The search for slow transients, and the effect of imperfect vertical alignment, in turbulent Rayleigh–Bénard convection

Abstract

We report experimental results for the influence of a tilt angle $\beta$ relative to gravity on turbulent Rayleigh–Bénard convection of cylindrical samples. The measurements were made at Rayleigh numbers $R$ up to $10^{11}$ with two samples of height $L$ equal to the diameter $D$ (aspect ratio $\Gamma \,{\equiv}\, D/L \,{\simeq}\, 1$), one with $L \,{\simeq}\, 0.5$ m (the ‘large’ sample) and the other with $L \,{\simeq}\, 0.25$m (the ‘medium’ sample). The fluid was water with a Prandtl number $\sigma \,{=}\, 4.38$.

In contrast to the experiences reported by Chillà Eur. Phys. J. B, vol. 40, 2004 p. 223) for a similar sample but with $\Gamma \,{\simeq}\, 0.5$ ($D\,{=}\,0.5$ and $L\,{=}\,1.0$m), we found no long relaxation times. For $R\,{=}\,9.4\,{\times}\,10^{10}$ we measured the Nusselt number $ \cal N$ as a function of $\beta$ and obtained a small $\beta$ dependence given by ${\cal N}(\beta)\,{=}\,{\cal N}_0 [1-(3.1\,{\pm}\,0.1)\,{\times}\, 10^{-2}|\beta|]$ when $\beta$ is in radians. This reduction of $\cal N$ is about a factor of 50 smaller than the result found by Chillà et al. (2004) for their $\Gamma\,{\simeq}\,0.5$ sample.

We measured sidewall temperatures at eight equally spaced azimuthal locations on the horizontal mid-plane of the sample and used them to obtain cross-correlation functions between opposite azimuthal locations. The correlation functions had Gaussian peaks centred about $t_1^{cc} \,{>}\, 0$ that corresponded to half a turnover time of the large-scale circulation (LSC) and yielded Reynolds numbers $\hbox{\it Re}^{cc}$ of the LSC. For the large sample and $R \,{=}\, 9.4\,{\times}\, 10^{10}$ we found $\hbox{\it Re}^{cc}(\beta) \,{=}\, \hbox{\it Re}^{cc}(0)\,{\times}\, [1 + (1.85\,{\pm}\, 0.21) |\beta| - (5.9\,{\pm}\, 1.7) \beta^2]$. Similar results were obtained from the auto-correlation functions of individual thermometers. These results are consistent with measurements of the amplitude $\delta$ of the azimuthal sidewall temperature variation at the mid-plane that gave $\delta(\beta) \,{=}\, \delta(0)\,{\times}\, [1 + (1.84 \,{\pm}\, 0.45) |\beta| - (3.1 \,{\pm}\, 3.9) \beta^2]$ for the same $R$. An important conclusion is that the increase of the speed (i.e. of $\hbox{\it Re}$) of the LSC with $\beta$ does not significantly influence the heat transport. Thus the heat transport must be determined primarily by the instability mechanism operative in the boundary layers, rather than by the rate at which ‘plumes’ are carried away by the LSC. This mechanism is apparently independent of $\beta$.

Over the range $10^9 \,{\lesssim}\, R \,{\lesssim}\, 10^{11}$ the enhancement of $\hbox{\it Re}^{cc}$ at constant $\beta$ due to the tilt could be described by a power law of $R$ with an exponent of $-1/6$, consistent with a simple model that balances the additional buoyancy due to the tilt angle by the shear stress across the boundary layers. Even a small tilt angle dramatically suppressed the azimuthal meandering and the sudden reorientations characteristic of the LSC in a sample with $\beta \,{=}\, 0$. For large $R$ the azimuthal mean of the temperature at the horizontal mid-plane differed significantly from the average of the top- and bottom-plate temperatures due to non-Boussinesq effects, but within our resolution was independent of $\beta$.