On the Transport of Vorticity in a Turbulent Field of Two Dimensional Diffusion

Abstract

Contents: § 1. introductory, & 2. the generation of vorticity and the transition from laminar to turbulent flow, §3. the freedom of eddying diffusion, § 4. the experimental fact supporting the existence of transverse resistance accompanying eddy motion, § 5. some criticisms on the theory of vorticitytransport in a turbulent field, § 6. the derivation of transverse force in a field of vorticity-transport, § 7. summary and concluding remarks. Abstract:-
In the second paragraph is discussed the problem of the generation of vorticity and the transition from laminar to to bulent flow.
In the well known theory of conservation of vorticity by Helmholtz four conditions are assum ed:
1. the fluid is free from viscosity.
2. the fluid is continuous in regard of velocity.
3. the fluid is under influence of force which has single valued potential.
4. the density of fluid is either uniform or a function of pressure only.
In ordinary fluid such as air or water vortices are commonly generated even when the conditions 3 and 4 are fulfilled. Hence in such cases the violence of the condition 1 or 2 must be responsible for the production of vortices. On this point there are at present two dominant theories:
One is the German school established by L. Prandtl and others in which the transition from laminar to turbulent flow or the generation of eddies is determined by the critical Reynolds number, the property of fluid being characterised by kinematic viscosity, but “es offen gelassen wird, ob sich die einzelnen Wirbel in der turbulenten Strömung entgegen der inneren Reibung durch Energieentnahme aus der Hauptströmung erhalten können, oder ob sie durch lokale äusseue Einflüsse, wie Wandrauhigkeit, immer erneut erregt werden (F. Noether)”.
The other is the Japaese school developed j by S. Fujiwhara and his collaborators, in which is advocated velocity discontinuity as necessary condition.
As already noticed by L. Prandtl and O. Tietjens it is necessary believed from theoretical stand-point of view to replace the “Differentialgleichungen” by “Differenzengleichungen” for the production of eddies and this coincides with the idea of discontinuity of the second kind termed by S. Fujiwhara. (Noether and Synge argue against this.) In the author's opinion the essential coincidence of the both theories is descriptively stated.
In the third paragraph the freedom of eddy motion is discussed. The measurement by Taylor and Fage shows the equipartition of eddying energy in a field free from boundary effect (Screse's observation contradicts to this result).
The above result is most important, because the transverse resistance derived by Sakakibara in mathematical form opposes against the theory of equipartition of eddying stress.
We find here that the transverse resistance is formed by the motion of bound eddies affected by the special configuration of boundary.
In the fourth paragraph are enumerated some observational facts supporting the existence of transverse resistance accompanying eddy motion.
In the polar front theory for cyclogenesis the two distinct air masses are in juxtaposition and the two counter currents characterising them are balanced by geostrophic force. As noticed by T. Bergeron and Swoboda the presence of bulge on a polar front is not sufficient in itself to bring about cyclogenesis, but the mechanism of cyclonic convergency is also difficult to explain by the theory of air mass.