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 106 to
1013 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=1013, 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 range from 1,000 to 200,000. Consistent with the
literature, we obtain scaling relations for Nusselt and Reynolds numbers as Nu∼Ra0.322 and Re∼Ra0.502. 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.Comment: 26 pages, 15 figures. Submitted to Journal of Fluid Mechanic