Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/248

§ 160 latter and not upon its angle, and for a given plane the thickness of the layer will be constant.

In the foregoing paragraph the term “thickness” is used somewhat loosely. It is evident that there is no definite point at which the influence ceases altogether; and this brings us to a convention which it is found advantageous to adopt.

Let us suppose that the plane be supported by a definite stratum of air to which a uniform downward motion is imparted (Fig. 107) ; let us term the vertical cross-section of this stream or stratum the “sweep” of the plane and denote its downward velocity by $$v$$.

Then it is clear that for similar planes the sweep will bear a definite constant relation to the area $$A ;$$ let us, as in § 109, denote the sweep by the symbol $$\kappa\ A$$ where $$\kappa$$ is a constant proper to the shape of the plane; in the case of rectangular planes a given value of $$\kappa$$ will correspond to some definite aspect ratio.

Now the mass of air handled per second will be $$= \rho\ \kappa\ A\ V ,$$ and the momentum $$= \rho\ \kappa\ A\ V\ v = \rho\ \kappa\ A\ V^2\ \sin \beta ,$$ which for small angles $$= \rho\ \kappa\ A\ V^2\ \beta,$$ where $$\beta$$ is in circular measure. We therefore have $$P_\beta = \rho\ \kappa\ V^2\ \beta,$$ that is, $$\frac{P_\beta}{P_{90}} = \frac{\rho\ \kappa\ V^2\ \beta}{C\ \rho\ V^2} = \frac{\kappa}{C}\ \beta$$ under the conditions of the present hypothesis.

But by § 159 we know from experiment that for small angles (such as under discussion) $$\frac{P_\beta}{P_{90}} = c\ \beta$$ where $$c$$ is a constant depending upon the plane form and aspect; thus our hypothesis leads us to an expression of the correct form.

If we endeavour to deduce the constant $$\kappa$$ from $$c$$ and $$C$$ (constants experimentally determined and known) from the resulting equation, $$\kappa = c\ C$$, we obtain a value far in excess of