Roughness as a Route to the Ultimate Regime of Thermal Convection

We use highly resolved numerical simulations to study turbulent Rayleigh-Bénard convection in a cell with sinusoidally rough upper and lower surfaces in two dimensions for $\mathrm{Pr}=1$ and $\mathrm{Ra}=[4\times 10^6,3\times 10^9]$. By varying the wavelength $\lambda$ at a fixed amplitude, we find an optimal wavelength ${\lambda }_{\mathrm{opt}}$ for which the Nusselt-Rayleigh scaling relation is $(\mathrm{Nu}-1\propto {\mathrm{Ra}}^{0.483})$, maximizing the heat flux. This is consistent with the upper bound of Goluskin and Doering [J. Fluid Mech. 804, 370 (2016)] who prove that Nu can grow no faster than $\mathcal{O}({\mathrm{Ra}}^{1/2})$ as $\mathrm{Ra}\to \infty$, and thus with the concept that roughness facilitates the attainment of the so-called ultimate regime. Our data nearly achieve the largest growth rate permitted by the bound. When $\lambda \ll {\lambda }_{\mathrm{opt}}$ and $\lambda \gg {\lambda }_{\mathrm{opt}}$, the planar case is recovered, demonstrating how controlling the wall geometry manipulates the interaction between the boundary layers and the core flow. Finally, for each Ra, we choose the maximum Nu among all $\lambda$, thus optimizing over all $\lambda$, to find ${\mathrm{Nu}}_{\mathrm{opt}}-1=0.01\times {\mathrm{Ra}}^{0.444}$.

Figure 1

The geometry of the rough surfaces and the equations of motion for our two-dimensional rectangular cell with $\mathrm{\Gamma }=2$.

Figure 2

The exponent in the scaling law $\mathrm{Nu}-1=A\times {\mathrm{Ra}}^{\beta }$ as a function of roughness wavelength $\lambda$ (here, we used $\lambda =0.03$, 0.05, 0.1, 0.154, 0.2, 0.286, 0.4, 0.5, 0.67 and 1.0.) Data from simulations are the circles and the line is a fit using $\beta =0.54x^{1.17}{e}^{-x}+0.28$, where $x=\lambda /{\lambda }_{\mathrm{opt}}$. At ${\lambda }_{\mathrm{opt}}=0.1$, we find a maximum ${\beta }_{\max }=0.483$. For $\lambda =1$, $\beta$ is slightly larger than 0.28 because of finite-size effects. See also Fig. 2 of the Supplemental Material [48].

Figure 3

Scaling relations for different $\lambda$. (a) Nu-Ra scaling relations for $\lambda ={\lambda }_{\mathrm{opt}}=0.1$. The linear least-squares fit is $\mathrm{Nu}-1=0.0043\times {\mathrm{Ra}}^{0.482}$. The dash-dotted line is the scaling fit $\mathrm{Nu}-1=0.034\times {\mathrm{Ra}}^{0.359}$ for single rough wall of $\lambda =0.154$ [40]. (b) The ($\lambda$, $\beta$) pairs in the order of increasing slope are (1.0, 0.296), (0.5, 0.319), (0.286, 0.393), and (0.1, 0.482). The remaining pairs [not shown in Fig. 3] are (0.03, 0.375), (0.05, 0.435), (0.154, 0.461), (0.2, 0.434), (0.4, 0.345), and (0.67, 0.297). The black line is the upper envelope as described by ${\mathrm{Nu}}_{\mathrm{opt}}-1=0.01\times {\mathrm{Ra}}^{0.444}$. See also Figs. 1 and 2 of the Supplemental Material [48].

Figure 4

A snapshot of the temperature field for $\lambda =0.1$ and $\mathrm{Ra}=2\times 10^9$. To see the effects of roughness, the flow field here can be contrasted with that in the smooth case studied by Johnston & Doering [28]. See also Fig. 3 of [40], which shows the transition from the planar to the rough flow field in the case of one rough wall.