Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/75

Rh 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.

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 $$\left ( \frac{\mbox{L}}{\mbox{T}^2} \right ) =$$ constant; the author has proved from aerodonetic considerations that the same law obtains in connection with aerial flight. In this law we have $$V$$ varies as the square root of $$l$$ so that the two laws are incompatible—that is to say, not capable of simultaneous fulfilment. This fact is well known in connection with model experiments relating to ship resistance, the results of experiment being subject to correction according to certain rules for frictional resistance, and similar correction will be required in the case of aerodrome experiments.

If it were possible, as by employing some different fluid, to alter the value of $$\nu$$ when experimenting with scale models, the necessity for applying a correction might be obviated; we have:—

By Fronde's law $$V^2 = c_1\ l ,$$ where $$c_1$$ is a constant. By Equation (7) $V = c\ \frac{v}{l} ,$ or $\sqrt{c_1\ l} = c\ \frac{v}{l} ,$ that is, $c_1^\frac{1}{2}\ l^\frac{3}{2} = c_\nu$

or, the kinematic viscosity is required to vary with the 3/2 power of the linear dimension.

We cannot always obtain fluids with viscosity to order, but if we select two fluids such as air and water, whose kinematic viscosities are, at 15° C., in the approximate ratio of 14:1, and if $$l_1$$ and $$l_2$$ represent the lengths of the two models, and $$\nu_1$$ and $$\nu_2$$ the values of the viscosities respectively, then,—

That is to say, that a model aerodrome, made to a $$\tfrac{1}{5.8}$$th scale