The aim of aerofoil theory is to explain and to predict the force experienced by an aerofoil, and a satisfactory theory has been developed in recent years for the lift force in the ordinary working range below the critical angle and for that part of the drag force which is independent of the viscosity of the air. Considerable insight has also been obtained into the nature of the viscous drag and into the behaviour of an aerofoil at and above the critical angle, but the theory remains at present in an incomplete state. The problem of the airscrew is essentially a part of aerofoil theory, since the blades of an airscrew are aerofoils which describe helical paths, and a satisfactory theory of the propulsive airscrew has been developed by extending the fundamental principles of aerofoil theory.

The object of this book is to give an account of aerofoil and airscrew theory in a form suitable for students who do not possess a previous knowledge of hydrodynamics. The first five chapters give a brief introduction to those aspects of hydrodynamics which are required for the development of aerofoil theory. The following chapters deal successively with the lift of an aerofoil in two dimensional motion, with the effect of viscosity and its bearing on aerofoil theory, and with the theory of aerofoils of finite span. The last three chapters are devoted to the development of airscrew theory.

An airscrew normally consists of a number of equally spaced identical radial arms, and the section of a blade at any radial distance r has the form of an aerofoil section whose chord is set at an angle θ to the plane of rotation. The blade angle θ and the camber of the aerofoil section decrease outwards along the blade. If the airscrew moved through the air as through a solid medium, the advance per revolution would be 2πr tan θ and this quantity would define the pitch of the screw. Actually this quantity will not have the same value for all radial elements of the blade and so it is customary to define as the geometrical pitch of the airscrew the value of 2πr tan θ at a radial distance of 70 per cent. of the tip radius. An airscrew rotates in a yielding fluid and in consequence the advance per revolution is not the same as the geometrical pitch and may in fact assume any value. The value of the advance per revolution for which the thrust of the airscrew vanishes is called the experimental mean pitch, and in many respects the characteristics of an airscrew are defined by the ratio of the experimental mean pitch to the diameter.

An ordinary propulsive airscrew experiences a torque or couple resisting its rotation and gives a thrust along its axis.

The theory of the lift force given by an aerofoil in two-dimensional motion has been developed by considering the flow of a perfect fluid governed by Joukowski's hypothesis that the flow leaves the trailing edge of the aerofoil smoothly. It is necessary now to examine the fundamental basis of this theory and the extent to which the assumed motion represents the actual conditions which occur with a viscous fluid.

All real fluids possess the property of viscosity and the conception of a perfect fluid should be such that it represents the limiting condition of a fluid whose viscosity has become indefinitely small. Now it is well known that the limit of a function f(x) as x tends to zero is not necessarily equal to the value of the function when x is equal to zero, and hence, to obtain the true conception of a perfect fluid, it is not sufficient to assume simply that the coefficient of viscosity is zero. The viscosity must be retained in the equations of motion and the flow for a perfect fluid must be obtained by making the viscosity indefinitely small.

Slip on the boundary.

The first point to be considered is the motion of the fluid at the surface of a body. In a viscous fluid the relative velocity at the surface of a body is zero and the body is surrounded by a narrow boundary layer in which the velocity rises rapidly from zero to a finite value.

Great advances in the theory of aeronautics have taken place since the first edition of this book by my late husband appeared in 1926, but the more fundamental parts of the theory, which are the subject of this book, remain in large measure unchanged. Particularly important advances have been made in the theory of viscous motion and of the flow in the boundary layer. At my request Mr H. B. Squire of the Royal Aircraft Establishment, Farnborough, who was a colleague of my husband, has prepared a set of notes which appear as an Appendix to the present edition and these notes indicate where important developments have taken place and where further information on the subject matter can be found. I am most grateful to Mr Squire for his assistance and desire to tender him my sincere thanks.

In preparing this second edition the opportunity has been taken to replace the non-dimensional k coefficients by the now more generally accepted C coefficients and my son, M. B. Glauert, has undertaken the necessary revision. One or two other minor changes have been made and a bibliography of some of the more important modern books on aerodynamics has been added.

A model airscrew rotating in a wind tunnel disturbs the uniform flow produced by the tunnel fan and causes variations of velocity which extend to a considerable distance from the airscrew. This flow is constrained by the presence of the tunnel walls and the uniform axial velocity V which occurs at a sufficient distance in front of the airscrew in the tunnel differs from that which would occur in free air. It is necessary therefore to determine an equivalent free airspeed V′, corresponding to the tunnel datum velocity V, at which the airscrew, rotating with the same angular velocity as in the tunnel, would produce the same thrust and torque. A theoretical solution of this problem can be obtained by extending the simple momentum theory to the case of an airscrew rotating in a wind tunnel. The equivalent free airspeed is defined as that which gives the same axial velocity through the airscrew disc as occurs in the tunnel, since this condition will maintain the same working conditions for the airscrew blades, provided the interference effects of the rotational velocity are negligible. The equivalent free airspeed for an airscrew in a closed jet wind tunnel is normally less than the tunnel datum velocity.

The assumption that there is no interference effect on the rotational velocity appears to be sound, but the representation of the interference effect by a change from the tunnel datum velocity to the equivalent free airspeed depends on the existence of the same axial velocity over the whole airscrew disc.

The flow pattern.

The deviation of the velocity at any point of the fluid from the undisturbed velocity V is due to the vortex system created by the aerofoil and can be calculated as the velocity field of this vortex system. The general nature of the vortex system, comprising the circulation round the aerofoil and the trailing vortices which spring from its trailing edge, has been discussed in 10·2, and the analysis of chapter XI provides a method of determining the strength of the vortex system associated with any monoplane aerofoil. The analysis is based on the assumption that the aerofoil can be replaced by a lifting line, and calculations based on this assumption will clearly be inadequate to determine the flow in the immediate neighbourhood of the aerofoil where the shape of the aerofoil sections will modify the form of the flow pattern. Also in the neighbourhood of the vortex wake it is necessary to consider the tendency of the trailing vortex sheet to roll up into a pair of finite vortices. Apart from these two limitations it is possible to obtain a satisfactory account of the flow pattern round an aerofoil from the simple assumption of a lifting line and of straight line vortices extending indefinitely down stream.

Stream lines and steady motion.

When a body moves through a fluid with uniform velocity V in a definite direction, the conditions of the flow are exactly the same as if the body were at rest in a uniform stream of velocity V, and it is usually more convenient to consider the problem in the second form. In general therefore the body will be regarded as fixed and the motion of the fluid will be determined relative to the body. A representation of the flow past a body at any instant can be obtained by drawing the stream lines, which are defined by the condition that the direction of a stream line at any point is the direction of motion of the fluid element at that point. In general, the form of the stream lines will vary with the time and so the stream lines are not identical with the paths of the fluid elements. Frequently, however, the flow pattern does not vary with the time and the velocity is constant in magnitude and direction at every point of the fluid. The fluid is then in steady motion past the body and the stream lines coincide with the paths of the fluid elements. The stream lines which pass through the circumference of a small closed curve form a cylindrical surface which is called a stream tube, and since the stream lines represent the direction of motion of the fluid there is no flow across the surface of a stream tube.

Note 1. (See p. 2.) It is now more usual to use the “quarter-chord point” as the point of reference for the measurement of moments. “The quarter-chord point” is the point on the chord line one quarter of the chord length from the leading edge.

Note 2. (See p. 39.) The contribution of the pressure and momentum integrals to the lift depends upon the shape of the large contour and the conclusion given on page 39 is not true for all shapes of contour; see Prandtl and Tietjens, Applied Hydro- and Aeromechanics, § 106.

Note 3. (See p. 95.) Since the publication of the first edition of this book a great deal of information on viscous flow and drag has been collected. This seems to show that vortex streets occupy a less significant place in the general picture than is indicated in Chap. VIII. For example, the wake of a circular cylinder takes the form of a vortex street in the range of Reynolds' numbers between 102 and 105, but at higher Reynolds' numbers the flow in the wake is turbulent but not periodic. Similarly, for aerofoils below the stalling incidence, a vortex street is only present in the wake for Reynolds' numbers below 105, which is outside the practical range. An account of modern work on this subject is given in Modern Developments in Fluid Dynamics (referred to elsewhere as FD).

The drag of a bluff body.

The theory of the two-dimensional motion of a perfect fluid has led to the determination of the lift of an aerofoil by means of the assumption of a circulation of the flow, but the solution is incomplete in several respects. The conditions which cause the circulation to develop at the commencement of the motion have not been investigated and the magnitude of the circulation is indeterminate except in the case of an aerofoil with a sharp trailing edge. Joukowski's hypothesis that the circulation must be such that the flow leaves the trailing edge smoothly also requires critical examination. Finally, the theory has not indicated the existence of any drag force on the aerofoil.

To examine these problems fully it is necessary to depart from the simple assumption of a perfect fluid and to determine the effects of the viscosity or internal friction, but some insight into the drag of a body can be obtained without introducing this complication. In developing the theory of the lift force it was convenient to consider the class of bodies which give a large lift force associated with a relatively small drag force, so that the latter might be neglected without modifying the essential conditions of the problem. Similarly in examining the drag force it is convenient to consider in the first place bodies of bluff form, symmetrical about the direction of motion, so that the lift force is zero and the drag force is large.

In order to obtain a more detailed knowledge of the behaviour of an airscrew than is given by the simple momentum theory, it is necessary to investigate the forces experienced by the airscrew blades and to regard each element of a blade as an aerofoil element moving in its appropriate manner. It is convenient, in developing the theory, to consider an ordinary propulsive airscrew under ordinary working conditions. The conditions for other types of airscrew and for other working conditions can then be examined as modifications of the main theory.

The airscrew will be assumed to have an angular velocity Ω about its axis and to be placed in a uniform stream of velocity V parallel to the axis of rotation. The sections of the blades of the airscrew have the form of aerofoil sections and the lift force experienced by a blade element in its motion relative to the fluid must be associated with circulation of the flow round the blade. Owing to the variation of this circulation along the blade from root to tip, trailing vortices will spring from the blade and pass downstream with the fluid in approximately helical paths. These vortices are concentrated mainly at the root and tips of the blades and so the slipstream of the airscrew consists of a region of fluid in rotation with a strong concentration of vorticity on the axis and on the boundary of the slipstream.