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 on heat transport in liquid metal convection with a Prandtl number of $Pr=0.029$ . The study covers 1/20≤ ≤ 1 experimentally and 1/50≤ ≤ 1 numerically, and a Rayleigh number $Ra$ range of 4× 10³ ≤ Ra ≤ 7× 10⁹ . It is found experimentally that the local effective heat transport scaling exponent γ changes with both $Ra$ and , attaining a -dependent maximum value before transition-to-turbulence and approaches γ =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 =H/(1+1.49 ⁻²)1/3 . Our numerical study shows Ra , i.e. $Ra$ based on , serves as a proper control parameter for heat transport above the onset with Nu-1=0.018(1+0.34/ ²)(Ra/Rac,-1) . Here Rac, represents the -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~ Ra^α , we propose a length scale = H/(1+1.49 ⁻²)1/[3(1-α )] . In the case of turbulent liquid metal convection with α =1/4 , our measurement shows that the heat transport will become weakly dependent on with Ra≡ Ra/(1+1.49 ⁻²)4/3 ≥ 7× 10⁵ . 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 =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.