A Non-normal-mode Marginal State of Convection in a Porous Box with Insulating End-Walls

  • Peder A. Tyvand
  • Jonas Kristiansen NølandEmail author


The 4th order Darcy–Bénard eigenvalue problem for the onset of thermal convection in a 3D rectangular porous box is investigated. We start from a recent 2D model Tyvand et al. (Transp Porous Med 128:633–651, 2019) for a rectangle with handpicked boundary conditions defying separation of variables so that the eigenfunctions are of non-normal mode type. In this paper, the previous 2D model (Tyvand et al. 2019) is extended to 3D by a Fourier component with wave number k in the horizontal y direction, due to insulating and impermeable sidewalls. As a result, the eigenvalue problem is 2D in the vertical xz-plane, with k as a parameter. The transition from a preferred 2D mode to 3D mode of convection onset is studied with a 2D non-normal mode eigenfunction. We study the 2D eigenfunctions for a unit width in the lateral y direction to compare the four lowest modes \(k_m = m \pi ~(m=0,1,2,3)\), to see whether the 2D mode \((m=0)\) or a 3D mode \((m\ge 1)\) is preferred. Further, a continuous spectrum is allowed for the lateral wave number k, searching for the global minimum Rayleigh number at \(k=k_c\) and the transition between 2D and 3D flow at \(k=k^*\). Finally, these wave numbers \(k_c\) and \(k^*\) are studied as functions of the aspect ratio.


Convection Non-normal mode Onset Porous box Rayleigh–Bénard problem 



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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical Sciences and TechnologyNorwegian University of Life SciencesÅsNorway
  2. 2.Faculty of Information Technology and Electrical EngineeringNorwegian University of Science and TechnologyTrondheimNorway

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