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# Nusselt number versus Rayleigh number, as obtained in the RBC experiments for different Rayleigh numbers, Pr ≈ 0.009, β = 0 • (open circles) with the effective scaling Nu ≈ 0.177Ra 0.215 (solid line); the RBC experiment for Ra = 1.42 × 10 7 , Pr = 0.0093, β = 0 • (filled circle; this run is the longest one (7 h) in the series of measurements. For the same Ra and Pr, the Nusselt numbers were also measured for different β, see table 3); the VC experiments for different Rayleigh numbers, Pr ≈ 0.009, β = 90 • (open squares) with the effective scaling Nu ≈ 0.178Ra 0.222 (dash-dotted line); the inset shows the compensated Nusselt number for RBC and VC experimental data; the DNS for Ra = 1.67 × 10 7 , Pr = 0.0094, β = 0 • (filled diamond); the DNS for Ra = 1.67 × 10 6 , Pr = 0.094, β = 0 • (cross); the LES for Ra = 1.5 × 10 7 , Pr = 0.0093, β = 0 • (open triangle). Results of the RBC DNS by Scheel & Schumacher (2017) for Pr = 0.005 (open diamonds) and for Pr = 0.025 (pluses) and predictions for Pr = 0.0094 of the Grossmann & Lohse (2000, 2001) theory considered with the pre-factors from Stevens et al. (2013) (dash line) are presented for comparison. Everywhere a cylindrical convection cell of aspect ratio 1 is considered.

Source publication

The influence of the cell inclination on the heat transport and large-scale circulation in liquid metal convection - Volume 884 - Lukas Zwirner, Ruslan Khalilov, Ilya Kolesnichenko, Andrey Mamykin, Sergei Mandrykin, Alexander Pavlinov, Alexander Shestakov, Andrei Teimurazov, Peter Frick, Olga Shishkina

## Contexts in source publication

**Context 1**

... heat and momentum transport First, we examine the classical case of RBC without inclination (β = 0 • ). The time-averaged mean heat fluxes, represented by the Nusselt numbers, are presented in figure 4. ...

**Context 2**

... is an excellent agreement of the DNS and LES data, for example, the Nusselt number deviates less than 1 % and the Reynolds number around 5 % for the RBC and VC cases. Also in figure 4 we compare our numerical results with the DNS by Scheel & Schumacher (2017), for Pr = 0.005 and Pr = 0.025. Our numerical results for Pr = 0.0094 and Pr = 0.0093 take place between the cited results by Scheel & Schumacher (2017) & Schumacher (2017) were conducted using completely different codes (the Nek5000 spectral element package in the latter case), but nevertheless lead to consistent results. ...

**Context 3**

... Pr = 0.0094, the predictions by the Grossmann & Lohse (2000, 2001) theory, with the prefactors from Stevens et al. (2013), are shown in figure 4 between the obtained experimental and numerical data. The experimental data for a certain Rarange around Ra = 10 7 , shown in figure 4, follow a scaling relation Nu ≈ (0.16 ± 0.01)Ra 0.22±0.06 ...

**Context 4**

... Pr = 0.0094, the predictions by the Grossmann & Lohse (2000, 2001) theory, with the prefactors from Stevens et al. (2013), are shown in figure 4 between the obtained experimental and numerical data. The experimental data for a certain Rarange around Ra = 10 7 , shown in figure 4, follow a scaling relation Nu ≈ (0.16 ± 0.01)Ra 0.22±0.06 (RBC, β = 0 • ). ...

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## Citations

... RB convection has attracted extensive investigations not only because it is a model system for studying buoyancy driving turbulence but also due to its importance in understanding the ubiquitous convection in nature, for example, convection in astrophysical and geophysical systems such as solar and mantle convection. 1 A fascinating feature of the turbulent RB convection is the emergence of well-defined coherent oscillation in the presence of a turbulent background. This robust oscillation has been observed in both the temperature field 8 and velocity field [9][10][11] and in convection systems with different working fluids 8,[12][13][14][15][16] and different geometries. [17][18][19][20] Qiu et al. 9 measured the velocity profile in a C ¼ 1 cell filled with water using laser Doppler velocimetry, and they found the mean flow field oscillates in a coherent manner. ...

... 35 Recently, Xie and Xia 36 experimentally studied the effect of the roughness surface on RBC, and they found that the roughness surface would increase the frequency of the temperature oscillation. Zwirner et al. 15 found that in a liquid metal convection system with low Pr, the twisting and sloshing motion also existed, and these two modes strongly influence the instantaneous heat transport. ...

We experimentally studied the effect of cell tilting on the temperature oscillation in turbulent Rayleigh–Bénard convection. We simultaneously measured the temperature using both in-fluid and in-wall thermistors for Ra=1.7×109 and 5.0×109 at Prandtl number Pr = 5.3. The tilt angles relative to gravity are set to 0°, 0.5°, 1°, 2°, and 7°. It is found that the temperature oscillation intensity measured in-fluid is much stronger than that measured in-wall, because the in-fluid thermistors measure both the large-scale circulation (LSC) and the plumes/plume clusters, while the in-wall thermistors only measure the LSC due to the filter effect of the sidewall. Despite the intensity difference, the obtained azimuthal profiles of the oscillation intensity measured by in-fluid and in-wall share similar spatial distribution, and the spatial distribution can be explained by the torsional motion near the top and bottom plates and the sloshing motion at the mid-height. With the in-fluid measurements, we find that with the increase in the tilt angle, the azimuthal profile of oscillation evolves toward a sawtooth-like profile and the intensity gets more prominent, which implies that the temperature oscillation becomes more coherent. Through a conditional sampling method based on the azimuthal position of LSC, we reveal that the uniformly distributed oscillation intensity in the level cell is the result of the superimposition of the random azimuthal motion and the sloshing motion. Tilting the cell can efficiently disentangle the above-mentioned two types of motions of LSC. Moreover, we found that the frequency of the temperature oscillation increases when the tilt angle increases, while the amplitude of the sloshing motion of the LSC remains unchanged, which is believed to be related to the confinement of the convection cell.

... There has been evidence that the heat transport in liquid metal turbulent RBC is influenced by the LSF structure (Zwirner et al. 2020;Schindler et al. 2022). However, how the flow state evolution mentioned above affects heat transport remains unexplored. ...

We present an experimental study of Rayleigh-B\'enard 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 $1.2\times10^{4} \le Ra \le 1.3\times10^{7}$. 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 structures near the onset of convection, which is in contrast with the case of $Pr\sim1$, where the cell structure is replaced by high-order flow modes transiently 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 (PDF) 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 PDF 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 PDF of the temperature fluctuation at the cell centre and a $Nu=0.19Ra^{0.25}$ scaling for the heat transport.

... Although these two models are extensively studied in the past decades, more generally, a significant misalignment exists between the global temperature gradient and gravity. The convection system with an arbitrary inclination angle is denoted as inclined convection [16][17][18][19][20][21][22][23] . ...

The influence of ratchets on inclined convection is explored within a rectangular cell (aspect ratio $\Gamma_{x}=1$ and $\Gamma_y=0.25$) by experiments and simulations. The measurements are conducted over a wide range of tilting angles ($0.056\leq\beta\leq \pi/2\,\si{\radian}$) at a constant Prandtl number ($\text{Pr}=4.3$) and Rayleigh number ($\text{Ra}=5.7\times10^9$). We found that the arrangement of ratchets on the conducting plate determines the dynamics of inclined convection, i.e., when the large-scale circulation (LSC) flows along the smaller slopes of the ratchets (case A), the change of the heat transport efficiency is smaller than $5\%$ as the tilting angle increases from 0 to $4\pi/9~\si{\radian}$; when the LSC moves towards the steeper slope side of the ratchets (case B), the heat transport efficiency decreases rapidly with the tilting angle larger than blue$\pi/9~\si{\radian}$. By analyzing the flow properties, we give a physical explanation for the observations. As the tilting angle increases, the heat carrier gradually changes from the thermal plumes to the LSC, resulting in different dynamical behavior. In addition, the distribution of the local heat transport also validates the explanation quantitatively. The present work gives insights into controlling inclined convection using asymmetric ratchet structures.

... There has been evidence that the heat transport in liquid metal turbulent RBC is influenced by the LSF structure (Zwirner et al. 2020;Schindler et al. 2022). However, how the flow state evolution mentioned above affects heat transport remains unexplored. ...

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.

... Some comparative simulations with a higher resolution of up to 2x10 6 elements showed a similar behaviour in the macroscopic flow dynamics of the LSC. In view of the high numerical costs typically connected with the simulation of liquid metal convection [10], we decided to use the 0.55x10 6 elements mesh for all following numerical simulations. ...

A possible explanation for the apparent phase stability of the 11.07-year Schwabe cycle of the solar dynamo was the subject of a series of recent papers. The synchronization of the helicity of an instability with azimuthal wavenumber m=1 by a tidal m=2 perturbation played a key role here. To analyze this type of interaction in a paradigmatic set-up, we study a thermally driven Rayleigh-B\'enard Convection (RBC) of a liquid metal under the influence of a tide-like electromagnetic forcing. As shown previously, the time-modulation of this forcing emerges as a peak frequency in the m=2 mode of the radial flow velocity component. In this paper we present new numerical results on the interplay between the Large Scale Circulation (LSC) of a RBC flow and the time modulated electromagnetic forcing.

... Even when these difficulties are circumvented, the lowest Prandtl number that can be explored is Pr ≈ 0.006 for liquid sodium (Horanyi, Krebs & Müller 1999), which is still some three orders of magnitude higher than in the solar convection zone. Direct numerical simulations (DNS) offer important tools but they, too, are hindered by demanding resolution requirements due to the highly inertial nature of low-Pr convection Schumacher, Götzfried & Scheel 2015;Pandey, Scheel & Schumacher 2018;Zwirner et al. 2020). The fact that the required computational power increases with increasing aspect ratio as Γ 2 further limits numerical investigations. ...

... Convection at low Prandtl numbers differs from its high-Pr counterpart by reduced heat transport and enhanced momentum transport Pandey et al. 2018;Zürner et al. 2019;Zwirner et al. 2020). Heat transport is quantified by the Nusselt number Nu, defined as the ratio of the total heat transport to that by conduction alone. ...

... The characteristic length scale of the dominant energy-containing structures is of the order of the size of the system when Γ ≈ 1, e.g. a large-scale circulation covering the entire domain Zürner et al. 2019;Zwirner et al. 2020). However, for Γ 1, mean circulation rolls with diameters larger than H are observed (Emran & Schumacher 2015); the resulting large-scale patterns of these rolls are termed turbulent superstructures of convection Stevens et al. 2018;Fonda et al. 2019;Green et al. 2020) as we stated already in § 1. ...

Horizontally extended turbulent convection, termed mesoscale convection in natural systems, remains a challenge to investigate in both experiments and simulations. This is particularly so for very low molecular Prandtl numbers, such as occur in stellar convection and in the Earth's outer core. The present study reports three-dimensional direct numerical simulations of turbulent Rayleigh–Bénard convection in square boxes of side length $L$ and height $H$ with the aspect ratio $\varGamma =L/H$ of 25, for Prandtl numbers that span almost 4 orders of magnitude, $10^{-3}\le Pr \le 7$ , and Rayleigh numbers $10^5 \le Ra \le 10^7$ , obtained by massively parallel computations on grids of up to $5.36\times 10^{11}$ points. The low end of this $Pr$ -range cannot be accessed in controlled laboratory measurements. We report the essential properties of the flow and their trends with the Rayleigh and Prandtl numbers, in particular, the global transport of momentum and heat – the latter decomposed into convective and diffusive contributions – across the convection layer, mean vertical profiles of the temperature and temperature fluctuations and the kinetic energy and thermal dissipation rates. We also explore the degree to which the turbulence in the bulk of the convection layer resembles classical homogeneous and isotropic turbulence in terms of spectra, increment moments and dissipative anomaly, and find close similarities. Finally, we show that a characteristic scale of the order of the mesoscale seems to saturate to a wavelength of $\lambda \gtrsim 3H$ for $Pr\lesssim 0.005$ . We briefly discuss possible implications of these results for the development of subgrid-scale parameterization of turbulent convection.

... The dominant force balance in low-Pr convection is between the inertial and pressure gradient forces, which leads to vigorous turbulence compared to those at moderate and high Pr [20]. Due to increasing availability of computational resources, low-Pr regime of convection, of great interest in geophysical and atmospheric flows, can now be computed [19,[21][22][23]. The velocity field in such flows exhibits a highly intermittent character, whereas the temperature field remains diffusive. ...

To understand turbulent convection at very high Rayleigh numbers typical of natural phenomena, computational studies in slender cells are an option if the needed resources have to be optimized within available limits. However, the accompanying horizontal confinement affects some properties of the flow. Here, we explore the characteristics of turbulent fluctuations in the velocity and temperature fields in a cylindrical convection cell of aspect ratio 0.1 by varying the Prandtl number Pr between 0.1 and 200 at a fixed Rayleigh number Ra=3×1010, and find that the fluctuations weaken with increasing Pr, quantitatively as in aspect ratio 25. The probability density function (PDF) of temperature fluctuations in the bulk region of the slender cell remains mostly Gaussian, but increasing departures occur as Pr increases beyond unity. We assess the intermittency of the velocity field by computing the PDFs of velocity derivatives and of the kinetic energy dissipation rate, and find increasing intermittency as Pr decreases. In the bulk region of convection, a common result applicable to the slender cell, large aspect ratio cells, as well as in 2D convection, is that the turbulent Prandtl number decreases as Pr−1/3.

... The dominant force balance in low-Pr convection is between the inertial and pressure gradient forces, which leads to vigorous turbulence compared to those at moderate and high Pr [20]. Due to increasing availability of computational resources, low-Pr regime of convection, of great interest in geophysical and atmospheric flows, can now be computed [19,21,22,23]. The velocity field in such flows exhibits a highly intermittent character, whereas the temperature field remains diffusive. ...

To understand turbulent convection at very high Rayleigh numbers typical of natural phenomena, computational studies in slender cells are an option if the needed resources have to be optimized within available limits. However, the accompanying horizontal confinement affects some properties of the flow. Here, we explore the characteristics of turbulent fluctuations in the velocity and temperature fields in a cylindrical convection cell of aspect ratio 0.1 by varying the Prandtl number $Pr$ between 0.1 and 200 at a fixed Rayleigh number $Ra = 3 \times 10^{10}$, and find that the fluctuations weaken with increasing $Pr$, quantitatively as in aspect ratio 25. The probability density function (PDF) of temperature fluctuations in the bulk region of the slender cell remains mostly Gaussian, but increasing departures occur as $Pr$ increases beyond unity. We assess the intermittency of the velocity field by computing the PDFs of velocity derivatives and of the kinetic energy dissipation rate, and find increasing intermittency as $Pr$ decreases. In the bulk region of convection, a common result applicable to the slender cell, large aspect ratio cells, as well as in 2D convection, is that the turbulent Prandtl number decreases as $Pr^{-1/3}$.

... We use water as the working fluid and the Prandtl number is kept at Pr = 4.3 throughout this work. For small Prandtl number working fluids (gases or liquid metals), the system may behave quite differently due to its relatively large thermal diffusivity (Zwirner & Shishkina 2018;Zürner et al. 2019;Zwirner et al. 2020). Combining experiments with DNS, we apply streamwise and spanwise confinements to turbulent thermal convection in the presence of an effective horizontal buoyancy using the platform of tilted RBC reported in Zhang et al. (2021). ...

We present an experimental and numerical study of turbulent thermal convection in the presence of an effective horizontal buoyancy that generates extra shear at the boundary. Geometrical confinements are also applied by varying the streamwise and spanwise aspect ratios of the convection cell to condense the plumes. With these, we systematically explore the effects of plume and shear on heat transfer. It is found that a streamwise confinement results in increased plume coverage but decreased shear compared with spanwise confinement. The fact that streamwise confinement leads to a higher vertical heat transfer efficiency than the spanwise confined case suggests that the increase of plume coverage is the dominant effect responsible for the enhanced heat transfer. Our results highlight the potential applications of coherent structure manipulation in efficient passive heat transfer control and thermal engineering. We also analyse the energetics of the present system and derive the expression of mixing efficiency accordingly. The mixing efficiency is found to increase with both the buoyancy ratio and streamwise dimension.

... [2,3,[15][16][17][18]. For Γ < 1, the LSC forms multiple rolls arranged on top of each other [4,[19][20][21]. The particular LSC configuration determines the magnitude of transferred heat which is quantified by the Nusselt number N u [17,21,22]. ...

The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between 4×10^{5} and ≤5×10^{9} and Prandtl numbers of 0.025≤Pr≤0.033. Our measurements are performed in two cylindrical samples of aspect ratios Γ=diameter/height=0.5 and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beam lines in different planes reveals a clear breakdown of coherence of the large-scale circulation for Γ=0.5. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number Re. The scaling exponent β of the power law Nu∝Ra^{β} crosses eventually over from β=0.221 to 0.124 when the liquid metal flow at Γ=0.5 reaches Ra≳2×10^{8} and the coherent large-scale flow is completely collapsed.