Airfoils and blade profiles: geometry and choosing the shape

An airfoil, the profile of a wing or propeller blade, is its cross-section, and its shape sets the compromise between what it does in the fluid and what it must withstand. Thin and lightly cambered, it glides well; thick, it resists bending better. This page describes the geometry of an airfoil, NACA airfoils, then the two levers of the lift-to-drag ratio: relative thickness (which acts on Cd) and camber (which acts on Cl).

On this page:

• Airfoil geometry
• NACA airfoils
• Choosing an airfoil: the lift-to-drag ratio

1: Airfoil geometry: chord, camber, thickness

The shape of the airfoil determines its aerodynamic or hydrodynamic performance, but also the mechanical strength of the blade. The compromise between the two is at the heart of designing a wing or a blade. On the fluid side, the thinner the airfoil, the less resistance it opposes to forward motion. A slight curve for lift, and the sheet of paper takes off effortlessly.

A slightly curved sheet of paper flies effortlessly; it could not carry a boat propeller.

On the mechanical strength side, the beam that gives the best ratio of weight to bending strength is the tube.

Making a tube fly is complicated (it can be done by spinning it, and we will come back to that). Asking a sheet of paper to take the thrust that propels a boat is just as hard. Between the tube and the sheet lies the airfoil. An airfoil is the outline obtained by cutting an aircraft wing, a helicopter blade, or a gas turbine or wind turbine blade vertically. Its shape strongly influences the characteristics and performance of the wing or blade.

The quantities of an airfoil: chord AB, mean camber line, camber f located at distance d from the leading edge, and forces L and D at angle of attack α.

On the figure, we read the following quantities:

• L, the lift, perpendicular to the fluid motion, in newtons;
• D, the drag, parallel to the fluid motion, in newtons;
• α, the angle of attack, or geometric incidence, between the chord and the fluid velocity V∞;
• A, the leading edge, and B, the trailing edge; AB is the reference chord, of length c;
• AMB, the upper surface (suction side), and ANB, the lower surface (pressure side);
• r, the leading-edge radius, and τ, the trailing-edge angle;
• APB, the mean camber line, made of the midpoints of the segments perpendicular to the chord, such as MN; f is its largest ordinate relative to AB, and d the position of this maximum camber;
• the ratio f/c is the relative camber; the maximum value of segment MN is the maximum thickness e_max, and e_max/c the maximum relative thickness.

Classical airfoils have a relative thickness of 6 to 20%, located at about a third of the chord (c/3). Their relative camber stays on the order of a few percent, and its position d/c varies from 15 to 50% of the chord.

2: NACA airfoils, and how to read a NACA 4417

Many airfoil families are catalogued around the world: Joukowski, Eppler, Wortmann, NACA, RAE, Göttingen, NLR, NASA/LRC, SANDIA. NACA airfoils are among the best known, and some are commonly used, in practice as well as to validate numerical methods. The NACA (National Advisory Committee for Aeronautics) is the predecessor of NASA. It studied four-digit and five-digit families, and laminar airfoils.

In the four-digit family, we read the name as follows: the first digit gives the maximum relative camber as a percentage of the chord, the second the position of this camber in tenths of the chord, the last two the maximum relative thickness as a percentage of the chord. The NACA 4417 therefore has:

• a maximum relative camber of 4%;
• located at 40% of the chord;
• a maximum relative thickness of 17%.

The four-digit NACA airfoil editor of Heliciel's interactive database, here set to the NACA 4417 (relative thickness 0.1687).

Calculated or measured airfoil performance is stored in the database as Cl (lift coefficient), Cd (drag coefficient) and Cm (moment coefficient). They are given for each angle of attack and each Reynolds number at which the airfoil was evaluated, with the normalized coordinates that define it.

3: How to choose an airfoil: the Cl/Cd lift-to-drag ratio

A propeller or a wing has a better or worse efficiency, and that efficiency is a relentless judge of our work. It is optimal only if the airfoils are well drawn. An airfoil is chosen for two opposing qualities: its lift-to-drag ratio and the strength that its thickness provides. Everything is therefore a matter of compromise.

Heliciel optimizes the airfoils automatically, selecting them by their maximum lift-to-drag ratio and by the thickness distribution. If you prefer to choose the airfoils of a blade or wing yourself, knowing the parameters that set this ratio will help.

The lift-to-drag ratio is the ratio of lift to drag of the airfoil: lift-to-drag ratio = Cl/Cd. A good Cl is high, a good Cd is low. We therefore have two levers.

First lever, Cd. The larger the relative thickness of the airfoil, the larger the Cd; reducing the relative thickness increases the ratio. Drag has two parts: pressure drag, with coefficient Cdp, and friction drag, with coefficient Cdf, so that Cd = Cdp + Cdf. Two examples show it.

Around a sphere, the pressure distribution gives the pressure drag, the sum of the pressure vectors in the figure. Its Cdp is 0.511. Friction is small next to it, so Cd = Cdp = 0.511.

Pressures around a sphere: they account for most of its drag.

Around an airship hull of the same volume, the sum of the pressure vectors gives a Cdp of 0.0331. Friction weighs more, because of the large contact surface. The measured total drag gives Cd = 0.1, hence a Cdf of about 0.067 (0.1 − 0.0331).

On an elongated hull, the pressures almost balance out: friction drag dominates.

The next figure, after Marchaj, shows how drag depends on relative thickness, for seven sections from 0.055 to 0.40 relative thickness, at Re = 4 × 10⁵. Total drag goes through a minimum for a relative thickness around 0.25 to 0.30, and rises on either side. At small relative thicknesses, pressure drag can become the minor part. To optimize a very thin wing or blade, we must then take care of the surface finish, which weighs heavily in friction drag.

Drag per unit length against relative thickness (Re = 4 × 10⁵): the total drag curve has a minimum. (The figure labels are in French: "Traînée" is drag.)

Second lever, Cl. We can also gain lift-to-drag ratio by raising the lift coefficient. The camber of the airfoil contributes a great deal, but one phenomenon disturbs the relation: stall.

Lift corresponds to the change of momentum of the fluid. It comes from our ability to make the fluid change direction: the more we deflect it downward, the greater the upward lift. Like a car in a bend, the faster the mass of fluid, the more inertia it has, and the less it accepts to change direction. If the bend is too sharp, the fluid separates and lift collapses. A slow fluid accepts a tight bend, hence strong camber; a fast fluid calls for low camber.

We keep three points:

• a change of direction, properly oriented, generates lift;
• camber causes, together with the angle of attack, the change of direction asked of the fluid, hence lift;
• at high speeds camber stays low; at low speeds it can be strong.

This graph, established for curved plates at Re = 3.14 × 10⁵, shows lift increasing with relative camber f/c, from 0 (flat plate) to 0.10. The theoretical lift slope is about 0.11 per degree.

Cl of curved plates against angle of attack: the stronger the camber f/c, the higher the curve climbs before stall. (Figure labels in French: "Coefficient de portance Cz" is lift coefficient.)

The following graphs compare three airfoils of different camber: the Göttingen 625 (thick airfoil), the Göttingen 417 (curved sheet) and the flat plate. They are drawn for four speeds, which the Reynolds number sums up together with the chord and the viscosity of the fluid. Both airfoils are in the Heliciel database.

The thick Göttingen 625 (relative thickness 0.2) and the thin curved-sheet Göttingen 417 (0.0668), as drawn by Heliciel's database.

Cl of the Göttingen 417a, the Göttingen 625 and the flat plate for four Reynolds numbers: their ranking changes with speed.

These curves show that camber is not enough to predict lift: speed and thickness play a role that no simple rule sums up. This is why Heliciel relies on polars per Reynolds number rather than on camber alone.

To go further, we can look at how to choose the airfoil of a blade, or how the strength of the blade limits thickness.