Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2014, Special Issue (2013), Article ID 695425, 10 pages.

Eddy Heat Conduction and Nonlinear Stability of a Darcy Lapwood System Analysed by the Finite Spectral Method

Jónas Elíasson

Full-text: Open access


A finite Fourier transform is used to perform both linear and nonlinear stability analyses of a Darcy-Lapwood system of convective rolls. The method shows how many modes are unstable, the wave number instability band within each mode, the maximum growth rate (most critical) wave numbers on each mode, and the nonlinear growth rates for each amplitude as a function of the porous Rayleigh number. Single amplitude controls the nonlinear growth rates and thereby the physical flow rate and fluid velocity, on each mode. They are called the flak amplitudes. A discrete Fourier transform is used for numerical simulations and here frequency combinations appear that the traditional cut-off infinite transforms do not have. The discrete show a stationary solution in the weak instability phase, but when carried past 2 unstable modes they show fluctuating motion where all amplitudes except the flak may be zero on the average. This leads to a flak amplitude scaling process of the heat conduction, producing an eddy heat conduction coefficient where a Nu-RaL relationship is found. It fits better to experiments than previously found solutions but is lower than experiments.

Article information

J. Appl. Math., Volume 2014, Special Issue (2013), Article ID 695425, 10 pages.

First available in Project Euclid: 1 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Elíasson, Jónas. Eddy Heat Conduction and Nonlinear Stability of a Darcy Lapwood System Analysed by the Finite Spectral Method. J. Appl. Math. 2014, Special Issue (2013), Article ID 695425, 10 pages. doi:10.1155/2014/695425.

Export citation


  • J. W. Elder, “Steady free convection in a porous medium heated from bellow,” Journal of Fluid Mechanics, vol. 27, part 1, p. 29, 1967.
  • K. I. Schneider, “Investigations of the influence of free convection on heat transfer through granular material,” in Proceedings of the 11th International Congress on Refrigeration, pp. 247–254, 1963.
  • R. A. Wooding, “Free convection of fluid in a vertical tube filled with porous material,” Journal of Fluid Mechanics, vol. 13, pp. 129–144, 1962.
  • E. Palm, J. E. Weber, and O. Kvernvold, “On steady convection in a porous medium,” Journal of Fluid Mechanics, vol. 54, part 1, p. 153, 1972.
  • A. A. Avramenko, A. V. Kuznetsov, B. I. Basok, and D. G. Blinov, “Investigation of stability of a laminar flow in a parallel-plate channel filled with a fluid saturated porous medium,” Physics of Fluids, vol. 17, no. 9, Article ID 094102, pp. 1–6, 2005.
  • J. Eliasson, “Convective ground water flow,” Series Paper 3, Institute of Hydrodynamics & Hydraulic Engineering, Technical University of Denmark, 1973.
  • N. Rudraiah, P. G. Siddheshwar, and T. Masuoka, “Nonlinear convection in porous media: a review,” Journal of Porous Media, vol. 6, no. 1, pp. 1–32, 2003.
  • N. Rudraiah, “Turbulent convection in porous media using spectral method,” in Proceedings of the 2nd Asian Congress of Fluid Mechanics, pp. 1015–1020, Seicha Press, Beijing, China, 1983.
  • N. Rudraiah and S. B. Rao, “Nonlinear cellular convection and heat transport in a porous medium,” Applied Scientific Research, vol. 39, no. 1, pp. 21–43, 1982.
  • E. Holzbecher, “Free and forced convection in porous media open at the top,” Heat and Mass Transfer, vol. 41, no. 7, pp. 606–614, 2005.
  • I. Sezai, “Flow patterns in a fluid-saturated porous cube heated from below,” Journal of Fluid Mechanics, vol. 523, pp. 393–410, 2005.
  • D. A. S. Rees and P. A. Tyvand, “Oscillatory convection in a two-dimensional porous box with asymmetric lateral boundary conditions,” Physics of Fluids, vol. 16, no. 10, pp. 3706–3714, 2004.
  • M.-H. Chang, “Thermal convection in superposed fluid and porous layers subjected to a horizontal plane Couette flow,” Physics of Fluids, vol. 17, no. 6, Article ID 064106, 2005.
  • M. Mamou, “Stability analysis of thermosolutal convection in a vertical packed porous enclosure,” Physics of Fluids, vol. 14, no. 12, pp. 4302–4314, 2002.
  • K. B. Haugen and P. A. Tyvand, “Onset of thermal convection in a vertical porous cylinder with conducting wall,” Physics of Fluids, vol. 15, no. 9, pp. 2661–2667, 2003.
  • E. R. Lapwood, “Convection of a fluid in a porous medium,” Proceedings of the Cambridge Philosophical Society, vol. 44, pp. 508–521, 1948.
  • J. Eliasson, “A note on the unsteady nonlinear convection in porous media,” Progress Reports 45, Technical University of Denmark, Institute of Hydrodynamics and Hydraulic Engineering, 1978.
  • M. Combarnous, “Natural convection in porous media and geothermal system,” in Proceedings of the 6th International Heat Transfer Conference, vol. 6, pp. 45–59, 1978.
  • J. M. Straus, “Large amplitude convection in porous media,” Journal of Fluid Mechanics, vol. 64, no. 1, pp. 51–63, 1974.
  • C. W. Fetter, Contaminant Hydrogeology, Macmillan, 1993.
  • I. Pop and D. B. Ingham, Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media, Pergamon Press, Oxford, UK, 2001.
  • A. Misirlioglu, A. C. Baytas, and I. Pop, “Free convection in a wavy cavity filled with a porous medium,” International Journal of Heat and Mass Transfer, vol. 48, no. 9, pp. 1840–1850, 2005.
  • M. F. El-Amin, N. A. Ebrahiem, A. Salama, and S. Sun, “Radiative mixed convection over an isothermal cone embedded in a porous medium with variable permeability,” Journal of Applied Mathematics, vol. 2011, Article ID 124590, 10 pages, 2011.
  • H. Beji, “Effets des non-linéarités et de la dispersion thermique sur la convection naturelle en milieu poreux confiné,” Journal de Physique III, pp. 267–284, 1933.
  • E. J. Braga and M. J. S. de Lemos, “Turbulent natural convection in a porous square cavity computed with a macroscopic $\kappa $-$\varepsilon $ model,” International Journal of Heat and Mass Transfer, vol. 47, no. 26, pp. 5639–5650, 2004.
  • D. R. Hewitt, J. A. Neufeld, and J. R. Lister, “Ultimate regime of high Rayleigh number convection in a porous medium,” Physical Review Letters, vol. 108, no. 22, Article ID 224503, 2012. \endinput