In order to render the deductions of theory more consonant with the results obtained in practice, J. S. Russell, Esq., of Edinburgh, conducted a series of experiments in the years 1834 and 1835, upon the Forth and Clyde canal, under circumstances more analogous to the cases of actual practice than those of previous experimenters. He has since given an account of these experiments in Vol. XIV. of the Transactions of the Royal Society of Edinburgh, of which we shall avail ourselves for a few extracts. The account is highly interesting, and likely to prove of great practical utility; Mr. Russell having ascertained the existence of certain phenomena not before adverted to, as well as the laws which regulate them; which throw great light on many intricate and perplexing points, and go far to account for the discrepancies which most accounts of extensive series of experiments exhibit.
Mr. Russell observes, that the law which connects the resistance of the fluid with the second power of the velocity, agrees very nearly with the motion of bodies that are wholly immersed, and with the motion of floating bodies that have certain velocities, and are placed in certain circumstances; but it is widely erroneous in its direct application to the motion of floating bodies at higher velocities and under different circumstances. In every large collection of experiments, examples are to be met, where the resistance, instead of following the law of the squares of the velocities directly, has been found to vary, not only with every different power of the velocities, from the first to the fourth power, but also in the inverse ratio of some of these powers. Two very striking illustrations of this are given in the following experiments by Mr. Russell; the one showing an increase of resistance, corresponding with a very high power of the velocity, and the other exhibiting a decrease of resistance, with an increase of velocity greater than the former.
They were made October 18, 1834, with a floating body weighing 12,579 lbs.:-
Velocity in feet per Second.
Resistance in lbs.
In the first two experiments it will be seen that the resistance increases in a greater ratio than the squares of the velocities, by nearly 15 per cent; whilst in the last two, in which the velocity increases from 8.76 to 14.28, the resistance actually becomes less, amounting to scarcely 1/3 of what the law of the squares of the velocities would indicate.
The result of Mr. Russell's investigations appears to establish the following conclusions: -
That the resistance does not follow the ratio of the squares of the velocities; except in those cases where the velocity is low, and the depth of the fluid considerable. That the increments are greater than those due to the squares of the velocities, as the velocity approaches a certain quantity, which is determined by the depth of the fluid. That at this point the resistance attains a first maximum; and that here, by certain elements of the form of the body, and of the dimensions of the fluid, they may become infinite. That immediately after this, there occurs a point of minimum, where the resistance becomes much less than that due to the square of the velocity; after which it continues to receive increments of which the ratio is less than that due to the square of the velocity. That according to the law of progression which has been established, the resistance will reach a second point of maximum when a velocity shall be obtained of 29 miles per hour, after which it will be rapidly diminished with every increase of velocity.
These singular deviations of the law of resistance from a uniformly progressive ratio arise principally from two causes, but slightly (if at all) adverted to by former experimenters. The first of these is an emersion of the floating body from the fluid, due to the velocity of the motion of the body, and by which the dynamic immersion is rendered less than the statical immersion of the body in the fluid. The second is the generation of waves by the floating body in the direction of its motion, which affect the form and surface of the fluid, the position of the floating body, and of the resistance. The velocity of these waves (which Mr. Russell calls the wave, or the solitary wave of progression) depends neither upon the form of the floating body, nor the velocity of its motion, but solely upon the depth of the fluid.
The first of these phenomena had been slightly noticed by some previous observers, but the fact was questioned by many writers. Mr. Russell, however, has established the fact experimentally, and has laid down the theory by which it is to be accounted for. To ascertain the fact of emersion, the following experiments were made. A slight skiff was fitted with 12 glass tubes, accurately graduated, which passed through holes in the bottom of the vessel, and were open at both ends, so as to allow the water to rise within the tubes to the level of the water outside the boat. The boat thus fitted was drawn along the canal at different velocities, and the height at which the water stood in the tubes was carefully noted by an observer seated in the boat. The immersion of the boat when at rest being 2.7 inches, the dynamic immersions were as under:-
MILES PER HOUR.
Having thus clearly established the fact by experiment, Mr. Russell gives the following proposition in explanation of it, viz., that the pressure downwards upon a fluid, by a body in motion at a given velocity, is diminished by a quantity equal to the pressure of a column of the fluid, having the height due to the velocity of the motion. The form of the floating body is no element in the formula of emersion, the law being a general one, and having for its foundation this simple principle, that gravity acting on a solid body during a given unit of time is a constant quantity, and that the displacement of the fluid by the weight of the body, being a quantity that increases both with the velocity and with the quantity of that displacement, must ultimately be equal in quantity, as it is opposite in direction, to the pressure of the solid downwards by gravity. When, however, the depth of the fluid is small, the results will be modified by the wave, and other elements of resistance.