Two-dimensional numerical computation for Rayleigh–Bènard convection with both the Navier–Stokes equation and the Boltzmann equation

Abstract

This paper compares the numerical solutions for Rayleigh–Bènard convection of rarefied air from the Boltzmann equation and from the Navier–Stokes equation. The Boltzmann equation was computed using the direct simulation Monte Carlo (DSMC) method. The maximum collision-number method was used to calculate collisions between molecules. The computation was carried out at the Knudsen number (Kn) = 0.029, 0.02, 0.01 and 0.005, and the equivalent Rayleigh number (Ra) = 2990. For the Navier–Stokes equation, the finite difference method was used to discretize governing equations, and the highly simplified marker-and-cell scheme was adopted to solve these equations. Computed profiles of the velocity and the temperature by the DSMC method approached those given by the Navier–Stokes equation as Kn decreased. The centre of the vortex approached the centre of the domain, in agreement with that obtained by the Navier–Stokes equation. The normalized temperature at the middle height and that averaged for the whole domain coincided with each other as Kn decreased and approached 0.5, which is equivalent to the value given by the Navier–Stokes equation.