Small-scale variation of convected quantities like temperature in turbulent fluid Part 2. The case of large conductivity

Abstract

The analysis reported in Part 1 is extended here to the case in which the conductivity κ is large compared with the viscosity ν, the conduction ‘cut-off’ to the θ-spectrum then being at wave-number (ε/κ³)^¼. It is shown, with a plausible and consistent hypothesis, that the convective supply of $\overline {\theta^2}$-stuff to Fourier components of θ with wave-numbers n in the range (ε/κ³)^¼ [Lt ] n [Gt ] (ε/ν³)^¼ is due primarily to motion on a length scale of order n^-1 acting on a uniform gradient of θ of magnitude $[(\overline {\nabla \theta)^2}]^{\frac {1}{2}}$. The consequent form of the theta;-spectrum within this same wave-number range is $\Gamma (n) = \frac {1}{3}C \chi \epsilon ^{\frac {2}{3}} k^{-3}n^ {-\frac {17} {3}}.$

The way in which conduction influences (and restricts) the effect of convection on the distribution of θ at these wave-numbers beyond the conduction cut-off is discussed.