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Inclined porous medium convection at large Rayleigh number

Published online by Cambridge University Press:  05 January 2018

Baole Wen Open the ORCID record for Baole Wen [Opens in a new window]  and
Gregory P. Chini Open the ORCID record for Gregory P. Chini [Opens in a new window]
Baole Wen
Affiliation:
Program in Integrated Applied Mathematics, University of New Hampshire, Durham, NH 03824, USA Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX 78712, USA
Gregory P. Chini*
Affiliation:
Program in Integrated Applied Mathematics, 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

High-Rayleigh-number ($Ra$) convection in an inclined two-dimensional porous layer is investigated using direct numerical simulations (DNS) and stability and variational upper-bound analyses. When the inclination angle $\unicode[STIX]{x1D719}$ of the layer satisfies $0^{\circ }<\unicode[STIX]{x1D719}\lesssim 25^{\circ }$, DNS confirm that the flow exhibits a three-region wall-normal asymptotic structure in accord with the strictly horizontal ($\unicode[STIX]{x1D719}=0^{\circ }$) case, except that as $\unicode[STIX]{x1D719}$ is increased the time-mean spacing between neighbouring interior plumes also increases substantially. Both DNS and upper-bound analysis indicate that the heat transport enhancement factor (i.e. the Nusselt number) $Nu\sim CRa$ with a $\unicode[STIX]{x1D719}$-dependent prefactor $C$. When $\unicode[STIX]{x1D719}>\unicode[STIX]{x1D719}_{t}$, however, where $30^{\circ }<\unicode[STIX]{x1D719}_{t}<32^{\circ }$ independently of $Ra$, the columnar flow structure is completely broken down: the flow transitions to a large-scale travelling-wave convective roll state, and the heat transport is significantly reduced. To better understand the physics of inclined porous medium convection at large $Ra$ and modest inclination angles, a spatial Floquet analysis is performed, yielding predictions of the linear stability of numerically computed, fully nonlinear steady convective states. The results show that there exist two types of instability when $\unicode[STIX]{x1D719}\neq 0^{\circ }$: a bulk-mode instability and a wall-mode instability, consistent with previous findings for $\unicode[STIX]{x1D719}=0^{\circ }$ (Wen et al., J. Fluid Mech., vol. 772, 2015, pp. 197–224). The background flow induced by the inclination of the layer intensifies the bulk-mode instability during its subsequent nonlinear evolution, thereby favouring increased spacing between the interior plumes relative to that observed in convection in a horizontal porous layer.

JFM classification

Convection: Convection in porous media Instability: Nonlinear instability Mathematical Foundations: Variational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 670 - 702
Copyright
© 2018 Cambridge University Press 

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