Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/324

hence we may take it as probably about 97${\tfrac {1}{2}}$ per cent, of its maximum. On the basis of § 204 this gives

For air	.  .  .  .  .	68.6 per cent.
For water	.  .  .  .  .	79.0 per cent.

We have no data of comparison in the case of air; in the case of water, so far as the author is informed, the highest actual efficiency recorded is somewhat over 70 per cent.

§ 207. Pressure Distribution.—It is evident that, according to the present theory, the propeller blade is amenable to precisely the same laws so far as its pressure-velocity relation is concerned as the ordinary aerofoil, and we presumably also have the two alternative types of fluid motion, the continuous and the discontinuous, according as the blade is given a pterygoid form (based on a helix) or whether a simple helical surface or sheet (the analogue of an aeroplane) is employed. We may read the appropriate pressure for air from either Table X. or XII., as the case may be.

A complication is introduced in the propeller blade by the fact that its different portions are moving at different velocities through the fluid, so that the pressures proper to least $\gamma$ vary at the different points along the length of the blade. This velocity, the $V$ of the propeller blade, will be given by the expression $V={\frac {\mbox{V}}{\sin \theta }},$ where ${\mbox{V}}$ is the velocity of the vessel, or, in terms of $r,$ we have

$V={\frac {\sqrt {(2\ \pi \ r)^{2}+p^{2}}}{p}}\ {\mbox{V}},$

where $p$ is the pitch, or

$V^{2}=(4\ \pi ^{2}\ r^{2}+p^{2})\ {\frac {{\mbox{V}}^{2}}{p^{2}}},$

and since $p$ and ${\mbox{V}}$ are constants the curve is of the form $y=x^{2}+$ const., when plotted (Fig. 128), where abscissae give radius in terms of pitch and ordinates give $V^{2}$ values. Now by § 185, for any aerofoil $P_{2}=V^{2}\times$ const., values of the constant being given in Table IX. Hence the curve (Fig. 128) may

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