Numerical study of natural convective heat transfer with large temperature differences

Abstract

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. No Boussinesq or low‐Mach number approximations of the Navier‐Stokes equations are used. Results for air are presented. The ideal‐gas law is used and viscosity is given by Sutherland’s law. An accurate low‐Mach number solver is developed. Here 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 the acceleration technique. Owing to the implicit treatment of the acoustic and the diffusive terms, the stiffness otherwise caused by high aspect ratio cells is removed. Low Mach number stiffness is treated by a preconditioning technique. By a combination of the preconditioning technique, the semi‐implicit discretization and the multigrid formulation 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 a variety of Rayleigh numbers.