THE AERODYNAMICS OE A SPINNING SHELL. 
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1.01. The Functional Form of the Drag Coefficient .—A careful consideration of the possible forms of the 
function / R , from the points of view of the kinetic theory of gases and the theory of dimensions, suggests 
that y, l/r, and <rjr should be possible arguments of / E , besides v/a. Here y is the ratio of the specific 
heats of the gas, l is the mean free path, and o- the effective diameter of its molecules. We may, if 
desired, replace Ijr by the more usual viscosity argument vr/v, where v is the kinematical coefficient of 
viscosity. Wind channel work on aerofoil and airscrew models shows that the argument vrjv is of great 
importance at low velocities. Its effects, however, in the case of shell models seem almost to have 
disappeared by the time a velocity of 40 f.s. (or at any rate 75 f.s.) is reached. Rayleigh* obtains 
formulae for the pressure on a piston moving in a pipe, which show the kind of way in which y, as well as 
v/a, might enter into the expression for / E . Valuations of y are, however, very small in practice. There 
is experimental evidence that some argument, other than v/a or y, has an appreciable effect in practice, 
and that this argument is probably not the viscosity term in the ordinary sense. It is not possible to 
pursue the question further here, or to assemble in detail the evidence, which is to be found in various 
minutes of the Ordnance Committee. 
So long as the stream lines of the flow remain unaltered by a change of velocity, the motion remains 
dynamically similar, the drag varies as v 2 , and the coefficient / R must be a constant. The drag is then 
said to obey the square law. Experiments with air screws, of high peripheral speed, appear to show that, 
up to values of v/a as great as O'7, there is no serious departure from the square law once a certain 
minimum velocity is exceeded, above which the ordinary viscosity effects become unimportant; this 
appears, from all the evidence, to be the case also for shells, the minimum velocity being of the order of 
50 f.s. As velocities of less than 100 f.s. may be ignored in ballistics, it is therefore customary to assume 
that the drag obeys the square law exactly for all velocities less than about 0‘7a. For all such velocities 
the stream lines of the flow will remain nearly unaltered and the motion will be dynamically similar. 
Above this velocity (O' 7 a.) the effects of the compressibility of the air become rapidly of great 
importance, and the whole nature of the air-flow changes as a, the velocity of sound, is reached and 
exceeded. These effects are represented by the variation of / E as a function of v/a. A good typical curve 
showing this variation is given by Cranz.I Another example will be found in fig. 4. 
We have so far ignored the fact that the shell is actually spinning about its axis of symmetry. There 
is no evidence to show that the drag, in the case of symmetry, is appreciably affected by the spin, and its 
neglect is probably justified. 
A more important question is the legitimacy of assuming, as we have tacitly done in (1.001), that the 
drag does not depend appreciably on the acceleration of the shell. With regard to the acceleration at low 
velocities, it is known that the effect of the air is to increase the virtual mass of any body by an amount 
of the order of the mass of air displaced. This is an increase of the order of 1 part in 2000, and is 
entirely negligible. At higher velocities, and on the general question, direct experimental evidence is 
unfortunately lacking. It is, however, difficult to see, by theoretical reasoning, how the past history of 
the shell can have any large effect, and there is sufficient general experimental evidence that (1.001) is, on 
the average, an adequate representation of the drag in the case of symmetry to be certain that the past 
history is of little importance, except conceivably for a very limited range of velocities, for example, in the 
neighbourhood of a, the velocity of sound. 
§ 1.1. r The Detailed Specification of the Complete Force System. 
The theory discussed in this paper treats the shell as a rigid body which is a solid 
of revolution, so that its axis of symmetry coincides with a principal axis of inertia. 
* “Aerial Plane Waves of Finite Amplitude,” ‘Scientific Papers,’ vol. V., or ‘Roy. Soc. Proc.,’ 
A, vol. LXXXIV. See in particular the last section of the paper, 
t ‘ Encyklop. der Math. Wiss.,’ vol. IV., Part II., p. 197. 
