Heat transfer in a quasi-one-dimensional Rayleigh–Bénard convection cell

Abstract

We report an experimental and numerical study on Rayleigh–Bénard convection in a slender rectangular geometry with the aspect ratio $\varGamma$ varying from 0.05 to 0.3 and a Rayleigh number range of $10^5\leqslant Ra\leqslant 3\times 10^9$ . The Prandtl number is fixed at $Pr=4.38$ . It is found that the onset of convection is postponed when the convection domain approaches the quasi-one-dimensional limit. The onset Rayleigh number shows a $Ra_c=328\varGamma ^{-4.18}$ scaling for the experiment and a $Ra_c=810\varGamma ^{-3.95}$ scaling for the simulation, both consistent with a theoretical prediction of $Ra_c\sim \varGamma ^{-4}$ . Moreover, the effective Nusselt–Rayleigh scaling exponent $\beta =\partial (\log Nu)/\partial (\log Ra)$ near the onset of convection also shows a rapid increase with decreasing $\varGamma$ . Power-law fits to the experimental and numerical data yield $\beta =0.290\varGamma ^{-0.90}$ and $\beta =0.564\varGamma ^{-0.92}$ , respectively. Near onset, the flow shows a stretched cell structure. In this regime, the velocity and temperature variations in a horizontal cross-section are found to be almost invariant with height in the core region of a slender domain. As the Rayleigh number increases, the system evolves from the viscous dominant regime to a plume-controlled one, a feature of which is enhancement in the heat transport efficiency. Upon further increase of $Ra$ , the flow comes back to the classical boundary-layer-controlled regime, in which the quasi-one-dimensional geometry has no apparent effect on the global heat transfer.