Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/342

§ 216 position of the propeller, the essential point being the extent to which the frictional wake is led into the periptery of the propeller blades. It is manifest that $$\mbox{v}_1$$ is limited to a value less than the sternward component of the impressed velocity (compare § 200).

In the hypothetical case chosen in § 200, where it is assumed that the whole of the wake current is utilised by the propeller, we have—

Phantom ship velocity $$= \mbox{V – v}_1 ,$$ or by § 198— $$E = \frac{\mbox{V – v}_1}{\mbox{V – v}_1 + \frac{\mbox{v}}{2}} =$$ $$\frac{\mbox{V – v}_1}{\mbox{V –}\ \frac{\mbox{v}_1}{2}}, \quad (\mbox{for}\ \mbox{v}_1 = \mbox{v}) ,$$

but for real ship $$E_1 = \frac{\mbox{V}}{\mbox{V – v}_1}\ E ,$$

or $$E_1 = \frac{\mbox{V}}{\mbox{V – v}_1} \times \frac{\mbox{V – v}_1}{\mbox{V –}\ \frac{\mbox{v}_1}{2}} = \frac{\mbox{V}}{\mbox{V –}\ \frac{\mbox{v}_1}{2}} ,$$

which is the result already deduced in § 200 by the direct application of the Newtonian principle.

The device of the phantom ship is in reality merely a method of expressing a simple problem in relative motion in a palatable form; it is obvious that the argument treats the wake current as a favourable tidal current, or as the flow of a river, the ship's motion being credited in respect of its change of position relatively to some fixed mark; the method of the “phantom ship” presents the problem in a clear and precise form.

The question of wake influence is probably of less importance in connection with aerial flight than it is in the problem of marine propulsion.

'''§ 217. The Hydrodynamic Standpoint. Superposed Cyclic Systems.'''—It is of interest to form a mental picture of the hydrodynamic system of flow that accompanies a screw propeller.

It is evident that according to peripteral theory each blade of the propeller forms the axial core of a cyclic system and that the