Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/77

Let $R_{1}$ be the resistance varying as $V,$ and $R_{2}$ be the resistance varying as $V^{2},$ then $R=R_{1}+R_{2},$ and we have

${\frac {dR}{dV}}={\frac {R_{1}+2\ R_{2}}{V}}=n\ {\frac {R}{V}},$

or $R_{1}+2\ R_{2}=n\ (R_{1}+R_{2}),$ from which

If we apply this to the case of a body obeying the normal law of skin-friction we have $n=1.5,$ or ${\frac {R_{2}}{R_{1}}}={\frac {.5}{.5}}=1,$ that is to say, the energy expended dynamically is equal to that expended in viscosity.

When the conditions are such that turbulence supervenes the expenditure of energy dynamically in the fluid disproportionately increases and consequently $R_{1}$ becomes greater than $R_{2},$ and in accordance with (9) the value of $n$ rises, until for very high velocities it approximates more and more closely to 2, when the law becomes more nearly $R$ varies as $V^{2}.$

The foregoing applies not only to the resistance of a plane moving tangentially through a fluid but to all cases of submerged fluid resistance; but at present the changes of the value of the index $n$ have been but imperfectly investigated.

 

§ 41. Resistance-Velocity Curve.—Let us suppose that a curve $a,\ a,\ a,\ a$ (Fig. 28) represents by its ordinates the resistance of a body of some particular geometrical form for different values of $V$ (abscissae), which we may suppose have been determined experimentally; then if $b,\ b,\ b,\ b$ be the curve for some other body of the same geometrical form but of different linear proportions, we shall, by the law of corresponding speeds, have for every given value of $R,\ V\propto {\frac {1}{l}},$ that is to say, the proportion $a\ c/b\ c$ is everywhere constant and the two curves are similar in relation to the axis of $y.$ Also we have the relation $a\ c/b\ c$ in the inverse ratio of the respective linear dimensions, so that a single curve may be employed to represent the velocity-resistance

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