An aerofoil or airfoil is the shape of the cross-section of an aeroplane wing, or any other structure used to increase lift, like helicopter rotors, or propellers, or turbine blades. Spoilers on cars, designed to decrease lift, are also in the shape of aerofoils.
Figure 1 shows a design for an aeroplane wing, and the cross-section of a spoiler from a high-performance sports car. Notice that the spoiler is the shape of an upside down wing.
In order to model air-flow over and around wings and spoilers, we need a mathe¬matical way of describing the aerofoil shape. One way is to start with a circle C in the complex plane which passes through the point z = 1 and for which the point z = -1 is in its interior.
For each point z ∈ C apply the mapping
w = z + 1/z
As you move around the circle, the collection of all the points to will form the shape of an aerofoil. This is illustrated in figure 2 overleaf. The black dots show some points on the circle, and the open dots show their positions as mapped onto the aerofoil.
1. The picture shown was generated using a circle with centre at (-0.11, 0.43). Using your calculator, or any other computational device, verify the computation: map enough points on the circle in order to be able to sketch an aerofoil shape.
2. Experiment with circles with other centres. Which centre, in your (group's) opin¬ion, gives the most realistic "wing-like" shape?
