Tailoring boundary geometry to optimize heat transport in turbulent convection

Turbulent Rayleigh-B\'enard convection between planar horizontal boundaries is a classical example of the challenge posed by multiple interacting scales in fluid dynamics. Here, by tailoring the geometry of the upper boundary we manipulate the boundary layer -- turbulent interior flow interaction, and study the turbulent transport of heat in two-dimensional Rayleigh-B\'enard convection with numerical simulations using the Lattice Boltzmann method. By fixing the roughness amplitude of the upper boundary and varying the wavelength $\lambda$, we find that the exponent $\beta$ in the Nusselt-Rayleigh scaling relation, $Nu-1 \propto Ra^\beta$, is maximized at $\lambda \equiv \lambda_{\text{max}} \approx (2 \pi)^{-1}$, but decays to the planar value in both the large ($\lambda \gg \lambda_{\text{max}}$) and small ($\lambda \ll \lambda_{\text{max}}$) wavelength limits. The changes in the exponent originate in the nature of the coupling between the boundary layer and the interior flow. We present a simple scaling argument embodying this coupling, which describes the maximal convective heat flux. Results from simulations with both top and bottom rough boundaries showing a further enhancement of heat transport will also be presented.