Oscillatory Non-normal-Mode Onset of Convection in a Porous Rectangle

Abstract

A special case of the fourth-order Darcy–Bénard problem in a two-dimensional (2D) rectangular porous box is investigated. The present eigenfunctions are of non-normal-mode type in the horizontal and vertical directions. They compose a time-periodic wave with one-way propagation out of the porous rectangle. Asymmetry in the horizontal direction generates an oscillatory time dependence of the marginal state, similar to Rees and Tyvand (Phys Fluids 16:3706–3714, 2004a). No analytical method is known for this non-degenerate eigenvalue problem. Therefore, the problem was solved numerically by the finite element method (FEM). Three boundaries of the rectangle are impermeable. The right-hand wall is fully penetrative. The lower boundary and the left-hand wall are heat conductors. The upper boundary has a given heat flux. The right-hand wall is thermally insulating. As a result, the computed eigenfunctions show complicated periodic time dependence. Finally, the critical Rayleigh number and the associated angular frequency are calculated as functions of the aspect ratio and compared against the case of normal modes in the vertical direction.

Oscillatory Non-normal-Mode Onset of Convection in a Porous Rectangle