Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/249

Rh that indicated by the experiment of the superposed plane (§ 154), hence it is evident that the hypothesis is insufficient.

§ 161. Extension of Hypothesis.—According to the principles laid down in Chaps. III. and IV., the neighbourhood of a plane or other aerofoil sustaining a load becomes the seat of a cyclic disturbance, and the air in advance of the aerofoil is in a state of upward motion; it has been shown that this up-current con- tributes to the supporting power of the plane or aerofoil, that is to say, its momentum contributes to the total load carried.

Let us represent this cyclic disturbance by supposing that in Fig. 107 the air stratum, instead of meeting the plane horizontally, has an upward component so that its motion (plotted relatively to the plane) be inclined at an angle $$a$$ (Fig. 108), so that its upward velocity will be $$V\ \sin a,$$ or for small angles $$V\ a .$$

Then the mass per second will be $$\rho\ \kappa\ A\ V$$ as before, and the momentum $$= \rho\ \kappa\ A\ V^2 (\alpha + \beta)$$ or  $$\frac{P_\beta}{P_{90}} = \frac{k}{C} (\alpha + \beta) .$$

But we know by § 159 that $$\frac{P_\beta}{P_{90}} = c\ \beta ,$$ so that we now have the equation—

or

Thus for any given plane, $$C$$ and $$c$$ being known experimentally, and $$\kappa$$ being estimated from trials of superposed planes, we can calculate the equivalent up-current due to the cyclic disturbance, within the limits of the present hypothesis. This qualifying phrase is necessary because the supposed motion of the fluid, as depicted in Fig. 108, is conventional, and it is only on this conventional basis that we have effected a solution. The theory on the present lines is more fully developed in Chap. VIII., where it is made to perform useful work. The author, however, does not regard it as by any means final; the theory of the future should be based on a more comprehensive