Analysis of unsteady heat transfer by natural convection in a two-dimensional square cavity using a high order finite-volume method.

Abstract

Unsteady heat transfer by natural convection in a closed square cavity is investigated numerically. A new finite-volume approach is developed and applied to the two-dimensional continuity, vorticity, and energy equations. The variation of the field variables is approximated by bi-quadratic interpolation formulas over the space occupied by the finite volume and the region surrounding it. These are used in the integral conservation laws for energy, vorticity and mass. The convective transport is modelled using a new upstream-weighting approach which uses volume averages for the vorticity and the energy transported across the boundaries of the finite volume. The weighting is dependent on the skewness of the velocity field to the surfaces of the finite volume as well as its strength. It is adaptive to local flow conditions. The velocities are obtained from the application of the velocity induction law. Use is made of an image system for the free vorticity of fluid. In this way, the no-penetration condition is enforced at the cavity boundaries, but at the same time it may allow a slip condition to exist. This is not permitted in a viscous flow analysis, and the slip velocity is reduced to zero by the production of free vorticity at the boundaries. Two test cases are treated which have exact solutions. The first is not new and involves a rotating shaft. The errors are less than.06% for this case. The second case is new and involves convection past a source and sink. The maximum error is 2.3%. For both test cases, the maximum error occurs at moderate values of the cell Peclet number and diminishes at the extreme low and high values. The time-development of the profiles of the vorticity, horizontal velocity, and temperature is examined at different locations within the cavity for Rayleigh numbers equal to 10³, 10⁴, and 10⁵. For these calculations, a 21 x 21 grid was used. The flow is found to approach a steady-state condition. The steady-state results are compared with a benchmark solution. In general, the agreement is excellent. The discrepancy is found to be less than 2% for the vast majority of the results for this relatively coarse grid.