Flow mode and global transport of liquid metal thermal convection in a cavity with $\mathrm{\Gamma }=1/3$

We report an experimental study of Rayleigh-Bénard convection of liquid metal thermal convection with a Prandtl number of $\mathrm{Pr}=0.029$ in the cuboid cell of aspect ratios $\mathrm{\Gamma }=\text{width/height}=1/3$. By arranging 48 thermistors and 6 ultrasonic sensors in the sidewall of the cell, the flow mode and corresponding flow and heat transfer characteristics are obtained with Rayleigh number Ra varying from $1.05\times {10}^6$ to $4.01\times {10}^7$. Three typical flow modes are defined from the results of temperature measurements, showing a double-roll mode (DRM) with two vertically stacked counterrotating rolls, a single-roll mode (SRM) that occupies the entire cell and a distorted single-roll mode existing in the transitions between DRM and SRM. Moreover, by gradually increasing the Ra one can observe the flow transition from DRM to SRM by the statistics of the lifetime for each flow mode. The complete Fourier analysis of temperature data shows that the origin of the flow modes can be attributed to the energy transformation between Fourier modes of different orders. When $\mathrm{Ra}\ge 1.19\times {10}^7$, the first Fourier mode energy has accounted for more than 75% of the total energy which is evidence that the SRM is fully established. Furthermore, the measured temperature profiles along the central axis of the cell follow a linear law in the bulk, and the slopes obey power-law scaling with Ra. Regarding the standard deviation of temperature profiles along the central axis of the cell which bulge in the bulk forming the second peak outside the thermal boundary layers at small Ra due to the complex three-dimensional flow structures. Finally, the similar scaling laws of the global heat and momentum transfer of the liquid metal thermal convection in a slender cuboid cell are observed, showing $\mathrm{Nu}\sim {\mathrm{Ra}}^{0.35}$ and $\mathrm{Re}\sim {\mathrm{Ra}}^{0.36}$ for heat and momentum transfer, respectively.

Figure 1

(a) Sketch of the convection apparatus and constant temperature system. (b) UDV transducers and thermistors locations on $l3$ layer. The UDV transducers measure the lateral $v_x$ and $v_y$ in the two orthogonal directions $x$ and $y$, respectively. The mean velocity ${\overline{v}}_x$ and ${\overline{v}}_y$ are averaged from the 1 cm near the crossing point. These two values are used as components of the horizontal velocity vector, denoted as $v_{\text{mid}}$ on this layer. ${\theta }_{l3}$ is measured by the eight thermistors on the sidewall based on the relative position of the hot ascending and cold descending plumes and used as the orientation of the LSC.

Figure 2

Example of the evolution of flow modes from SRM to DRM at $\mathrm{Ra}=6.23\times {10}^6$ and the free-fall time ${\tau }_{\text{ff}}\approx 5.0\mathrm{s}$. Time trace of (a) the flow orientation differences between the other four horizontal layers and the midheight layer $l3$. The hottest (coldest) temperature points on the sidewall are connected with red (blue) lines at each layer to sketch the flow modes (b) SRM, (c) TM, and (d) DRM, respectively. The color bar represents the degree of the normalized temperature fluctuations. (e) Time trace of the normalized flow strength.

Figure 3

(a) Percentage of the total run time during which the large-scale flow was in the SRM (circles), the TM (squares), and the DRM (triangles) as a function of Ra. The vertical dashed line indicates the critical Ra value, ${\mathrm{Ra}}_{\mathrm{c}}=1.19\times {10}^7$. (b) Average lifetimes of the SRM, the TM, and the DRM.

Figure 4

(a) Parameter $\overline{S_k}$ as a function of Ra for each horizontal layer. Panels (b) and (c) are the time series of the ratio of the first Fourier mode energy and the high-order Fourier mode energy to the total Fourier mode energy at $\mathrm{Ra}=6.23\times {10}^6$ and $\mathrm{Ra}=3.23\times {10}^7$, respectively. The horizontal dashed lines indicate a value of 0.75.

Figure 5

(a) Time-averaged temperature along the central axis of the cell as a function of the height $z$ at various Ra. (b) Example of the time-averaged temperature profile along the entire height of the cell (open symbols) and the mean temperature of the five horizontal layers (solid symbols) at $\mathrm{Ra}=3.97\times {10}^7$. The bulk flow temperature ${\mathrm{T}}_c$ and the sidewall temperature ${\mathrm{T}}_s$ are fitted by two linear functions, respectively, with respect to the height of the cell z. The slope of each linear fitting versus Ra is shown in panel (c). (d) Normalized vertical temperature gradients along the sidewall as a function of Ra.

Figure 6

The temperature standard deviations along the central axis of the cell as a function of the normalized height $z/H$ at various Ra.

Figure 7

(a)–(f) Time series of the temperature fluctuation at the cell center for different values of Ra. (g) PDF of the corresponding time series in panels (a)–(f).

Figure 8

(a) Nusselt number Nu as a function of the Rayleigh number Ra on logarithmic scales. The solid line is the best power-law fit for all the data. The inset plots the standard deviation of Nu vs Ra. (b) Normalized conditional average of Nu, $\langle \mathrm{Nu}{|}_{\mathrm{mode}}\rangle /\langle \mathrm{Nu}\rangle$ for the SRM (circles), the TM (squares) and the DRM (triangles), respectively, where $\langle \mathrm{Nu}\rangle$ is Nu averaged over the entire run.

Figure 9

Reynolds number Re as a function of the Rayleigh number Ra on logarithmic scales.