Category Aircraft Flight

The dependence of drag on lift

The lift produced by a wing is dependent on the flow speed and the circulation, which is related to the strength of the vortex system. In level flight, the lift is equal to the weight. Thus, at constant altitude and aircraft weight, the required vortex strength is reduced as the speed increases. Since the trailing vortex drag also depends on the strength of the vortex system, the trailing vortex drag also reduces with increasing speed. In fact, the drag coefficient for trailing vortex drag is proportional to CL2, and it may be remembered that for level flight the CL value required reduces with increasing speed.

In contrast, the boundary layer normal pressure and surface friction drag rise roughly as the square of the speed. From Fig. 4.21 we see that as a result, there is a minimum value for the overall drag, and this minimum occurs when the trailing vortex drag is equal to the boundary layer drag. There is, therefore, a disadvantage in trying to fly any aircraft too slowly. The implications of this, in terms of performance and stability, are discussed in later chapters.

It is important to note that trailing vortex drag is not the only drag con­tribution that is lift-dependent. If a symmetrical wing section is set at zero angle of attack to a stream of air, the boundary layers on both upper and lower surfaces will be identical, but once the angle of attack is increased, and lift is generated, the boundary layers will alter, together with the amount of drag produced. Thus, it will be seen that some of the boundary layer (profile) drag is also lift-dependent.

For further information on drag, the reader is referred to Hoerner (1965), who gives an excellent detailed treatise on the subject.

Recommended further reading

Lachmann, G. V., (editor), Boundary layer and flow control, Vols I & II, Pergamon Press, 1961.

Hoerner, S. F., Fluid dynamic drag, Hoerner, New Jersey, 1965.

The dependence of drag on lift

Choice of powerplant

Although the jet engine has now been around for more than half a century, vir­tually all small private aircraft are still powered by reciprocating petrol (gasoline) engines driving propellers. Large commercial transport and military aircraft are predominantly propelled by turbo-jet or turbo-fan engines, while for the intermediate size of civil aircraft, ranging from small executive transports to short-haul feeder airliners, a gas-turbine driving a propeller is frequently chosen. The reason for these divisions will be seen from the descriptions that follow.

Reciprocating engines

Small reciprocating engines can produce a surprisingly large amount of power in relationship to their size. A large model aircraft engine can produce about 373 W (0.5 bhp), which is rather more than the power that a good human athlete can sustain for periods exceeding one minute. A problem arises, how­ever, when an engine is required for a very large or fast aircraft, where a con­siderable amount of power is required.

If we simply tried to scale up a typical light aircraft engine, the stresses due to the inertia of the reciprocating parts would also increase with scale, and we would very soon find that there was no material capable of withstanding such stresses. This is because the volume, and hence, the mass and inertia of the rotating parts, increases as the cube of the size, whereas, the cross-sectional area only increases as the square.

The way to overcome this problem is to keep the cylinder size small, but to increase the number of cylinders. Thus, whereas a light aircraft will normally have four or six cylinders, the larger piston-engined airliners of the 1940s and 1950s frequently used four engines with as many as 28 cylinders each. The complexity of such arrangements leads to high costs, both in initial outlay, and in servicing. To get some idea, try working out how long it would take you to change the sparking plugs in an aircraft with four engines of 28 cylinders each, and two plugs per cylinder. Do not forget to allow some time for moving the ladders.

Many arrangements of cylinders were tried, in order to devise a convenient, compact and well balanced configuration. By the end of the era of the large piston engines, in the early 1950s, two types predominated; the air-cooled radial, and the in-line water-cooled V-12. The latter is typified by the Rolls – Royce Merlin (Fig. 6.15), which was used on many famous allied aircraft of the Second World War, including the Spitfire and Mustang. Water-cooling was used on the in-line V-12 engines, because of the difficulties of producing even cooling of all cylinders with air. Modern light aircraft engines are normally air­cooled, with four or six cylinders arranged in a flat configuration.

Choice of powerplant

Fig. 6.15 The classic liquid-cooled V-12 Rolls-Royce Merlin, which propelled many famous allied aircraft during the Second World War, including the Spitfire and Mustang. A large supercharger is fitted

The economics of high speed

In the discussion above and in Chapter 6 we discovered that the jet engine’s performance in terms of efficiency improves with speed, eventually becoming higher than that of the piston engine/propeller combination. As we increase the cruising speed, or Mach number, so we can employ power plants having a steadily improving efficiency.

An idea of the overall efficiency of the airframe/engine combination can be obtained by multiplying the airframe efficiency (best lift/drag ratio) by the engine efficiency. The result of this is shown in Fig. 7.9 and indicates that this overall efficiency can be kept surprisingly constant with speed. Thus for a given journey we can in principle construct an aircraft which will cruise at high speed and only use the same amount of fuel as its low speed competitor.

The above argument does not imply that the overall efficiency of an indi­vidual aircraft does not vary with speed. It merely means that we can design particular configurations intended for operation at widely differing speeds, with similar overall efficiencies.

The economical operation of a commercial aircraft is not just a matter of the amount of fuel used per passenger on a given flight. Aircraft cost a great deal and must complete as many flights per day as possible to pay their way. Crews have to be paid by the hour; and the airline which can provide the fastest service will generally attract the most passengers – other factors being equal. These factors clearly make the high speed aircraft a very attractive option.

Again we must beware of making too sweeping conclusions from such an argument. The design of a particular aircraft to fill a particular slot in the market is very complicated. Development costs, especially for supersonic and


Fig. 7.9 Overall aircraft efficiency

This figure represents best achievable figures. As airframe efficiency declines achievable propulsive efficiency rises to compensate

hypersonic configurations where little previous experience is available, are very high. We also have to remember that we have only considered the problem assuming we can cruise our aircraft at its optimum speed throughout the flight. The Concorde, which is an example of a supersonic transport, had to spend a substantial part of the flight cruising at subsonic speed to avoid creating too much disturbance on the ground with its shock waves. It may also have to spend some time queueing to land. These factors may significantly increase the fuel usage over the flight and a comparatively small change in fuel prices may nullify the other commercial advantages described above. In spite of these difficulties Concorde showed a good operating profit.

We also find certain ‘natural breaks’ in the scale of economical cruising speeds. At a flight Mach number in the region of unity we know that there is a rapid increase in the drag which can be achieved for a given lift. It is some time before the improved engine efficiency makes up for this. It is for this reason that there is a gap in the cruising speed of transport aircraft between the majority of aircraft which cruise at flight Mach numbers of approximately 0.8 and Concorde which cruises at a Mach number of 2.

Concorde represents another limit, that imposed by kinetic heating. Above this Mach number serious problems begin to be encountered with conventional light alloy materials and greatly increased development and construction costs must be accepted.

However, the more general argument for high speed, aimed at very long-term developments, is of interest in sorting out the practical from the pipedream. As Dietrich Kuchemann (1978), an aerodynamicist who has contributed much to the development of high speed aircraft, points out, the semi-orbital hypersonic airliner travelling to Australia from the UK in a couple of hours may well be a sensible long-term goal.

Dealing with the wing centre section

We mentioned above that the wing centre section posed problems as well as the tip region. Although in the real aircraft there will, in general, be a fuselage, we can get a useful insight into the basic problem by first considering the wing in isolation.

The problem is, in some ways, very similar to the tip problem that we have already discussed. We see a similar reduction in sweep of the isobars to that encountered at the tip (Fig. 9.16(a)). However because of the mutual influence of the wing sections further outboard and the influence of the trailing vortex system, the loading at the centre section becomes less, rather than more peaky (Fig. 9.14), and, in addition, the overall loading in the centre section becomes lower because of this effect.

Neither of these effects is particularly welcome. If the centre section loading is less ‘peaky’ there will be an even greater tendency for stalling to take place first at the tip region, which, as we have already seen is undesirable. The loss of overall load in the centre region is also undesirable because this means that the wing will have a reduced overall efficiency. There is also a structural implica­tion because the bending moment on the wing will be increased if the load is concentrated towards the tips.

The same methods can be used to solve the problems at the centre section as were used for the tips – we can alter the aerofoil thickness, or its camber or we can twist the wing to alter the local angle of attack. We can also change the planform in this region, but this again is frought with structural and other problems which we will examine later.

By introducing local changes in the section we aim to make the load distri­bution approach, as far as possible, the distribution which is obtained on the infinite sheared wing. In order to maintain the sweep of the lines of constant pressure (isobars) at the centre section, the point of maximum thickness can be moved forwards on the section. At the same time a local negative camber is used which again shifts the centre of loading towards the front of the section. By these means we can, at the design condition, achieve a reasonably efficient load distribution while, at the same time, encouraging stall to occur at the inner section before the tip region.

Static stability

Solving the problems

The precise analysis of aircraft stability is an extremely complicated process. For conventional straight-winged aircraft in the pre-jet age, it was found that by making a few simplifying assumptions, the problems could be reduced to a form where they could be solved by traditional analysis and hand calculations. Some aspects of this approach are still perpetuated in introductory texts and courses, because the simplification can act as an aid to understanding. The increasing aerodynamic complexity of aircraft has, however, rendered many of the assumptions inappropriate, and for industrial purposes, a more complete solution of the stability equations is normally attempted. This direct approach has been made practical by the advent of the digital computer, but despite the advances that have been made in theoretical methods, the analysis of air­craft stability still represents a considerable challenge, particularly for uncon­ventional types such as the forward-swept X-29 shown in Fig. 9.20.

Although we shall not attempt to describe the process of stability analysis, we can at least explain some of the principles and design features involved in producing a stable and controllable aircraft.

The requirements for trim and stability

For steady flight, the forces acting on an aircraft must be in balance, and there must be no resultant turning moment about any axis. When this condition is achieved, the aircraft is said to be trimmed. In Fig. 11.1 we show an aircraft that is trimmed about its pitching axis.

An aircraft is said to be statically stable if it tends to return to its initial flight conditions; attitude, speed etc., after being disturbed by a gust or a small


For aircraft to be trimmed L„x a – M0 = L, x b

Fig. 11.1 Forces on an aircraft trimmed for steady level flight

The movements about the centre of gravity due to wing lift, the tail downforce and the pitching couple are exactly in balance

In this simple example we have chosen a case where the thrust and drag forces are on the same line. This is not generally true, and thrust and drag forces normally affect the trim. Fuselage effects have also been ignored impulsive input from the controls. Normally, for steady flight, we require the aircraft to be both trimmed and stable.

There is frequently considerable confusion about the difference between balanced or trimmed, and stable. If you balance a ball on the end of your finger, it may temporarily be perfectly balanced, but it is certainly not in a stable position.

In general, the more stable we make an aircraft, the less manoeuvrable it becomes. A very stable aircraft always tends to continue on its existing path, so excessive stability must be avoided.

We can quickly get some idea of how stable an aircraft is by ignoring inertia or time-dependent effects, and just looking at the balance of the forces and moments acting on the aircraft; in other words, by treating the problem as if it were one of statics. Once it is established that an aircraft is statically stable it is then necessary to go on to investigate the inertia and time-dependent effects; the so-called dynamic stability described in the next chapter. This approach was part of the traditional method of breaking down the complex problem of aircraft stability, and although computational techniques have to some extent rendered it unnecessary, it is still useful, particularly when introducing the subject.

Approach and landing

The landing is the most difficult task the pilot has to undertake. It requires an accurate approach to position the aircraft correctly in relation to the runway, together with precise control during touch-down which may be complicated by winds blowing across the flightpath.

Figure 13.6 shows the stages from initial approach to touch-down. Some way out from the runway the aircraft speed is reduced and high lift devices extended to reduce the minimum flying speed. A typical landing configuration is shown in Fig. 13.7. Comparing this with the corresponding take-off con­figuration it can be seen that a lot more trailing-edge flap is used because extra drag is, of course, a positive advantage during landing, both from the point of view of the final deceleration of the aircraft and because a high drag configura­tion leads to easier speed control.

At the start of the landing manoeuvre the aircraft is aligned with the runway and put into a steady descent along the ‘glide path’. As the runway threshold is reached the angle of attack is increased so that the rate of descent is reduced and the aircraft is ‘flared’ so that it flies just above and nearly parallel to the runway until the touchdown point is reached. At this point the aim is to stop as quickly and safely as possible. In order to provide aerodynamic braking and

Fig. 13.7 Landing configuration

The BAe 146 with everything deployed. Double flaps fully extended. Lift dumpers deployed above the wings to increase drag and destroy lift, and rear airbrake doors wide open

to sit the aircraft firmly on the runway ‘lift dumpers’, or spoilers, may be used Fig. 13.7. Jet aircraft frequently use thrust reversers (Fig. 6.32) to provide further deceleration and to relieve the wheel brake requirement. Some military aircraft even resort to the use of a braking parachute to shorten the landing run.

Pressure and lift

Figure 1.15 shows how the pressure varies around an aerofoil section. The shaded area represents pressures greater than the general surrounding or ‘ambient’ air pressure, and the unshaded region represents low pressures. It will be seen that the difference in pressures between upper and lower surfaces is greatest over the front portion of the aerofoil, and therefore most of the lift force must come from that region. This effect was quite pronounced on older wing sections, but nowadays the trend is to design aerofoil sections to give a

fairly constant low pressure over a large proportion of the top surface. This produces a more uniform distribution of lift along the section, giving both structural and aerodynamic advantages, as we shall describe at various points later.

Since the relative flow speed reduces to zero at the stagnation position, it follows that the pressure there must have its highest possible value. This maximum value is therefore called the stagnation pressure. Stagnation pressure should not be confused with static pressure defined earlier. Unfortunately, and for obvious reasons, it often is! Static pressure is just the air pressure. Stagnation pressure is the pressure at a stagnation position; a position where there is no relative motion between the air and the surface.

From Fig. 1.15 we see that the pressure falls rapidly as the air accelerates and flows away from the stagnation position, becoming extremely low around the leading edge. This low leading edge pressure is again contrary to expecta­tion, but is linked to the fact that the stagnation position is behind the leading edge on the underside. Thus, the air taking the upper-surface route has to flow forward, and then negotiate a fairly sharp curve. In order for the air to do this, rather than carry on in a straight line, there must be a low pressure on the leading edge, to pull the flow into a curved path: i. e. to provide the necessary centripetal acceleration.

Favourable and unfavourable conditions

As described above, separation tends to occur when air flows from a low pres­sure to a high one. This is therefore known as an adverse pressure gradient. Conversely, flow from a high pressure to a low one is called a favourable pres­sure gradient.

A favourable pressure gradient not only inhibits separation, but slows down the rate of boundary layer growth, and delays transition. In the next chapter, we will show how we can exploit this factor to produce low-drag aerofoil sec­tion shapes.

Leading-edge separation

Flow separation is particularly likely to occur when the air tries to go round a very sharp bend, as on the nose of the thin aerofoil. For air to travel around a curve, the pressure on the outside of the curve must be greater than on the inside, in order to provide the necessary ‘cornering’ (centripetal) force. Thus, the pressure on the leading edge of an aerofoil is often locally very low. On the upper surface, the pressure initially rises again rapidly with distance from the leading edge. A strong adverse pressure gradient (flow from a low pressure to a high one) is therefore produced, and the flow tends to separate at, or very near the leading edge.

When such leading-edge separation occurs, the stall or loss of lift may be both sudden and severe. Aerofoils with a large radius leading edge are less prone to producing leading-edge separation, and therefore tend to have a more progressive and safer stall characteristic. As we shall see later, however, there are various reasons why it is sometimes advantageous to use an aerofoil with a sharp leading edge.

It is a common mistake to confuse separation and transition. Transition is where the boundary layer changes from laminar to turbulent. Separation is where the flow ceases to follow the contours of the surface. The fact that separation is normally accompanied by large-scale turbulence is probably the source of the confusion.

Favourable and unfavourable conditions

Fig. 3.4 Sometimes the separated boundary layer may reattach forming a ‘bubble’ of recirculating air

The pros and cons of high lift devices

The high lift coefficients obtained with both leading and trailing-edge devices incur a penalty in terms of drag, but this may be acceptable or even useful in landing, as described in Chapter 13. Note the extreme amount of curvature used on the flaps of the Andover shown in Fig. 3.16.

For take-off, it is normal to use a configuration giving lower CL and less drag. Smaller flap angles are almost invariably used.

There are many versions of slot, slat and flap, in addition to the examples illustrated in Figure 3.13. Their effectiveness depends on the precise geo­metry of the device, and on the type of aerofoil section used. It is therefore impractical to try to indicate a figure for the order of improvement in CL for competing designs. Generally, and unfortunately, the most effective devices tend to be the most complicated and heaviest.

Transonic drag rise and centre of pressure shift

The dramatic change in flow from subsonic to supersonic conditions is, as might be expected, accompanied by marked loading changes on the aerofoil. One important consequence of this is a rearward shift in the centre of lift.

The formation of the shock waves as the flow develops in the transonic speed range leads to the formation of a large separated wake (Fig. 5.18(b)). This in turn leads to a very rapid drag rise over a small Mach number range.

Transonic drag rise and centre of pressure shift


This shock wave is a reflection from the tunnel wall






Fig. 5.18 Shock wave development on a conventional aerofoil

(a) Subsonic flow with no shocks (b) Transonic flow. The approaching flow is subsonic, but patches of supersonic flow develop downstream of the leading edge, terminating in a shock wave on both upper and lower surfaces (c) Supersonic approach flow. Oblique shock waves initiated at the leading edge slow the flow to a lower Mach number than the approach. The flow then accelerates to a higher Mach number, and is finally reduced again via a second pair of shock waves at the trailing edge


Transonic drag rise and centre of pressure shiftTransonic drag rise and centre of pressure shift

Transonic drag rise and centre of pressure shift

Fig. 5.19 Effect of Mach number on lift and drag coefficients at constant angle of attack

Shock induced separation causes a rapid increase in drag coefficient in transonic region

The drag rises much more rapidly than the dynamic pressure so that the drag coefficient rises. The drag coefficient falls again as the fully supersonic flow pattern is established and Fig. 5.19 shows the typical transonic drag coefficient peak which is of great importance in the design of both transonic and super­sonic aircraft as we shall see in later chapters.

Figure 5.19 also shows that the lift coefficient varies significantly as the speed of sound is approached. It should be noted that Fig. 5.19 shows the vari­ation of lift and drag coefficients at constant angle of attack. If the angle of attack is varied as the flight speed is changed in order to keep the overall lift (rather than the lift coefficient) constant, as would be the case in cruising flight, then a slight fall in the drag coefficients is frequently experienced just prior to the rapid rise as the speed of sound is approached. This occurs because the increase in lift coefficient means that the angle of attack can be reduced. This local reduction in drag coefficient can be usefully exploited in design.