Dimensions of Units

For sufficiently slow speeds the force and speed are propor tional to each other; hence F= constant X • v = const. X iaDv P which is Stokes' Law. For a sphere, experiment and dynamical theory indicate that the constant is 3r. The motion is non turbulent. For very rapid motion F is found to vary more nearly as the square of the velocity and the equation for it must then approach the form which is independent of the viscosity. To illustrate the use of these results in connection with small-scale models consider cases in which it is required to find the force in air of a machine of linear dimension D. by means of measurements in water with a model of linear dimensions D.. The data required are: Ratio of viscosities of air and water 1/6o approx: Ratio of densities of air and water 0.0013 approx.

Ratio of

µ/p of air and water io approx.

Ratio of

AV p of air and water o.6 approx.

Hence going back to the general equation the condition for dynamic similarity is that = io and is then o-6.

The object gained by driving the model in water is to diminish the speed necessary to about one-tenth the very high speed that would be required in air. The numbers given are only approximate since both ,u and p vary with temperature. A rise of 25 degrees C diminishes the viscosity of water to one-half so that particular attention must be paid to its temperature during the experiments.

If however the model is used in the same medium as the main apparatus—for example in air for all aeroplane work—it is usually impossible to work with velocities that satisfy the principle of similitude. It is necessary then to make experi ments with as large a variation of velocity as possible and to extrapolate the curve of F plotted against vD, the values ofµ and p being now constants; or, what is more usual, to take the left side of the equation in the alternative form already given, and to plot against vD (at high speeds F varies nearly as and the extrapolation is safer if this choice of ordinate is made). A useful guide in design is thereby provided which is subsequently tested out on the full scale machine.

Plates.

Difficulties arise in many cases. Experiments on square plates placed in a stream of fluid do not accord with the deductions from the above equations and have given rise to a great deal of discussion. In applying the deductions it must, of course, be borne in mind that it is only the group of plates whose thickness varies in the same proportion as the other lengths that are comparable. According to the formula a reduction in D should carry with it the same effect as an increase in ,u/p whereas the opposite effect has been observed. The only possible con

clusion is that phenomena arise which are not contemplated in the dimensional specification. Appeal has been made by Bairstow and Booth to the compressibility of air in front of the plate but examination shows that this does not provide a large enough effect at the usual velocities concerned. They show, however, that all the reliable results can be represented by a formula of the type F = a for values of vD from i to 35o (foot-second units). This would fit in with the requirements for different fluids provided acc and b cc (p but this does not seem as yet to have been proved. Stanton (Proc. Inst. Civil. Eng. 171) has brought forward evidence that a negative pressure on the leeward side of the plate is responsible for the complication.

Influence of Boundary.

When the fluid is bounded by a tube and the obstacle is on its axis the mean velocity at which turbulence begins is changed. For example, it has been found experimentally by Hisamitu Nisi, Phil Mag. XLVI. 754 (using in succession two tubes of square section of 2.6 and 2.o cm. lengths of sides and containing, in any single experiment, a cylinder at right angles to the length of tube or a small sphere), that the velocity of air at which turbulence begins in the tube can be expressed very closely by the formula n• where A is the diameter of the tube, n = A 8.i 5, 68.2 for spheres, and 2.65, for cylinders. The form of the equation was chosen to suit dimensional requirements but for the index n resort had to be made to experiment.

Floating Bodies.

In the case of a floating body gravity is a determining factor. Since is a non-dimensional term we can allow for it most generally by writing Dynamical similarity requires that the right hand side should be a constant and this requires that vDp/µ = const. and = const. If both the small and the large scale body are intended to float in water, A, p and g are all constants; therefore we require that vD= const. and const. simultaneously; but this is impossible, No similitude can exist therefore between bodies of different sizes. Viscosity however, only plays a small part, owing to the force varying nearly as the square of the velocity. The right hand side must, to this approximation, be independent of and therefore of vDphl and we need only consider the term which must be the same both for model and original and therefore so must F/D. That is to say the forces are directly as the linear dimensions when the velocities are as the square roots of the linear dimensions. The assumption throughout is, as usual, that the bodies compared are geometrically similar.