Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/246

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§ 159

AERODYNAMICS.

or,

W\propto \beta \ A\ V^{2}\quad \beta \propto {\frac {W}{A\ V^{2}}}

(4)

and by (3) and (4),

$y=\beta \ W={\frac {W^{2}}{c\ C\ \rho \ A\ V^{2}}}$

or,

y\propto {\frac {W^{2}}{A\ V^{2}}}

(5)

or for given value of $W,$

y\propto {\frac {1}{A\ V^{2}}}

(5a)

Work done aerodynamically per second

=y\ V,

or

\ \propto {\frac {W^{2}}{A\ V}},

or for given values of

W

and

A

, Power (h.p.)

\propto {\frac {1}{V}}.

(6)

We may interpret and summarise the above as follows:—

(1) The normal reaction of any given plane is proportional to its angle. The constant connecting the quantities depends upon the aspect ratio, and increases with the aspect ratio according to a law not at present known.

(2) The weight supported is sensibly equal to the normal reaction.

(3) Neglecting skin-friction and edge effect, the resistance in the line of flight varies as the angle multiplied by the weight sustained.

(4) Other things being equal, the weight supported varies as the square of the velocity.

(5) Neglecting skin-friction and edge effect, the resistance in the line of flight is directly as the square of the weight sustained and inversely as the area and the square of the velocity.

(6) Neglecting skin-friction and edge effect, the work done per unit time, i.e., the power required for a given weight sustained and a given area, varies inversely as the velocity of flight.

§ 160. The Newtonian Theory Modified; the Hypothesis of Constant “Sweep.”—In the theory of the Newtonian medium, for a given velocity the mass of fluid dealt with is proportional to the sine of the angle </math>\beta ;</math> in a real fluid it is evident that the particles cannot cross each other's paths as depicted in Fig. 92, but will be

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