Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-17T23:17:08.352Z Has data issue: false hasContentIssue false
  • English
  • Français

Unicellular natural circulation in a shallow horizontal porous layer heated from below by a constant flux

Published online by Cambridge University Press:  26 April 2006

S. Kimura ,
M. Vynnycky  and
F. Alavyoon
S. Kimura
Affiliation:
Tohoku National Industrial Research Institute, Agency of Industrial Science and Technology, Ministry of International Trade and Industry, 4-2-1 Nigatake, Miyagino-Ku, Sendai, 983, Japan Present address: Department of Mechanical Engineering, Kanazawa University 2-40-20 Kodatsumo, Kanazawa, 920, Japan
M. Vynnycky
Affiliation:
Tohoku National Industrial Research Institute, Agency of Industrial Science and Technology, Ministry of International Trade and Industry, 4-2-1 Nigatake, Miyagino-Ku, Sendai, 983, Japan
F. Alavyoon
Affiliation:
Vattenfall Utveckling AB, S-810 70 Älvkarleby, Sweden
Get access Rights & Permissions [Opens in a new window]

Abstract

Natural convection in a saturated horizontal porous layer heated from below and cooled at the top with a constant flux is studied both analytically and numerically. Linear stability analysis indicates that unicellular recirculation remains a stable mode of flow as the aspect ratio (A) of the layer is increased, in contrast to the situation for an isothermally heated and cooled layer. An analytical solution is presented for fully developed counterflow in the infinite-aspect-ratio limit; this flow is found to be linearly stable to transverse disturbances for Rayleigh number (Ra) as high as 506, at which point a Hopf bifurcation sets in; however, further analysis indicates that an exchange of stability due to longitudinal disturbances will occur much sooner at Ra ≈ 311.53. The velocity and temperature profiles of the counterflow solution, whilst not strictly speaking valid in the extreme end regions of the layer, otherwise agree very well with full numerical computations conducted for the ranges 25 [les ] Ra [les ] 1050, 2 [les ] A [les ] 10. However, for sufficiently high Rayleigh number (Ra between 630 and 650 for A = 8 and Ra between 730 and 750 for A = 4, for example), the computations indicate transition from steady unicellular to oscillatory flow, in line with the Hopf bifurcation predicted by the linear stability analysis for infinite aspect ratio.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 294 , 10 July 1995 , pp. 231 - 257
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aidun, C. K. & Steen, P. H. 1987 Transition to oscillatory heat transfer in a fluid saturated porous medium. J. Thermophys. Heat Transfer 1, 268–273.Google Scholar
Alavyoon, F. 1992 The effects of porous separators on free convection and mass transfer in electrochemical systems - application to recharging lead-acid cells. In Heat and Mass Transfer in Porous Media (ed. M. Quintard & M. Todorović), pp. 349–379. Elsevier.
Alavyoon, F. 1993 On natural convection in vertical porous enclosures due to prescribed fluxes of heat and mass at the vertical boundaries. Intl J. Heat Mass Transfer 36, 2479–2498.Google Scholar
Bark, F. H., Alavyoon, F. & Dahlkild, A. A. 1992 On unsteady free convection in vertical slots due to prescribed fluxes of heat and mass at the vertical walls. J. Fluid Mech. 235, 665–689.Google Scholar
Bejan, A. 1983 The boundary layer regime in a porous layer with uniform heat flux from the side. Intl J. Heat Mass Transfer 26, 1339–1346.Google Scholar
Caltagirone, J. P. 1975 Thermoconvective instabilities in a horizontal porous layer. J. Fluid Mech. 72, 268–287.Google Scholar
Gary, J. & Kassoy, D. R. 1981 Computation of steady and oscillatory convection in saturated porous media. J. Comput. Phys. 40, 120–142.Google Scholar
Horne, R. N. & O'Sullivan, M. J. 1974 Oscillatory convection in a porous medium heated from below. J. Fluid Mech. 66, 339–352.Google Scholar
Kassoy, D. R. & Cotte, B. 1985 The effects of sidewall heat loss on convection in a saturated porous vertical slab. J. Fluid Mech. 152, 361–378.Google Scholar
Kimura, S. & Pop, I. 1992 Conjugate natural convection between horizontal concentric cylinders filled with porous medium. Wärme- und Stoffübertr. 27, 85–91.Google Scholar
Kimura, S., Schubert, G. & Strauss, J. M. 1986 Route to chaos in porous-medium thermal convection. J. Fluid Mech. 166, 305–324.Google Scholar
Kimura, S., Schubert, G. & Straus, J. M. 1989 Time-dependent convection in a fluid-saturated porous cube heated from below. J. Fluid Mech. 207, 153–189.Google Scholar
Lowell, R. P. & Shyu, C. T. 1978 On the onset of convection in a water-saturated porous box: effect of conducting walls. Lett. Heat Mass Transfer 5, 371–378.Google Scholar
Murphy, H. D. 1979 Convective instabilities in vertical fractures and faults. J. Geophys. Res. 84, 6121–6130.Google Scholar
Nield, D. A. 1968 Onset of thermohaline convection in a porous medium. Water Resources Res. 4, 553–560.Google Scholar
Nield, D. A. 1991 Convection in a porous medium with inclined temperature gradient. Intl J. Heat Mass Transfer 34, 87–92.Google Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. Hemisphere.
Riley, D. S. & Winters, K. H. 1989 Modal exchange mechanisms in Lapwood convection. J. Fluid Mech. 204, 325–358.Google Scholar
Riley, D. S. & Winters, K. H. 1991 Time periodic convection in porous media: the evolution of Hopf bifurcations with aspect ratio. J. Fluid Mech. 223, 457–474.Google Scholar
Schubert, G. & Straus, J. M. 1979 Three-dimensional and multi-cellular steady and unsteady convection in fluid-saturated porous media at high Rayleigh numbers. J. Fluid Mech. 94, 25–36.Google Scholar
Schubert, G. & Straus, J. M. 1982 Transitions in time-dependent thermal convection in fluid-saturated porous media. J. Fluid Mech. 121, 301–313.Google Scholar
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, 2097–2113.Google Scholar
Steen, P. H. 1983 Pattern selection for finite-amplitude convection states in boxes of porous media. J. Fluid Mech. 136, 219–242.Google Scholar
Sutton, F. 1970 Onset of convection in a porous channel with net through flow. Phys. Fluids 13, 1931–1934.Google Scholar
Vasseur, P., Satish, M. G. & Robillard, L. 1987 Natural convection in a thin inclined porous layer exposed to a constant heat flux. Intl J. Heat Mass Transfer 30, 537–549.Google Scholar
Wang, M., Kassoy, D. R. & Weidman, P. D. 1987 Onset of convection in a vertical slab of porous media between two impermeable conducting blocks. Intl J. Heat Mass Transfer 30, 1331–1341.Google Scholar
Weber, J. E. 1974 Convection in a porous medium with horizontal and vertical temperature gradients. Intl J. Heat Mass Transfer 17, 241–248.Google Scholar
Weidman, P. D. & Kassoy, D. R. 1986 The influence of side wall heat transfer on convection in a confined saturated porous medium. Phys. Fluids 29, 349–355.Google Scholar