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.

Introduction

Our principal aim so far has been to lay down the foundations of surface vorticity analysis for a series of progressively more advanced turbomachinery flow problems. Although a brief outline of threedimensional flow analysis was presented in Chapter 1, specific applications have been limited to problems which are twodimensional in the strict mathematical sense. Unlike the source panel method, which has been extensively applied to threedimensional flows, serious application of the surface vorticity analysis has been limited to few such engine problems. The aim of the first part of this chapter will be to expand on the basic foundation theory for dealing with the flow past three-dimensional objects by surface vorticity modelling and to consider two such problems in turbine engines which have received some attention. These will include the prediction of engine cowl intake performance at angle of attack and the behaviour of turbine cascades exhibiting sweep.

As discussed in Chapter 3 the flow through turbomachinery blade passages is in general three-dimensional, although the design or analysis problem may be tackled in a practical way by reference to a series of superimposed equivalent interacting two-dimensional flows. The two models usually adopted, which are equivalent in some respects, are the S-1, S-2 surfaces of Wu (1952) and the superposition of blade-to-blade (S-2 type) flows upon an assumed axisymmetric meridional flow. We concluded Chapter 5 with a derivation of the meridional flow equations for ducted propellers, indicating that the blade-to-blade/meridional interactions result in vorticity production within the mainstream.

Introduction

Over the next three chapters we shall develop analyses to deal with progressively more complex problems in the fields of ducted propellers or fans and turbomachine meridional flows. As illustrated in Chapter 3, a design strategy frequently adopted for such devices involves representation of the fully three-dimensional flow as a series of superimposed and connected two-dimensional flows. These are of two main types, blade-to-blade and meridional flow. Having dealt with the first of these, we now turn our attention to the second principal turbomachine problem, calculation of the meridional flow. Turbomachine annuli, Fig. 4.1, are of many different configurations but are usually axisymmetric. For design purposes meridional through-flows are likewise often assumed to be axisymmetric. In general it is important to build into meridional analysis the interactions of the blade-to-blade flow which results in vortex shedding and stagnation pressure or enthalpy gradients. These matters will be dealt with in Chapters 5 and 6, including extension to a consideration of some three-dimensional flows which have been studied by surface vorticity modelling. In the present chapter the foundations will simply be laid for the analysis of axisymmetric potential flows by the surface vorticity method with applications to bodies of revolution, engine or ducted propeller cowls, wind tunnel contractions and turbomachinery annuli.

Axisymmetric flows are in fact two-dimensional in the mathematical sense, even in the presence of circumferentially uniform swirling velocities.

Introduction

The early contributors to the surface vorticity method such as Martensen (1959), Jacob & Riegels (1963) and Wilkinson (1967a) were concerned primarily with the development of a flexible numerical method for the solution of potential flows. Preceding chapters testify to the scope and power of this conceptually simple technique and to the imagination and creativity of a host of later research workers who have extended the method to deal with a wide range of engineering potential flow problems. Although the broader physical significance of the surface vorticity model, as expounded in Chapter 1, has always been realised, only recently has this been more fully explored by attempts to model the rotational fluid motion of real fluids including both boundary layer and wake simulations. The remaining chapters will lay down progressively the essential fundamentals of this work which the reader requires to proceed to practical computational schemes, employing what has come to be known as the ‘vortex cloud’ or ‘discrete vortex’ method.

All real flows involve rotational activity developed in the regions adjacent to flow surfaces or in the rear wake region in the case of bluff bodies. Some flows also exhibit spontaneous boundary layer separation or stall behaviour while in other situations flow separation occurs inevitably from sharp corners.

Introduction

A range of flow computational techniques has been developed over many years to meet the design and analysis requirements of a wide range of rotodynamic machines, some of which were illustrated by Fig. 4.1. For dealing with turbomachine meridional or through flows, which are usually completely confined within a continuous duct annulus such as that of the mixed-flow fan depicted in Fig. 4.1(a), surface vorticity or panel methods have been proved less attractive in competition with grid based analyses such as the matrix through-flow method of Wu (1952) and Marsh (1966) and the more recent time marching analyses such as those of Denton (1974), (1982). Although the annulus boundary shape exercises important control over the flow through the blade regions, in all turbomachines complex fluid dynamic processes occur throughout the whole flow field due to interactions between the S-1 and S-2 flows which were referred to in Chapter 3. Boundary integral methods based solely upon potential flow equations such as we have considered so far obviously cannot handle these interactions between the blade-to-blade and meridional flows, which involve detailed field calculations and spatial variations of properties best dealt with by the introduction of a grid strategically distributed throughout the annulus. Some attempts to achieve this with extended vortex boundary integral analysis will be outlined in Chapter 6, but generally speaking channel grid methods such as those referred to above have proved more fruitful to date for turbomachinery meridional analysis.

Introduction

The main objective of this chapter is to present the reader with a practical numerical approach to vortex cloud modelling of bluff body flows, drawing upon the techniques developed earlier in the book and especially the treatments of vortex dynamics and viscous diffusion considered in Chapters 8 and 9. Reporting on Euromech 17, which was entirely devoted to bluff bodies and vortex shedding, Mair & Maull (1971) remarked upon the preponderance of experimental work at that time and the need for more theoretical studies to be attempted, since there was little discussion of numerical techniques. It was felt, on the other hand, that since such flows showed marked three-dimensional characteristics (e.g. a circular cylinder von Karman street wake will not in general be correlated along its length for L/D ratios in excess of 2.0), two-dimensional computations, whilst being of interest, would not be very useful. It was admitted however that ‘with an increase in the size of computers a useful three-dimensional calculation could become a reality’. By the time of the next Euromech 119 on this subject, Bearman & Graham (1979), one third of the papers focused on theoretical methods, the majority based upon the Discrete Vortex Method (DVM). Various reviews of the rapid subsequent progress with DVM were given by Clements & Maull (1975), Graham (1985a) and Roberts & Christiansen (1972) and a fairly comprehensive recent review of U.K.

Introduction

So far we have considered only the case of fully attached inviscid steady flows, for which the introduction of a surface vorticity sheet of appropriate strength and of infinitesimal thickness, together with related trailing vorticity in three-dimensional flows, is completely adequate for a true representation. As pointed out in Chapter 1, where the justification of this model was argued from physical considerations, the surface vorticity method is representative of the infinite Reynolds number flow of a real fluid in all but one important respect, namely the problem of boundary layer separation. Real boundary layers involve complex mechanisms characterised by the influence of viscous shear stresses and vorticity convections and eddy formation on the free stream side. Depending upon the balance between these mechanisms and the consequent transfer of energy across a boundary layer, flow separation may occur when entering a rising pressure gradient, even at very high Reynolds numbers. Flow separation at a sharp corner will most certainly occur as in the case of flow past a flat plate held normal to the mainstream direction.

For a decade or so the development of computational fluid dynamic techniques to try to model these natural phenomena has attracted much attention and proceeded with remarkable success. The context of a good deal of this work has fallen rather more into the realm of classical methods than that of surface vorticity modelling, and is often classified by the generic title Vortex Dynamics.