Category Aircraft Flight

Variation of lift with angle of attack and camber

As shown in Fig. 1.17, the lift coefficient is directly proportional to the angle of attack for small angles.

Figure 1.17 also shows the effect of camber on lift coefficient. It will be seen that the influences of angle of attack and camber are largely independent: that is, the increase in lift coefficient due to camber is the same at all angles of attack.

Cambered aerofoils can produce higher maximum lift coefficients than sym­metrical ones. Also, as shown in Fig. 1.17, they produce lift at zero angle of attack. The angle at which no lift is generated is therefore negative, and is known as the zero-lift angle.

Variation of lift with angle of attack and camber

Fig. 1.17 Variation of lift coefficient with angle of attack and camber

The increase in lift coefficient due to camber is almost independent of the angle of attack

The shape of the camber or mean line is important as well, as it affects the position of the line of action of the resultant lift force. Later on, we shall describe how variations in camber can be used to control the aerodynamic properties of a wing.

The generation of lift and the formation of the starting vortex

Figure 3.5 shows what happens when an aerofoil is rapidly accelerated from rest. At first, when virtually no lift is generated, the streamlines show an almost anti-symmetrical pattern, with a rear dividing position situated on the upper surface near the trailing edge. This pattern is similar to that given by earlier ver­sions of the classical theory, where no lift was predicted (see Fig. 1.6(b)).

As the flow speed increases, the boundary layer starts to separate at the trailing edge, due to the adverse pressure gradient, and a vortex begins to form, as shown in Fig. 3.5(b).

The vortex grows, moving rearwards, until it eventually leaves the surface and proceeds downstream, as in Fig. 3.5(c). This detached vortex is the starting vortex that we described in Chapter 2. We can see that it is the production of this starting vortex that destroys the anti-symmetry of the flow, resulting in dif­ferences in pressure and speed between the upper and lower surfaces. Thus, it is viscosity, working through the mechanism of boundary layer separation and starting vortex formation, that is ultimately responsible for the generation of lift.

The upper and lower surface flows rejoin at the trailing edge with no abrupt change of direction; the Kutta condition mentioned in Chapter 1. The upper and lower surface boundary layers join to form a wake of air moving more slowly than the surrounding air stream.

The generation of lift and the formation of the starting vortex

Fig. 3.5 The formation of the starting vortex

The generation of lift and the formation of the starting vortex

Fig. 3.6 The velocity variation in the boundary layer is rather like that in a wheel rolling along a surface, and may similarly be thought of as being a combination of rotational and translational movement

In Chapter 1 we showed how the difference in the speeds above and below the wing could be represented as being equivalent to superimposing a circulat­ing vortex type of flow on the main stream. By similar reasoning, we can say that, since the flow speed in the boundary layer is faster at the outside than at the surface, it too can be represented by a combination of rotation and translation, as illustrated in Fig. 3.6. Once again, it should be noted that no air particle actually goes round in circles. The flow in the boundary layer merely

has rotational tendency superimposed on its translational motion. However, if a speck of dust enters the boundary layer, it will rotate as it moves along.

Importance of speed of sound – Mach number

It was mentioned above that an aircraft travelling at supersonic speed does not affect the state of the air ahead of the aircraft, while at subsonic speed the disturbance is propagated far upstream. In order to understand the reason for this we need to take a look at how the aircraft is able to make its presence felt as it travels through the air.

Figure 5.2(a) shows the nose of an aircraft flying at subsonic speed. As the flow approaches the nose of the aircraft it slows and the pressure locally increases. The influence of this region of increased pressure is transmitted upstream against the oncoming flow at the speed of sound (approximately 340 m/s at sea level). If the flow approaching the aircraft is subsonic then the disturbance will be transmitted faster than the oncoming flow and the aircraft will be able to make its presence felt infinitely far upstream.

Figure 5.2(b) shows what happens in supersonic flight. The disturbance can only make headway through an area near the nose where the flow is locally subsonic. The flow upstream is separated from this localised region by a shock wave, and is completely uninfluenced by the presence of the aircraft.

As the speed of the flow increases, so the region of subsonic flow at the nose gets smaller and the shock wave gets stronger (i. e. the pressure, density and temperature jumps all become larger).

This is why the speed of the aircraft relative to the speed of sound is the important factor in determining the flow characteristics. This ratio is known as the flight Mach number.

Flight Mach No. = Aircraft speed/speed of sound

Importance of speed of sound - Mach number
Importance of speed of sound - Mach number
Importance of speed of sound - Mach number
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Подпись: 'Subsonic' flow S'
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Importance of speed of sound - Mach numberFig. 5.2 Propagation of pressure disturbances

(a) At subsonic speeds pressure disturbances generated at the nose travel at speed of sound and can make headway against oncoming flow (b) At supersonic speed the disturbances can only propagate through the locally subsonic region near the nose

When the flight Mach number is greater than one, then the aircraft is flying supersonically. When it is less than one then it is flying subsonically.

When an aircraft is flying supersonically we have seen that there may be local areas, such as the region near the nose, where the flow speed is locally reduced. Not only is the speed reduced, but the local temperature will rise, thus increasing the local speed of sound. Because of this there will be regions where the flow is locally subsonic (Fig. 5.3(a)).

Conversely, the regions on an aircraft where the flow speed is locally increased, such as the top of the wings, may lead to localised patches of super­sonic flow (Fig. 5.3(b)) even when the flight Mach number is subsonic. Thus we need to define a local Mach number for different areas of the flow.

Local Mach No. = Local flow speed/local speed of sound.

Importance of speed of sound - Mach numberSupersonic ‘patch’ due to local speeding up of flow I (Local M>1)

Подпись:Подпись: wave

Importance of speed of sound - Mach number

-"f’] Shock

Fig. 5.4 Change of speed along wind-tunnel duct

If there is a small pressure difference between ends of duct, speed rises to maximum at throat and then decreases

For larger pressure differences speed becomes supersonic downstream of throat

The need for an alternative

Attempts to build really large piston-engined aircraft were thwarted by the lack of power. The Bristol Brabazon (Fig. 6.16), which had eight large engines coupled in pairs through massive gearboxes to contra-rotating propellers, was a good example of the impractical result of such attempts. Imagine chang­ing the sparking plugs on that lot! It was designed to carry around a mere

The need for an alternative

Fig. 6.16 Piston-engined power

The massive Bristol Brabazon 1 used eight large piston engines coupled in pairs to four sets of contra-rotating propellers. Intended as a non-stop transatlantic aerial luxury-liner, it was rendered obsolete by the faster more comfortable jet-propelled airliners. Even the turbo-prop Brabazon 2 was abandoned before completion. Piston-engined transports continued to be used for several years for freight and second-class passenger transport (Photo courtesy of British Aerospace (Bristol))



100 passengers; rather less than a typical modern small feeder-liner such as the BAe 146 (Fig. 6.26).

Endurance with piston engine

We saw above that the piston engine/propeller combinations give approxim­ately constant power over the typical operating speed range of the aircraft for a given fuel flow rate. Thus, as far as the engine is concerned, we will get the best endurance when operating at as low a power rating as possible. Fortunately this coincides with the airframe requirement and so we operate at the minimum power speed (Fig. 7.10).

Let us now examine further the implications for the operation of the aircraft as we did for the case of best economy. Because we are interested in low power, we need to minimise the required power not only with respect to the cruising speed, but also with respect to the cruising altitude. As we saw earlier in this chapter, the required power (equal to drag times air speed) gets greater with increasing height because of the higher air speed required for a given drag. Thus, on our simplified picture of things, we will obtain the best endurance for this type of power plant by operating at low altitude.

In order to reduce the required power still further we can use a low wing loading to reduce the speed for minimum power. Thus a piston-engined air­craft designed for endurance will tend to have a relatively large wing area.

Transonic area rule

We saw in the previous chapter that the cross-sectional area distribution of the complete aircraft was very important from the point of view of reducing the wave drag due to volume. The same is true in the transonic speed range, if the area distribution is not smooth then the transonic drag rise can be greatly increased.

Because we are concerned with Mach numbers near the speed of sound, the direction of the Mach waves in any region where the flow is supersonic will be normal to the direction of motion and so the transonic area rule is concerned with cross-sectional area normal to the centreline, unlike the supersonic case (Chapter 8).

The way in which a satisfactory distribution of cross-sectional area can be obtained varies according to the requirements of the design. In a passenger­carrying aircraft it is usually inconvenient to depart from a basically cylin­drical fuselage, and the influence of the area rule on the cross-sectional area distribution is not readily apparent unless the variation of the area along the length of the aircraft is examined in detail. In other cases, however, such as the Rockwell B1 (Fig. 9.18), the fuselage design is not restricted in this way and the influence of the area rule is clearly shown by the waisted fuselage.

The longitudinal static stability of conventional aircraft

The movement of the centre of lift on a cambered wing has a destabilising effect. Figure 11.4 shows a cambered-section wing that is balanced or trimmed



Aerodynamic centre

Fig. 11.2 Centre of pressure and aerodynamic centre

(a) The centre of lift or centre of pressure moves forward as the wing angle of attack increases.

(b) For an aerofoil, the situation shown in (a) above can be represented by a constant moment or couple (shown here in its true directional sense) and a lift force acting through a fixed point known as the aerodynamic centre.

N. B. The sign convention is that nose-up pitching is positive. A cambered aerofoil as shown above therefore gives a mathematically negative moment about the aerodynamic centre

at one angle of attack, by arranging the line of action of the lift to pass through the centre of gravity. If the angle of attack increases due to some upset, then the lift force will move forward, ahead of the centre of gravity, as shown. This will tend to make the front of the wing pitch upwards. The more it does so, the greater will be the upsetting moment. Such a wing is, therefore, not inherently stable on its own.

The condition for longitudinal static stability is that a positive (nose-up) change of angle of attack should produce a negative (nose-down) change in pitching moment.

where M is the pitching moment, and a is the pitch angle.

The conventional method for making an aircraft longitudinally stable, is to introduce a secondary surface, which is called a tailplane in the British con­vention, or more aptly, a horizontal stabiliser in American terminology.

To give some idea of how the tailplane works, we will consider two simple cases, one very stable, and the other highly unstable. Fig. 11.5(a) shows an aircraft trimmed for steady level flight. For simplicity we consider a case where the thrust and drag forces both pass through the centre of gravity, and thus produce no moment. We also ignore the forces and moments on the fuselage. The tail is initially producing a downward force, and hence, a nose-up pitching moment about the centre of gravity, whereas the wing lift produces a nose – down moment. The couple Mo is drawn nose up in Fig. 11.5, which is the normal mathematical convention.

If the aircraft is tipped nose-up by some disturbance as in Fig. 11.5(b), then the tail downforce and its moment will decrease, and the wing lift and its moment will increase. The moments are, therefore, no longer in balance, and

Fig. 11.5 Longitudinal static stability – a highly stable case

In this simplified example, the thrust and drag forces pass through the centre of gravity, and effects due to the fuselage are ignored.

(a) Aircraft trimmed in pitch LW x a – Mo = Lt x b

(b) If the angle of attack increases due to some upset, the tail-down force will decrease, and wing lift will increase. This will result in a nose-down pitching moment, tending to restore the aircraft to its original attitude.

N. B. Mo is shown here in the mathematically positive (rather than true) sense

there is a net nose-down moment, which will try to restore the aircraft to its original attitude. This aircraft is thus longitudinally statically stable.

Note, that in the simplified description above, we have ignored the inertia of the aircraft, and we have neglected the flexibility of the aircraft and its controls. Most importantly, we have ignored the effect of the fuselage which normally has a significant destabilising effect. It should further be noted that our simple example does not include any effects due to wing sweep. We should also have taken account of the fact that as the wing angle of attack and lift increases, so will the downwash at the tail. The increased downwash at the tail means that the tail downforce does not fall off as sharply with changing angle of attack as would otherwise have been expected. The restoring force is thus weakened.

Fig. 11.6 Longitudinally unstable arrangement – negative longitudinal dihedral necessitated by having centre of gravity much too far aft

Although the aircraft was initially trimmed, any increase in angle of attack will produce an unstable nose-up pitching moment.

(a) Aircraft trimmed with wing at 2°. Angle of attack and tail at 4°.

(b) Aircraft attitude increased by 2°. Wing now at 4° and tail at 6°. The wing lift will double, but the tail lift will only increase by 50%. The aircraft becomes untrimmed, with a nose-up pitching moment, and it will diverge from its initial attitude.

N. B. Just having the centre of gravity behind the wing aerodynamic centre does not make the aircraft unstable; it depends on how far aft the CG is

Wing downwash on the tail generally has a destabilising effect. The influence can be reduced by mounting the tail high relative to the wing.

Figure 11.6 shows a case of an aircraft which is trimmed, yet in a longitudin­ally unstable condition. The wing is initially at 2° angle of attack, and the tail is at 4° angle of attack. If we look at what happens when the angle of attack increases by 2°, due to a disturbance, then we see that the wing angle of attack will double. Since lift is directly proportional to angle of attack, it follows that the wing lift will double. In contrast, increasing the tail angle of attack by 2° from 4° to 6° represents only a 50% increase, with a corresponding 50% increase in tail lift (which is further reduced by the effects of downwash). The resulting forces will, therefore, produce a nose-up pitching moment, and the aircraft will continue to diverge from its original attitude.

The flare and touch-down

The final stages of the landing also offer the pilot a choice of techniques. Two alternatives are illustrated in Fig. 13.8. In the first the angle of attack is increased over a comparatively short period to arrest the descent, a manoeuvre known as the flare. The aircraft then flies parallel to the runway as the speed falls further and finally sinks onto the undercarriage. In the now less-common tail-wheel undercarriage, the final touch-down can either be on the main wheels only, or, with a greater amount of pilot skill, the aircraft can be brought to the angle of attack which results in all three wheels touching simultaneously, the so-called three-point landing.

Fig. 13.8 Alternative landing techniques

(a) Rapid ‘flare’ following straight ‘glide’ (b) Gradual ’round out’

(b) is easier than (a) but gives poorer obstacle clearance

An alternative method is to reduce the glide angle more progressively and to fly the aircraft along an almost circular path onto the runway. This type of approach is less demanding on the pilot, but results in slightly worse ability to clear obstacles near the threshold.

The generation of lift by a wing

In order to understand how the planform of the wing affects lift and drag, we need to look at the three-dimensional nature of the air flow near a wing.

You may remember, that we described in Chapter 1, how the wing pro­duced a circulatory effect; behaving like a vortex. A major breakthrough in the

The generation of lift by a wing

Fig. 2.1 Wing geometry understanding of aircraft aerodynamics came at the end of the nineteenth century, when the English engineer F. W. Lanchester reasoned that if a wing or lifting surface acts like a vortex, then it should possess all the general prop­erties of a vortex. Long before the Wright Brothers’ first flight, a theory of vortex behaviour had been developed which indicated that a vortex could only persist if it either terminated in a wall at each end, or formed a closed ring like a smoke ring. In Fig. 2.3 we show in very simplified form, how this requirement of forming a closed circuit is met. In the diagram we see that the circulatory effect of the wing, which is known as the wing-bound vortex, turns at its ends to form a pair of real vortices, trailing from near the wing tips. The ring is com­pleted by a so-called starting vortex downstream.

These vortices do exist in reality and we can easily detect the trailing vortices in a wind tunnel by using a wool tuft which will rotate rapidly if placed in the appropriate position behind a model. On a real aircraft they can sometimes be seen as fine lines of vapour streaming from near the wing tips, as seen in Fig. 2.4. This often occurs at airshows, particularly on damp days. They are most likely to be seen when an aircraft is pulling out of a dive, and is therefore

The generation of lift by a wing

Fig. 2.2 High aspect ratio on the powered glider version of the Europa (lowest aircraft)

(Photo courtesy of Europa Aircraft Ltd)

The generation of lift by a wing

Fig. 2.3 Simplified wing vortex system

The generation of lift by a wing

Fig. 2.4 Trailing vortices originating at the wing tips of the late-lamented TSR-2, made visible by atmospheric vapour condensation (Photo courtesy of British Aerospace)

generating a large amount of lift, so that the wing circulation and trailing vortices are strong.

In the wing trailing vortex, as in a whirlwind or whirlpool, the speed of the rotating fluid decreases with distance from the central core. From the Bernoulli relationship, we can see that, since the air speed in the centre of the vortex is high, the pressure is low. The low pressure at the centre is accompanied by a low temperature, and any water vapour in the air tends to condense and become visible in the centre of the trailing vortex lines, as in Fig. 2.4. Note that the vapour trails frequently seen behind high-flying aircraft are normally formed by condensation of the water vapour from the engine exhausts, and not from the trailing vortices. Figure 2.5 is a flow-visualisation picture showing the trailing vortices forming at the wing tips.

Boundary layer scale effect – model testing

In Fig. 3.18 we show two thin almost flat wing sections, a full-size one and a scale model, placed at zero angle of attack in a stream of air. In this situation, the position of transition from laminar to turbulent flow will be roughly the same distance from the leading edge in both cases, as illustrated.

From the diagram, you will see that the scale model will therefore have a greater proportion of laminar boundary layer, and consequently a lower drag per unit of area than for the larger one. So the drag per unit area measured on the model is not representative of full scale.

To correct for the effect of scale, the model could be placed in a stream of air moving faster than that for the larger section. This would increase the Reynolds number, and move the position of transition forward. If the speed were sufficiently high, transition could be moved to a position corresponding to that of the full-scale section.

Boundary layer scale effect - model testing


Fig. 3.18 On a thin flat plate at zero angle of attack, the transition position would be at roughly the same distance from the leading edge for both model and full-size plates. The model would therefore have a higher proportion of laminar boundary layer

The same principle applies to all shapes, and to obtain similar flow patterns between model and full scale, it is necessary to ensure that the Reynolds num­ber in the model test is the same as for the full-size aircraft in flight.

The Wright brothers and other early experimenters were either unaware of this fact, or did not bother about it. Their simple wind-tunnel tests conducted on very small models at low speeds indicated that thin plate-like wings gave a better ratio of lift to drag than ones with a thicker aerofoil type of section. Thus, early aircraft had thin plate-like wings. It was Prandtl who spotted the error, and found that when the Reynolds number of the tests was increased by running the tunnel faster, or using larger models, thicker wing sections pro­duced a better lift-to-drag ratio than curved or flat plates.

The reason for the poor performance of thick aerofoil sections at very low Reynolds numbers (small models at low speeds), is that the flow will be laminar over most of the surface and thus will separate very easily. A thin plate with a sharp leading edge generates turbulence at the leading edge, and the resulting turbulent boundary layer is better able to stay attached. Model aircraft often perform better when equipped with means of turbulating the boundary layer, and require quite different wing section shapes from full-size aircraft, as described by Simons (1999).