A multigrid method for natural convective heat transfer with large temperature differences

Steady-state two-dimensional solutions to the full compressible Navier–Stokes equations are computed for laminar convective motion of a gas in a square cavity with large horizontal temperature differences. Results for air are presented. The ideal-gas law is used and viscosity is given by Sutherlandʹs law. An accurate low-Mach number solver is developed. Hereby an explicit third-order discretization for the convective part and a line-implicit central discretization for the acoustic part and for the diffusive part are used. The semi-implicit line method is formulated in multistage form. Multigrid is used as acceleration technique. A convergence behaviour is obtained which is independent of grid size, grid aspect ratio, Mach number and Rayleigh number. Grid converged results are shown for Rayleigh numbers between 102 and 107.