Petrov-Galerkin Finite Element Approach Using Exponential Test Functions for Three-Dimensional Natural Convection in Cubic Enclosure.

Abstract

Recently, we presented a finite element scheme based on the Petrov-Galerkin weak formulation using exponential test functions, for solving the unsteady two-dimensional natural convection up to high Rayleigh numbers. The incompressible Navier-Stokes equations and energy equation are discretized by a semi-explicit scheme in which velocity and temperature are treated explicitly and pressure is treated implicitly with respect to the time variable. As the time-marching scheme, the fractional step method is also adopted in the work. The resulting numerical solutions demonstrate that the method is capable of solving the set of equations accurately and in a stable manner up to high Rayleigh numbers. The purpose of this paper is to extend the Petrov-Galerkin finite element method using exponential test functions, to three-dimensional natural convection problems. The validity of the present method is shown for unsteady flows in a cubic enclosure through comparison with other existing numerical solutions.