Heat transport in liquid metal convection from onset to turbulence: the effect of small aspect ratio

We present a systematic study on the effects of small aspect ratios $\varGamma$ on heat transport in liquid metal convection with a Prandtl number of $Pr=0.029$ . The study covers $1/20\le \varGamma \le 1$ experimentally and $1/50\le \varGamma \le 1$ numerically, and a Rayleigh number $Ra$ range of $4\times 10^3 \le Ra \le 7\times 10^{9}$ . It is found experimentally that the local effective heat transport scaling exponent $\gamma$ changes with both $Ra$ and $\varGamma$ , attaining a $\varGamma$ -dependent maximum value before transition-to-turbulence and approaches $\gamma =0.25$ in the turbulence state as $Ra$ increases. Just above the onset of convection, Shishkina ( Phys. Rev. Fluids , vol 6, 2021, 090502) derived a length scale $\ell =H/(1+1.49\varGamma ^{-2})^{1/3}$ . Our numerical study shows $Ra_{\ell }$ , i.e. $Ra$ based on $\ell$ , serves as a proper control parameter for heat transport above the onset with $Nu-1=0.018(1+0.34/\varGamma ^2)(Ra/Ra_{c,\varGamma }-1)$ . Here $Ra_{c,\varGamma }$ represents the $\varGamma$ -dependent critical $Ra$ for the onset of convection and $Nu$ is the Nusselt number. In the turbulent state, for a general scaling law of $Nu-1\sim Ra^\alpha$ , we propose a length scale $\ell = H/(1+1.49\varGamma ^{-2})^{1/[3(1-\alpha )]}$ . In the case of turbulent liquid metal convection with $\alpha =1/4$ , our measurement shows that the heat transport will become weakly dependent on $\varGamma$ with $Ra_{\ell }\equiv Ra/(1+1.49\varGamma ^{-2})^{4/3} \ge 7\times 10^5$ . Finally, once the flow becomes time-dependent, the growth rate of $Nu$ with $Ra$ declines compared with the linear growth rate in the convection state. A hysteresis is observed in a $\varGamma =1/3$ cell when the flow becomes time-dependent. Measurements of the large-scale circulation suggest the hysteresis is caused by the system switching from a single-roll-mode to a double-roll-mode in an oscillation state.