Supersonic aerofoils

We have spent some time considering the way in which lift is produced in subsonic flow (Chapter 1). There are some similarities in supersonic flow. The lift is produced by a difference in pressure between the top and bottom sur­faces, and this requires a high speed on the top surface and a reduced speed on the lower surface whether the flow is subsonic or supersonic.

However, although the two cases have this much in common, there are con­siderable differences between the flow patterns of the high and low speed cases. For example shock wave generation is an important factor at high speed, and suitable design to minimise the drag caused by the formation of these shock waves is extremely important. With these points in mind, it is likely that the aerofoil sections which perform best in supersonic conditions may look con­siderably different from their low speed cousins.

Fig. 8.3 Canard and delta The Eurofighter Typhoon is required to fulfil a variety of roles, but a simpler fixed planform has been chosen, as this reduces weight, complexity and cost

In Chapter 5 we examined the changing flow over a typical subsonic type of aerofoil as the upstream Mach number increases (Fig. 5.18). The flow is characterised by the development of shock wave systems at the leading and trailing edges. In the supersonic flow regime, the flow field is entirely super­sonic, with the exception of a small patch of subsonic flow on the blunt lead­ing edge in the region of the stagnation point.

The wave drag associated with this type of aerofoil is high because of the strong bow shock wave. Such an aerofoil is therefore not suitable for use in supersonic flow unless the wing is swept to reduce the effective approach velocity (see Chapter 2).

In order to reduce the strength of the bow shock wave it is desirable to make the leading edge of the aerofoil sharp. This will remove the region of near nor­mal shock associated with the blunt leading edge, with a consequent reduction in wave drag. Figure 8.5 shows a particularly simple form of supersonic aero­foil, the ‘double wedge’ section. We met this briefly in Chapter 5 and now look at its suitability for practical application.

Figure 8.5 also gives a comparison of the surface pressure distribution on the double wedge aerofoil at subsonic and supersonic speeds for small angles of attack. In the subsonic case we would expect to get the typical suction peak near the leading edge on the upper surface followed by a recompression as we move towards the trailing edge. On the bottom surface we will obtain a

Pressure lower than surrounding atmospheric

Pressure higher than surrounding atmospheric

Pressure higher than surrounding atmospheric

Pressure lower than surrounding atmospheric

Fig. 8.5 Pressure distribution on double wedge aerofoil

(a) Subsonic (very low angle of attack) (b) Supersonic stagnation point, and the higher pressure on the undersurface will also con­tribute to the overall lift.

The pressure distribution on the aerofoil in a supersonic air stream is very much simpler, each of the four faces of the diamond cross-section experiencing virtually constant pressure. This follows from the fact that the flow over the two forward-facing surfaces is uniform as the bow shock waves simply deflect the entire flow until it becomes parallel with the surface direction (Chapter 5). Similarly the expansion fans generated from the apexes on the upper and lower surfaces turn the flow so that it is parallel to the rearward-facing surfaces. This results in a uniform pressure over these surfaces as well.

It is when we increase the angle of attack that the biggest surprise occurs, however. We already know that, for low speeds, thin aerofoils and, even worse, those with sharp leading edges, will stall at relatively low angles of attack. Even if the flow were to successfully negotiate the sharp leading edge we would not do all that well. The sudden change in surface direction at the junction between the front and rear surface would again lead to separation; this time over the rear part of the aerofoil (Fig. 8.6(a)).

SUPERSONIC AEROFOILS 221

Fig. 8.6 Double wedge aerofoil at low and high speeds (a) Low speed – increased angle of attack (b) Low speed – angle of attack further increased (c) Supersonic flow – flow unseparated

When we look at the supersonic flow, however, we find that the flow deflection caused by the bow shock waves removes any problem at the sharp leading edge. The flow now divides right at the leading edge rather than at an undersurface stagnation point as is the case with subsonic flow.

The flow is also quite happy to negotiate the subsequent abrupt change in surface direction by means of the expansion fan (Fig. 8.6(c)), because, as we saw in Chapter 5, the local pressure gradient is favourable at supersonic speeds (i. e. pressure reduces in the direction of motion). At subsonic speed, however, there is a locally unfavourable gradient, and so the boundary layer would separate here, even if the angle of attack were low enough to prevent earlier separation at the sharp nose.

Thus we find that aerofoils with sharp leading edges and abrupt changes in surface slope, factors which would lead to disastrous performance at low

speed, perform quite well in the supersonic speed range. Compared to a typical low speed aerofoil, for which L/D ratios in the order of 40 can be obtained, their performance does not look all that exciting. The comparatively poor performance is, of course, due to the wave drag which has to be overcome. This penalty may, however, be acceptable in many military applications where speed is of prime importance. For civil transport aircraft, too, the poor lift-to-drag ratio may be acceptable. The increased cruising speed allows better utilisation of the aircraft and a better measure of overall efficiency may be the cruising speed times the lift-to-drag ratio (Chapter 7).

The only trouble with all this is that although these simple aerofoil sections are good at supersonic speed, their performance, as we have seen, is hopeless at low speed. There is, however, a class of ‘aircraft’ which is not called upon to fly at low speeds at all; air-to-air missiles. Such aerofoil sections are therefore very often employed for these devices.

The problem of poor maximum lift coefficient at low speed is not the only difficulty encountered in the aerodynamic design of high speed aircraft. In Chapter 5 we saw how the centre of pressure moves rearwards on an aerofoil as the supersonic flow pattern is established. This change in centre of pressure position results in a large change in longitudinal trim which must either be accommodated by the provision of large tail surfaces, or by other means. For example the Concorde used fuel transfer fore and aft to change the position of the aircraft centre of gravity as is mentioned in Chapter 10.

If we want to take off or land our aircraft from conventional runways and to have a reasonable subsonic performance, as well as operate at supersonic speeds, we need to employ a wing with acceptable low speed and high speed performance and which does not have any violent change in flow character­istics as the aircraft accelerates through its speed range. It is the precise nature of this compromise which is responsible for the large variety of solutions which are found in practice.