Skip to main content
SpringerLink
Log in

Oscillatory Non-normal-Mode Onset of Convection in a Porous Rectangle

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

A special case of the fourth-order Darcy–Bénard problem in a two-dimensional (2D) rectangular porous box is investigated. The present eigenfunctions are of non-normal-mode type in the horizontal and vertical directions. They compose a time-periodic wave with one-way propagation out of the porous rectangle. Asymmetry in the horizontal direction generates an oscillatory time dependence of the marginal state, similar to Rees and Tyvand (Phys Fluids 16:3706–3714, 2004a). No analytical method is known for this non-degenerate eigenvalue problem. Therefore, the problem was solved numerically by the finite element method (FEM). Three boundaries of the rectangle are impermeable. The right-hand wall is fully penetrative. The lower boundary and the left-hand wall are heat conductors. The upper boundary has a given heat flux. The right-hand wall is thermally insulating. As a result, the computed eigenfunctions show complicated periodic time dependence. Finally, the critical Rayleigh number and the associated angular frequency are calculated as functions of the aspect ratio and compared against the case of normal modes in the vertical direction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Barletta, A., Storesletten, L.: Onset of convection in a vertical porous cylinder with a permeable and conducting side boundary. Int. J. Therm. Sci. 97, 9–16 (2015)

    Article  Google Scholar 

  • Beck, J.L.: Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 1377–1383 (1972)

    Article  Google Scholar 

  • Horton, C.W., Rogers, F.T.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945)

    Article  Google Scholar 

  • Lapwood, E.R.: Convection of a fluid in a porous medium. Proc. Camb. Philos. Soc. 44, 508–521 (1948)

    Article  Google Scholar 

  • Nield, D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 11, 553–560 (1968)

    Article  Google Scholar 

  • Nilsen, T., Storesletten, L.: An analytical study on natural convection in isotropic and anisotropic porous channels. ASME J. Heat Transf. 112, 396–401 (1990)

    Article  Google Scholar 

  • Nygård, H.S., Tyvand, P.A.: Onset of convection in a porous box with partly conducting and partly penetrative sidewalls. Transp. Porous Med. 84, 55–73 (2010)

    Article  Google Scholar 

  • Rees, D.A.S., Lage, J.L.: The effect of thermal stratification on natural convection in a vertical porous insulation layer. Int. J. Heat Mass Transf. 40, 111–121 (1996)

    Article  Google Scholar 

  • Rees, D.A.S., Tyvand, P.A.: Oscillatory convection in a two-dimensional porous box with asymmetric lateral boundary conditions. Phys. Fluids 16, 3706–3714 (2004a). (Referred to in the text as the RT paper)

    Article  Google Scholar 

  • Rees, D.A.S., Tyvand, P.A.: The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries. J. Eng. Math. 490, 181–193 (2004b)

    Article  Google Scholar 

  • Storesletten, L., Tveitereid, M.: Natural convection in a horizontal porous cylinder. Int. J. Heat Mass Transf. 34, 1959–1968 (1991)

    Article  Google Scholar 

  • Tyvand, P.A., Storesletten, L.: Degenerate onset of convection in vertical porous cylinders. J. Eng. Math. 113, 165–181 (2018)

    Article  Google Scholar 

  • Tyvand, P.A., Nøland, J.K., Storesletten, L.: A non-normal-mode marginal state of convection in a porous rectangle. Transp. Porous Med. 128, 633–651 (2019)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Drøbakveien 31, 1432 Ås, Norway

    Peder A. Tyvand

  2. Faculty of Information Technology and Electrical Engineering, Norwegian University of Science and Technology, 7034, Trondheim, Norway

    Jonas Kristiansen Nøland

Authors
  1. Peder A. Tyvand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonas Kristiansen Nøland
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonas Kristiansen Nøland.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tyvand, P.A., Nøland, J.K. Oscillatory Non-normal-Mode Onset of Convection in a Porous Rectangle. Transp Porous Med 129, 955–974 (2019). https://doi.org/10.1007/s11242-019-01314-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-019-01314-x

Keywords

Search

Navigation