Parameter-free symmetry-preserving regularization modeling of a turbulent differentially heated cavity

Since direct numerical simulations of buoyancy driven flows cannot be computed at high Rayleigh numbers, a dynamically less complex mathematical formulation is sought. In the quest for such a formulation, we consider regularizations (smooth approximations) of the non-linearity: the convective term is altered to reduce the production of small scales of motion by means of vortex stretching. In doing so, we propose to preserve the symmetry and conservation properties of the convective terms exactly. This requirement yielded a novel class of regularizations that restrain the convective production of smaller and smaller scales of motion in an unconditionally stable manner, meaning that the velocity cannot blow up in the energy-norm (in 2D also: enstrophy-norm). The numerical algorithm used to solve the governing equations preserves the symmetry and conservation properties too. In the present work, a criterion to determine dynamically the regularization parameter (local filter length) is proposed: it is based on the requirement that the vortex stretching must stop at the scale set by the grid. Therefore, the proposed method constitutes a parameter-free turbulence model. The resulting regularization method is tested for a 3D natural convection flow in an air-filled (Pr = 0.71) differentially heated cavity of height aspect ratio 4. Direct comparison with DNS results at Rayleigh number 6.4 × 10^8 ≤ Ra ≤ 10^11 shows fairly good agreement even for very coarse grids. Finally, the robustness of the method is tested by performing simulations with Ra up to 10^17. A 2/7 scaling law of Nusselt number has been obtained for the investigated range of Ra.