Research
Ultimate Scaling
Asymptotic Derivation
Under Kraichnan-Spiegel theory, without the logarithmic corrections of Kraichnan, the scaling argument associated with it is quite elegant. Since it is assumed that the BLs no longer control transport, one can consider Nu for the full vertical length scale, \(L\), of the convective cell while neglecting the conductive term in the total heat transport. This means,
\[\mathrm{Nu} \sim \frac{\rho c_{p}\left\langle\overline{w^{\prime} \theta}\right\rangle}{k \Delta T / L}=\frac{\left\langle\overline{w^{\prime} \theta}\right\rangle}{\kappa \Delta T / L}\]
Thus, under proper scaling for turbulent eddies, it follows that
\[\mathrm{Nu} \sim \frac{U L \Delta T}{\kappa \Delta T}=\frac{L \sqrt{g \alpha \Delta T L}}{\kappa}=\operatorname{Pr}^{1 / 2} \operatorname{Ra}^{1 / 2}\]
Work by Kraichnan introduced logarithmic corrections for this scaling law of the form \(\mathrm{Nu} \sim \mathrm{Ra}^{1 / 2}(\log \mathrm{Ra})^{-3 / 2}\).