Time evolution equation for advective heat transport as a constraint for optimal bounds in Rayleigh-Bénard convection

Upper bounds on the heat transport and other quantities of interest in Rayleigh-Bénard convection are derived in previous work from constraints resulting from the equations of time evolution for kinetic energy, the root mean square of temperature, and the temperature averaged over horizontal planes. Here we investigate the effect of a constraint derived from the time evolution equation for the advective heat transport. This additional constraint leads to improved bounds on the toroidal dissipation.

Figure 1

The bound $D$ as a function of $\mathrm{Pr}$ for $\mathrm{Ra}=1.6\times {10}^4$ (left panel) and $8.192\times {10}^6$ (right panel).

Figure 2

The Prandtl number ${\mathrm{Pr}}_a$ that has to be exceeded for constraint (10) to reduce the bound $D$, plotted as function of $\mathrm{Ra}$. The straight line is given by $1.18\times {\mathrm{Ra}}^{1/2}$.

Figure 3

$(D_0-D_{\infty })/D_0$ as a function of $\mathrm{Ra}$.

Figure 4

The integral $J=\int d^3\mathit{r}\left|{\partial }_{z}G(\mathit{r},{\mathit{r}}^{\prime })\right|$ as a function of $z^{\prime }$, the $z$ coordinate of ${\mathit{r}}^{\prime }$.

Figure 5

Contour plot of $g\left(\mathit{r}\right)$ in a cross section of the layer.