Frederick William Lanchester

“ The viscosity of a substance is measured by the tangential force on the unit area of either of two horizontal planes at a unit distance apart one of which is fixed while the other moves with the unit of velocity, the space between being filled with the viscous distance. ”

“ Corresponding Speed.—The above expression enables us to formulate at once a law of corresponding speed for motion in any viscous fluid, for, if the physical properties of the fluid do not vary in any way will be constant and we have that is to say, for submerged model experiments, in which the condition of acceleration = constant does not apply, the smaller the model the higher the speed, in the direct proportion of the linear dimension—a rather unexpected result. ”

“ It may therefore be deduced that the form of flow will be more modified for low than for high velocities, in which case the form of resistance we are now discussing will not vary exactly in the direct ratio of the velocity.
Bodies other than of streamline form will also be affected by this type of viscous resistance, when it will appear as an added resistance. The only exception is found in the case of a plane moving tangentially, the consideration of which introduces the important subject of skin-friction. ”

“ It is probable that the departure from the V2 law is less than is indicated by the balanced plane experiment, owing to the smaller plane being unduly affected in each case by its proximity to the larger one. It is conceivable that the smaller plane being situated in the counterwake of the larger, will in effect be surrounded by air moving with above the normal wind velocity, and so show a fictitiously high pressure value. ”

“ Turbulence.—The steady state of viscous motion depicted in Figs. 26 and 27, on which the laws of viscosity and skin-friction have been based, is found in practice to obtain over a moderate range of velocity only. When a certain critical velocity is exceeded the continuity becomes broken and the phenomenon of turbulence manifests itself. ”

“ Thus, if the curve be plotted for a one foot diameter circular plane and one foot per second velocity is represented by one inch, then for a two foot circular plane a one foot per second velocity will be represented by two inches, that is to say, a given pressure will be developed at one-half the velocity. ”

“ Froude found that an increase in area is not accompanied by a proportionate increase in resistance; he also found that the index connecting resistance and velocity is in general less than 2, the mean result of several experiments giving 1.92. Colonel Beaufoy, also experimenting in sea water, gives the value 1.7 to 1.8.
Dines, experimenting in the open air, obtained results that have some interest from the present standpoint. In spite of some conflicting evidence, it would, in the main, appear that, under the conditions of experiment, the V2 law is a very close approximation to the truth. ”

“ For very small values of we allow that quantities varying as and become negligible, and the curve will be of the form and leave the origin as an inclined straight line. When the velocity is very great resistances that vary as the lower powers of the velocity will be negligible in comparison to those that vary as and consequently the curve will approach asymptotically to the form It is questionable whether the stage can exist when the viscous reaction of the fluid is due wholly to its own inertia ”

“ Dines specifically states that he finds the wind pressure on the normal plane and on bodies generally varies strictly as the square of the velocity, a result which it is difficult, in view of dimensional theory, to harmonise with the above experiments. ”

“ One of the least satisfactory results of dimensional theory, so far as revealed by a comparison with conclusions that would be naturally formed from experience, is the inverse relation that exists for homomorphous motion between and It would appear that for bodies of similar form any state of motion—say the state when discontinuity sets in—is reached when their respective velocities are in the inverse ratio of the linear dimension. ”

“ The law of corresponding speeds employed in naval architecture is primarily influenced by considerations of wave-making, in which (as shown later in the present work) the dimensional basis is acceleration constant ”

“ Fig. 30.
quantity of momentum and an infinite quantity of energy [9] ; in other words, the steady state involves a force applied for an infinite time through an infinite distance; it also constitutes a violation of the principle of no momentum of § 5.
If we suppose that the stress, due to the propulsion or to the resistance of the body, be transmitted by viscosity to the walls of the vessel, as when the body is moving quite slowly, or when the thickness of intervening fluid is small; then the resistance will ”

“ In this Dines agrees with the previous experiments of Newton. Hutton, and others, and with the contemporary work of Langley.
It is to be inferred that in cases of direct resistance the Stokes (R ∝ V) and Allen (R ∝V1.5) stages are confined to bodies of very small size and very low velocity. The bodies employed by Dines varied from some few square inches to some few square feet area. ”