Natural convection in a horizontal fluid layer with a periodic array of internal square cylinders - Need for very large aspect ratio 2D domains

We consider the problem of natural convection in a horizontal layer of fluid, bounded between two infinite parallel plates. The present configuration includes a periodic array of square cylinders embedded within the layer, which introduces an externally imposed horizontal length scale. The central question we address here is the effect of the horizontal extent of the computational domain (and the number of square cylinders contained within the domain) on the flow dynamics and overall heat transfer. Computations that use small aspect ratio domains, which contain only one or two of the square cylinders, severely constrain the plume dynamics. Only large aspect ratio computational domains (AR ≥ 10) allow multiple plumes and their complex dynamics, and begin to approach the results of an infinite layer. Here at a modest Rayleigh number of 106, where the flow is unsteady, we systematically vary the horizontal extent of the computational domain by progressively including more of the internal periodic square cylinders. The approach to approximating an infinite layer is observed to be quite slow and even with wide aspect ratio computational domains that include as many as 12 internal periodic square cylinders the asymptotic state is not closely approximated. The need for very large aspect ratio computational domain in such problems is clear