Variational Multiscale Large Eddy Simulations of Rayleigh-B\'{e}nard Convection at High $Ra$

Large eddy simulations (LES) of two- and three-dimensional Rayleigh-B\'{e}nard convection are performed up to $Ra=10^{14}$ for $Pr=1$ and $Pr=7$. These simulations are performed using novel LES models based on the variational multiscale (VMS) formulation. The new model is presented as a mixed model which combines the VMS formulation for Rayleigh-B\'{e}nard convection with an eddy viscosity model (EVM) to capture the effects of the Reynolds stresses in high $Ra$ convection. In the present work, the Wall-Adapting Local Eddy-viscosity (WALE) model is used as the EVM. The new models were implemented in the finite element code Drekar and simulations were performed using continuous, piecewise linear finite elements. The two dimensional simulations were performed in a domain with aspect ratio $2$ while the three-dimensional simulations were performed in a cylinder of aspect ratio $1/4$. Least squares fits to the $Nu$-$Ra$ data show $Nu_{_{2D}} = 0.127 Ra^{0.284}$ and $Nu_{_{3D}} = 0.104 Ra^{0.310}$.