Turbulent convection against rough walls: Manipulating a broken symmetry

We present results from well resolved numerical simulations of turbulent convection in a cell with rough walls in two dimensions. In order to examine hypotheses regarding the interaction between the boundary layers and the interior of the flow, we study classical Rayleigh-Benard convection with the top plate having a sinusoidal roughness distribution. The amplitude of the roughness is such that it is always larger than the thermal boundary layer thickness for all Rayleigh numbers ($Ra$) considered here. The lattice Boltzmann method is used to model the Navier-Stokes and heat transport equations within the Boussinesq approximation. We observe a scaling law, for the Nusselt number ($Nu$), $Nu \sim Ra^{1/3}$ over three decades in $Ra$, from $Ra = 10^6$ to $Ra = 10^9$, at a Prandtl number of $1$. The scaling law obtained is in good agreement with recent experiments. We discuss the effects of the additional top-down broken symmetry on the mean temperature in the core-flow region, the plumes generated at the rough surface, and the two-point temperature correlation function at different heights. It is found that the correlation length scales as the wavelength of the roughness distribution.