A Chebyshev collocation algorithm for 2-D non-boussinesq convection

A Chebyshev collocation algorithm is developed to integrate the time-dependent Navier-Stokes equations for natural convection flow with large temperature differences. The working fluid is assumed to be a perfect gas and its thermophysical properties vary with temperature according to Sutherland laws. The governing equations do not allow for acoustic waves. The generalized Helmholtz and Uzawa operators which arise from time discretization are solved iteratively and the performances of several types of preconditioners and iterative schemes are examined. The algorithm is validated by computing almost Boussinesq flows and by comparing with previous results obtained with a finite difference algorithm. We investigate the effects of the temperature difference and of total mass contained within the cavity on the transition to unsteadiness in a cavity of aspect ratio 8. It is shown that these parameters have, indeed, a significant effect on the value of Rayleigh number at which unsteadiness is triggered. We also discuss the nature of the time-periodic solution which is obtained for Ra slightly supercritical values.