Large-eddy simulation of Rayleigh-B\'enard convection at extreme Rayleigh numbers

We adopt the stretched spiral vortex sub-grid model for large-eddy simulation (LES) of turbulent convection at extreme Rayleigh numbers. We simulate Rayleigh-B\'enard convection (RBC) for Rayleigh numbers ranging from $10^6$ to $10^15$ and for Prandtl numbers 0.768 and 1. We choose a box of dimensions 1:1:10 to reduce computational cost. Our LES yields Nusselt and Reynolds numbers that are in good agreement with the direct-numerical simulation (DNS) results of Iyer et al. (Proc. Natl. Acad. Sci., vol 117 (14), 2020, pp. 7594-7598), albeit with a smaller grid size and at significantly reduced computational expense. For example, in our simulations at $Ra = 10^13$, we use grids that are 1/120 times the grid-resolution as that of the DNS (Iyer et al., Proc. Natl. Acad. Sci., vol 117 (14), 2020, pp. 7594-7598). The Reynolds numbers in our simulations span 3 orders of magnitude from 1,000 to 1,700,000. Consistent with the literature, we obtain scaling relations for Nusselt and Reynolds numbers as $Nu \sim Ra^{0.317}$ and $Re \sim Ra^{0.497}$. We also perform LES of RBC with periodic side-walls, for which we obtain the corresponding scaling exponents as 0.343 and 0.477 respectively. Our LES is a promising tool to push simulations of thermal convection to extreme Rayleigh numbers, and hence enable us to test the transition to ultimate convection regime.