Finite element calculations of very high Rayleigh number thermal convection

Abstract

Summary. A finite element method with uniform and variable resolution meshes is used to model very high Rayleigh number Ra thermal convection in a square box of infinite Prandtl number, Boussinesq fluid with constant viscosity and thermodynamic properties. Heating is either entirely from below or mostly from within and the boundaries are stress free. The variable mesh is coarse in the interior of the convection cell and it is fine in the very thin boundary layers and plumes surrounding the core. The highest resolution variable mesh has a dimensionless grid spacing of 0.027 in the core and 0.0017 in the boundary layers. The boundary layers contain about 10 mesh points even at the highest values of Ra considered and are thus highly resolved. The variable mesh approach is shown to yield reliable simulations of convection as long as the aspect ratio of the most elongated boundary layer elements is not too large; values of about 4 to 6 work well. This aspect ratio also measures the increase in resolution in the boundary layers as compared with the central core. Steady single-cell rolls are computed for bottom heating and Ra up to 5 × 105 times the marginal instability value of the Rayleigh number Racr. One and two-cell roll solutions are calculated for f= 1, 0.8 and 0.6, where f is the fraction of the heat escaping through the top of the box that is generated internally. The values of Racr for f= 1, 0.8 and 0.6 are 1296, 1024 and 864, respectively. The largest of Ra/Racr at which unicellular convection is stable (steady) are approximately 390, 610 and 970, for f= 1, 0.8 and 0.6.