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

Abstract

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.