Category Aircraft Flight

More about shock waves – normal and oblique shocks

Let us look once more at the nose of our supersonic aircraft. We saw how the shock waves formed in front of it, slowing the air down almost instantaneously and providing a subsonic patch through which the pressure information could propagate a limited distance upstream at the speed of sound (Fig. 5.2). It should be noted that the shock wave itself is able to make headway against the oncoming stream above the speed of sound. Only weak pressure disturbances travel at the speed of sound. The stronger the shock wave is, the faster it can travel through the air.

Considering the problem from the point of view of a stream of air approach­ing a stationary aircraft, this means that the faster the oncoming stream, the stronger the shock wave at the nose becomes. Thus the changes in pressure, density, temperature and velocity which occur through the shock wave all increase with increasing air speed upstream of the shock wave. A mathematical analysis of the problem shows that the strength of the shock wave, expressed as the ratio of the pressure in front of the wave to that behind, depends solely on the Mach number of the approaching air stream.

If we now stand further back from the aircraft we see that the bow shock wave which forms over the nose is, in fact, curved (Fig. 5.3(a)). As we get fur­ther from the nose tip so the shock wave becomes inclined to the direction of the oncoming flow. In this region the shock wave is said to be oblique. At the nose, where it is at right angles to the oncoming flow, it is said to be a normal shock wave.

The oblique shock wave acts in the same way as the normal wave except that it only affects the component of velocity at right angles to itself. The com­ponent of velocity parallel to the wave is completely unaffected. This means that the direction of the flow is changed by an oblique shock (Fig. 5.6) whereas

More about shock waves - normal and oblique shocks

Fig. 5.6 Flow deflection by oblique shock wave

Tangential component Vt remains unchanged but V„2 < Vn

More about shock waves - normal and oblique shocks

Fig. 5.7 Flow deflection through bow shock wave

Deflection reaches a maximum and then reduces again

it is unaffected by a normal shock. In both cases, however, the magnitude of the velocity is reduced as the flow passes through the shock wave.

Looking more carefully at the effect of the bow shock wave (Fig. 5.7) we see that, in general, the same flow deflection can be obtained by two possible angles of oblique wave. The reason for this is given in Fig. 5.8. The wave of greater angle at A is stronger because the velocity component normal to the wave front is greater. It therefore changes the oncoming velocity component more than the weaker wave at point B.

Adding the resulting velocity components immediately downstream of the shock waves at two points (Fig. 5.8) shows how a particular point B (where the shock wave is weak) can be chosen with exactly the same flow deflection as at A (with a strong shock wave).

It should also be noted that for a normal shock wave the downstream flow is always subsonic, as it is for most strong oblique waves. The fact that the

More about shock waves - normal and oblique shocks

Fig. 5.8 Weak and strong shock waves

Strong shock at A gives same deflection as weak shock at B, but greater pressure jump since V2 < V2

velocity component parallel to the wave is not changed means, however, that the flow downstream of the weak oblique wave is supersonic.

Thermodynamic efficiency

In the gas turbine, the burning process causes the air to be heated at virtually constant pressure, in constrast to the piston engine, where the air is heated in an almost constant volume with rapidly rising pressure. The (thermodynamic) efficiency of both types of engine can be shown to depend on the pressure ratio during the initial compression process. Increasing the pressure ratio increases the maximum temperature, and the efficiency is, therefore, limited by the maximum temperature that the materials of the hottest part of the engine can withstand.

The temperature limitation is rather more severe in the gas turbine, since the maximum temperature is sustained continuously, whereas in the piston engine, it is only reached for a fraction of a second during each cycle. For a long time, this factor led to a belief that the gas turbine was so inherently inefficient in comparison with a reciprocating engine, that it was not worth bothering with.

At high altitude, the atmospheric air temperature is reduced, so for a given compressor outlet temperature, a greater temperature and pressure ratio between inlet and outlet can be allowed. Thus, the thermodynamic efficiency tends to rise with increasing altitude. This factor, coupled with the advantages of high altitude flight, described in Chapter 7, makes the high speed turbo-jet – propelled aircraft a surprisingly efficient form of transport. In fact, as we show in Chapter 7, for long-range subsonic jet-propelled transport, there is no eco­nomic advantage in using an aircraft designed to fly slowly.

The thermodynamic efficiency of gas turbines improved dramatically during the first three decades of development mainly because of progress in producing materials capable of sustaining high temperatures, improvements in the cool­ing of critical components, and better aerodynamic design of compressors and turbines.

Maximum angle of climb

Figure 7.11 shows the forces acting on an aircraft in a steady climb. If the climb is steady then there can be no net force acting on the aircraft either along the flightpath, or at right angles to it. If we consider the forces acting along the flightpath we can see (Fig. 7.11) that the sine of the climb angle is given by the difference between thrust and drag divided by the aircraft weight. Thus to operate at the maximum angle of climb possible we need the biggest possible value of thrust minus drag.

If the thrust minus the drag is equal to the weight we have a vertical climb, e. g. the Harrier (Fig. 7.12). If thrust minus drag is greater than the weight then the aircraft will be in an accelerating, rather than a steady climb.

If, however the difference between thrust and drag is less than the aircraft weight, some lift must still be provided by the wings. To be able to climb at all the aircraft must be operating at a height at which the engine is capable of producing more thrust than the drag of the aircraft.

If, for instance, the aircraft is flying straight and level initially we can plot the now familiar variation of drag with flying speed. Let us suppose that the

Fig. 7.13 Climbing flight

Increased throttle setting gives excess of thrust over drag for climb Best climb angle is obtained when thrust minus drag is maximum

aircraft is operating at point A on this curve. An increase in throttle setting will give an available thrust-minus-drag difference for climb as shown (Fig. 7.13). If we know the engine characteristics at the new throttle setting we can optim­ise the airspeed to give the best possible thrust/drag difference.

Here we must turn our attention to the type of powerplant being used once again. If we are dealing with a turbo-jet and thrust will not vary very much with speed in the operating range we are considering. All we need to do therefore is to gratefully accept the maximum thrust that the engine will give and fly at the speed which produces the least amount of drag (point A in Fig. 7.14).

If we are using a piston engine/propeller combination, we have already seen that the thrust falls with increasing speed and so we must reach a compromise between the requirements of airframe and powerplant and operate at a speed somewhat lower than the minimum drag speed in order to achieve the max­imum angle of climb (Fig. 7.15).

At this point a word of caution is necessary. We have estimated the best climbing angle using the drag curves derived for straight and level flight. When the aircraft is climbing examination of the forces normal to the flightpath (Fig. 7.11) shows that the lift developed by the wing will be reduced by a factor equal to the cosine of the climb angle and is thus no longer equal to the aircraft weight. Our drag curve will therefore need to be modified and this, in turn, may change the best speed for climb.

A large number of aircraft, such as civil airliners and military transport air­craft, are not required to indulge in particularly violent manouevres. Although the rate of climb might be quite high, because the forward speed is also high, the angle of climb is frequently not very great. In such cases our original approximation will not be too far from the truth.

Flying wings and blended wing-fuselage concepts

It has long been the dream of aircraft designers to produce civil airliners with no separate tail or fuselage, as with the B2 Spirit bomber (Fig. 4.19). The advantages would include much lower aerodynamic drag, and reduced weight. There are, however, several problems. Much of the structural load on a civil aircraft derives from the stresses due to pressurisation of the cabin, and by far the most efficient cross-sectional shape is a circle. Horizontally-arranged double or multiple bubble arrangements may be used, but passenger access between the bubbles then becomes an issue. Longitudinal stability considera­tions mean that the range of centre of gravity positions is relatively restricted, so passenger movements might need to be controlled. There are also difficulties involved in access and in the placing of passenger external view windows. None of these problems is insuperable, but the real constraint would be the very high costs of such a radical development.

Stability of canard aircraft

The stability criteria for a canard or tail-first configuration aircraft (Fig. 11.8) are essentially the same as for a conventional one. When the aircraft is trimmed, the forward wing (foreplane) should be arranged to generate a higher lift coefficient than the rearward wing (main-plane). The foreplane is therefore usually set at a higher geometric incidence than the main-plane, thus giving lon­gitudinal dihedral. On a canard it is the larger rear wing surface that generates

Fig. 11.8 A stable canard arrangement

The aircraft has to be trimmed with the foreplane generating a higher lift coefficient than the main-plane. The foreplane is therefore normally set at a higher incidence.

most of the lift, so it follows that on a stable canard, both surfaces must be producing lift.

Since both surfaces on a canard produce positive lift, the overall wing area, total weight, and drag can all be lower than for the conventional arrangement. Also, as we have already mentioned, pitch control is achieved by lifting the nose by increasing the foreplane lift, rather than by pushing the tail down. This shortens the take-off run, and generally improves the pitch control character­istics. The manoeuvrability of the canard configuration is one of the features that makes it attractive for interceptor aircraft (see Figs 10.1 and 10.8).

Another claimed advantage of the canard is, that since the foreplane is at a higher angle of attack than the main-plane, the foreplane will stall before the main-plane, thus making such aircraft virtually unstallable. Unfortunately in violent manoeuvres, or highly turbulent conditions this may not be true, and once both planes stall, recovery may be impossible, because neither surface can be used to produce any control effect.

The main problems with the canard configuration stem from interference effects between the foreplane wake and the main wing. In particular, the down – wash from the foreplane tilts the main wing resultant force vector backwards, thus increasing the drag. By careful design, however, the advantages can be made to outweigh the disadvantages, and highly successful canard designs by Burt Rutan such as the Vari-Eze shown in Fig. 4.20 provoked renewed interest in the concept.

For forward-swept wings, as on the X-29 shown in Fig. 9.20, the foreplane interference can be a positive benefit, as the downwash suppresses the tendency of the inboard wing section to stall at high angles of attack.

For pressurised passenger aircraft, the canard arrangement has the added advantage that the main wing spar can pass behind the pressure cabin, as in the Beech Starship shown in Fig. 4.10. A problem remains in that, unless there is a rearward extension of the fuselage, the fin (vertical stabiliser) may have to be large to compensate for the fact that it is not very far aft of the centre of gravity.

Unusual landing requirements

Thus far we have considered the landing manoeuvre for aircraft operating from conventional runways. Within this group we include special short take-off and landing (STOL) aircraft such as the C-17 (Fig. 10.20), since the techniques employed are essentially similar.

Sometimes aircraft are required to have a shorter landing run than is obtain­able by conventional means, as for example in carrier landing. Although the carrier can help by sailing into the wind as fast as possible, the deck is short, and additional deceleration has to be provided by an arrester hook which

Fig. 13.12 The A380 landing

Note the large number of wheels required because of the massive weight (Photo courtesy of R. Wilkinson)

engages with a wire across the deck. The ultimate in landing performance is of course provided by the vertical take-off and landing (VTOL) Harrier (Fig. 7.12) or Osprey (Fig. 1.30).

At the other extreme the Space Shuttle (Fig. 8.19) commenced its approach without power at hypersonic speed. We looked at the high speed part of the landing manoeuvre in Chapter 8. The final approach, however, was very sim­ilar to those we have already dealt with, except that there was no longer the option to fly down the glide path under power. The lack of this ability means that it was not possible to be nearly so precise in achieving a particular touch­down point, with the result that a long runway was needed. Since the whole of the re-entry and landing manoeuvre was unpowered accurate computer control was needed right from the point of re-entry if the Shuttle was to end up in the right continent, let alone the right airfield.


To sustain an aircraft in the air in steady and level flight, it is necessary to gen­erate an upward lift force which must exactly balance the weight, as illustrated in Fig. 1.1. Aircraft do not always fly steady and level, however, and it is often


Fig. 1.1 Forces on an aircraft in steady level flight

The lift exactly balances the weight, and the engine thrust is equal to the drag


Fig. 1.2 The direction of the aerodynamic forces

The lift force is at right angles to the direction of flight relative to the air and to the wing axis, and is therefore not always vertically upwards. Note that as in the case illustrated, an aircraft does not normally point in exactly the same direction as it is travelling

necessary to generate a force that is not equal to the weight, and not acting vertically upwards, as for example, when pulling out of a dive. Therefore, as illustrated in Fig. 1.2, we define lift more generally, as a force at right angles to the direction of flight. Only in steady level flight is the lift force exactly equal in magnitude to the weight, and directed vertically upwards. It should also be remembered that, as shown in Fig. 1.2, an aircraft does not always point in the direction that it is travelling.

Downwash and its importance

The trailing vortices are not just a mildly interesting by-product of wing lift. Their influence on the flow extends well beyond their central core, modifying the whole flow pattern. In particular, they alter the flow direction and speed in the vicinity of the wing and tail surfaces. The trailing vortices thus have a strong influence on the lift, drag and handling properties of the aircraft.

Referring to Fig. 2.7, we see that the air behind the wing is drawn down­wards. This effect, which is known as downwash, is apparent not only behind the wing, but also influences the approaching air, and the flow over the wing itself. Figure 2.8 shows that the downwash causes the air to be deflected down­wards as it flows past the wing.

There are several important consequences of this deflection. Firstly, as we can see from the diagrams, the angle of attack relative to the modified local airstream direction, is reduced. This reduction in effective angle of attack means that less lift will be generated, unless we tilt the wing at a greater angle to compensate.

The second, and more important consequence may be explained by further reference to Fig. 2.8. It will be seen that, since the air flow direction in the

Downwash and its importance



Downwash and its importance

Fig. 2.8 The effect of downwash on lift and drag

(a) Lift force in two-dimensional flow with no downwash effect

(b) Downwash changes local approach flow direction. The resultant force is tilted backwards relative to the flight direction, and has a rearward (trailing vortex) drag component with reduced lift due to reduction in the effective angle of attack

(c) To restore the lift to its value in two-dimensional flow, the angle of attack must be increased. The drag component will increase correspondingly


Downwash and its importance

vicinity of the wing is changed, what was previously the lift force vector, is now tilted backwards relative to the flight direction. There is therefore a rearward drag component of this force.

This type of drag force was at one time called induced drag, but the more descriptive term trailing vortex drag is now usually preferred. We shall deal with drag forces in more detail in Chapter 4.

Another consequence of downwash is that the air flow approaching the tailplane is deflected downwards, so that the effective angle of attack of the tailplane is reduced. The downwash depends on the wing circulation and there­fore varies with flight conditions.

It is often thought that the downwash is entirely responsible for the lift, by the principle of momentum change. This is not so. What is invariably forgot­ten is that the trailing vortices also produce a large upwash outboard of the wing tips. The upward momentum change thus produced cancels out the down­ward momentum change of the downwash. If we sandwich a wing between the walls of a wind-tunnel, so that there are no trailing vortices, air particles behind the wing will return roughly to their original height, and yet the lift is greater than when downwash is present. In calculating lift, it is always necessary to consider forces due to pressure as well as momentum. A detailed discussion of the concepts involved is however beyond the scope of this book.

Drag coefficient

As with lift, it is convenient to refer to a drag coefficient CD defined, in a sim­ilar way to lift coefficient, by

Drag = Dynamic pressure x wing area x CD or D = pV2 x S x CD

where S is the wing plan area.

For an aircraft, a major contribution to the overall drag comes from the wing, and is largely dependent on the plan area. We therefore wish to find ways of minimising the drag for a given wing plan area, and it is sensible to relate CD to the plan area, as in the expression above. Note, however, that for cars, CD is based on the frontal area. Drag coefficient values for cars cannot, therefore, be compared directly with values for aircraft. The drag coefficient of missiles is also normally based on the body frontal area.

The wing drag coefficient depends on the angle of attack, the Reynolds num­ber (air density x speed x mean wing chord/viscosity coefficient), and on the Mach number (speed/speed of sound). For many shapes, the dependence of CD on Reynolds number is weak over a wide range, and for simple estimations, the dependence on Reynolds number is often ignored. For speeds up to about half the speed of sound, the variation with Mach number is normally negligible, and so, for early low speed aircraft, it was customary to treat CD as being dependent only on the angle of attack and geometric shape of the aircraft. However, as we described in the last chapter, ignoring the effects of Reynolds number can lead to serious errors. For high speed aircraft, the effect of Mach number becomes extremely important.

The production of thrust forces by a jet engine

The change in the speed of the air between inlet and outlet means that its momentum has been increased, so thrust is obviously produced, but where? At first sight, air flowing through a hollow tube might be expected to produce nothing more than friction drag. In fact, the thrust force is mainly produced by pressure differences between rearward-facing and forward-facing surfaces. In Fig. 6.2, the contributions to thrust and drag of a typical jet engine are shown. Note how the net output thrust is only a small proportion of the total thrust produced internally, indicating that there are very large internal stresses. The case shown relates to a stationary engine. In flight, much of the thrust may come from the pressure distribution in the intake duct system.

The actual distribution of forces in and around the engine varies with its design and the operating conditions. There are many contributions, and it is no simple matter to assess them all accurately. However, we can conveniently measure the total thrust by determining the overall momentum change and pressure difference across the engine.

The overall net thrust is partly related to the air flow round the outside of the engine. The external flow mostly produces drag, but round the leading edge (the
rim) of the intake, the flow speed is high, so the pressure is low, and under some conditions this may produce a significant forward thrust component. The aero­dynamic design of the intake, ducting and engine nacelle is thus very important.