This is the first study reporting a full three-dimensional (3D) non-normal-mode onset of convection in a porous medium. In this paper, the onset of thermal convection in a vertical porous cylinder is investigated theoretically. In particular, the contribution includes the following novelties. The homogeneous cylinder has a circular cross section. The eigenvalue problem is non-separable in space because of a constant heat flux condition at the upper boundary. In addition, a partly conducting cylinder wall is represented by a Robin parameter a . All boundaries are impermeable. The eigenvalue problem is solved numerically in the COMSOL Multiphysics environment. The critical Rayleigh number for the lowest onset modes is reported as a function of the ratio of the cylinder radius and its height. The only mode number, m , represents azimuthal dependency. The axisymmetric mode $$m=0$$ m = 0 corresponds to the preferred mode of convection for a small-cylinder radii. The numerical onset criterion is validated with the well-known analytical limit case, $$a \rightarrow \infty$$ a → ∞ . Finally, we performed a visual comparison of the thermo-mechanical eigenfunctions against an established problem, where the only difference is the thermal condition at the upper boundary. The present 3D analysis is a step toward a full adequate modeling of experimental reality for convection onset, beyond the standard constraints of mathematical convenience.

中文翻译：

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垂直多孔圆柱体中对流的非正常模式开始

这是第一项报告多孔介质中全三维 (3D) 非正常模式对流开始的研究。在本文中，从理论上研究了垂直多孔圆柱体中热对流的发生。特别是，贡献包括以下新颖性。均质圆柱体具有圆形横截面。由于上边界处的热通量条件恒定，特征值问题在空间中是不可分的。此外，部分导电的气缸壁由 Robin 参数 a 表示。所有的界限都是不可渗透的。特征值问题在 COMSOL Multiphysics 环境中以数值方式求解。最低起始模式的临界瑞利数报告为圆柱半径与其高度之比的函数。唯一的模式编号 m 表示方位角相关性。轴对称模式 $$m=0$$ m = 0 对应于小圆柱半径的首选对流模式。数值起始标准用众所周知的分析极限情况验证， $$a \rightarrow \infty$$ a → ∞ 。最后，我们对热机械本征函数与既定问题进行了视觉比较，其中唯一的区别是上边界的热条件。目前的 3D 分析是朝着对流开始的实验现实的完全充分建模迈出的一步，超出了数学便利的标准限制。我们针对已建立的问题对热机械本征函数进行了视觉比较，其中唯一的区别是上边界的热条件。目前的 3D 分析是朝着对流开始的实验现实的完全充分建模迈出的一步，超出了数学便利的标准限制。我们针对已建立的问题对热机械本征函数进行了视觉比较，其中唯一的区别是上边界的热条件。目前的 3D 分析是朝着对流开始的实验现实的完全充分建模迈出的一步，超出了数学便利的标准限制。