Mathematical modeling for low-Mach-number compressible thermal convection analysis in a square enclosure

Abstract

We derived a set of low-Mach-number compressible fluid-flow equations for analyzing thermal convection under large temperature differences and performed corresponding numerical simulations. This research has two key aspects. First, we propose a formulation for the perturbation of the equation of state. When considering the perturbation of the equation of state, we investigate the effects of a term that appears under high-temperature-difference conditions. Furthermore, we evaluate the analysis performance of the proposed formulation under the conventional low-temperature-difference conditions. Second, we perform calculations using a fully explicit algorithm, regardless of the temperature difference. This method eliminates the need for matrix calculations by artificially increasing the Mach number for the perturbation of the equation of state, thereby improving computational efficiency. This artificially adjusted Mach number is directly correlated with the calculation efficiency. However, since the maximum Mach number at which a physically appropriate flow field can be obtained depends on the computational conditions, the Mach number is also discussed. The present study confirms that the higher the temperature difference, the higher the adjustable range of the Mach number. Conversely, the lower the temperature difference, the lower the Mach number range, and the more limited the time step size. Furthermore, we confirmed that the temperature-difference conditions determine the efficiency of the incompressible fluid-flow analysis, encompassing solving the conventional pressure Poisson equation. Additionally, as in many studies, in natural convection in a square enclosure with a high-temperature-difference, the flow field exhibited an asymmetric distribution that differed from that obtained by the Boussinesq approximation.