Effect of Uniform and Non-uniform wall heating on Three-Dimensional Magneto-Hydrodynamics Natural Convection and Entropy Generation: A computational study using New Higher Order Super Compact Scheme

TL;DR

This study tackles 3D MHD natural convection in a cubic cavity filled with molten lithium under uniform and nonuniform wall heating, employing a novel Higher-Order Super Compact (HOSC) finite-difference scheme. The method achieves $2^{nd}$-order temporal and $4^{th}$-order spatial accuracy on a compact $7$-point stencil, with pressure handled by a modified compressibility approach and solved via Hybrid Biconjugate Gradient Stabilized iterations. Key findings show that increasing Rayleigh number $Ra$ enhances convection and heat transfer, while increasing Hartmann number $Ha$ dampens flow and reduces heat transfer; entropy generation analyses reveal Bejan-number dynamics and a transition from heat-transfer-dominated to viscous/magnetic irreversibilities at high $Ra$, varying with boundary conditions. The work validates the HOSC approach against literature and demonstrates its capability to capture complex 3D MHD convection and thermodynamics, offering design insights for thermal management and magnetized-fluid systems. Overall, the paper contributes a novel numerical tool and provides nuanced understanding of how $Ha$, $Ra$, and wall-heat patterns govern 3D MHD natural convection and entropy production in molten-metal applications.

Abstract

Current research work deals with the effect of uniform and non-uniform wall heating on magnetohydrodynamic (MHD) natural convection within a three-dimensional (3D) cavity filled with molten lithium. A new Higher-Order Super Compact (HOSC) finite difference scheme is used to analyze the thermal behavior under both heating scenarios. After the quantitative and qualitative validations, the computed results are analyzed for a range of Hartman number ($Ha = 25, 50, 100, 150$) and Rayleigh number ($Ra = 10^3, 10^4, 10^5$) with fixed $Pr=0.065$ (molten lithium). Three distinct heating scenarios, i.e., uniform heating ($T_h = 1$), $y$-dependent non-uniform heating ($T_h = sin(πy$)), and a combination of $y$ and $z$-dependent non-uniform heating ($T_h = sin(πy)sin(πz)$) are investigated on the left wall ($x=0$) of the cubic cavity. It is found that variations in the $Ha$ and $Ra$, along with distinct thermal boundary conditions, exert significant effects on both the temperature distribution and flow field inside the 3D cubical cavity. Specifically, an increase in $Ra$ corresponds to enhanced heat transfer, highlighting the dominance of convection. Conversely, an increase in $Ha$ leads to a reduction in heat transfer due to the deceleration of fluid velocity. The scenario in which walls are uniformly heated exhibits the most significant total entropy generation. It is observed that with an increase in the $Ra$, the Bejan number ($Be$) decreases, which ultimately leads to an increase in total entropy generation. The implementation of the new HOSC scheme in this analysis showcases its effectiveness in capturing the complexities of 3D MHD-driven natural convection and entropy generation. This study offers significant information that might help improve the optimization and design of relevant engineering systems. Thus, our work stands out as genuinely novel and pioneering in its approach.

Figures (26)

• Figure 1: (a-c) Schematic view of the problem for three different cases (d) Visualization of the grids with $51\times51\times51$ grid size.
• Figure 2: The super-compact unsteady stencil
• Figure 3: Comparison of stream‐function and isotherm contours (at $Ra = 10^5$) between (a-d) existing Chamkha_2012 and (e–h) present computed results
• Figure 4: Comparison of Isotherms (at $Ra=10^3$, $Ha=90$) for natural convection on symmetry plane ($z=0.5$): Moderres et al. (2023) Moderres_2023 vs. present result
• Figure 5: Case 1. Effect of different $Ha$ (Column-wise) and $Ra$ (Row-wise) on streamlines pattern at different $xy$ planes $(z=0.05, z=0.5, z=0.95)$ with fixed $Pr = 0.065$