Abstract
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 \(m=0\) 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, \(a \rightarrow \infty\). 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.
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Tyvand, P.A., Nøland, J.K. Non-normal-Mode Onset of Convection in a Vertical Porous Cylinder. Transp Porous Med 136, 319–341 (2021). https://doi.org/10.1007/s11242-020-01513-x
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DOI: https://doi.org/10.1007/s11242-020-01513-x