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We report a numerical investigation of a previously noticed but less explored flow state transition in two-dimensional turbulent Rayleigh–Bénard convection. The simulations are performed in a square domain over a Rayleigh number range of 
$10^7 \leq Ra \leq 2 \times 10^{11}$ and a Prandtl number range of 
$0.25 \leq Pr \leq 20$. The transition is characterized by the emergence of multiple satellite eddies with increasing 
$Ra$, which orbit around and interact with the main vortex roll in the system. Consequently, the main roll is squeezed to a smaller size compared with the domain and wanders around in the bulk region irregularly and extensively. This is in sharp contrast to the flow state before the transition, which is featured by a domain-sized circulatory roll with its vortex centre ‘condensed’ near the domain's centre. Detailed velocity field analysis reveals that there exists an abrupt increase in the energy fluctuations of the Fourier modes during the transition. Based on this phase-transition-like signal, the critical condition for the transition is found to follow a scaling relation as 
$Ra_t \sim Pr^{1.41}$ where 
$Ra_t$ is the critical Rayleigh number for the transition. This scaling relation is quantitatively explained by a phenomenological model grounded on the bistability behaviour (i.e. spontaneous and stochastic switching between the two flow states) observed at the edge of the transition. The model can also account for the effects of aspect ratio on the transition reported in the literature (van der Poel et al., Phys. Fluids, vol. 24, 2012).
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.
Using thermal convection in liquid metal, we show that strong spatial confinement not only delays the onset Rayleigh number 
$Ra_c$ of Rayleigh–Bénard instability but also postpones the various flow-state transitions. The 
$Ra_c$ and the transition to fully developed turbulence Rayleigh number 
$Ra_f$ depend on the aspect ratio 
$\varGamma$ with 
$Ra_c\sim \varGamma ^{-4.05}$ and 
$Ra_f\sim \varGamma ^{-3.01}$, implying that the stabilization effects caused by the strong spatial confinement are weaker on the transition to fully developed turbulence when compared with that on the onset. When the flow state is characterized by the supercritical Rayleigh number 
$Ra/Ra_{c}$ (
$Ra$ is the Rayleigh number), our study shows that the transition to fully developed turbulence in strongly confined geometries is advanced. For example, while the flow becomes fully developed turbulence at 
$Ra\approx 200Ra_c$ in a 
$\varGamma =1$ cell, the same transition in a 
$\varGamma =1/20$ cell only requires 
$Ra\approx 3Ra_c$. Direct numerical simulation and linear stability analysis show that in the strongly confined regime, multiple vertically stacked roll structures appear just above the onset of convection. With an increase of the driving strength, the flow switches between different-roll states stochastically, resulting in no well-defined large-scale coherent flow. Owing to this new mechanism that only exists in systems with 
$\varGamma <1$, the flow becomes turbulent in a much earlier stage. These findings shed new light on how turbulence is generated in strongly confined geometries.
We present an experimental study of Rayleigh–Bénard convection using liquid metal alloy gallium-indium-tin as the working fluid with a Prandtl number of 
$Pr=0.029$. The flow state and the heat transport were measured in a Rayleigh number range of 
$1.2\times 10^{4} \le Ra \le 1.3\times 10^{7}$. The temperature fluctuation at the cell centre is used as a proxy for the flow state. It is found that, as 
$Ra$ increases from the lower end of the parameter range, the flow evolves from a convection state to an oscillation state, a chaotic state and finally a turbulent state for 
$Ra>10^5$. The study suggests that the large-scale circulation in the turbulent state is a residual of the cell structure near the onset of convection, which is in contrast with the case of 
$Pr\sim 1$, where the cell structure is transiently replaced by high order flow modes before the emergence of the large-scale circulation in the turbulent state. The evolution of the flow state is also reflected by the heat transport characterised by the Nusselt number 
$Nu$ and the probability density function (p.d.f.) of the temperature fluctuation at the cell centre. It is found that the effective local heat transport scaling exponent 
$\gamma$, i.e. 
$Nu\sim Ra^{\gamma }$, changes continuously from 
$\gamma =0.49$ at 
$Ra\sim 10^4$ to 
$\gamma =0.25$ for 
$Ra>10^6$. Meanwhile, the p.d.f. at the cell centre gradually evolves from a Gaussian-like shape before the transition to turbulence to an exponential-like shape in the turbulent state. For 
$Ra>10^6$, the flow shows self-similar behaviour, which is revealed by the universal shape of the p.d.f. of the temperature fluctuation at the cell centre and a 
$Nu=0.19Ra^{0.25}$ scaling for the heat transport.
