Lesson 1 · 35 min
Airfoil Geometry and Families
Module 3 sized the business jet from a sketch, a few statistics and the Breguet equations. Before the sketch can become a drawing, the wing needs a shape: first its cross-section, the airfoil, and then its planform. This lesson describes airfoils the way designers do, by a few numbers that fix their shape, and introduces the families those numbers come from.
Learning objectives
- Name the parts of an airfoil: chord, mean camber line, thickness and leading-edge radius.
- Build a NACA four-digit airfoil from its designation, and find its coordinates at any point.
- Read five-digit, six-series and supercritical designations.
- Explain what laminar-flow and supercritical airfoils were designed to do.
The parts of an airfoil
An airfoil is the shape of a wing cut by a plane parallel to the flight direction. Four numbers describe most of it:
- The chord \(c\): the straight line from the leading edge to the trailing edge, and its length. Everything else is measured as a fraction of it.
- The mean camber line: the curve halfway between the upper and lower surfaces. Its greatest height above the chord is the camber; a symmetric airfoil has none.
- The thickness: the distance between the surfaces, measured perpendicular to the camber line. Its greatest value, as a fraction of the chord, is the thickness ratio \(t/c\).
- The leading-edge radius: the radius of the circle that fits the rounded nose. A sharp nose stalls abruptly; a blunt one gently (Lesson 2).
Camber mostly sets how much lift the airfoil gives at a given angle, and its pitching moment. Thickness mostly sets the structure and fuel the wing can hold, the maximum lift, and how early the flow becomes supersonic. The two are designed separately and then added: the upper and lower surfaces are the thickness distribution laid on either side of the camber line.
The NACA four-digit family
In the 1930s the National Advisory Committee for Aeronautics (NACA) built families of airfoils from simple formulas, so that one shape could be compared with another by changing one number at a time. The four-digit family is the simplest. In NACA 2412:
- the first digit is the maximum camber \(m\) in percent of the chord: 2%;
- the second digit is its position \(p\) in tenths of the chord: 40%;
- the last two digits are the thickness ratio in percent: 12%.
A symmetric airfoil has zeros for the first two digits: NACA 0012 is 12% thick with no camber.
NACA four-digit airfoil (\(x\) and \(y\) as fractions of the chord)
\[ y_c = \frac{m}{p^2}\left(2px - x^2\right) \quad (x \le p) \] \[ y_c = \frac{m}{(1-p)^2}\left[(1 - 2p) + 2px - x^2\right] \quad (x > p) \] \[ y_t = 5t\left(0.2969\sqrt{x} - 0.1260x - 0.3516x^2 + 0.2843x^3 - 0.1015x^4\right) \]The camber line \(y_c\) is two parabolas that meet at \(x = p\). The half-thickness \(y_t\) is the same for every four-digit airfoil, scaled by \(t\); it is greatest at \(x = 0.30\), where it equals \(t/2\). The leading-edge radius is \(r = 1.1019\,t^2c\).
The surfaces are found by stepping \(y_t\) out on either side of the camber line, perpendicular to it. With \(\theta = \arctan(\dd y_c/\dd x)\), the slope angle of the camber line:
Upper and lower surfaces
\[ x_u = x - y_t\sin\theta, \qquad y_u = y_c + y_t\cos\theta \] \[ x_l = x + y_t\sin\theta, \qquad y_l = y_c - y_t\cos\theta \]For thin, lightly cambered airfoils \(\theta\) is small, so \(y_u \approx y_c + y_t\) and \(y_l \approx y_c - y_t\).
Example 1.1 — A point on the NACA 2412
For the NACA 2412, find the camber, the half-thickness and the upper-surface point at \(x = 0.30\), and the leading-edge radius for a chord of \(2.0\ \text{m}\).
Show solution
Here \(m = 0.02\), \(p = 0.4\), \(t = 0.12\), and \(x = 0.30 < p\), so the forward parabola applies:
\[ y_c = \frac{0.02}{0.16}\left[2(0.4)(0.30) - 0.30^2\right] = 0.125(0.15) = 0.01875 \] \[ \frac{\dd y_c}{\dd x} = \frac{2m}{p^2}(p - x) = 0.25(0.10) = 0.025, \qquad \theta = 1.432^\circ \] \[ y_t = 5(0.12)(0.16262 - 0.03780 - 0.03164 + 0.00768 - 0.00082) = 0.06002 \] \[ x_u = 0.30 - 0.06002\sin 1.432^\circ = 0.2985 \] \[ y_u = 0.01875 + 0.06002\cos 1.432^\circ = 0.07875 \] \[ r = 1.1019(0.12)^2(2.0) = 0.0317\ \text{m} \]The half-thickness at 30% of the chord is 6%: half of the 12% thickness, as it must be at the thickest point. On the 2.0 m chord the upper surface is 0.158 m above the chord line there.
Other families
Later families kept the idea of a designation that encodes the shape, but tuned the shape for a purpose:
| Family | Example | What the digits mean | Designed for |
|---|---|---|---|
| NACA four-digit (1930s) | NACA 2412 | camber 2%, at 40% of the chord, 12% thick | simple, gentle stall; light aircraft, tails |
| NACA five-digit (1930s) | NACA 23012 | design lift coefficient \(0.15 \times 2 = 0.3\); maximum camber at \(30/2 = 15\%\) of the chord; 12% thick | more lift, less pitching moment; maximum camber far forward |
| NACA six-series (1940s) | NACA \(64_2\)-215 | six-series, minimum pressure at 40% of the chord; low drag for \(c_l\) within \(\pm 0.2\) of the design value; design \(c_l = 0.2\); 15% thick | laminar flow: a "drag bucket" of low drag around the design lift coefficient |
| Supercritical (1970s) | NASA SC(2)-0410 | second-generation supercritical; design \(c_l = 0.4\); 10% thick | high subsonic speed: a flat top delays and weakens the shock |
Laminar-flow airfoils move the point of minimum pressure, and with it the start of turbulent flow, well aft. A boundary layer that stays laminar has much less skin friction, so within a narrow range of lift coefficients the drag drops sharply. Outside that range, or on a surface spoiled by rivets, insects or rain, the bucket disappears.
Supercritical airfoils address the problem of the jet age. At high subsonic speeds the flow over the upper surface becomes supersonic and ends in a shock wave, which adds drag. A supercritical airfoil has a flatter upper surface, so the supersonic region is weaker and the shock forms later, and its camber is concentrated near the trailing edge to recover the lift lost by the flat top. For the same speed it can be thicker, which saves weight and holds more fuel (Lesson 3).
Today most new aircraft use airfoils designed by computer for their own wing: the numbers are no longer standard, but the ideas are the same. The four-digit family remains the best way to learn what each feature does, which is why this module uses it.
Check your understanding
Key takeaways
- An airfoil is a mean camber line with a thickness distribution laid on either side of it.
- NACA \(mptt\): maximum camber \(m\)% at \(p\) tenths of the chord, thickness \(tt\)% (greatest at 30% of the chord).
- Five-digit: first digit \(\times 0.15\) is the design \(c_l\). Six-series and supercritical designations also give the design \(c_l\) and \(t/c\).
- Laminar-flow airfoils cut friction near their design \(c_l\); supercritical airfoils delay the transonic drag rise.
- Next, Lesson 2 shows what camber and thickness do to the lift, drag and pitching moment.