Dynamical Determination of the Scales of Turbulence.

Abstract : Integral scales for fully developed turbulent flows may be defined as moments of the two-point tensor correlations among the dynamical variables which characterize the flow, for example velocity and temperature. We show in this paper that this concept of integral scales allows for a rigorous derivation of rate equations for the integral scales (at a single point) if two major assumptions are made. The first assumption is that the integral scales are sufficiently small (compared with the scale of spatial variations of the mean quantities) to enable one to neglect the difference between convection at two points separated by less than the integral scale itself. This assumption is necessary to obtain convections of the integral scale (which is a one-point function) at a single point and thus represents the most general physical requirement for the very existence of closed equations for the integral scales. The second major assumption that we find necessary for a rigorous derivation of rate equations for the integral scales stems from a coupling between the evolution of the Reynolds stress and other one-poing correlations to the evolution of the integral scale. We observe that we can obtain from the two-point correlations at every space point and at every instant of time, both the Reynolds stress (by collapsing the two points to one), and the integral scale (by a suitable integration in relative separation).