Natural convection in a differentially heated corner region

This is a fundamental study of the phenomenon of natural convection in the region formed by a vertical warm wall rising above a cold horizontal wall, or in the region between a cold vertical wall extending downward from a warm horizontal surface. The study consists of scale analysis, numerical simulations, and an asymptotic solution of the low Rayleigh number limit. The scale analysis predicts the persistence of a single cell in the corner region, regardless of Rayleigh number. The cell migrates toward the corner as the Rayleigh number Ra_H increases: the flow rate and the net heat transfer rate vary as Ra^1/7_H. The scale analysis is verified qualitatively and quantitatively by means of numerical experiments in the range Ra_H =10³–10⁷, Pr=0.7–7, H/L=1–4, where Pr is the Prandtl number and H/L is the height/length ratio of the corner region. Additional numerical simulations are presented for cases where one or both walls have uniform heat flux; in these cases, the heat transfer rate shows nearly the same behavior as when the corner walls are both isothermal. The asymptotic solution for the Ra_H→0 limit shows that the flow field is relatively insensitive to whether the wall temperature varies continuously or discontinuously through the corner point.