Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/268



§ 172. Introductory.—At some future period it may be found possible to rationalise the treatment of the theory concerned with the form of the aerofoil on a comprehensive basis, so that the sectional form at every point shall be correlated to the pressure reaction and the strength of the cyclic disturbance. At present we are compelled to take our stand on a simplified and somewhat conventional hypothesis.

In the case of the aeroplane, in respect of which a certain amount of experimental data is available, we can at once proceed to apply the fundamental propositions of the preceding chapter, to determine the angle of least resistance, thus:—

Let, as before (§ 163), $$x + y$$ be the total resistance in the line of flight, where $$x$$ is the direct resistance (due to skin friction, etc.) and $$y$$ that due to work expended dynamically.

Then the condition of least resistance is that $$x = y .$$

Now $$x = \xi\ A\ P_{90} ,$$ and $$y = \beta\ W = c\ \beta^2\ A\ P_{90}$$ (for small values of $$\beta$$), or $$\xi = c\ \beta^2$$, that is, $$\beta = \sqrt{\frac{\xi}{c}}$$ where $$\beta$$ is the angle of inclination in radians; $$\xi$$ is the coefficient of skin friction (§ 157), and $$c$$ is the constant according to § 159.

If $$\beta^\circ$$ be the value of the angle expressed in degrees, the expression becomes $$\beta_c = \frac{180}{\pi}\ \sqrt{\frac{\xi}{c}} .$$

Taking for example the case of a square plane for which the value of $$c$$ is 2, and taking $$\xi =$$ .02, we have—