Longitudinal Versus Lateral Eddy Length-Scale (Second Note on a Vorticity-Transfer Hypothesis of Turbulence Theory)

Abstract

A previously introduced vorticity-transfer-and-adaptation hypothesis is briefly summarized and supplemented by consideration of certain spatial derivatives of the components of the eddy displacement vector. The conventional length-scale of turbulence, redefined in the preceding note as a lateral co-variance = (x?z??)½, is supplemented by the new concept of a longitudinal length-scale, defined as the variance L = (x?2?)½). In correspondence to the Karman constant (which concerns the rate of change of l in the direction lateral to the mean flow) there is a new constant which concerns the rate of change of L in the longitudinal direction. It is proposed to refer to this as the Reichardt constant, because the supplements are of direct importance for turbulence in free flows (such as submerged jets), and because the new hypothesis offers an improvement in comparison with ?Reichardt's momentum transfer law? or ?inductive theory? of jet-diffusion. It is shown that the structure of mean profiles in two-dimensional jets can be very satisfactorily predicted by the new theory. The empirically known discrepancy between lateral distributions of heat and momentum is now definitely explained as the result of vorticity transfer; specifically, by the fact that momentum follows from vorticity transfer, and therefore does not diffuse in the same manner as a scalar property such as heat per unit mass. The ratio of eddy diffusivities for heat and momentum in the two-dimensional jet has the minimum value of 4/3 at the axis. Finally, profiles of cross-stream velocity components in free flow are discussed and a new explanation for an old and thus far unresolved discrepancy between empirical and theoretical results is given.