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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...
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... 16 Since then, VC has become a prevalent topic in experimental, theoretical, and numerical studies. [17][18][19][20][21][22][23][24][25][26][27] The VC system is usually applied to sidewall heating or cooling situations, for example, ice melting and reactor cooling. 28,29 The study of VC systems can also help us understand more complex thermal convection phenomena. ...
We employ the direct numerical simulation to study the heat transfer behavior and flow structures in a vertical convection system with rough sidewalls. The parameters are chosen with Rayleigh number spanning the range of 1×108≤Ra≤3×1010 and Prandtl number fixed at 1.0. The results reveal that the impact of rough walls on the Nusselt number Nu and the Reynolds number Re is influenced by the height of the rough element h. When h is not sufficiently high, the roughness impedes the flows within the boundary layer and traps massive heat between rough elements, and both Nu and Re are lower than those in the smooth-wall case. However, the extent of the Nu and Re reduction regimes decreases as Ra increases. For sufficiently large Ra, the reduction regime for both Nu and Re may vanish, and roughness breaks up the limitation of the thermal boundary layer and facilitates the eruption of thermal plumes from roughness tips, resulting in the enhancement of both Nu and Re. Based on these results, the critical heights hc for Nu and hcr for Re are identified. Both exhibit similar scaling behavior with Ra, with hc consistently being larger than hcr for the same value of Ra.
... This setup makes DHVC an invaluable tool in investigating thermal patterns, fluid motion, and their underlying interaction, thereby attracting significant attention from experimental, theoretical, and numerical researchers for decades. [1][2][3][4] The scientific inquiry into DHVC began with Batchelor's pioneering work, 1 which first addressed the steady-state heat transport across the double-glazed windows. Subsequent studies by Elder 5,6 explored laminar and turbulent free convection in vertical slots, providing key insights into flow dynamics and heat transport mechanisms. ...
... This study has extended our comprehension of DHVC dynamics by confirming scaling relations of Nu $ Ra 0:25 and Re $ Ra 0:37 in laminar DHVCs, particularly within a Ra range of 10 8 -10 10 . Recently, Zwirner et al. 2 numerically and experimentally explored turbulent thermal convection of liquid metal with small Pr % 0:03 in a boxshaped container with various aspect ratios. They developed a method to accurately derive the wind-based Re values from experimental Doppler-velocimetry data, utilizing 2D autocorrelation of vertical velocity time-space data. ...
This study explores the non-Oberbeck–Boussinesq (NOB) effects on hydrodynamics and heat transport in two-dimensional glycerol-filled differentially heated vertical cavity (DHVC). The simulations span Rayleigh numbers (Ra) from 2 × 10 3 to 5 × 10 9 and temperature difference ( Δ θ ̃) up to 50 K at a Prandtl number (Pr) of 2547. We showed the emergence of stratified flow structures, delineated the NOB effects on temperature distribution symmetry, and analyzed the scaling behaviors of the Nusselt number (Nu), Reynolds number (Re), and thermal boundary layer (BL) thicknesses ( λ ¯ h θ and λ ¯ c θ) against Ra. For R a ≥ 3 × 10 5, the stratification number (S) shows reduced sensitivity to changes in Ra, stabilizing around 0.5. Additionally, the center temperature ( θ cen) appears to be unaffected by Ra and increases linearly with Δ θ ̃ for R a > 10 6, satisfying θ cen ≈ 2.99 × 10 − 3 K − 1 Δ θ ̃. Our results also revealed that N u ∼ R a γ Nu and R e ∼ R a γ Re with 0.2649 ≤ γ Nu ≤ 0.2654 and 0.3633 ≤ γ Re ≤ 0.3643, respectively, where γ Nu and γ Re exhibit a monotonic decrease as NOB effects intensify. For all investigated Ra values, N u NOB / N u OB < 1 and R e NOB / R e OB > 1 hold consistently, with deviations from OB predictions capped at 6.38% and 2.63% for R a ≥ 10 8, respectively. The analysis of thermal BL thickness reveals distinct scaling behaviors, characterized by λ ¯ h , c θ ∼ R a γ λ ¯ h , c, with scaling exponents ranging from −0.2690 to −0.2669 for both OB and NOB scenarios. Notably, it reveals a divergence from water-based DHVC trends, showing linear decreases in the hot wall's scaling exponent and increases for the cold wall.
... With this configuration, research is still concerned to identify proper scaling laws from the setup's characteristic Rayleigh number Ra to the heat transport and boundary layer dimensions. In Zwirner et al. [48] it was shown by simulations and experiments, that the plate size is a much more dominating factor than the aspect ratio in contrast to the horizontal setup. ...
... Interestingly, for Ra below 1 × 10 5 the velocity profile in the viscous boundary layer can be detected. According to Zwirner et al. [48] the boundary layer should be in the range of 2.5 to 1.5 mm, of which ULM is here shown to detect. For increased Ra the boundary layer is further pushed to the wall and the velocity gradient increases. ...
Convection of liquid metals drives large natural processes and is important in technical processes. Model experiments are conducted for research purposes where simulations are expensive and the clarification of open questions requires novel flow mapping methods with an increased spatial resolution. In this work, the method of Ultrasound Localization Microscopy (ULM) is investigated for this purpose. Known from microvasculature imaging, this method provides an increased spatial resolution beyond the diffraction limit. Its applicability in liquid metal flows is promising, however the realization and reliability is challenging, as artificial scattering particles or microbubbles cannot be utilized. To solve this issue an approach using nonlinear adaptive beamforming is proposed. This allowed the reliable tracking of particles of which super-resolved flow maps can be deduced. Furthermore, the application in fluid physics requires quantified results. Therefore, an uncertainty quantification model based on the spatial resolution, velocity gradient and measurement parameters is proposed, which allows to estimate the flow maps validity under experimental conditions. The proposed method is demonstrated in magnetohydrodynamic convection experiments. In some occasions, ULM was able to measure velocity vectors within the boundary layer of the flow, which will help for future in-depth flow studies. Furthermore, the proposed uncertainty model of ULM is of generic use in other applications.
... Previous works (George & Capp 1979) also pointed out this possibility. Finally, our results show the same 1/5 Prandtl exponent dependence predicted by Shishkina (2016) and recently emphasized by Zwirner et al. (2022). However, we recall here that we have not checked this Prandtl scaling with our numerical simulations, which were all performed at P r = 0.1. ...
Motivated by nuclear safety issues, we study the heat transfers in a thin cylindrical fluid layer with imposed fluxes at the bottom and top surfaces (not necessarily equal) and a fixed temperature on the sides. We combine direct numerical simulations and a theoretical approach to derive scaling laws for the mean temperature and for the temperature difference between the top and bottom of the system. We find two asymptotic scaling laws depending on the flux ratio between the upper and lower boundaries. The first one is controlled by heat transfer to the side, for which we recover scaling laws characteristic of natural convection (Batchelor, Q. Appl. Maths , vol. 12, 1954, pp. 209–233). The second one is driven by vertical heat transfers analogous to Rayleigh–Bénard convection (Grossmann & Lohse, J. Fluid Mech. , vol. 407, 2000, pp. 27–56). We show that the system is inherently inhomogeneous, and that the heat transfer results from a superposition of both asymptotic regimes. Keeping in mind nuclear safety models, we also derive a one-dimensional model of the radial temperature profile based on a detailed analysis of the flow structure, hence providing a way to relate this profile to the imposed boundary conditions.
... . 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 and ) 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 () and Rayleigh number (). 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 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 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 . The horizontal velocity components were measured at a horizontal cross-section at half height. The Rayleigh numbers were in the range . The effect of rotation is quantified by the Ekman number, which was in the range 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 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.
