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Structure and stability of steady porous medium convection at large Rayleigh number

Published online by Cambridge University Press:  05 May 2015

Baole Wen ,
Lindsey T. Corson  and
Gregory P. Chini
Baole Wen
Affiliation:
Program in Integrated Applied Mathematics, University of New Hampshire, Durham, NH 03824, USA Center for Fluid Physics, University of New Hampshire, Durham, NH 03824, USA
Lindsey T. Corson
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
Gregory P. Chini*
Affiliation:
Program in Integrated Applied Mathematics, University of New Hampshire, Durham, NH 03824, USA Center for Fluid Physics, University of New Hampshire, Durham, NH 03824, USA Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, USA
*
†Email address for correspondence: greg.chini@unh.edu
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Abstract

A systematic investigation of unstable steady-state solutions of the Darcy–Oberbeck–Boussinesq equations at large values of the Rayleigh number $\mathit{Ra}$ is performed to gain insight into two-dimensional porous medium convection in domains of varying aspect ratio $L$. The steady convective states are shown to transport less heat than the statistically steady ‘turbulent’ flow realised at the same parameter values: the Nusselt number $\mathit{Nu}\sim \mathit{Ra}$ for turbulent porous medium convection, while $\mathit{Nu}\sim \mathit{Ra}^{0.6}$ for the maximum heat-transporting steady solutions. A key finding is that the lateral scale of the heat-flux-maximising solutions shrinks roughly as $L\sim \mathit{Ra}^{-0.5}$, reminiscent of the decrease of the mean inter-plume spacing observed in turbulent porous medium convection as the thermal forcing is increased. A spatial Floquet analysis is performed to investigate the linear stability of the fully nonlinear steady convective states, extending a recent study by Hewitt et al. (J. Fluid Mech., vol. 737, 2013, pp. 205–231) by treating a base convective state, and secondary stability modes, that satisfy appropriate boundary conditions along plane parallel walls. As in that study, a bulk instability mode is found for sufficiently small-aspect-ratio base states. However, the growth rate of this bulk mode is shown to be significantly reduced by the presence of the walls. Beyond a certain critical $\mathit{Ra}$-dependent aspect ratio, the base state is most strongly unstable to a secondary mode that is localised near the heated and cooled walls. Direct numerical simulations, strategically initialised to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically steady porous medium convection results from a balance between the competing effects of these two types of instability.

JFM classification

Convection: Convection in porous media Instability: Instability Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 772 , 10 June 2015 , pp. 197 - 224
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© 2015 Cambridge University Press 

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