Introduction

Atmospheric pressure, temperature, humidity, wind speed, and sea surface temperature are fundamental parameters for determining the weather. If it were possible to measure these five parameters simultaneously across the globe at regular time, distance and height intervals, the problems of medium- and long-range weather forecasting would be greatly reduced.

Before the advent of satellites, global weather measurements on such a scale would have been totally impracticable. Systematic meteorological observations used to be extremely sparse. Although most industrialized countries maintained a network of ground-level observation stations, vertical sounding was limited to launching occasional balloons from land and sea, and receiving sporadic reports from commercial aircraft. These horizontal and vertical measurements covered only a small portion of the earth, forcing meteorologists to bridge the gap with educated guesses.

In the late 1940s, sounding rockets equipped with cameras took pictures of the earth from altitudes of 100 km and more. These early photographs revealed a whole new family of physical relationships in the atmosphere. In the years that followed the launch of Sputnik in 1957, much effort was devoted to developing television cameras and radiometers suitable for satellite meteorology. Throughout the 1960s, the United States and the Soviet Union deployed a host of increasingly powerful weather satellites which transmitted visible and infrared images to ground stations on the earth. The first steps towards making global weather observations had been taken.

Low-orbiting Satellites

The early meteorological satellites were launched into low earth orbit by necessity, since rockets in those days had limited lifting capacity.

The limited extent of the stream of air in a wind tunnel, whether of open or of closed working section, imposes certain restrictions on the flow past an aerofoil or other body under test, and the determination of the magnitude of this interference is of considerable importance, since it is found that certain corrections must be applied to the aerodynamic characteristics of an aerofoil tested in a wind tunnel before they are applicable to free air conditions. This interference correction is independent of and additional to any correction which may be necessary to allow for the change of scale from a model aerofoil to an actual aeroplane wing.

The theory of the interference has been developed by Prandtl in his second aerofoil paper by considering the conditions which must be satisfied at the boundary of the stream. The continental wind tunnels usually have an open working section and the condition of constant pressure must be satisfied at the boundary of the stream. British wind tunnels, on the other hand, have a closed working section of square or rectangular cross section, and the boundary condition takes the form that the component of the velocity normal to the tunnel walls must be zero. This boundary condition can be satisfied analytically by the introduction of a suitable series of images of the model, and the interference experienced by the model is the induced velocity corresponding to the vortex systems of these images.

Circulation and vorticity.

The definition of the circulation round a closed curve in two dimensions (see 4·1) as the integral of the tangential component of the velocity round the circumference of the curve can be extended at once to the more general case of motion in three dimensions by removing the restriction that the curve must lie in a single plane. Also by dividing any surface bounded by this curve into a network by a series of intersecting lines it can be shown that the circulation round the curve is equal to the sum of the circulations round the elementary areas formed by the network.

The vorticity of a fluid element in two-dimensional motion was defined (see 4·3) as twice the angular velocity of the element. This definition is retained in the more general case of three-dimensional motion but the axis of rotation of the fluid element may now point in any direction. By following the direction of the axis of rotation of successive fluid elements it is possible to construct a curved line whose direction coincides at every point of its length with the axis of rotation of the corresponding fluid element. Such a line is called a vortex line.

The vortex lines which pass through the points of the circumference of a small closed curve C will form the surface of a vortex tube, of which the curve C is a cross section.

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.