Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity

TLDR

External large‑scale forcing of a passive scalar in a turbulent fluid generates small‑scale variations of that scalar. The study theoretically investigates the high‑wave‑number spectrum of a passive scalar, accounting for convection and molecular diffusion, and connects the results to Townsend’s vorticity model. Using Kolmogorov‑type hypotheses for small‑scale velocity fluctuations, the authors derive the scalar spectrum by incorporating convection and diffusion effects. Both analytical approaches yield the same power‑law spectrum for θ, as given in equation (4.8).

Abstract

When some external agency imposes on a fluid large-scale variations of some dynamically passive, conserved, scalar quantity θ like temperature or concentration of solute, turbulent motion of the fluid generates small-scale variations of θ. This paper describes a theoretical investigation of the form of the spectrum of θ at large wave-numbers, taking into account the two effects of convection with the fluid and molecular diffusion with diffusivity k. Hypotheses of the kind made by Kolmogoroff for the small-scale variations of velocity in a turbulent motion at high Reynolds number are assumed to apply also to small-scale variations of θ.Previous contributions to the problem are reviewed. These have established that the spectrum of θ varies as , the result being given by (4.8). The same result is obtained, using essentially the same approximation about the velocity field, from a different kind of analysis in terms of velocity and θ correlations. Finally, the relation between this work and Townsend's model of the small-scale variations of vorticity in a turbulent fluid is discussed.