Abstract
The analytical theory on Darcy–Bénard convection is dominated by normal-mode approaches, which essentially reduce the spatial order from four to two. This paper goes beyond the normal-mode paradigm of convection onset in a porous rectangle. A handpicked case where all four corners of the rectangle are non-analytical is therefore investigated. The marginal state is oscillatory with one-way horizontal wave propagation. The time-periodic convection pattern has no spatial periodicity and requires heavy numerical computation by the finite element method. The critical Rayleigh number at convection onset is computed, with its associated frequency of oscillation. Snapshots of the 2D eigenfunctions for the flow field and temperature field are plotted. Detailed local gradient analyses near two corners indicate that they hide logarithmic singularities, where the displayed eigenfunctions may represent outer solutions in matched asymptotic expansions. The results are validated with respect to the asymptotic limit of Nield (Water Resour Res 11:553–560, 1968).
Highlights
Normal modes are in common use as analytical tools for solving eigenvalue problems of the fourth order and higher
Normal modes for the thermo-mechanical problem of convection onset has qualitative differences between the horizontal and vertical directions
The Darcy–Bénard onset problem of convection in a porous medium with rectangular geometry enjoys the status of being a highly mature problem, where little is left for further analysis
Summary
Normal modes are in common use as analytical tools for solving eigenvalue problems of the fourth order and higher. The first non-normal-mode solutions for the horizontal direction were presented by Nilsen and Storesletten (1990) They studied the HRL problem in 2D with conducting and impermeable sidewalls. If the boundary conditions for a HRL problem are compatible with normal modes in all spatial directions except for one, the eigenvalue problem can in most cases be solved analytically. The rectangular shape of all cell walls in the RT paper is a trivial fact, following from the separation of variables in the horizontal and vertical directions, being dictated by the normal-mode conditions at the lower and upper boundary. The highest possible complexity of a HRL eigenvalue problem in 2D will be exposed, with the simple combinations of either Dirichlet or Neumann conditions at all four walls of a porous rectangle with horizontal and vertical sides
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