Parameter-free modelling of a turbulent differentially heated cavity with Rayleigh number up to 1011

Abstract

Since direct numerical simulations of natural convection in a differentially heated cavity cannot be performed 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 nonlinearity. The regularization method basically alters the convective terms 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 by means of vortex stretching in an unconditional 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 we propose to determine the filter length dynamically with the requirement that the vortex stretching must be stopped at the scale set by the grid. Finally, the proposed parameter-free regularization model is successfully tested for a turbulent natural convection flow in an air-filled differentially heated cavity of aspect ratio 4 with Rayleigh number up to 10 11.