Multigrid finite difference computation of natural convection past sharp-edged horizontal cylinders

Abstract

Laminar natural convection over sharp-edged horizontal isothermal objects has been studied numerically for a range of parametric values of interest. Finite-difference solutions to the Boussinesq equations were obtained for three different isothermal two-dimensional configurations placed in an infinite fluid medium: a finite flat plate, a square bar and an external right corner. In addition, the natural convection heat transfer in an enclosure between an inner square bar and an outer concentric circular cylinder was considered to study the Rayleigh number effect and the possibility of flow separation past the sharp edges. Difficulty associated with the complex physical flow domain was overcome by applying the body-fitted coordinate transformation as well as a variable mesh system. The basic numerical methods used in the present work were the alternating direction implicit (ADI) method and the successive overrelaxation (SOR) method. In order to accelerate the finite difference computation, a multigrid method applied to the ADI scheme was developed. The present multigrid method was first tested on a linear elliptic partial differential equation as well as on a cavity flow with incompressible Novier-Stokes equations. Throughout these numerical experiments the present method turned out to be robust. The optimum time step on each grid level for the linear problem was obtained by a stability analysis. Extension of the method to the natural convective flow in the enclosure using body-fitted coordinates was carried out successfully. In the natural convective flow on a flat plate in an infinite fluid medium with Rayleigh number in the range $10^3$<Ra<$10^5$, no indication of flow separation was found. For the square bar, however, the boundary layer separated easily at the upper sharp edges for Ra>5×$10^3$. In this case, well-defined twin vortices exist above the upper horizontal surface. Comparison of the numerical results with the earlier experimental data obtained un...