Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-12-01T04:39:09.112Z Has data issue: false hasContentIssue false
  • English
  • Français

On Hadley flow in a porous layer with vertical heterogeneity

Published online by Cambridge University Press:  29 August 2012

A. Barletta  and
D. A. Nield
A. Barletta*
Affiliation:
DIENCA, Alma Mater Studiorum – Università di Bologna, Viale Risorgimento 2, I-40136 Bologna, Italy
D. A. Nield
Affiliation:
Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The onset of thermoconvective instability in a horizontal porous layer with a basic Hadley flow is studied, under the assumption of weak vertical heterogeneity. Hadley flow is a single-cell convective circulation induced by horizontal linear changes of the layer boundary temperatures. When combined with heating from below, these thermal boundary conditions yield a temperature gradient inclined to the vertical, in the basic state. The linear stability of the basic state is studied by considering small-amplitude disturbances of the velocity field and the temperature field. The linearized governing equations for the disturbances are then solved both by Galerkin’s method of weighted residuals and by a combined use of the Runge–Kutta method and the shooting method. The effect of weak heterogeneity of the permeability and the effective thermal conductivity of the porous medium is studied with respect to neutral stability conditions. It is shown that, among the normal mode disturbances, the most unstable are longitudinal rolls, that is, plane waves with a wave vector perpendicular to the imposed horizontal temperature gradient. The effect of heterogeneity becomes important only for high values of the horizontal Rayleigh number, associated with the horizontal temperature gradient, approximately greater than 60. In this regime, the effect of heterogeneity is destabilizing. It is shown that heterogeneity with respect to thermal conductivity is of major importance in the onset of instability.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection in porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 710 , 10 November 2012 , pp. 304 - 323
Copyright
Copyright © Cambridge University Press 2012

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

1. Barletta, A., Celli, M. & Kuznetsov, A. V. 2012 Heterogeneity and onset of instability in Darcy’s flow with a prescribed horizontal temperature gradient. Trans. ASME: J. Heat Transfer 134, 042602.CrossRefGoogle Scholar
2. Barletta, A. & Nield, D. A. 2010 Instability of Hadley–Prats flow with viscous heating in a horizontal porous layer. Trans. Porous Med. 84, 241–256.CrossRefGoogle Scholar
3. Barletta, A. & Rees, D. A. S. 2012 Local thermal non-equilibrium effects in the Darcy–Bénard instability with isoflux boundary conditions. Intl J. Heat Mass Transfer 55, 384–394.CrossRefGoogle Scholar
4. Brevdo, L. 2009 Three-dimensional absolute and convective instabilities at the onset of convection in a porous medium with inclined temperature gradient and vertical throughflow. J. Fluid Mech. 641, 475–487.CrossRefGoogle Scholar
5. Brevdo, L. & Ruderman, M. 2009a On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part 1. Normal modes. Trans. Porous Med. 80, 137–151.CrossRefGoogle Scholar
6. Brevdo, L. & Ruderman, M. 2009b On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part 2. Absolute and convective instabilities, and spatially amplifying waves. Trans. Porous Med. 80, 153–172.CrossRefGoogle Scholar
7. Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
8. Diaz, E. & Brevdo, L. 2011 Absolute/convective instability dichotomy at the onset of convection in a porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. J. Fluid Mech. 681, 567–596.CrossRefGoogle Scholar
9. Finlayson, B. A. 1972 Method of Weighted Residuals and Variational Principles. Academic.Google Scholar
10. Finlayson, B. A. & Scriven, L. E. 1966 The method of weighted residuals: a review. Appl. Mech. Rev. 19, 735–748.Google Scholar
11. Kaloni, P. N. & Qiao, Z. 1997 Non-linear stability of convection in a porous medium with inclined temperature gradient. Intl J. Heat Mass Transfer 40, 1611–1615.CrossRefGoogle Scholar
12. Manole, D. M. & Lage, J. L. 1995 Numerical simulation of supercritical Hadley circulation, within a porous layer, induced by inclined temperature gradients. Intl J. Heat Mass Transfer 38, 2583–2593.CrossRefGoogle Scholar
13. Manole, D. M., Lage, J. L. & Antohe, B. V. 1995 Supercritical Hadley circulation within a layer of fluid saturated porous medium: bifurcation to traveling wave. ASME HTD 309, 23–29.Google Scholar
14. Manole, D. M., Lage, J. L. & Nield, D. A. 1994 Convection induced by inclined thermal and solutal gradients, with horizontal mass flow, in a shallow horizontal layer of a porous medium. Intl J. Heat Mass Transfer 37, 2047–2057.CrossRefGoogle Scholar
15. Narayana, P. A. L., Murthy, P. V. S. N. & Gorla, R. S. R. 2008 Soret-driven thermosolutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. J. Fluid Mech. 612, 1–19.CrossRefGoogle Scholar
16. Nield, D. A. 1990 Convection in a porous medium with inclined temperature gradient and horizontal mass flow. In Heat Transfer 1990: Proceedings of the Ninth International Heat Transfer Conference, Jerusalem, Israel (ed. Hetsroni, G. ), vol. 5, pp. 153–158. Hemisphere.Google Scholar
17. Nield, D. A. 1991 Convection in a porous medium with inclined temperature gradient. Intl J. Heat Mass Transfer 34, 87–92.CrossRefGoogle Scholar
18. Nield, D. A. 1998 Convection in a porous medium with inclined temperature gradient and vertical throughflow. Intl J. Heat Mass Transfer 41, 241–243.CrossRefGoogle Scholar
19. Nield, D. A. 2008 General heterogeneity effects on the onset of convection in a porous medium. In Emerging Topics in Heat and Mass Transfer in Porous Media (ed. Vadász, P. & Bear, J. ). Theory and Applications of Transport in Porous Media , vol. 22, pp. 63–84. Springer.CrossRefGoogle Scholar
20. Nield, D. A. & Bejan, A. 2006 Convection in Porous Media, third edition. Springer.Google Scholar
21. Nield, D. A. & Kuznetsov, A. V. 2007 The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium. Intl J. Heat Mass Transfer 50, 1909–1915.CrossRefGoogle Scholar
22. Nield, D. A. & Kuznetsov, A. V. 2011 The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium with horizontal throughflow. Intl J. Heat Mass Transfer 54, 5595–5601.CrossRefGoogle Scholar
23. Nield, D. A., Manole, D. M. & Lage, J. L. 1993 Convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. J. Fluid Mech. 257, 559–574.CrossRefGoogle Scholar
24. Straughan, B. 2010 Stability and Wave Motion in Porous Media. Springer.Google Scholar
25. Weber, J. E. 1974 Convection in a porous medium with horizontal and vertical temperature gradients. Intl J. Heat Mass Transfer 17, 241–248.CrossRefGoogle Scholar