Abstract In this paper, a numerical investigation of natural convection in a porous medium confined by two horizontal eccentric cylinders is presented. The cylinders are impermeable to fluid motion and retained at uniform different temperatures. While, the annular porous layer is packed with glass spheres and fully-saturated with air, and the cylindrical packed bed is under the condition of local thermal non-equilibrium. The mathematical model describing the thermal and hydrodynamic phenomena consists of the two-phase energy model coupled by the Brinkman-Forchheimer-extended Darcy model under the Boussinesq approximation. The non-dimensional derived system of formulations is numerically discretised and solved using the spectral-element method. The investigation is conducted for a constant cylinder/particle diameter ratio ( D i / d ) = 30, porosity ( e ) = 0.5, and solid/fluid thermal conductivity ratio ( k r ) = 38.6. The effects of the vertical, horizontal and diagonal heat source eccentricity (−0.8 ⩽ e ⩽ 0.8) and the annulus radius ratio (1.5 ⩽ RR ⩽ 5.0) on the temperature and velocity distributions as well as the overall heat dissipation within both the fluid and solid phases, for a broad range of Rayleigh number (104 ⩽ Ra ⩽ 8 × 10 7 ). The results show that uni-cellular, bi-cellular and tri-cellular flow regimes appear in the vertical eccentric annulus at the higher positive eccentricity e = 0.8 as Rayleigh number increases. However, in the diagonal eccentric annulus, the multi-cellular flow regimes are shown to be deformed and the isotherms are particularly distorted when Rayleigh number increases. In contrast, in the horizontal eccentric annulus, it is found that whatever the Rayleigh number is only an uni-cellular flow regime is seen. In addition, it is shown that the fluid flow is always unstable in the diagonal eccentric geometry at e = 0.8 for moderate and higher Rayleigh numbers. However, it loses its stability in the vertical eccentric geometry only at two particular cases, while it is always stable in the horizontal eccentric geometry, for all eccentricities and Rayleigh numbers.

中文翻译：

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含有球形颗粒的偏心环中的非达西贝纳对流

摘要 本文对由两个水平偏心圆柱体约束的多孔介质中的自然对流进行了数值研究。圆柱体对流体运动是不可渗透的，并保持在均匀的不同温度下。而环形多孔层被玻璃球填充并充满空气，圆柱形填充床处于局部热非平衡状态。描述热力学和流体动力学现象的数学模型由在 Boussinesq 近似下的 Brinkman-Forchheimer 扩展达西模型耦合的两相能量模型组成。配方的无量纲派生系统在数值上离散并使用谱元方法求解。研究是针对恒定的圆柱体/颗粒直径比 (D i / d ) = 30 进行的，孔隙率 (e) = 0.5，固体/流体热导率比 (kr) = 38.6。垂直、水平和对角线热源偏心率 (−0.8 ⩽ e ⩽ 0.8) 和环空半径比 (1.5 ⩽ RR ⩽ 5.0) 对温度和速度分布以及流体和流体内整体散热的影响固相，适用于广泛的瑞利数 (104 ⩽ Ra ⩽ 8 × 10 7 )。结果表明，随着瑞利数的增加，在较高的正偏心度 e = 0.8 下，垂直偏心环中出现单细胞、双细胞和三细胞流动状态。然而，在对角偏心环中，当瑞利数增加时，多细胞流动状态显示变形并且等温线特别扭曲。相比之下，在水平偏心环带中，发现无论瑞利数如何，都只能看到单细胞流动状态。此外，结果表明，对于中等和较高瑞利数，在 e = 0.8 的对角偏心几何中，流体流动总是不稳定的。然而，它只在两种特殊情况下在垂直偏心几何中失去稳定性，而在水平偏心几何中，对于所有偏心和瑞利数，它总是稳定的。