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

Abstract

Some numerical results for the two- and three-dimensional de Vahl Davis benchmark are presented. This benchmark describes thermal convection in a square (cubic) cavity with vertical heated walls in a wide range of Rayleigh numbers (104 to 1014), which covers both laminar and highly turbulent f lows. Turbulent f lows are usually described using a turbulence model with parameters that depend on the Rayleigh number and require adjustment. 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 an incomplete resolution, which, in some cases, are able to provide acceptable results without resolving Kolmogorov scales. On the basis of this approach, the so-called parameter-free computational techniques have been developed. These methods cover a 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 the Navier–Stokes equations in the Boussinesq approximation. This technique does not involve a turbulence model or any tuning parameters and has a second-order approximation scheme in time and space on uniform and nonuniform grids with a minimal computational stencil. Testing the technique on the de Vahl Davis benchmark and a sequence of refined grids shows that the method yields integral heat f luxes with a high degree of accuracy for both laminar and highly turbulent f lows. For Rayleigh numbers up to 1014, a several percent accuracy is achieved on an extremely coarse grid consisting of 20 × 20 cells refined toward the boundary. No definite or comprehensive explanation of this computational phenomenon has been given. Cautious optimism is expressed regarding the perspectives of using the new method for thermal convection computations at low Prandtl numbers typical of liquid metals.