Abstract
The onset of thermal convection in two-dimensional porous cavities heated from below is studied theoretically. An open (constant-pressure) boundary is assumed, with zero perturbation temperature (thermally conducting). The resulting eigenvalue problem is a full fourth-order problem without degeneracies. Numerical results are presented for rectangular and elliptical cavities, with the circle as a special case. The analytical solution for an upright rectangle confirms the numerical results. Streamlines penetrating the open cavities are plotted, together with the isotherms for the associated closed thermal cells. Isobars forming pressure cells are depicted for the perturbation pressure. The critical Rayleigh number is calculated as a function of geometric parameters, including the tilt angle of the rectangle and ellipse. An improved physical scaling of the Darcy–Bénard problem is suggested. Its significance is indicated by the ratio of maximal vertical velocity to maximal temperature perturbation.
Highlights
The Darcy–Bénard problem for free convection in a porous layer is being recognized as a classical problem of hydrodynamic stability
The classical HRL onset problem was pioneered by Horton and Rogers (1945) and Lapwood (1948), who established the standard of searching for normal-mode solutions of this eigenvalue problem
This paper deals with the fourth-order Darcy–Bénard eigenvalue problem, which is of basic interest in mathematical physics
Summary
The Darcy–Bénard problem for free convection in a porous layer is being recognized as a classical problem of hydrodynamic stability. Nield was first out to challenge the normal-mode paradigm of the Darcy–Bénard onset problem (Nield 1968) He extended the classical normal-mode type HRL solution (Horton and Rogers 1945; Lapwood 1948) by considering all possible combinations of Dirichlet or Neumann conditions for the perturbation temperature and the vertical velocity at the upper and lower boundaries of the horizontal porous layer. The nice analytical properties of the Dirichlet-type onset problem for two-dimensional cavities (Rees and Tyvand 2004) are lost as we replace the mechanical Dirichlet condition with a Neumann condition, bringing us back to a fourth-order thermomechanical problem without degeneracy. We will show that these two problems have the same onset criterion because we relate it to our model with conducting and open walls
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