Petrov-Galerkin Finite Element Analysis Using Exponential Functions for Natural Convection in a Square Cavity.

Abstract

In our previous work, a novel finite element scheme for solving the unsteady incompressible Navier-Stokes equations up to high Reynolds numbers was developed. This scheme is based on the Petrov-Galerkin formulation using exponential-type test functions. The numerical results obtained in the previous work demonstrate that the method is capable of solving the incompressible Navier-Stokes equations accurately and in a stable manner for a wide range of Reynolds numbers. In this paper, the Petrov-Galerkin finite element method using exponential functions is extended to natural convection in a two-dimensional enclosed square cavity. The unsteady incompressible Navier-Stokes equations and energy equation are discretized by a semi-explicit scheme with respect to the time variable. As the time-marching scheme, the fractional step method is used effectively in this study. The validity of the present method is shown for unsteady flows at high Rayleigh numbers through comparison with other existing numerical solutions.