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Stability of columnar convection in a porous medium

  • Duncan R. Hewitt (a1), Jerome A. Neufeld (a1) (a2) (a3) and John R. Lister (a1)

Convection in a porous medium at high Rayleigh number $\mathit{Ra}$ exhibits a striking quasisteady columnar structure with a well-defined and $\mathit{Ra}$ -dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ‘heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber $k$ and amplitude $\widehat{A}$ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter $A= \widehat{A}\mathit{Ra}/ k$ . A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of $A$ . The growth rate of the most unstable mode scales with ${A}^{4/ 9} $ for $A\gg 1$ , and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh–Bénard (Rayleigh–Darcy) cell at high $\mathit{Ra}$ , and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers $k$ greater than $k\sim {\mathit{Ra}}^{5/ 14} $ as $\mathit{Ra}\rightarrow \infty $ . This stability criterion is consistent with hitherto unexplained numerical measurements of $k$ in a Rayleigh–Darcy cell.

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Journal of Fluid Mechanics
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