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.