one optimized for the conditions in each section of the wing.

File:PSU-90-125.PNG
An airfoil designed for winglets (PSU 90-125WL)

## Airfoil terminology

The various terms related to airfoils are defined below:[1]

• The mean camber line is a line drawn midway between the upper and lower surfaces.
• The chord line is a straight line connecting the leading and trailing edges of the airfoil, at the ends of the mean camber line.
• The chord is the length of the chord line and is the characteristic dimension of the airfoil section.
• The maximum thickness and the location of maximum thickness are expressed as a percentage of the chord.
• For symmetrical aerofoils both mean camber line and chord line pass from centre of gravity of the aerofoil and they touch at leading and trailing edge of the aerofoil.
File:Denney.kitfox.g-foxc.arp.jpg
An airfoil section is nicely displayed at the tip of this Denney Kitfox aircraft (G-FOXC), built in 1991.

## Thin Airfoil Theory

A simple mathematical theory of two-dimensional (i.e. with infinite span) thin airfoils was devised by Ludwig Prandtl and others in the 1920s.

The airfoil is modeled as a thin lifting mean-line (camber line). The mean-line, y(x), is considered to produce a distribution of vorticity $\gamma (s)$ along the line, s. By the Kutta condition, the vorticity is zero at the trailing edge. Since the airfoil is thin, x (chord position) can be used instead of s, and all angles can be approximated as small.

From the Biot-Savart law, this vorticity produces a flow field $w(s)$ where

$w(x) = \frac{1} {(2 \pi)} \int_{0}^{c} \frac {\gamma (x')}{(x-x')} dx'$

where x is the location at which induced velocity is produced, x' is the location of the vortex element producing the velocity and c is the chord length of the airfoil.

Since there is no flow normal to the curved surface of the airfoil, w(x) balances that from the component of main flow V which is locally normal to the plate - the main flow is locally inclined to the plate by an angle $\alpha - dy/dx$. That is

$V . (\alpha - dy/dx) = w(x) = \frac{1} {(2 \pi)} \int_{0}^{c} \frac {\gamma (x')}{(x-x')} dx'$

This integral equation can by solved for $\gamma(x)$, after replacing x by

$\ x = c(1 - cos (\theta ))/2$ ,

as a Fourier series in $A_n sin(n \theta)$ with a modified lead term $A_0 (1+cos (\theta)) /sin(\theta)$

That is $\frac{\gamma(\theta)} {(2V)} = A_0 \frac {(1+cos(\theta))} {sin(\theta)} + \sum A_n . sin (n \theta))$

(These terms are known as the Glauert integral).

The coefficients are given by $A_0 = \alpha - \frac {1}{\pi} \int_{0}^{\pi} ((dy/dx) . d\theta$

and $A_n = \frac {2}{\pi} \int_{0}^{\pi} cos (n \theta) (dy/dx) . d\theta$

By the Kutta-Joukowski theorem, the total lift force F is proportional to

$\rho V \int_{0}^{c} \gamma (x). dx$

and its moment M about the leading edge to $\rho V \int_{0}^{c} x.\gamma (x) . dx$

The calculated Lift coefficient depends only on the first two terms of the Fourier series, as

$\ C_L = 2 \pi (A_0 + A_1/2)$

The moment M about the leading edge depends only on $A_0, A_1 and A_2$ , as

$\ C_M = - 0.5 \pi (A_0+A_1-A_2/2)$

The moment about the 1/4 chord point will thus be,

$\ C_M(1/4c) = - \pi /4 (A_1 - A_2)$.

From this it follows that the center of lift is aft of the 'quarter-chord' point 0.25 c, by

$\ \Delta x /c = \pi /4 ((A_1-A_2)/C_L)$

The aerodynamic center, AC, is at the quarter-chord point. The AC is where the pitching moment M' does not vary with angle of attack, i.e.

$\frac { \partial (C_{M'}) }{ \partial (C_L)} = 0$

## References (thin airfoil/aerofoil theory)

1. http://www.desktopaero.com/appliedaero/airfoils1/tatderivation.html
2. http://www.aeromech.usyd.edu.au/aero/thinaero/
3. Batchelor G K (1967), An Introduction to Fluid Dynamics, Cambridge UP, pp467-471