On the Concept of Similiarity in the Theory of Isotropic Turbulence

Abstract

Much recent work has been done in the study of isotropic turbulence, particularly from the point of view of its spectrum. But the underlying concept is still the assumption of the similarity of the spectrum during the process of decay, which is equivalent to the idea of self-preservation of the correlation functions introduced by the senior author. It is however generally recognized that the correlation function does change its shape during the process of decay, and hence the concept of self-preservation or similarity must be interpreted with suitable restrictions. Under the limitation to low Reynolds numbers of turbulence, the original idea of Kármán-Howarth has been confirmed. Then the decay consists essentially of viscous dissipation of energy separately in each individual frequency interval. However, when turbulent diffusion of energy, i.e., transfer of energy between frequency intervals, occurs at a significant rate, the interpretation of the decay process and the spectral distribution is quite varied. This an be seen by a comparison of Heisenberg, Batchelor, Frenkiel, and the present authors. The purpose of the present paper is an attempt to clarify this situation.