Progress in AERONAUTICAL SCIENCES Volume 3 Edited by ANTONIO FERRI Professor of Aerodynamics, Polytechnic Institute of Brooklyn, U.S.A. D. KÜCHEMANN Royal Aircraft Establishment, Farnborough, England L. H. G. STERNE Training Center for Experimental Aerodynamics, Belgium PERGAMON PRESS OXFORD · LONDON · NEW YORK · PARIS 1962 PERGAMON PRESS LTD. Headington Hill Hall, Oxford. 4 and 5 Fitzroy Square, London W.l PERGAMON PRESS INC. 122 East 55th Street, New York 22, N.T. PERGAMON PRESS S.A.R.L. 24 Rue des Écoles, Paris Ve PERGAMON PRESS G.m.b.H., Kaiserstrasse 75, Frankfurt am Main Copyright © 1962 PERGAMON PRESS LTD. Library of Congress Card Number 60-15351 Set in Monotype Modern No. 7 ll/12pt. and Printed in Great Britain by J. W. Arrowsmith Ltd., Bristol. PREFACE THE third volume in this series contains four articles dealing with various topics mainly in the field of aerodynamics. It is certain that the last word has yet to be written on any one of these topics and certainly some of the information offered here, though not ephemeral, will merit amplification in a few years' time. As is only to be expected in a series of this kind, the individual articles are not interconnected but report on fields where advances happen to have been made. As such, they give perhaps some indication of the vigorous activity and continuing growth of the aeronautical sciences. Certainly there would appear to be no dearth of material for an annual volume in this series. In fact, this third volume appears only a few months after the second volume: what was originally intended to form one volume was split into two for the greater convenience of the reader. The first paper is a review of the aerodynamics of swept-winged air­ craft and as such is a contribution to the solution of the problem of present-day aerodynamic design which, as noted in Volume I, has emerged as a consequence of developments which have gone beyond the classical concepts of aircraft design. This paper is a first review of part of the research work, which was a co-operative effort on a large scale by British aircraft firms and research establishments, aimed at the development of supersonic airliners. Not all the individual refer­ ence papers concerned in the study have yet been published and it is hoped to report on other aspects of this research work in later volumes in this series. The second paper of this volume is concerned with ducted propellers which in recent years have attracted considerable interest owing to their potential application to a variety of aircraft types, notably the important category of those able to take off vertically or requiring a short take-off run. The paper gives an exhaustive survey of the widely dispersed literature in a way which does not ignore practical appli­ cations. The third paper, on hypersonic facilities, provides the reader with a balanced review of the, to some, bewildering multiplicity of means for experimenting with hypersonic flows. Even though development still proceeds at a rapid pace, the overall outlines and the usefulness of particular facilities for particular purposes begin to become more firmly established and a survey is, therefore, both opportune and wel­ come. vii vin PREFACE The last paper gives an account of gust research—that fascinating field of work where meteorology, fluid dynamics and aircraft dynamics meet. The subject will be seen to be of great interest and prolific in its many aspects, one of which is, of course, the comfort and safety of ourselves as airline passengers. SOME AERODYNAMIC PRINCIPLES FOR THE DESIGN OF SWEPT WINGS J. A. BAGLEY Aerodynamics Department, Royal Aircraft Establishment, Farnborough Contents LIST OF SYMBOLS 2 1. INTRODUCTION 5 2. THE CLASSICAL AIRCRAFT AND THE EXTENSION TO SWEPT WINGS 10 3. THE SHEARED WING OF INFINITE SPAN 14 4. SWEPT WINGS OF FINITE ASPECT RATIO 22 5. THE DESIGN OF SWEPT WINGS 30 5.1. Aerodynamic features of the flow over swept wings 30 5.2. Design principles 38 6. CALCULATION METHODS AVAILABLE FOR DESIGN PURPOSES 43 6.1. Methods of designing the basic wing section 45 6.1.1. Incompressible flow 45 6.1.2. Compressible flow 48 6.2. Design methods for the finite wing, at subsonic speeds 51 6.2.1. Planform, camber and twist 51 6.2.2. Thickness 57 6.2.3. Compressible flow 61 6.2.4. Thickness and lift combined 62 6.3. Design methods for the wing-fuselage combinations at subsonic speeds 63 6.4. Design methods for the wing-fuselage combinations at supersonic speeds 67 6.4.1. Thickness problems 68 6.4.2. Lifting configurations 75 ACKNOWLEDGEMENTS 79 LIST OF REFERENCES 80 1 SOME AERODYNAMIC PRINCIPLES FOR THE DESIGN OF SWEPT WINGS J. A. BAGLEY Aerodynamics Department, Royal Aircraft Establishment, Farnborough Summary—The design of swept wings has now reached the stage where a coherent set of aerodynamic principles has emerged. The pur­ pose of this paper is to summarize these principles, and to indicate methods of designing wings in accordance with them. It is important to design wings so that the type of flow obtained in practice is the same as that assumed in the design theory, and so that it is a flow which is usable—i.e. which can be predicted and controlled. It is shown that these requirements lead to the concept of a sub-critical flow which can be obtained on certain swept wings. These are restricted to a fairly narrow band of sweep angles, depending on the design Mach number, and the aspect ratio and thickness of the wings are correspond­ ingly limited. Practical design methods are discussed in order to illustrate the physical principles used in design, but a critical comparison of different calculation methods is not attempted. List of Principal Symbols a Local speed of sound a(y) = CL(y)l*e(y) Sectional lift slope on swept wing a Speed of sound in free stream (except in Eq. 59A) 0 A Aspect ratio of wing A a Aspect ratio of analogous wing A\, A Constants in Eqs. 61 and 62 2 B = {l-Jfo2[(l-Opi)cos2 -Opi(l-|/i|)sin2 ]}* Çt 9t c(y) Wing chord c Geometric mean chord CD Overall drag coefficient ÖD(y) Sectional drag coefficient @DB(y) Sectional drag coefficient due to boundary layer effects CD-F Drag coefficient due to skin friction (complete wing) C = Do/qS D0 ^Di(y) Sectional drag coefficient due to thickness ODV Vortex drag coefficient (complete wing, or sectional) CW Wave drag coefficient due to lift GL Overall lift coefficient CL(2/) Sectional lift coefficient CLCT Cruising lift coefficient Value of CL for (L/D) m C Overall pitching moment of wing m 2 Aerodynamic Principles for the Design of Swept Wings 3 Cp Pressure coefficient Cp* Critical pressure coefficient, denned by Eq. (17) Cpi Pressure coefficient at zero lift in incompressible flow (Eq. 47) (7 T Pressure coefficient at trailing edge P C u(%) Pressure coefficient on upper surface of wing V Cpi(x) Pressure coefficient due to wing thickness Op2(#) Pressure coefficient due to wing lift D Total drag of wing or wing-fuselage combination DL Drag due to lift of wing or combination Do = D —D . Drag independent of the lift distribution L DT Wave drag due to volume of configuration Dw Wave drag due to lift of configuration 1 + sin φ /(φ) =log- r— 1 — sin φ Q(B) See Eq. (83) Η(θ) See Eq. (84) K = π^4((7z)v+Crz)w)/CrL2. Drag due to lift factor K\(y) Spanwise interpolation factor used in Eqs. (68) and (76) Ky = TTACDYJGL2' Vortex drag factor K = TT.4Z2CW/2J32$2CL2. Wave drag due to lift factor w I Overall length of configuration, except in Eqs. (83) and (84) l(x), l(x, y) Local load coefficient, — Δ(7 Ρ L Overall lift on wing or combination L(x) Lift on section of combination intercepted by plane x = constant L(x, Θ) Lift on section of combination intercepted by oblique Mach plane (see Eq. (84)) (L/D)m Maximum lift-drag ratio (L/D)cr Cruising lift^-drag ratio M = V/a Local Mach number Mo = Vo/ao Free-stream Mach number Merit Critical Mach number, i.e. value of Mo when M = 1 first on body Mes Sheaved wing critical Mach number, i.e. value of M when 0 Eq. (15) is first satisfied n Chordwise loading parameter, defined by Eqs. (57), (57A) n = n(y , <p) a & a no Chordwise loading parameter, defined by Eq. (57) q = %poVo2 Free-stream kinetic pressure q(x) Source distribution on aerofoil r Radial co-ordinate r(x) Radius of fuselage R Mean radius of fuselage s Semi-span of wing (except in Eq. (64)) S Area of wing planform S(x) Cross-sectional area of combination intercepted by plane x = constant S(x, Θ) Area of combination intercepted by oblique Mach plane (see Eq. (83)) 4 J. A. BAGLEY SF(X) Cross-sectional area of fuselage S(i)(x),SW(x),SM(x) Defined in Eqs. (19), (37), (38), and (41) SW(x) = dz/dx t SW{x) = dzs/dx t Wing maximum thickness Vx, Vy> v, Vf, ν Velocity increments in x-, y-, z-, ξ- and ^-directions z η vx vy> Az Velocity increments in incompressible flow on analogous 9 wing vq, v Velocity increments due to source distributions X zq νγ, νγ Velocity increments due to vortex distributions χ ζ V Local flow velocity Vo Free-stream velocity; flight velocity of aeroplane Vx* Vy Vzy Vf, V Components of V parallel to #-, y-, z-, f-, and η-directions t v Vo Vzo Components of Vo parallel to x- and z-directions x f TTp, Wi Final and initial weights of aeroplane x Stream wise co-ordinate xn End of pressure "plateau" on rooftop aerofoil section y Spanwise co-ordinate y & ~ ßy Spanwise co-ordinate on analogous wing z Vertical co-ordinate z(x) Aerofoil surface Zs{x), z(x, y) Wing camber line or surface s zt{x)y zt{x, y) Aerofoil or wing thickness distribution. a Wing incidence o.(y) Sectional incidence <£e(y) Sectional effective incidence, defined by Eq. (50) oii(y) Sectional induced incidence, defined as ωαι(2/) 0 αΐο(2/) Downwash at infinity, defined by Eq. (49) ß = {\l-M *\}i 0 ßx = {l-ifo2(co82 ~(7pi)P 9 ß = {l-lfo2cos2<p}* 2 y Ratio of specific heats, (y = 1-4 in numerical examples) y{x)y γ(χ, y) Vortex distributions representing lift distribution on aero­ foil or wing Δ(7 Difference between pressure coefficients on upper and Ρ lower surfaces Δα(ι/) Effective change of sectional incidence due to boundary layer effects η = x sin φ -f y cos φ Co-ordinate parallel to wing sweep lines (except in Eq. (82)) Θ Inclination of oblique cutting plane to stream direction (in Eqs. (83), (84)) Θ Defined by Eq. (71) K(y) Spanwise interpolation factor in Eqs. (68) and (76) \(y) Spanwise interpolation factor for load distribution defined by Eq. (58) K = A(Va) M (y) Spanwise interpolation factor for velocity increments due to thickness—see Eq. (67) Aerodynamic Principles for the Design of Swept Wings 5 ξ = x cos φ — y sin φ Co-ordinate normal to wing sweep lines p Air density (except in Eq. (79)) po Free stream air density a Defined by Eq. (63) r(y) Spanwise interpolation factor in Eqs. (68) and (76) Ta = T(y ) φ Wing sweep angle 9a Sweep of analogous wing <Pc/ a Sweep of mid-chord line 9t Sweep of thickness line φο Leading-edge sweep φι Trailing-edge sweep Φ Velocity potential Φ = ΒΦ/dx, Φ = δΦ/δι/, etc. ζ ν ω Downwash factor, see Section 6.2.1 1. Introduction AEBOPLANES with swept wings have been designed to meet various requirements since the earliest days of powered flight, but during the last twenty years swept wings have been employed primarily for high­ speed aircraft. During this period, the problem of designing wings to have high aerodynamic efficiency in the cruising condition has been studied, initially for subsonic speeds and subsequently for supersonic speeds. At the present stage, a coherent set of aerodynamic principles has emerged from this work which seem to be generally accepted, and the time appears to be opportune for a survey of this field. The purpose of this paper is to summarize these aerodynamic principles, and to bring into perspective methods of designing wings in accordance with these ideas. The basic premise of the design approach described here is that to attain high aerodynamic efficiency it is necessary to aim for a flow pattern which will be stable and predictable, and will have a lowjdrag. A type of flow which meets these requirements, and can apparently be obtained on sweptback wings of the kind considered here, is that found on a two-dimensional aerofoil at low speeds. This type of flow is utilized on the '"classical" aircraft, and its essential features have been described by Maskell1. Flow separation takes place only along the trailing edge, so that extensive regions of separated flow on the aerofoil surface itself are avoided, and viscous effects are confined to a thin boundary layer. Furthermore, the flow is smooth everywhere and there are no discon­ tinuities such as shock waves. Maskell1 and Küchemann2 have shown how the prescription of a particular type of flow pattern is in itself a powerful factor in defining the shape of a practical aeroplane. This approach is adopted in Section 2 of this paper in order to define the class of wings to be considered. 6 J. A. BAGLEY The requirement for high aerodynamic efficiency in the cruising condi­ tion is interpreted as a requirement for a specified lift-drag ratio, and it is shown that practical aeroplanes to meet this requirement fall into a fairly limited range of geometries. Consequently, the later sections are concerned mainly with the drag of wings and wing-fuselage com­ binations which fall into this limited family. The drag forces on any flying body can be separated into those due to skin friction and those arising from the summation of the pressures on the surface of the body. Methods of estimating these normal-pressure forces, or of designing body shapes on which they will be small, can be divided into two categories. One group depends on the calculation of the surface pressure distribution itself, which can then be integrated to give the pressure drag; this may be categorized as the "near field" approach to design. The second group utilizes what may be called the "far field" approach, where the drag is identified with the transport of momentum through a large control surface surrounding the body. The momentum transport can then be related to the distributions of volume and lift on the body, without considering the details of the surface pres­ sure distribution. By considering as the control surface enclosing the configuration a cylinder of large radius with its axis parallel to the stream, it can be shown that at subsonic speeds the only transport of momentum is through the downstream end of the cylinder (see, for example, Heaslet and Lomax, Section D.14 of Ref. 3). This surface can be taken infinitely far downstream of the body ; it then becomes the so-called Trefftz plane, and the drag corresponding to the momentum transport through this surface in in viscid flow is the vortex drag of the configuration associated with the lift distribution. (In a real viscous flow, there may be an addi­ tional drag term, due to the influence of the boundary layer. See page 32). The "far field" approach thus leads, at subsonic speeds, to the well- known relations between the spanwise distribution of lift, the down- wash behind the wing, and the vortex drag. In particular, if the trailing vortex sheet is assumed to be planar, it leads to the familiar theorem which states that the vortex drag has its minimum value, for given total lift and span, when the spanwise load distribution is elliptical. At supersonic flight speeds, there is an additional transport of mo­ mentum through the curved surfaces of the cylinder, which gives rise to the so-called "wave drag" of the configuration. It is manifested physically as a change in the flow pattern, usually giving rise to shock waves near the nose and tail of the configuration, and frequently at other places also. The wave drag might be evaluated, in principle, by considering the energy losses through these shocks, or by integration of the pressure distribution over the whole surface of the configuration,