The thrust F produced by the engines is of great importance in almost every phase of flight because it counteracts the drag and enables the aircraft to climb if required. The maximum available thrust Fm depends on the height and speed of the aircraft and is limited by the approved ‘rating’ for the appropriate phase of the flight. The three ratings that are important in relation to aircraft performance calculations are those specified for take-off, climb and cruise, and the rated thrust for each of these is the maximum available. In any phase of flight the thrust can of course be reduced by the pilot below the rated value, usually by moving a single control lever which is commonly known as a ‘throttle’ lever, even though it may act on a complex engine control system.

For almost all aspects of aircraft performance calculation it is necessary to know how Fm varies with the speed and height of the aircraft. In addition, for calculations of range, endurance and operating cost, a knowledge is required of the rate of consumption of fuel and the way in which this varies with flight speed, height and engine throttle setting. In this chapter the principles governing these variations will be discussed and approximate equations will be introduced for representing the variations in calculations of aircraft performance. For this purpose the rate of consumption of fuel will be expressed as the ratio of the rate of consumption to either the thrust or the shaft power of the engine.

The performance of an aircraft is essentially a statement of its capabilities and a different selection of these will normally be specified for the various categories such as transport, military and light aircraft, even though several common performance factors will feature in every such selection. For the engineer involved in the creation of a new design, these performance features serve as design criteria or at least desirable objectives, whereas late in the design and development stages the sales staff will quote the performance features as the basis for the commercial strength of the emerging aircraft. For either reason the performance will be stated in terms of quantities such as direct operating cost (DOC), maximum range for various payloads and fuel loads, cruising speed and airport requirements for landing and take-off. While the sales and design attitudes will be distinct, although related, this book addresses the early stages of the design process which must also bear considerable allegiance to performance as viewed by a potential customer.

The estimation of performance proceeds in stages, starting with parametric studies based on simple assumptions and progressing to more refined calculations as the main features of the design become established and the confidence in data grows. Estimation techniques are important not only because they allow the engineering team to proceed while data are crude or speculative, but also because construction of the new aeroplane will begin well in advance of the engineering refinements, and if there is accuracy in the early estimations this will be rewarded by a reduction in modifications as the fabrication and assembly effort progresses toward regular production.

In the design of a civil aircraft the condition of steady level cruise is of prime importance because improved fuel economy in this flight regime makes a direct and valuable contribution to the reduction of operating costs. Performance in the climb is often less important, but it cannot be ignored because a climb is always needed to reach the required cruising height after take-off and Air Traffic Control may also require the aircraft to change height during the cruise. For military aircraft, performance in the climb may be a primary design requirement because there is often a need to reach a specified height and speed in the shortest possible time, either from take-off or from some other prescribed initial conditions of height and speed.

The quantities that are of most interest in calculations of climbing performance are the rate of climb VC = V sin γ and the time required and fuel used in climbing from one specified height to another. In many cases there is a change of speed during the climb, so that the aircraft is accelerating, but it will be shown that a correction can easily be made for the effect of the acceleration on the rate of climb. The angle of climb is also of some interest, although it is important mainly at low altitudes where there may be obstacles to be cleared or where a large angle of climb may be required for reasons of noise abatement.

In earlier chapters, except in the discussions of the landing flare and the take-off transition immediately after lift-off, it has been assumed that the flight path is straight, so that there is no component of acceleration normal to the flight path. In this chapter flight in a curved path will be considered, concentrating on the usual form of banked turn as shown in Figure 8.1, in which the angle of bank is adjusted so that there is no sideslip and therefore no component of aerodynamic force normal to the plane of symmetry of the aircraft. In such a turn the required lift is greater than the weight, thus CL is greater than it would be in straight and level flight at the same speed and consequently the drag is also greater. This raises the requirement for thrust, even to maintain level flight, and thus the rate of climb obtainable with the maximum available thrust is reduced and may become negative. As the turn becomes tighter and the normal acceleration V2/R is increased, due to either a high speed or a small turn radius, or both, there will be increased demands for CL and for thrust to maintain height, with the consequence that limitations may be imposed by stalling or buffeting or by the engine rating.

This chapter addresses the interdependences among speed, rate of turn, rate of climb and additional ‘g-load’ on the pilot, as well as the limitations on one or other of these when some are fixed.

The estimation of the performance of an aircraft requires calculations of quantities such as rate of climb, maximum speed, distance travelled while burning a given mass of fuel and length of runway required for take-off or landing. The aim of this book is to explain the principles governing the relations between quantities of this kind and the properties of the aircraft and its power plant. Thus the emphasis is on the development of simple analytical expressions which depend only on the basic aircraft properties such as mass, lift and drag coefficients and engine thrust characteristics. Although extensive numerical data are required for the most accurate estimates of performance in the later stages of a design, the use of such data is not considered here and the data required for use in the simple expressions to be derived are of the kind that would be readily available at the preliminary design stage of an aircraft. Only fixed wing aircraft are considered and the measurement of performance in flight is not discussed.

One of the authors (WAM) has given for many years a short course of lectures on aircraft performance to engineering students at the University of Cambridge. Experience with these lectures has drawn attention to the shortcomings of existing books and to the need for a new book with the aim stated above. The book follows the same approach as the lectures, although it covers a greater range of topics and these are examined in much greater detail. Little previous knowledge of aircraft is assumed and the level of mathematics required should be well within the capabilities of engineering students, even in their first year.

The first chapter has given an introduction to the characteristics of atmospheric air and has provided a valid basis for the expression of the aerodynamic force developed during flight in that air, but there has not yet been any attempt to consider the balance of forces necessary to satisfy the laws of mechanics. Except for Chapters 6 and 8 and parts of Chapters 4 and 10, this book is directed mainly towards flight with zero or negligible acceleration so that the equations to be developed are those of statics, not dynamics.

Consideration of the effects of varying speed and altitude on the aerodynamic force on an aircraft can be greatly simplified by examining the dependence of the lift and drag coefficients on the Reynolds and Mach numbers. This dependence has already been mentioned briefly and is discussed further in this chapter, where it is shown that for a given aircraft the variations of Reynolds number caused by changing speed and altitude are likely to have only small effects. With increasing Mach number in the high subsonic range there is usually a large increase of drag coefficient and this important effect is introduced briefly, deferring a more detailed account until Chapter 10.

An important measure of the aerodynamic efficiency of an aircraft is the ratio L/D of lift to drag, since there is always a desire to create lift with as little cost in drag as possible. In this chapter the effects of this ratio (or its reciprocal D/L) on some important performance parameters are examined and it is shown that there is a minimum value of D/L which is especially important.

An aircraft with vectored thrust is defined here as one in which the pilot is able to vary the direction of the engine thrust over a wide range, usually at least 90°. The main advantage of this facility is that if the maximum available thrust is greater than the weight, the aircraft is able to take-off and land vertically, i.e. with zero ground run. If the thrust is large but still less than the take-off weight it may be possible to use thrust vectoring to give a substantial reduction in the distance required for take-off and in this case the landing weight may be less than the available thrust so that a vertical landing may be possible. Aircraft with vectored thrust are commonly known as V/STOL aircraft because they are capable of vertical or short take-off and landing.

V/STOL aircraft have been designed with several distinct configurations, the best known being that used in the Harrier as shown in Figure 9.1 and described by Fozard (1986), where the propulsive jets can be deflected downward by movable nozzles. Other forms have been reviewed by Poisson-Quinton (1968) and include the tilt rotor, where lifting rotors of the kind used in helicopters are tilted forward to operate like normal propellers in forward flight, and the tilt wing where the rotor axes remain fixed in the wing and the whole wing–rotor assembly is rotated relative to the fuselage. An example of a tilt-rotor aircraft is the Bell Textron Osprey shown in Figure 9.2.

In most of the traditional areas of performance studies the central effort is directed towards a refinement of aerodynamic design in order to ensure efficiency of flight. In studying take-off and landing performance attention is directed more to the capacity of the engines to accelerate the aircraft in a condition of high drag coefficient when the available distance is limited, and to the braking capacity during landing for the same reason. There is limited opportunity for refinement of aerodynamics, although in recent years some progress has been made in reducing the drag of an aircraft in the take-off configuration. In the crucial periods close to lift-off and touchdown the behaviour of the aircraft is strongly dependent on piloting technique and there is a need to define standard procedures for these two manoeuvres, based on reference speeds which are used in defining the criteria laid down in airworthiness regulations for safe operations. The lengths of the ground run and the airborne sector in a take-off are both strongly influenced by engine performance and the effective drag polar, whereas in landing there is an airborne sector which is critically dependent on piloting technique to produce a tangential flare to touchdown, followed by a ground run which depends on braking capacity.

The take-off performance cannot be defined so simply when allowance is made for failure of an engine and this chapter considers not only the ideal performance when the manoeuvre proceeds as planned, but also the reduced performance obtained after an engine failure.

The range of an aircraft is the distance that can be flown while burning a specified mass of fuel and is one of the few simple performance parameters by which the commercial value of an aircraft may be judged. Any change in design which leads to an increase in range is always desirable because it gives either

1. (i) a reduction in the fuel needed for a given distance, or

2. (ii) an increased distance for a given fuel load.

The reduced fuel of (i) reduces the cost of fuel for a specified flight and may also allow more payload to be carried for a specified total aircraft weight, provided there is the necessary space in the aircraft. The increased distance of (ii) allows the aircraft to fly over a long distance with fewer stops for refuelling and for a military aircraft it gives a valuable increase in the radius of action.

Any flight involves take-off, climb, cruise, descent and landing but except for short flights the greater part of the fuel is used in the cruise and the emphasis in this chapter is on cruising range, although the fuel used in climb and descent is also considered. The speeds and heights that should be used in the cruise to give maximum range are discussed in some detail, taking account of the constraints that are usually imposed by Air Traffic Control. It is assumed that the aircraft is flying in still air, except in §7.12 where it is shown that a wind affects not only the range relative to the ground but also the optimum speed for achieving maximum ground range.

On an aircraft in supersonic flight shock waves are always present, extending to a great distance from the aircraft. As mentioned in §3.1.2 they are the cause of an additional component of drag, the wave drag, which has been neglected in the preceding chapters but which must be considered in transonic and supersonic flight as an important component of the total drag. The coefficient of wave drag of an aircraft depends on the lift coefficient and on the Mach number M and hence, with this component of drag included, the drag polar relating CD to CL now depends on M. The drag polar may still be represented approximately by the simple parabolic drag law as given by Equation (3.6) but now the coefficients K1 and K2 are functions of Mach number.

In the preceding chapters where wave drag was neglected, a curve relating β to Ve, as in Figure 2.9, could be applied to all heights, for a given aircraft weight, but with wave drag included this is no longer valid, because for any given value of Ve the Mach number varies with height. Thus no simplification can be obtained by the use of Ve, rather than V or M, and consequently the use of Ve* and the speed ratio ν is no longer helpful in the calculation of performance. In contrast, the quantity (ƒ − β) is still of prime importance in determining rate of climb and acceleration, as it is at lower speeds, and in general this quantity must be calculated for each combination of speed and height.

The drag acting on an aircraft is of supreme importance in determining either the performance obtainable with a given thrust or the thrust required to achieve a specified performance, the latter being important because for a given speed the rate of consumption of fuel is approximately proportional to thrust. This chapter gives a brief account of the principal components of drag and the essential flow mechanisms on which they depend. Particular attention is paid to the nomenclature used for the drag components and to the conditions for various minima, because the undisciplined use of some of the terms has led to confusion in the past.

The drag polar for an aircraft has been discussed in Chapter 2 and in this chapter the representation of the polar by a simple mathematical expression is considered. The commonly used simple parabolic drag law has the great advantage of allowing many aspects of performance to be expressed in terms of simple equations, but the limitations of this drag law are emphasised and some alternative laws are introduced. In using either the simple parabolic law or any of the alternatives it is important to ensure that the constants in the equations are chosen to give the best possible agreement with the real drag polar over the range of CL that is important.

The flight conditions for minimum drag and for minimum drag power have been discussed briefly in Chapter 2 and in this chapter they are examined in more detail, making use of the parabolic equation for the drag polar mentioned earlier.

Components of drag

The total drag of an aircraft may be regarded as the sum of several components.