Skip to main content Accessibility help

Inclined porous medium convection at large Rayleigh number

  • Baole Wen (a1) (a2) and Gregory P. Chini (a1) (a3)


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.


Corresponding author

Email address for correspondence:


Hide All
Aidun, C. K. & Steen, P. H. 1987 Transition to oscillatory convective heat transfer in a fluid-saturated porous medium. J. Thermophys. Heat Transfer 1, 268273.
Bories, S. A. & Combarnous, M. A. 1973 Natural convection in a sloping porous layer. J. Fluid Mech. 57, 6379.
Bories, S. A., Combarnous, M. A. & Jaffrenou, J. Y. 1972 Observations des différentes formes d’écoulements thermoconvectifs dans une couche poreuse inclinée. C. R. Acad. Sci. Paris A 275, 857860.
Bories, S. A. & Monferran, L. 1972 Condition de stabilité et échange thermique par convection naturelle dans une couche poreuse inclinée de grande extension. C. R. Acad. Sci. Paris  B 274, 47.
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.
Caltagirone, J. P. & Bories, S. 1985 Solutions and stability criteria of natural convective flow in an inclined porous layer. J. Fluid Mech. 155, 267287.
Caltagirone, J. P., Cloupeau, M. & Combarnous, M. 1971 Convection naturelle fluctuante dans une couche poreuse horizontale. C. R. Acad. Sci. Paris B 273, 833836.
Doering, C. R. & Constantin, P. 1998 Bounds for heat transport in a porous layer. J. Fluid Mech. 376, 263296.
Fu, X., Cueto-Felgueroso, L. & Juanes, R. 2013 Pattern formation and coarsening dynamics in three-dimensional convective mixing in porous media. Phil. Trans. R. Soc. Lond. A 371, 20120355.
Gill, A. E. 1969 A proof that convection in a porous vertical slab is stable. J. Fluid Mech. 35, 545547.
Graham, M. D. & Steen, P. H. 1992 Strongly interacting traveling waves and quasiperiodic dynamics in porous medium convection. Physica D 54, 331350.
Graham, M. D. & Steen, P. H. 1994 Plume formation and resonant bifurcations in porous-media convection. J. Fluid Mech. 272, 6790.
Hewitt, D. R. & Lister, J. R. 2017 Stability of three-dimensional columnar convection in a porous medium. J. Fluid Mech. 829, 89111.
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2012 Ultimate regime of high Rayleigh number convection in a porous medium. Phys. Rev. Lett. 108, 224503.
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2013 Stability of columnar convection in a porous medium. J. Fluid Mech. 737, 205231.
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2014 High rayleigh number convection in a three-dimensional porous medium. J. Fluid Mech. 748, 879895.
Horton, C. W. & Rogers, F. T. 1945 Convection currents in a porous medium. J. Appl. Phys. 16, 367370.
Kaneko, T.1972 An experimental investigation of natural convection in porous media. MSc thesis, University of Calgary.
Kaneko, T., Mohtadi, M. F. & Aziz, K. 1974 An experimental study of natural convection in inclined porous media. Intl J. Heat Mass Transfer 17, 485496.
Kimura, S., Schubert, G. & Straus, J. M. 1986 Route to chaos in porous-medium thermal convection. J. Fluid Mech. 166, 305324.
Kimura, S., Schubert, G. & Straus, J. M. 1987 Instabilities of steady, periodic, and quasi-periodic modes of convection in porous media. Trans. ASME J. Heat Transfer 109, 350355.
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508521.
MacMinn, C. W. & Juanes, R. 2013 Buoyant currents arrested by convective dissolution. Geophys. Res. Lett. 40, 20172022.
Metz, B., Davidson, O., de Coninck, H., Loos, M. & Meyer, L. 2005 IPCC Special Report on Carbon Dioxide Capture and Storage. Cambridge University Press.
Moya, S. L., Ramos, E. & Sen, M. 1987 Numerical study of natural convection in a tilted rectangular porous material. Intl J. Heat Mass Transfer 30, 741756.
Nield, D. A. & Bejan, A. 2013 Convection in Porous Media, 4th edn. Springer.
Nikitin, N. 2006 Third-order-accurate semi-implicit Runge–Kutta scheme for incompressible Navier–Stokes equations. Intl J. Numer. Meth. Fluids 51, 221233.
Otero, J., Dontcheva, L. A., Johnston, H., Worthing, R. A., Kurganov, A., Petrova, G. & Doering, C. R. 2004 High-Rayleigh-number convection in a fluid-saturated porous layer. J. Fluid Mech. 500, 263281.
Pau, G. S. H., Bell, J. B., Pruess, K., Almgren, A. S., Lijewski, M. J. & Zhang, K. 2010 High-resolution simulation and characterization of density-driven flow in CO2 storage in saline aquifers. Adv. Water Resour. 33, 443455.
Peyret, Roger 2002 Spectral Methods for Incompressible Viscous Flow. Springer.
Phillips, O. M. 1991 Flow and Reactions in Permeable Rocks. Cambridge University Press.
Phillips, O. M. 2009 Geological Fluid Dynamics: Sub-surface Flow and Reactions. Cambridge University Press.
Rees, D. A. S. & Bassom, A. P. 2000 The onset of Darcy–Bénard convection in an inclined layer heated from below. Acta Mech. 144, 103118.
Schubert, G. & Straus, J. M. 1982 Transitions in time-dependent thermal convection in fluid-saturated porous media. J. Fluid Mech. 121, 301313.
Sen, M., Vasseur, P. & Robillard, L. 1987 Multiple steady states for unicellular natural convection in an inclined porous layer. Intl J. Heat Mass Transfer 30, 20972113.
Trefethen, L. N. & Bau, D. III 1997 Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM).
Tsai, P. A., Riesing, K. & Stone, H. A. 2013 Density-driven convection enhanced by an inclined boundary: implications for geological CO 2 storage. Phys. Rev. E 87, 011003.
Voss, C. I., Simmons, C. T. & Robinson, N. I. 2010 Three-dimensional benchmark for variable-density flow and transport simulation: matching semi-analytic stability modes for steady unstable convection in an inclined porous box. Hydrogeol. J. 18, 523.
Wen, B.2015 Porous medium convection at large Rayleigh number: Studies of coherent structure, transport, and reduced dynamics. PhD thesis, University of New Hampshire.
Wen, B., Chini, G. P., Dianati, N. & Doering, C. R. 2013 Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porous medium convection. Phys. Lett. A 377, 29312938.
Wen, B., Chini, G. P., Kerswell, R. R. & Doering, C. R. 2015a Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 92, 043012.
Wen, B., Corson, L. T. & Chini, G. P. 2015b Structure and stability of steady porous medium convection at large Rayleigh number. J. Fluid Mech. 772, 197224.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Inclined porous medium convection at large Rayleigh number

  • Baole Wen (a1) (a2) and Gregory P. Chini (a1) (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.