Effects of wall conductivities on magnetoconvection in a cube

This research article delves into the natural convection of liquid metal in a three-dimensional cavity with varying magnetic fields and wall conductivities using direct numerical simulation. Two opposite sidewalls are heated/cooled, while the magnetic field is perpendicular to the main circulation. Our primary focus is on examining flows within the Grashof number, $\text{Gr}⩽{10}^8$, the Hartmann number, $\text{Ha}⩽400$, and the wall conductance ratio, $C_w=0.01$–1. It is found that weakly conducting walls experience a significant enhancement in convection within a specific range of magnetic field strength, whereas highly conducting walls exhibit pronounced flow attenuation. The applied horizontal magnetic field alters the plume's topology and dynamics, generating a more coherent and energetic large-scale flow structure, while it weakens convection by consuming buoyant potential energy through Joule dissipation. This results in a competition between the rectifying effect and the damping effect to determine whether the magnetic field has a positive or negative feedback on heat transfer, with the quasi-two-dimensional state serving as a critical point. Additionally, varying wall conductivities transform the current distribution within the parallel layer, influencing the flow's response to changes in field strength. The formation of corner vortices can be considered by the curvature of the boundary layer that undergoes turning at corners. Furthermore, the effects of wall shear and plume transport on heat transfer are systematically investigated. The analysis reveals that while the plume area remains almost constant, the condensation of coherent structures facilitates greater horizontal heat transport per unit area of the plume, contributing significantly to the overall heat transfer enhancement. Finally, the computed Nusselt number Nu and Reynolds number Re can be correlated as functions of ${\text{Ha/Gr}}^{1/3}$. The critical conductance ratio referring to the complete suppression of convection conforms to the scaling of $2.31{\left({\text{Ha/Gr}}^{1/3}\right)}^{-5}$, where heat transfer occurs solely by thermal conduction once it exceeds this value.

Figure 1

Schematic diagram of the studied physical model.

Figure 2

(a) Temperature monitoring for $\text{Gr}={10}^8,\text{Ha}=200$, and $C_w=0.1$ at the probe location $(0.45,0,0)$. (b) Comparison of Nu between Gajbhiye and Eswaran [26] and the present simulations at $\text{Pr}=0.01$ and $C_w=0.01$.

Figure 3

Buoyant convection for $\text{Gr}=3\times {10}^7,\text{Ha}=200$, and $C_w=0.1$.

Figure 4

The distribution of the mean vertical velocity along the $x$-axis near the cold wall for $\text{Gr}={10}^8$.

Figure 5

Streamline plots at the midplane $z=0$ for $\text{Gr}=8\times {10}^7$.

Figure 6

The isosurface of $Q_{3\text{D}}={0.5*\left(\right|\left|A\right||}_F^2-{\left|\right|S\left|\right|}_F^2)$ (the second invariant of the velocity gradient tensor, $S$ is a symmetric tensor and $A$ is an antisymmetric tensor) represents the roll structures for $\text{Gr}=8\times {10}^7$, colored by local temperature.

Figure 7

The distribution of vertical Lorentz force along the $x$-axis and electric current in solid-fluid domain for $\text{Gr}=3\times {10}^7$.

Figure 8

Flow rate estimation of the horizontal jet in the cavity.

Figure 9

The spanwise distribution of physical quantities at the peak of jet for $\text{Gr}={10}^7$ and $\text{Ha}=400$.

Figure 10

Time series of temperature fluctuations monitored at the thermal boundary layer and the PSD for $\text{Gr}=1\times {10}^8$.

Figure 11

The distribution of the mean temperature along the $x$-axis for $\text{Gr}={10}^8$.

Figure 12

The isothermal surface inside the fluid zone for $\text{Gr}=8\times {10}^7$.

Figure 13

The evolution of stratification parameter $S_{\theta }$ vs ${\text{Ha/Gr}}^{1/2}$.

Figure 14

Contours of pressure distribution with streamlines (black lines) and isothermals (white lines) at the midplane $z=0$ for $\text{Gr}=8\times {10}^7$.

Figure 15

Horizontal and vertical boundary layer thicknesses calculated using surface-time-averaged velocity profiles for $C_w=0.01$ and 1 $[{\delta }_u=U/(d\overline{u}/dx{|}_{\text{wall}})]$.

Figure 16

Phase diagram representing the corner vortex morphology in the cavity at each parameter. Squares represent the presence of four corner vortices, circles denote only two vortices, lower left and upper right, and triangles signify the absence of vortices. The symbols are colored by the ratio of the thickness of the horizontal and vertical boundary layers.

Figure 17

Normalized global heat and momentum transports as functions of the control parameter ${\text{Ha/Gr}}^{1/3}$.

Figure 18

Cross-sections of various quantities at $\text{Gr}=1\times {10}^8$. The parameters from left to right are as follows: $\text{Ha}=0;C_w=0.01,\text{Ha}=100;C_w=0.05,\text{Ha}=100;C_w=0.1,\text{Ha}=100;C_w=1,\text{Ha}=100$.

Figure 19

Statistics of dimensionless physical quantities related to heat transfer. The $C_w$ corresponding to the parameter points in the figures are 0.01, 0.05, 0.1, and 1 from left to right, using a logarithmic treatment. (a) Volume-averaged Nusselt number ${\text{Nu/Nu}}_0$ and mean shear rate $|\overline{{\tau }_f}|/|\overline{{\tau }_f}{|}_0$ at the heated wall $x=0.5$. (b) Plume area $S/S_0$ in the mid-$yz$ plane extracted from the mean field and the actual heat flux content $Q_p/Q_{p,0}$ transported within the plume region.

Figure 20

The critical conditions to approach heat conduction for various wall conductance ratios. The inset represents the threshold $C_w^{*}$ as a function of Ha and Gr. If $C_w>C_w^{*}$, heat transfer degenerates into thermal conduction, as shown in the shaded area.