Circulation without Rotation

In chapter 2 I introduced the idea of describing a flow by means of a stream function. What is the stream function for a vortex? From the definition of a vortex, the flow is in concentric circles of radius r so there is no veloc­
ity component u’ along the radius, but there is a tangential velocity com­ponent V along the circumference of the circles. It will be recalled that the speed of flow is given by the rate of change of the stream function y. In polar coordinates the two relevant equations relating v’ and u’ to the stream func­tion are

1 dy

—-— = u = 0 and r дв

ду = ,= Г dr Inr

Integration of the second of these equations answers the question posed ear­lier and gives the stream function for the vortex as

w = —— logr.

2n

Both Lanchester and his critics made repeated references to flows with circulation but that were also described as “irrotational.” The flow around a simple circular vortex of the kind I have been describing is an example of circulation without rotation. It may seem puzzling to say that the air in such a vortex is not rotating. The problem lies with a divergence between technical and common usage. The difficulty is resolved, at least in part, by distinguishing between the overall path of the fluid element, that is, its cir­cular orbit around the vortex, and its behavior while going round that path. The distinction is sometimes illustrated in textbooks by reference to a Ferris wheel, where the chairs are carried around but always hang vertically and so maintain the same orientation, that is, the chairs “circulate” but do not “rotate.” Circulation is also compatible with an absence of rotation when a vortex exists as one component in a flow. Thus the combination of a uniform wind and a vortex motion, of the kind assumed in the circulatory theory of lift, can also be conceived as an irrotational motion with circulation.

One final feature of vorticity and circulation should be mentioned before introducing Lanchester’s own use of these ideas. In 1869 Kelvin established a remarkable theorem about circulation in an ideal fluid. He proved that it does not and cannot change with time.9 In an ideal fluid obeying the Euler equations the value of the circulation cannot change as a result of any pro­cesses such as the movement of a body through the fluid. Circulation can exist, but once in existence it can be neither augmented nor diminished. Thus if the circulation starts as zero, it stays zero. If the amount of circulation is symbolized by Г, then Kelvin’s theorem is expressed by saying that the rate of change of Г with time is zero. In mathematical symbols,

d г

The proof of Kelvin’s theorem depends on the condition that the con­tour of integration used to establish the circulation is one that moves along with the fluid. The contour must be made up of the same elements of fluid at all times. Such a contour is sometimes called a material loop and stands in contrast to the kind of contour mentioned previously, which is a purely geometrical entity and can be selected on grounds of mathematical or ex­perimental convenience. This restriction of the theorem to the circulation around material loops has consequences that are both important and subtle. It made the precise implications of the theorem difficult to decide. Later we see that Kelvin’s theorem was interpreted in different ways by different groups of experts. The divergence of opinion had a significant impact on the discus­sion of how (or whether) a wing can generate lift by generating circulation. The generation of circulation by the movement of a wing through the air would involve a variation of circulation with time. Is it ruled out by Kelvin’s theorem? If it is, then what does this mean for the circulation theory of lift? I shall come back to these questions, but now, having laid the foundations, I turn to Lanchester’s pioneering statement of the circulation theory.