An Implicit Scheme for the Numerical Simulation of Time-dependent Convection in Side-heated Cavities

The time-dependent, buoyancy-driven flow in cavities heated and cooled from the side, with adiabatic horizontal walls, is investigated with an implicit, two-dimensional finite volume scheme. The numerical algorithm integrates the Navier-Stokes equations, in primitive form, with a second-order accurate time-stepping scheme, and it is characterized by absolute, to machine accuracy, mass conservation. The results are presented for a Prandtl number, Pr, of 0.005, and corresponding values of Grashof number, Gr, of 1x10^6 and 3x10^6 and a value of Pr of 0.71, with a value of the Rayleigh number, Ra, of 2x10^9. For the results with Pr = 0.005, the effects of time-stepping order and spatial and temporal resolutions are identified and discussed. For the Pr = 0.71 case, the results are qualitatively compared with spectral-code predictions with favorable agreement. It is concluded that first-order time-stepping schemes should not be used when dealing with time-dependent buoyant flows, and that the proposed algorithm is capable of performing accurate simulations of unsteady, and possibly chaotic, convective flows in side-heated cavities.