Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/215

point to the probability of a lower value than .40 for the Helmholtz constant, in view of the probability of higher resistances being experienced with greater depths of rim. It is worthy of remark that Dines obtained, for a hemispherical cup, pressures about 16 per cent, greater than for a plane circular disc.

In general, if we neglect the influence of rotational motion within the stream, let, as before, $C$ be the experimentally ascertained constant for any plane, and $c$, the pressure constant on the Helmholtz hypothesis, and let $C+n\ C$ he the total augmented pressure for the same plane fitted with a deep rim, we shall have the relation:—

$c_{1}=.50\ {\boldsymbol {-}}\ n\ C\quad$ or, $\quad n\ C=.50\ {\boldsymbol {-}}\ c_{1}.$

If we apply this to the two-dimensional case of the infinite strip, we have Kirchhoff's determination of the Helmholtz constant $c_{1}=.440,$ and Dines' experimental result $C=.76,$ so that—

$.76\ n=.50\ {\boldsymbol {-}}\ .44\quad$ or, $\quad n=.08.$

That is to say, the maximum possible addition to the pressure is, in this case, about 8 per cent.

Dealing with this problem in an analytical investigation, Love^[1] has shown that if $\epsilon$ be the ratio of height of lips to breadth of plane, the pressure will be increased approximately by an amount $={\frac {6}{5}}{\sqrt {\epsilon }}$ of that for the same plane without the lips. It is evident that this expression only holds good for small or moderate values of $\epsilon$, for the limiting value would otherwise be exceeded. This would occur when ${\sqrt {\epsilon }}={\frac {5}{6}}\times .08=.0666$  or, $\ \epsilon =.258,$ so that Love's approximate equation will be perceptibly in error for some considerably smaller value.

§ 142. Planes of Intermediate Proportion.—We have so far dealt with the two extreme cases of contour form typical of two-

1. “Theory of Discontinuous Fluid Motion,” Proc. Camb. Phil. Soc, VII., 1891.