Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/157

the turbulence as initially taking the form of a series of vortex filaments following each other in rapid succession and acting as rollers between the live fluid and dead water (on this point Kelvin does not differ materially from Helmholtz), and if we represent the resulting system of flow as in Fig. 56, in which the motion is given diagrammatically relatively to the vortex rollers, so that the apparent motion of the fluid on the two sides is opposite, then, making certain assumptions, we can obtain some results from dimensional theory.

Let it be granted that for different values of $V$ and $\nu$ the size of the individual rollers may vary, but the form of the disturbance is homomorphous.

Let $\omega$ be the angular velocity of a roller taken at some stated point on some specified line of flow; then,

	$\omega =(F)\ V,\ \nu .$
As in § 38, let us write	$\omega ^{x}=V^{y},\ \nu ^{z}.$
Dimensionally	${\frac {1}{T^{x}}}={\frac {L^{y}}{T^{y}}}\ {\frac {L^{2z}}{T^{z}}}$
Thence we have	$x=y+z,$
and	$y={\boldsymbol {-}}2\ z.$

Taking $x=1,$ we obtain $y=2;\;z={\boldsymbol {-}}1.$

Hence the expression becomes $\omega =V^{2}\ \nu ^{-1}\ \times$ constant, or $V^{2}=\omega \nu \ \times$ constant. Taking $r$ for the radius of the roller at the point chosen, we can write this expression in the form,—

$r={\frac {\nu }{V}}\ \times$ constant.^[1]

§ 106. Conclusions from Dimensional Theory.—From the above expression the following conclusions may be drawn:—

In different fluids ceteris paribus the size (diameter) of the rollers will vary directly as the kinematic viscosity. Hence in an inviscid fluid the rollers will become of vanishingly small diameter, or the surface containing them will be a surface of gyration of Helmholtz, that is a surface of discontinuity.

1. This is, as it evidently should be, the same expression as determined generally for homomorphous motion in § 38, $r$ being the linear dimension.

137