Non-normal-Mode Onset of Convection in a Vertical Porous Cylinder
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Non‑normal‑Mode Onset of Convection in a Vertical Porous Cylinder Peder A. Tyvand1 · Jonas Kristiansen Nøland2 Received: 16 December 2019 / Accepted: 13 November 2020 © Springer Nature B.V. 2020
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 → ∞ . 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. Keywords Convection · Onset · Non-normal modes · Porous cylinder
1 Introduction The onset of convection in a horizontal porous layer heated from below was first investigated in two pioneering papers (Horton and Rogers 1945; Lapwood 1948). The lower and upper boundaries were assumed impermeable and thermally conducting, leading * Jonas Kristiansen Nøland [email protected] Peder A. Tyvand [email protected] 1
Faculty of Mathematical Sciences and Technology, Norwegian University of Life Sciences, 1432 Ås, Norway
2
Faculty of Information Technology and Electrical Engineering, Norwegian University of Science and Technology, 7034 Trondheim, Norway
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P. A. Tyvand, J. K. Nøland
to normal-mode-type eigenfunctions in the vertical direction. With infinite horizontal extent, a Fourier decomposition in the horizontal direction implies normal-mode dependency. Beck (1972) generalized the original theory to a rectangular porous cavity with impermeable and thermally insulating walls, which preserves the normal-mode type of solution in the horizontal directions. Tyvand and Storesletten (2018) pointed out that normal modes in a spatial direction represent a degeneracy of the eigenvalue problem, reducing it from a fourth-order problem to an essentially second-order problem. While Wooding (1959) had shown how normal modes could be applied to the onset problem for any vertical cylinder with insulating and
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