Comparisons of Galerkin and finite difference methods for solving highly nonlinear thermally driven flows
Abstract
The equations of motion for a high Prandtl number Boussinesq fluid in a square 2D cavity with sidewail heating and cooling and perfectly conducting end walls have been solved by means of Galerkin as well as ADI (alternatingdirectionimplicit) finite difference methods for Rayleigh numbers up to 8 × 10 ^{6} and two angles of tilt. The finite difference solutions for the conductive flux at the heated wall converge monotonically from above with increasing numbers of mesh points; whereas, the Galerkin solutions converge from below and undergo modest oscillations with long period as additional terms are included. The nearly quiescent core and associated hydrodynamic boundary layers are, for given numbers of mesh points/terms, better represented by the finite difference method. With increasing precision in the wall heat flux and/or shear, the computational costs for both methods become comparable; however, for errors in excess of 23%, the Galerkin method is more economical.
 Publication:

Journal of Computational Physics
 Pub Date:
 November 1974
 DOI:
 10.1016/00219991(74)900953
 Bibcode:
 1974JCoPh..16..271D
 Keywords:

 Conductive Heat Transfer;
 Equations Of Motion;
 Finite Difference Theory;
 Galerkin Method;
 Nonlinear Equations;
 Wall Flow;
 Boundary Layer Flow;
 Convective Flow;
 Elliptic Differential Equations;
 Partial Differential Equations;
 Prandtl Number;
 Rayleigh Number;
 Fluid Mechanics and Heat Transfer