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# The parameter a from the relation δ u /L = a √ Re (cf. Prandtl 1905) vs the Reynolds number. All data from DNS at different aspect ratios Γ for the Reynolds number based on the wind velocity (2.7) and Re L .

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Using complementary experiments and direct numerical simulations, we study turbulent thermal convection of a liquid metal (Prandtl number $\textit {Pr}\approx 0.03$ ) in a box-shaped container, where two opposite square sidewalls are heated/cooled. The global response characteristics like the Nusselt number ${\textit {Nu}}$ and the Reynolds number...

## Citations

... . This dependency in DHVCs has recently attracted increasing attention. 8,[14][15][16][17][18][19] Depending on the type of the flow regime, the scaling exponent of Nu $ Ra c Nu varies from 1/4 to 1/3. Shishkina 15 theoretically derived the dependencies of Nu and Re on Ra and Pr by studying similarity solutions for boundary layer (BL) equations in laminar DHVCs. ...

... They discovered that the scaling relationships suggest that Nu $ Ra 0:25 and Re $ Ra 0:37 in the laminar regime. Zwirner et al. 18 investigated the turbulent thermal convection of a liquid metal with a very small Prandtl number (Pr % 0:03) in a box-shaped container using a combination of experiments and DNS, where 5 Â 10 3 Ra 10 8 . They introduced a novel method for extracting wind-based Re values from the 2D autocorrelation of the time-space data of the vertical velocity. ...

In this study, we examined non-Oberbeck–Boussinesq (NOB) effects on a water-filled differentially heated vertical cavity through two-dimensional direct numerical simulations. The simulations encompassed a Rayleigh number (Ra) span of 107–1010, temperature difference (Δθ̃) up to 60 K, and a Prandtl number (Pr) fixed at 4.4. The center temperature (θcen) was found to be independent of Ra and to increase linearly with Δθ̃, as presented by θcen≈1.18×10−3 K−1Δθ̃. The thermal boundary layer (BL) thicknesses near the hot and cold walls (λ¯hθ and λ¯cθ, respectively) are found to scale as λ¯h,cθ∼Raγ λ¯h,c, where the scaling exponent γ λ¯h,c ranges from −0.264 to −0.262. For more detail, the scaling exponent γ λ¯h displays an increasing trend, while γ λ¯c demonstrates a decreasing trend. However, the sum of the hot and cold thermal BL thicknesses was found to be constant at a fixed Ra in the presence of NOB effects. Our detailed investigation of the Nusselt number (Nu) and Reynolds number (Re) revealed that Nu∼Ra0.258 and Re∼Ra0.364, showing insensitivity to NOB effects. These exponents were smaller than those for Rayleigh–Bénard convection. The NOB modifications on Nu and Re were less than 1.2% and 2.5%, respectively, even at Δθ̃=60 K. Our results also revealed that key parameters such as θcen and normalized ratios [(λ¯NOBθ/λ¯OBθ)h,c, NuNOB/NuOB, and ReNOB/ReOB] exhibit universal correlations with Δθ̃. Remarkably, these relationships are consistent across varying Ra values. This observation underscored the influence of NOB effects on these parameters could be confidently forecasted using just the temperature difference (Δθ̃) for Ra∈[107,1010].

... Nowadays techniques for the individual measurement of velocities and temperatures in LMs exist but simultaneous and fast measurements of these two quantities in such environment have not been published yet. In fact, most of the work addressing liquid metal measurements are focused on average quantities [10], finding correlations in a particular setup [11] or largescale dynamics [12]. ...

... A schematic of the setup is shown in Fig. 2. In differentially heated cavities, a constant temperature difference is maintained between two opposite side walls while all other sides of the cavity are insulated, i.e. ideally adiabatic. This configuration usually gives a flow with similar characteristics to the isothermal vertical plate on its active walls and for this reason it is also called vertical convection (VC) [11] Inside the cavity, the imposed temperature difference triggers a natural convection loop rotating in the direction from the warmer side towards the colder side [30]. The main features of the flow change according to the working fluid but in all cases the main non-dimensional parameters that control the flow and the turbulence phenomena inside the cavity are the Prandtl number (Pr), the Grashof number (Gr) and the Rayleigh number (Ra = Gr*Pr). ...

Turbulent heat fluxes (THFs) estimation is of paramount importance in the determination of heat transfer in fluids. Numerical models for low Prandtl number fluids are still unreliable and their experimental evaluation is a challenging task since it requires simultaneous measurement of fast velocity and temperature fluctuations. In nuclear applications, a better understanding of THFs in liquid metals could lead to more precise predictions of the primary coolant temperature for the evaluation of nominal operation (forced convection regime) and accidental conditions (mixed and/or natural convection regime). The first part of this work focuses on the selection of the measurement techniques suitable for water, GaInSn and LBE and the thorough literature review required. Tests in different setups led to the choice of sheathed type K thermocouples and fiber Bragg gratings for temperature measurements and Ultrasound Doppler Velocimetry and Hot Wire Anemometry for velocity measurements. The comparison carried out among the different techniques underlines advantages and limitations of each of them. Calibration of each technique is performed and cross-effects of temperature and velocity are evaluated. Uncertainty analyses are also carried out. To conclude, first results obtained in a differentially heated cavity made of stainless steel 316L with an edge of 60 mm are presented. DNS numerical simulations are performed to know the ranges of the quantities to be measured and to have results available for comparison with experiments.

... The non dimensional Biot (Bi) number represents the flow of heat through the fluid-wall interface. Thus, this last boundary condition corresponds to the limit Bi → ∞, which means a very good conducting wall in comparison with the fluid as used in the case of convection in a cylinder under damped thermal flux [2] and magnetoconvection in a cylinder [3,4]. On the other hand, the ideal fixed heat flux condition at the boundary, which corresponds to a bad conducting or adiabatic wall in the limit Bi → 0, has also been given some attention [5,6,7,8,9,10]. ...

... In blanket, liquid metal flows at a slow speed (usually a few mm/s) [9] and works in an environment of large temperature differences and strong magnetic field, so the buoyancy effect becomes very important. The canonical configurations for studying the buoyant convection are the Rayleigh-Bernard convection (RBC) [10][11][12][13][14], where the liquid metal is confined between a cooled top plate and a heated bottom plate, and vertical convection (VC) [15,16], where the liquid metal is confined between two differently heated isothermal vertical walls. Shishkina [15] derived the scaling laws of the Reynolds number and Nusselt number in VC without magnetic field for fluids with the Prandtl numbers that range from 10 -2 to 30 based on the boundary layer theory; liquid metals with the smaller Prandtl number have better heat transport performance than fluids with the larger Prandtl number. ...

In the fusion reactor, the conducting liquid metals usually work in an environment of large temperature differences and strong magnetic field. The flow driven by the interaction of the Seebeck effect and magnetic field enlightens a promising approach to enhance heat transfer under strong magnetic field. Liquid metal thermal convection affected by the Seebeck effect and magnetic field is simulated using the partitioned iteration algorithm with liquid lithium as working fluid. It is found that the Seebeck effect can change energy transport pattern and greatly improve the heat transfer efficiency under strong magnetic field. With the increase of magnetic field intensity, the flow changes from steady vertical circulation to unsteady horizontal circulation and finally to steady horizontal circulation. The flow regime diagram based on the two dimensionless parameters, Gr / Te and Ha 2 / Te , can reflect the characteristics of different energy transport patterns. The flow generated by the Seebeck effect is most remarkable when O Ha 2 / Te ≈ 1 . The Nusselt numbers at different flow regimes show that the Seebeck effect can enhance the heat transfer efficiency of liquid metal under strong magnetic field about 50% and 90%, respectively, under different Glashof numbers.

... Rayleigh-Bénard convection (RBC), a fluid layer heated from below and cooled from above, is a classical model to study turbulent convection [7][8][9][10] . Another ongoing interesting system in the study of buoyancy-driven turbulent flows is vertical natural convection (VC), in which the temperature gradient is perpendicular to the gravity [11][12][13][14][15] . 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 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.

... 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.

... 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 ...

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 $.

Numerical case studies for natural convection were conducted for triangular and hexagonal enclosures as a follow-up to a previous study of natural convection for a baffled cylinder in a square enclosure. For a Prandtl number of 0.71, a range of the length ratio corresponding to the spatial ratio was varied from 0.4 to 0.9, and Rayleigh numbers from 2.5 × 10³ to 5.0 × 10⁷ was considered. To model the buoyancy force, the Boussinesq approximation was adopted. The boundary condition of constant wall temperature was applied to the cylinder and enclosure surfaces. Overall trends of the heat transfer performance were separated into conduction- and convection-dominated regimes, similar to the previous study. Because of the wider space of the triangular enclosure, however, an advantage of the convection effect was observed in the triangular enclosure. Due to the short distance between the cylinder and enclosure, the hexagonal enclosure had a more conduction-dominated characteristic than that of the triangular enclosure. A strong correlation was identified between the location of the high-energy vortex and the Nusselt number distribution. Using the calculation results, an improved average Nusselt number correlation is proposed to include triangular, square and hexagonal enclosures.

Kurzfassung
Mit der kontaktlosen induktiven Strömungstomografie (CIFT) lassen sich Geschwindigkeitsfelder in elektrisch leitfähigen Flüssigkeiten global bestimmen. Kenntnisse über den Strömungszustand in Metallschmelzen sind für industrielle Prozesse, wie das Stranggießen von Stahl, von immenser Bedeutung und können auch in der Grundlagenforschung nutzbringend angewendet werden, z.B. zur Analyse von konvektiven Flüssigmetallströmungen als Modellsysteme des Wärmetransportes. Das Verfahren beruht auf der präzisen Messung kleinster Magnetfeldänderungen durch geeignete Sonden und der nachfolgenden Rekonstruktion der Strömungsstruktur durch die Lösung eines linearen inversen Problems. In dieser Veröffentlichung geben wir einen Überblick über die Entwicklungen der letzten Dekade und diskutierten je einen Anwendungsfall für CIFT aus der grundlegenden und der angewandten Fluiddynamik.