Ultimate State of Thermal Convection

The ultimate regime of thermal convection, the so-called Kraichnan regime [R. H. Kraichnan, Phys. Fluids 5, 1374 (1962)], hitherto has been elusive. Here numerical evidence for that regime is presented by performing simulations of the bulk of turbulence only, eliminating the thermal and kinetic boundary layers and replacing them with periodic boundary conditions.

Figure 1

(a) $\mathrm{N}\mathrm{u}$ (squares, lower curve) and $\mathrm{R}\mathrm{e}$ (circles, upper curve) as a function of $\mathrm{R}\mathrm{a}$ from our simulation. The error bars are based on the variances of measured quantities which have been obtained by splitting the total time series into smaller pieces. Note that in this way we could also check that our simulations are statistically stationary. The straight lines are linear regressions giving $\mathrm{N}\mathrm{u}\sim {\mathrm{R}\mathrm{a}}^{0.51\pm 0.06}$ and $\mathrm{R}\mathrm{e}\sim {\mathrm{R}\mathrm{a}}^{0.55\pm 0.02}$. (b) Same as in (a), but now compensated by ${\mathrm{R}\mathrm{a}}^{1/2}$.