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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.

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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 • ). ...
Context 5
... figure 4 we also present the measured scaling relations for Nu versus Ra for the case of VC (β = 90 • ). The scaling relation in VC is found to be quite similar to that in RBC, namely ...

## Citations

... 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). ...
Article
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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]. ...
Article
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.
... Thermal convection inevitably arises in the case of a horizontal temperature gradient. It is also worth noting that the VC and RBC are the two limiting cases in the inclined convection [6][7][8][9]. ...
... The work devoted to the study of natural convection in liquid gallium for the crystal growth applications can be distinguished as one of the early ones [20]. In some studies on inclined convection, the position of a cylindrical container at the extreme horizontal point will also correspond to the case of vertical convection [8,9] The most complete study from the fluid mechanics point of view on vertical convection of liquid metal in a box-shaped container was published in [21]. ...
... Lx/H = 8; Ly/H = 4 -∼ 10 2 − 10 4 [19] 10 −2 < P r < 30 10 5 < Ra < 10 10 L/D= 1 ∼ 10 0 − 10 2 < 1.8 [26] 10 −3 < P r < 10 10 3 < Gr < 5 · 10 7 L/H = 10/6 -- [9] P r ∼ 0.009 Ra > 10 7 L/D = 1 ∼ 6 − 7 ∼ 10 4 [5] P r = 10 10 7 < Ra < 10 14 H/L = 1 17 − 1908 ∼ 10 1 − 10 4 [21] P r = 0.03 5 · 10 3 < Ra < 10 8 1, 2, 3 and 5. ∼ 1 − 20 ∼ 10 2 − 2 · 10 4 ...
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Heat and momentum transfer of low-Prandtl-number fluid ($Pr=0.029$) in a closed rectangular cavity ($100\times60\times10$ mm$^3$) heated at one side and cooled at the opposite side are analyzed. The electromagnetic forces into the liquid metal are generated by the travelling magnetic field inductor and directed towards buoyancy forces. Large eddy simulations are performed with the Grashof number $Gr$ from $1.9\cdot 10^5$ to $7.6\cdot 10^7$ and the electromagnetic forcing parameter $F$ from $2.6\cdot 10^4$ to $2.6\cdot10^6$. An experimental validation of the simulation results of vertical convection and electromagnetically driven flow using GaInSn alloy has been performed. Three types of flow patterns are obtained for different interaction parameters $N = F / Gr$: counterclockwise flow, clockwise flow, and coexistence of two vortices. Analysis of the Reynolds number shows that the transition zone from natural convection to electromagnetic stirring lies in the range $0.02<F/Gr<0.07$ and two braking modes are found. The transition point between the convective heat transfer regimes is found for $F / Gr$ around 1. The analysis of isotherms deformation showed that in such convective systems it is possible to achieve minimum deviation of the isotherm shape from a straight line in the range of $0.05 <F/Gr <0.2$.
... The situation is improved if some stabilizing external factor is artificially introduced, by means of which the large-scale circulation is "fixed" in a certain azimuthal position. For example, in a number of experimental [28,[34][35][36] and numerical [37][38][39] works, it was shown that the tilt of the container by a small angle almost completely "fixes" the CS in a certain position. In the paper, we show the results of our numerical simulation of the Rayleigh-Bénard convection in a cylinder, both tilted and non-tiled, and perform the AIM analysis of the large-scale coherent structure. ...
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The revealing of the turbulence archetypes is one of the fundamental problems in the study of turbulence, which is important not only from the fundamental point of view but also for practical applications, e.g., in geophysics of ocean and lakes. The paper is devoted to the study of the emergence of coherent structures and the identification of their turbulent archetypes, typical for the free convective flows of the Rayleigh-Bénard type. Using Direct Numerical Simulation, we perform a numerical study of two refined convective flows: convection in a cylinder heated from below and internally heated convection in a layer. The main purpose of the study is identifying coherent structures (CS), investigating its main features and properties, and determining the turbulence archetypes using the anisotropy invariant map (AIM). We show that, in both configurations considered, CS takes place. In a cylinder, CS is a single large-scale vortex that can rotate azimuthally in non-titled container, but is almost “fixed” in the case of slightly tilted cylinder; in a layer, CS is a quasi-2D vortex, which can arise, exist for some time, disrupt, and then re-emerge again in the orthogonal direction. Nevertheless, the turbulence archetypes represented by the AIM are quite similar for both cases, and there are the distinct CS fingerprints on AIM.
... The previous analysis focused on fluids with P r = 1, but thermal convection is relevant in nature in a wide variety of fluids and many experiments are conducted in water (P r ≈ 4) or in liquid metals (P r 1) (Zwirner et al. 2020). Therefore, we now explore the P r parameter space with P r = 0.1, 1 and 10 for Ra up to 10 9 in the 2D RBC setup. ...
Preprint
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This work addresses the effects of different thermal sidewall boundary conditions on the formation of flow states and heat transport in two- and three-dimensional Rayleigh--B\'enard convection (RBC) by means of direct numerical simulations and steady-state analysis for Rayleigh numbers $Ra$ up to $4\times10^{10}$ and Prandtl numbers $Pr=0.1,1$ and $10$. We show that a linear temperature profile imposed at the conductive sidewall leads to a premature collapse of the single-roll state, whereas a sidewall maintained at a constant temperature enhances its stability. The collapse is caused by accelerated growth of the corner rolls with two distinct growth rate regimes determined by diffusion or convection for small or large $Ra$, respectively. Above the collapse of the single-roll state, we find the emergence of a double-roll state in two-dimensional RBC and a double-toroidal state in three-dimensional cylindrical RBC. These states are most prominent in RBC with conductive sidewalls. The different states are reflected in the global heat transport, so that the different thermal conditions at the sidewall lead to significant differences in the Nusselt number for small to moderate $Ra$. However, for larger $Ra$, heat transport and flow dynamics become increasingly alike for different sidewalls and are almost indistinguishable for $Ra>10^9$. This suggests that the influence of imperfectly insulated sidewalls in RBC experiments is insignificant at very high $Ra$ - provided that the mean sidewall temperature is controlled.
... [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]. ...
Preprint
Full-text available
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 $4 \times 10^5$ and $\leq 5 \times 10^9$ and Prandtl numbers of $0.025~\leq~Pr~\leq ~0.033$. Our measurements are performed in two cylindrical samples of aspect ratios $\Gamma =$ 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 beamlines in different planes reveals a clear breakdown of the large-scale circulation for $\Gamma = 0.5$. 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 $\beta$ of the power law $Nu\propto Ra^{\beta}$ crosses over from $\beta=0.221$ to 0.124 when the liquid metal flow at $\Gamma=0.5$ reaches $Ra\gtrsim 2\times 10^8$.
... For slow rotation, an LSC spanning the entire cell with two secondary corner rolls is observed in P whereas a four-roll structure is seen in P ⊥ , typical of classical RBC at large Ra and for Γ ∼ 1 (see e.g. Shishkina, Wagner & Horn 2014;Zwirner et al. 2020). Near the plates, the LSC and the secondary corner flows move the fluid towards the sidewall (figure 1b) so the Coriolis acceleration (−2Ωe z × u) induces anticyclonic fluid motion close to the plates. ...
... At γ = 1ork 1 = 2/7, we obtain the result (35) directly from (34); substituting (35)into(32) then gives f ξ (ξ m ) = e −1/3 √ 2/2, which is equal to the limiting value of (36)a sk 1 tends to 2/7. Using (35) and (36), we can calculate G ξξ (ξ m ) = 2 f (ξ m ) f ξ (ξ m ) and obtain ...
... Evaluating this result at ξ = ξ m and using (35), we obtain ...
... To compare our analytical results with numerical results, we use data obtained in previously conducted DNS, details of which were reported in [9,14,33,35]. More precisely, the DNS data for Pr = 0.005, 0.021, and 0.7 at various Ra are taken from [33], for Pr = 0.1 and Ra = 10 8 , the data are from [14], for Pr = 0.0094 and Ra = 1.67 × 10 7 , the data are from [35], and all other DNS data, analyzed and presented here, are taken from [9]. ...
Article
Using a closed set of boundary layer equations [E. S. C. Ching et al., Phys. Rev. Research 1, 033037 (2019)] for turbulent Rayleigh-Bénard convection, we derive analytical results for the dependence of the heat flux, measured by the Nusselt number (Nu), on the Reynolds (Re) and Prandtl (Pr) numbers and two parameters that measure fluctuations in the regime where the horizontal pressure gradient is negligible. This regime is expected to be reached at sufficiently high Rayleigh numbers for a fluid of any given Prandtl number. In the high-Pr limit, Nu=F1(k1)Re1/2Pr1/3 and, in the low-Pr limit, Nu tends to π−1/2Re1/2Pr1/2, where F1(k1) has a weak dependence on the parameter k1 in the eddy viscosity that measures velocity fluctuations. These theoretical results further reveal a close resemblance of the scaling dependencies of heat flux in steady forced convection and turbulent Rayleigh-Bénard convection and this finding solves a puzzle in our present understanding of heat transfer in turbulent Rayleigh-Bénard convection.
... In this section, we therefore want to investigate how spatial temperature variations very close to the walls, which are probed by temperature gradients, differ for the three different sets of BCs. Without discussing further details, we point here to an alternative experimental set-up that establishes a better temperature equilibration in the plates by Zwirner et al. (2020). ...
... Experiments with mercury in cylinders with H ¼ D and H ¼ 2D revealed a power law relation with c ¼ 2=7 % 0:286 over the range 10 5 Ra 10 11 [51]. Recent experiments with liquid sodium (Pr ¼ 0:0094) in a cell of unit aspect ratio (H ¼ D) showed a slightly lower value of the scaling exponent: c % 0:22 for 4 Â 10 6 < Ra < 2 Â 10 7 [52]. ...
... Direct numerical simulations of liquid metal RBC have been limited to moderate Ra, with the exception of the recent study [53] where Ra 10 9 were considered. They demonstrate consistent results with good agreement with experiments in terms of the Nu versus Ra scaling [52,54,55]. At the same time, the experimentally measured values of Nu tend to be lower than those obtained in computations. ...
... As a first illustration of the nonlinear and complex effect of the tilt angle, we will briefly review the recent results obtained for natural convection in inclined cells [52,55,[278][279][280][281]. It has been found that the cell inclination enhances large-scale circulation and modifies small-scale turbulence, affecting the global convective heat and mass transfer. ...
Article
An imposed strong magnetic field suppresses turbulence and profoundly changes the nature of the flow of an electrically conducting fluid. We consider this effect for the case of mixed convection flows in pipes and ducts, in which unique regimes characterized by extreme temperature gradients and high-amplitude fluctuations (the so-called magnetoconvective fluctuations) have been recently discovered. The configuration is directly relevant to the design of the liquid-metal components of future nuclear fusion reactors. This review presents the general picture of the flow transformation emerging from the recent studies, illustrates the key known facts, and outlines the remaining open questions. Implications for fusion reactor technology and novel experimental and numerical methods are also discussed.