Parameter-free numerical method for modeling thermal convection in square cavities in a wide range of Rayleigh numbers

Some computational results for the two and three-dimensional Davis benchmark are presented. This benchmark represents thermal convection in a square (cubical) cavity with vertical active walls in a wide range of Rayleigh numbers (10 4 to 10 14), which covers both laminar and highly turbulent flows. A turbulence model with parameters that depend on a Rayleigh number and require adjustment is usually used to describe turbulent flows. An alternative is Direct Numerical Simulation (DNS) methods, but they demand extremely large computational grids. Recently there has been an increasing interest in DNS methods with incomplete resolution, which are able to provide in some cases acceptable results without resolving Kolmogorov scales. On the basis of such an approach the so-called parameter-free computational techniques have been developed. These methods cover wide range of Rayleigh numbers and allow computing various integral properties of heat transport on relatively coarse computational grids. In this paper, a new numerical method based on the CABARET scheme is proposed for solving Navier-Stokes equations with Boussinesq approximation. This turbulent model-free technique includes no additional parameters and has a second-order approximation scheme in time and space on uniform and non-uniform computational grids with minimal computational stencil. Testing of the technique on the Davis benchmark and the sequence of refined grids shows that the method allows one to compute integral heat fluxes with a high degree of accuracy both for laminar and highly turbulent flows. For the Rayleigh numbers up to 10 14, a several percent accuracy has been achieved on an extremely coarse grid consisting of 20×20 cells refined toward boundary. There is no a definite and comprehensive explanation of this computational phenomenon. Cautious optimism exists regarding the perspectives of using the new method of thermal convection computations for low Prandtl numbers typical of liquid metals.