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.

It is likely that I was introduced to the work of Philip Doddridge several years before I encountered that of any other of the subjects of this book, for in the village chapel in which I was christened, almost all baptismal services began with the singing of Doddridge's hymn ‘See Israel's gentle shepherd stand’. Indeed, it is for his hymns that Doddridge is now best remembered. Like Pepys, Doddridge was no mathematician, yet he was one of the leaders in an educational movement that did much to secure a place for mathematics in the curriculum at secondary and tertiary level. Moreover, his own mathematical education is well documented and it is for these reasons that he has been selected to represent his period and the dissenting academies with which his name is so closely associated.

THE PURITAN REVOLUTION AND ITS AFTERMATH

In the time of the Commonwealth increased emphasis came to be placed on education. The forces of conservatism were, for the time being, driven underground and advancement came to depend more on merit than on social class. In such a climate, established doctrines and ways were more readily challenged. In particular, the curricula of the grammar schools and universities were attacked as lacking social utility and relevance, and demands were made for increased mathematical and scientific studies (1). Attempts were also made to break the monopoly of Oxford and Cambridge in the field of university education in England.

In the previous chapter we described how mathematics established itself within the curriculum of the older endowed schools and how the AIGT came to be established. Meanwhile, state intervention in education had sought to ensure that a uniform brand of arithmetic was taught in those elementary schools which received public money, and the State had also, somewhat unwillingly, become involved in a form of secondary education. During the period which we now describe, secondary education, too, was to be accepted as a state responsibility and mathematics teaching within that sector was to be reformed. Moreover, not only was the practice of mathematics education to be reviewed, but considerable attention was also to be directed at the study of mathematics education. Whereas, for example, J. M. Wilson was largely happy to practise as a mathematics teacher, Godfrey and his contemporaries, such as Branford, Carson and Nunn, attempted to found a discipline of mathematics education.

SCHOOL AND RAWDON LEVETT

Many, if not all, mathematicians are greatly indebted to their first mathematics teacher and this was certainly true of Charles Godfrey. His master at King Edward's School, Birmingham, was Rawdon Levett, one of the founder members of the AIGT (pp. 134f above). Levett, later described as ‘probably the best schoolmaster I ever knew’ (1), was clearly the kind of teacher profoundly to influence a young pupil. He was a Cambridge mathematician {2} who had come to King Edward's in 1869 after four years teaching at Rossall School.