Aspect ratio dependence of heat transport by turbulent Rayleigh–Bénard convection in rectangular cells

Abstract

AbstractWe report high-precision measurements of the Nusselt number $Nu$ as a function of the Rayleigh number $Ra$ in water-filled rectangular Rayleigh–Bénard convection cells. The horizontal length $L$ and width $W$ of the cells are 50.0 and 15.0 cm, respectively, and the heights $H= 49. 9$, 25.0, 12.5, 6.9, 3.5, and 2.4 cm, corresponding to the aspect ratios $({\Gamma }_{x} \equiv L/ H, {\Gamma }_{y} \equiv W/ H)= (1, 0. 3)$, $(2, 0. 6)$, $(4, 1. 2)$, $(7. 3, 2. 2)$, $(14. 3, 4. 3)$, and $(20. 8, 6. 3)$. The measurements were carried out over the Rayleigh number range $6\ensuremath{\times} 1{0}^{5} \lesssim Ra\lesssim 1{0}^{11} $ and the Prandtl number range $5. 2\lesssim Pr\lesssim 7$. Our results show that for rectangular geometry turbulent heat transport is independent of the cells’ aspect ratios and hence is insensitive to the nature and structures of the large-scale mean flows of the system. This is slightly different from the observations in cylindrical cells where $Nu$ is found to be in general a decreasing function of $\Gamma $, at least for $\Gamma = 1$ and larger. Such a difference is probably a manifestation of the finite plate conductivity effect. Corrections for the influence of the finite conductivity of the top and bottom plates are made to obtain the estimates of $N{u}_{\infty } $ for plates with perfect conductivity. The local scaling exponents ${\ensuremath{\beta} }_{l} $ of $N{u}_{\infty } \ensuremath{\sim} R{a}^{{\ensuremath{\beta} }_{l} } $ are calculated and found to increase from 0.243 at $Ra\simeq 9\ensuremath{\times} 1{0}^{5} $ to 0.327 at $Ra\simeq 4\ensuremath{\times} 1{0}^{10} $.