Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/341

Rh § 216. The Influence of the Frictional Wake.— It has already been shown that the efficiency of a propeller of any kind is increased by the fact of its operating on the frictional wake.

The efficiency that we have been discussing, the $$E$$ of the screw propeller, represents the efficiency on the basis of § 198; that is to say, the propeller is supposed to act on virgin water, and the towing efficiency is that taken as unity.

When we consider the wake as influencing the efficiency we have to adopt a convention. It is known that the wake is in reality a very disturbed region whose velocity varies greatly from point to point. Mr. R. E. Froude has shown that the mean wake velocity over the area swept by the propeller may be taken as its effective velocity without serious error, and he has also introduced the useful conception of a phantom ship having a speed equal to the actual velocity minus the mean wake velocity, that is, $$\mbox{V – v}_1 .$$

The resistance of the phantom ship is supposed, at its velocity $$\mbox{V – v}_1 ,$$ to be equal to that of the real ship at its velocity $$= \mbox{V} ,$$ so that the propeller designed for the phantom ship on the basis of simple theory will be correct for the real ship to work in its frictional wake. The important fact is thus rendered apparent, that the form of the propeller proper to highest efficiency is independent of the existence or otherwise of a frictional wake.

Now the useful work is proportional to the actual velocity of the vessel, since the resistance (and therefore the thrust) is the same in both the phantom and the reality; consequently the useful work is in the relation—

$$\frac{\mbox{Real}}{\mbox{Phantom}} = \frac{\mbox{V}}{\mbox{V – v}_1} .$$

But the total work done in propulsion is the same in both cases, therefore if $$E_1$$ is the efficiency under real conditions, we have—

$$E_1 = \frac{\mbox{V}}{\mbox{V – v}_1}\ E .$$

The value of $$\mbox{v}_1$$ depends upon the lines of the vessel and A.F.