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# Critical Ra c;Γ for the onset of convection: Linear growth rates (colored vertically elongated boxes) from the linearized DNS approach (GOLDFISH) compared to the neutral stability curves (blue lines) from the eigenvalue LSA for (a) 2D box with isothermal sidewalls, (b) 2D box with adiabatic sidewalls, and (c) cylinder with adiabatic sidewall. Black lines show Ra c;Γ ¼ 1708ð1 þ C=Γ 2 Þ 2 with a best-fit C for the linearized DNS data (dashed lines) and with theoretical C for isothermal sidewall (solid line). Pluses in (c) show Ra c;Γ from the nonlinearized DNS data (AFiD) [4]. Temperature contours near the onset of convection are shown for some Γ, as obtained from the linearized DNS. See details in [5-8] and the Supplemental Material [9].

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While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of...

## Contexts in source publication

**Context 1**

... LSA.-We have verified the estimate (14) for the Γ dependence of the critical Ra c;Γ for the onset of convection with linearized DNSs for the 2D and 3D cases and with the eigenspectrum LSA for the 2D case. The growth rates obtained with both methods are in a very good agreement, see Figs. 1(a) and 1(b). The numerically obtained Ra c;Γ as function of Γ [Eq. (14)] for the isothermal sidewalls are in excellent agreement with the analytical estimates. Equation (14) captures the trend and reflects well also the shape of the neutral curve for the case of adiabatic sidewalls. The best-fit constants C (C ≈ 0.52 for the 2D domain and C ≈ 0.77 ...

**Context 2**

... neutral curve for the case of adiabatic sidewalls. The best-fit constants C (C ≈ 0.52 for the 2D domain and C ≈ 0.77 for the cylinder) are, however, smaller than the theoretical predictions for the isothermal sidewalls, see Figs. 1(b) and 1(c). Isosurfaces of the temperature of the flow fields near the onset of convection are shown for some Γ in Fig. 1 as well. The azimuthalmode transition found for the cylinder between Γ ¼ 1 and 2 is consistent with the experiments ...

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... The resulting dynamical system is governed by the Rayleigh number Ra = gα∆T H 3 /(νκ) and the Prandtl number Pr = ν/κ which are defined by acceleration due to gravity g, the thermal expansion coefficient α, the temperature difference between heating and cooling plate ∆T , the domain height H, the kinematic viscosity ν and the thermal diffusivity κ of the fluid. Additionally, the aspect ratio Γ = W/H as the ratio of domain width W and height H and the container's shape affects the flow [39,40]. In the present large aspect ratio experiment, so-called turbulent superstructures emerge [41][42][43][44]. ...

The measurement of the transport of scalar quantities within flows is oftentimes laborious, difficult or even unfeasible. On the other hand, velocity measurement techniques are very advanced and give high-resolution, high-fidelity experimental data. Hence, we explore the capabilities of a deep learning model to predict the scalar quantity, in our case temperature, from measured velocity data. Our method is purely data-driven and based on the u-net architecture and, therefore, well suited for planar experimental data. We demonstrate the applicability of the u-net on experimental temperature and velocity data, measured in large aspect ratio Rayleigh-Bénard convection at Pr = 7.1 and Ra = 2 × 10 5 , 4 × 10 5 , 7 × 10 5 . We conduct a hyper-parameter optimization and ablation study to ensure appropriate training convergence and test different architectural variations for the u-net. We test two application scenarios that are of interest to experimentalists. One, in which the u-net is trained with data of the same experimental run and one in which the u-net is trained on data of different Ra . Our analysis shows that the u-net can predict temperature fields similar to the measurement data and preserves typical spatial structure sizes. Moreover, the analysis of the heat transfer associated with the temperature showed good agreement when the u-net is trained with data of the same experimental run. The relative difference between measured and reconstructed local heat transfer of the system characterized by the Nusselt number Nu is between 0.3% and 14.1% depending on Ra . We conclude that deep learning has the potential to supplement measurements and can partially alleviate the expense of additional measurement of the scalar quantity.

... For any given Pr and Ra-range, the theory provides accurate predictions of the value of γ, for containers of aspect ratio Γ ≳ 1. For Γ ≪ 1, the data can be rescaled according to the method suggested in [22,1], which we do not discuss here, as in the present study Γ = 1. ...

In magnetoconvection, the flow of electromagnetically conductive fluid is driven by a combination of buoyancy forces, which create the fluid motion due to thermal expansion and contraction, and Lorentz forces, which distort the convective flow structure in the presence of a magnetic field. The differences in the global flow structures in the buoyancy-dominated and Lorentz-force-dominated regimes lead to different heat transport properties in these regimes, reflected in distinct dimensionless scaling relations of the global heat flux (Nusselt number $\textrm{Nu}$) versus the strength of buoyancy (Rayleigh number $\textrm{Ra}$) and electromagnetic forces (Hartmann number $\textrm{Ha}$). Here, we propose a theoretical model for the transition between these two regimes for the case of a quasistatic vertical magnetic field applied to a convective fluid layer confined between two isothermal, a lower warmer and an upper colder, horizontal surfaces. The model suggests that the scaling exponents $\gamma$ in the buoyancy-dominated regime, $\textrm{Nu}\sim\textrm{Ra}^\gamma$, and $\xi$ in the Lorentz-force-dominated regime, $\textrm{Nu}\sim(\textrm{Ha}^{-2}\textrm{Ra})^\xi$, are related as $\xi=\gamma/(1-2\gamma)$, and the onset of the transition scales with $\textrm{Ha}^{-1/\gamma}\textrm{Ra}$. These theoretical results are supported by our Direct Numerical Simulations for $10\leq \textrm{Ha}\leq2000$, Prandtl number $\textrm{Pr}=0.025$ and $\textrm{Ra}$ up to $10^9$ and data from the literature.

... ± 0.04) × 10 −4 mol/m 3 Pa and P 0 = 1.0 bar, D = (1.85 ± 0.02) × 10 −9 m 2 /s the diffusion coefficient of CO 2 in water, H = 17.6 ± 0.35 mm the height of the liquid barrier, and ν = 9.5 × 10 −7 m 2 /s the kinematic viscosity of water [8,30,31]. We obtain Ra H ≈ (8.8 ± 0.5) × 10 6 , which is well above the critical Rayleigh number, Ra H,c = 1.29 × 10 6 , based on the minimal aspect ratio (Γ max = d/H = 0.17) of our experimental setup [32]. ...

... ± 0.6) × 10 3 , by taking the average critical value for the twelve experiments shown in figure 6. We compare this value to the critical Rayleigh number from Ahlers et al. for Rayleigh-Bénard convection in a cylinder with adiabatic sidewalls, which we believe to be the closest available approximation to our system [32]: ...

... intersect, is exactly δ = δ * = 1.13 mm, which is reasonably close to z 1 , with a corresponding Ra δ * = Ra c = 2.51 × 10 3 . This further emphasises the difficulty in defining the thickness for the self-similar diffusion boundary layer, as by selecting a lower intensity threshold, and thus higher concentration cut-off C δ , we could have reproduced the prediction from Ahlers et al. [32]. ...

The dissolution and subsequent mass transfer of carbon dioxide gas into liquid barriers plays a vital role in many environmental and industrial applications. In this work, we study the downward dissolution and propagation dynamics of CO2 into a vertical water barrier confined to a narrow vertical glass cylinder, using both experiments and direct numerical simulations. Initially, the dissolution of CO2 results in the formation of a CO2-rich water layer, which is denser in comparison to pure water, at the top gas-liquid interface. Continued dissolution of CO2 into the water barrier results in the layer becoming gravitationally unstable, leading to the onset of buoyancy driven convection and, consequently, the shedding of a buoyant plume. By adding sodium fluorescein, a pH-sensitive fluorophore, we directly visualise the dissolution and propagation of the CO2 across the liquid barrier. Tracking the CO2 front propagation in time results in the discovery of two distinct transport regimes, a purely diffusive regime and an enhanced diffusive regime. Using direct numerical simulations, we are able to successfully explain the propagation dynamics of these two transport regimes in this laterally strongly confined geometry, namely by disentangling the contributions of diffusion and convection to the propagation of the CO2 front.

... In a recent meta study by Ahlers et al. (2022), data from a great number of experiments were compiled and a correction with respect to the aspect ratio of the convection cell was proposed, which improved the collapse of the data. The data were divided into two sets, where Pr ≈ 4.4 in the first and Pr ≈ 0.8 in the second. ...

... Only after the data have been corrected the curves show a common point of transition. Nevertheless, Ahlers et al. (2022) ...

... The thickness of the thermal boundary layers can be estimated from the prefactor in Nu = aRa 1/3 , as δ T ≈ 0.5a −3/4 η B . With a ≈ 0.05 (Iyer et al. 2020;Ahlers et al. 2022), we obtain δ T ≈ 5η B , which seems reasonable. Sun, Cheung & Xia (2008) report δ T = 0.58 mm and η = 0.4 mm from an experiment at Ra = 2.5 × 10 10 with water (Pr = 4.3). ...

We consider the Nusselt–Rayleigh number problem of Rayleigh–Bénard convection and make the hypothesis that the velocity and thermal boundary layer widths, $\delta _u$ and $\delta _T$ , in the absence of a strong mean flow are controlled by the dissipation scales of the turbulence outside the boundary layers and, therefore, are of the order of the Kolmogorov and Batchelor scales, respectively. Under this assumption, we derive $Nu \sim Ra^{1/3}$ in the high $Ra$ limit, independent of the Prandtl number, $\delta _T/L \sim Ra^{-1/3}$ and $\delta _u/L \sim Ra^{-1/3} Pr^{1/2}$ , where $L$ is the height of the convection cell. The scaling relations are valid as long as the Prandtl number is not too far from unity. For $Pr \sim 1$ , we make a more general ansatz, $\delta _u \sim \nu ^{\alpha }$ , where $\nu$ is the kinematic viscosity and assume that the dissipation scales as $\sim u^3/L$ , where $u$ is a characteristic turbulent velocity. Under these assumptions we show that $Nu \sim Ra^{\alpha /(3-\alpha )}$ , implying that $Nu \sim Ra^{1/5}$ if $\delta _u$ were scaling as in a Blasius boundary layer and $Nu \sim Ra^{1/2}$ (with some logarithmic correction) if it were scaling as in a standard turbulent shear boundary layer. It is argued that the boundary layers will retain the intermediate scaling $\alpha = 3/4$ in the limit of high $Ra$ .

... The phase diagram in Figure 6 is obtained in rectangular convection cells by narrowing the width only. If the geometrical confinement is applied to both lateral directions, a recent study [42] showed that the critical Ra number for the onset of convection follows a power law of Ra c ∼ 1708(1 + C/Γ 2 ) 2 under Oberbeck-Boussinesq conditions, where C is a constant that depends on the shape of the convection cells. It is not clear at this stage how the boundaries of the plume-controlled regime (dotted and dashed lines in Figure 6) will be reshaped in convection systems with other geometries. ...

Tuning transport properties through the manipulation of elementary structures has received a great success in many areas, such as condensed matter physics. However, the ability to manipulate coherent structures in turbulent flows is much less explored. This article reviews a recently discovered mechanism of tuning turbulent heat transport via coherent structure manipulation. We first show how this mechanism can be realized by applying simple geometrical confinement to a classical thermally-driven turbulence, which leads to the condensation of elementary coherent structures and significant heat-transport enhancement, despite the resultant slower flow. Some potential applications of this new paradigm in passive heat management are also discussed. We then explain how the heat transport behaviors in seemingly different turbulence systems can be understood by this unified framework of coherent structure manipulation. Several future directions in this research area are also outlined.

... The heat transport properties are then related to the scaling behavior between the dimensionless heat flux (characterized by the Nusselt number Nu) and the dimensionless temperature difference (characterized by the Rayleigh number Ra), i.e., Nu ∼ Ra β where β is the scaling exponent. Decades of studies on RB setup show the emergence of universal scaling exponent in the constitutive law [9][10][11][12][13][14][15][16][17][18] . Typically, one theoretically arguments β = 1/3 from the elegant theory of marginal stability 9,16,18 , or β ≈ 0.3 from experimental observations 13 in the classical regime, and β = 1/2 in the ultimate regime predicted by a mixing length model assuming that the heat flux is fully controlled by turbulence 11,12 . ...

... Decades of studies on RB setup show the emergence of universal scaling exponent in the constitutive law [9][10][11][12][13][14][15][16][17][18] . Typically, one theoretically arguments β = 1/3 from the elegant theory of marginal stability 9,16,18 , or β ≈ 0.3 from experimental observations 13 in the classical regime, and β = 1/2 in the ultimate regime predicted by a mixing length model assuming that the heat flux is fully controlled by turbulence 11,12 . Both heat transport scaling relations are extensively examined by various experimental and numerical investigations [13][14][15][16][17][18] . ...

... Typically, one theoretically arguments β = 1/3 from the elegant theory of marginal stability 9,16,18 , or β ≈ 0.3 from experimental observations 13 in the classical regime, and β = 1/2 in the ultimate regime predicted by a mixing length model assuming that the heat flux is fully controlled by turbulence 11,12 . Both heat transport scaling relations are extensively examined by various experimental and numerical investigations [13][14][15][16][17][18] . ...

The emergence of unified constitutive law is a hallmark of convective turbulence, i.e., $Nu \sim Ra^\beta$ with $\beta \approx 0.3$ in the classical and $\beta=1/2$ in the ultimate regime, where the Nusselt number $Nu$ measures the global heat transport and the Rayleigh number $Ra$ quantifies the strength of thermal forcing. In recent years, vibroconvective flows have been attractive due to its ability to drive flow instability and generate ``artificial gravity'', which have potential to effective heat and mass transport in microgravity. However, the existence of constitutive laws in vibroconvective turbulence remains unclear. To address this issue, we carry out direct numerical simulations in a wide range of frequencies and amplitudes, and report that the heat transport exhibits a universal scaling law $Nu \sim a^{-1} Re_\mathrm{os}^\beta$ where $a$ is the vibration amplitude, $Re_\mathrm{os}$ is the oscillational Reynolds number, and $\beta$ is the universal exponent. We find that the dynamics of boundary layers plays an essential role in vibroconvective heat transport, and the $Nu$-scaling exponent $\beta$ is determined by the competition between the thermal boundary layer (TBL) and vibration-induced oscillating boundary layer (OBL). Then a physical model is proposed to explain the change of scaling exponent from $\beta=2$ in the OBL-dominant regime to $\beta = 4/3$ in the TBL-dominant regime. We conclude that vibroconvective turbulence in microgravity defines a distinct universality class of convective turbulence. This work elucidates the emergence of universal constitutive laws in vibroconvective turbulence, and opens up a new avenue for generating a controllable effective heat transport under microgravity or even microfluidic environment in which gravity is nearly absent.

... Thus the onset Rayleigh number for convection Ra c cannot be determined directly. Following Wei (2021), we determine Ra c from the normalised time-averaged flow strength δ / T. When convection sets in, the first unstable mode is the azimuthal m = 1 mode in a cylindrical cell with Γ = 1 (Hébert et al. 2010;Ahlers et al. 2022). Thus Ra c can be determined once δ / T is larger than the experimentally detectable temperature differences. ...

... It is seen that E 1 / E t is always larger than 0.85 in liquid metal convection, suggesting the LSF is in the form of a single-roll structure. When Ra < Ra t , the LSF is the cell structure observed just beyond the onset of convection, i.e. the m = 1 azimuthal mode (Hébert et al. 2010;Ahlers et al. 2022 liquid metal convection is a residual of the cell structure near the onset of convection. ...

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.

... He et al. (2012) reported a transition region covering 10 13 Ra c 5 × 10 14 and our estimates using both methods fall within this range. It is also worth mentioning that Ra c has been reported to be strongly dependent on the aspect ratio Γ of the convection cell (Roche et al. 2010;He, Bodenschatz & Ahlers 2020;Ahlers et al. 2022). ...

We report an experimental study of the viscous boundary layer (BL) properties of turbulent Rayleigh–Bénard convection in a cylindrical cell. The velocity profile with all three components was measured from the centre of the bottom plate by an integrated home-made particle image velocimetry system. The Rayleigh number $Ra$ varied in the range $1.82 \times 10^8 \le Ra \le 5.26 \times 10^9$ and the Prandtl number $Pr$ was fixed at $Pr = 4.34$ . The probability density function of the wall-shear stress indicates that using the velocity component in the mean large-scale circulation (LSC) plane alone may not be sufficient to characterise the viscous BL. Based on a dynamic wall-shear frame, we propose a method to reconstruct the measured full velocity profile which eliminates the effects of complex dynamics of the LSC. Various BL properties including the eddy viscosity are then obtained and analysed. It is found that, in the dynamic wall-shear frame, the eddy viscosity profiles along the centre line of the convection cell at different $Ra$ all collapse on a single master curve described by $\nu _t^d / \nu = 0.81 (z / \delta _u^d) ^{3.10 \pm 0.05}$ . The Rayleigh number dependencies of several BL quantities are also determined in the dynamic frame, including the BL thickness $\delta _u^d$ ( ${\sim } Ra^{-0.21}$ ), the Reynolds number $Re^d$ ( ${\sim }Ra^{-0.46}$ ) and the shear Reynolds number $Re_s^d$ ( ${\sim } Ra^{0.24}$ ). Within the experimental uncertainty, these scaling exponents are the same as those obtained in the static laboratory frame. Finally, with the measured full velocity profile, we obtain the energy dissipation rate at the centre of the bottom plate $\varepsilon _{w}$ , which is found to follow $\langle \varepsilon _{w} \rangle _t \sim Ra^{1.25}$ .

... In this regime, the heat transport is mostly insensitive to Γ −1 and adapts the value from the unconfined case. By reducing the horizontal extent the flow first enters a plume-controlled regime for moderate confinement (Γ −1 1), in which the heat transport is enhanced, before, in the severely confined regime (Γ −1 1), the heat transport is strongly reduced (Chong et al. 2015;Chong & Xia 2016 Ahlers et al. 2022). Similar to the rotation-controlled regime in rotating RBC, vertically coherent structures form within the plume-controlled regime in confined RBC (Chong et al. 2015;Hartmann et al. 2021). ...

Moderate rotation and moderate horizontal confinement similarly enhance the heat transport in Rayleigh–Bénard convection (RBC). Here, we systematically investigate how these two types of flow stabilization together affect the heat transport. We conduct direct numerical simulations of confined-rotating RBC in a cylindrical set-up at Prandtl number $\textit {Pr}=4.38$ , and various Rayleigh numbers $2\times 10^{8}\leqslant {\textit {Ra}}\leqslant 7\times 10^{9}$ . Within the parameter space of rotation (given as inverse Rossby number $0\leqslant {\textit {Ro}}^{-1}\leqslant 40$ ) and confinement (given as height-to-diameter aspect ratio $2\leqslant \varGamma ^{-1}\leqslant 32$ ), we observe three heat transport maxima. At lower $ {\textit {Ra}}$ , the combination of rotation and confinement can achieve larger heat transport than either rotation or confinement individually, whereas at higher $ {\textit {Ra}}$ , confinement alone is most effective in enhancing the heat transport. Further, we identify two effects enhancing the heat transport: (i) the ratio of kinetic and thermal boundary layer thicknesses controlling the efficiency of Ekman pumping, and (ii) the formation of a stable domain-spanning flow for an efficient vertical transport of the heat through the bulk. Their interfering efficiencies generate the multiple heat transport maxima.

... Due to rotational symmetry, most experiments and many numerical investigations have been conducted in upright cylinders, hence the aspect ratio Γ = D/H between cylinder diameter D = 2R and height H is a parameter quantifying the geometrical constraints. The height H is a good length scale in RBC only for sufficiently large Γ because only then is Nu independent of Γ (Ahlers et al. 2022;Zwirner et al. 2021). Nevertheless, most experiments are conducted in cylinders of Γ close to 1 in order to maximize H, and in this way Ra. ...

We report on the presence of the boundary zonal flow in rotating Rayleigh–Bénard convection evidenced by two-dimensional particle image velocimetry . Experiments were conducted in a cylindrical cell of aspect ratio $\varGamma =D/H=1$ between its diameter ( $D$ ) and height ( $H$ ). As the working fluid, we used various mixtures of water and glycerol, leading to Prandtl numbers in the range $6.6 \lesssim \textit {Pr} \lesssim 76$ . The horizontal velocity components were measured at a horizontal cross-section at half height. The Rayleigh numbers were in the range $10^8 \leq \textit {Ra} \leq 3\times 10^9$ . The effect of rotation is quantified by the Ekman number, which was in the range $1.5\times 10^{-5}\leq \textit {Ek} \leq 1.2\times 10^{-3}$ in our experiment. With our results we show the first direct measurements of the boundary zonal flow (BZF) that develops near the sidewall and was discovered recently in numerical simulations as well as in sparse and localized temperature measurements. We analyse the thickness $\delta _0$ of the BZF as well as its maximal velocity as a function of Pr , Ra and Ek , and compare these results with previous results from direct numerical simulations.