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

Abstract

The 4th order Darcy-Benard eigenvalue problem for the onset of thermal convection in a 3D rectangular porous box is investigated. We start from a recent 2D model for a rectangle with hand-picked boundary conditions defying separation of variables so that the eigenfunctions are of non-normal mode type. In this paper, the previous 2D model is extended to 3D by a Fourier component with wavenumber $k$ 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 >= 1) is preferred. Further, a continuous spectrum is allowed for the lateral wavenumber $k$, searching for the global minimum Rayleigh number at k=k_c and the transition between 2D and 3D flow at k=k*. Finally, these wavenumbers k_c and k* are studied as functions of the aspect ratio.

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