Transport in Porous Media

, 80:153 | Cite as

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part II. Absolute and Convective Instabilities, and Spatially Amplifying Waves

Article

Abstract

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number Rh < 49, the instability is absolute implying that, as the vertical Rayleigh number, Rv, increases passing through its critical value, Rvc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as Rv, increases beyond the critical value, while the horizontal Rayleigh number, Rh, and the Péclet number, Qv, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, Rv, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient.

Keywords

Saturated porous layer Darcy flow Inclined thermal gradient and vertical through flow Transverse disturbances Absolute and convective instabilities and spatially amplifying waves 

References

  1. Ashpis D.E., Reshotko E.: The vibrating ribbon problem revisited. J. Fluid Mech. 213, 531–547 (1990)CrossRefGoogle Scholar
  2. Bers, A.: Theory of absolute and convective instabilities. In: Auer, G., Cap, F. (eds.) International Congress on Waves and Instabilities in Plasmas, pp. B1–B52. Inst. Fuer Theoret. Physik (Innsbruck), Austria (1973).Google Scholar
  3. Brevdo L.: A study of absolute and convective instabilities with an application to the Eady model. Geophys. Astrophys. Fluid Dyn. 40, 1–92 (1988)CrossRefGoogle Scholar
  4. Brevdo L.: Spatially amplifying waves in plane Poiseuille flow. Z. Angew. Math. Mech. 72(3), 163–174 (1992)CrossRefGoogle Scholar
  5. Brevdo L.: Stability problem for the Blasius boundary layer. Z. Angew. Math. Mech. 75(5), 371–378 (1995)Google Scholar
  6. Brevdo L.: Convective instability of and signalling in spatially developing unbounded flows and media. Lectures Series Monographs 2006–07: Thermo-hydraulic instabilities. von Karman Institute for Fluid Dynamics Rhode Saint Genèse, Belgium (2007)Google Scholar
  7. Brevdo, L., Ruderman, M.S.: On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part I. Normal modes. Transp. Porous. Media (2009). doi:10.1007/s11242-009-9348-7
  8. Brevdo L., Laure P., Dias F., Bridges T.J.: Linear pulse structure and signalling in a film flow on an inclined plane. J. Fluid Mech. 396, 37–71 (1999)CrossRefGoogle Scholar
  9. Bridges T.J., Morris P.J.: Differential eigenvalue problem in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437–460 (1984)CrossRefGoogle Scholar
  10. Briggs R.J.: Electron–Stream Interaction With Plasmas. MIT Press, Cambridge (1964)Google Scholar
  11. Coddington, E. A., Levinson N.: Theory of Ordinary Differential equations. McGraw-Hill, New York (1955)Google Scholar
  12. Drazin P.G., Reid W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)Google Scholar
  13. EPRI. In: Bose, J.E. (ed.) Soil and Rock Classification for the Design of Ground-Coupled Heat Pump Systems—Field Manual. Electric Power Research Institute Special Report, EPRI CU-6600. (1989).Google Scholar
  14. Lin C.C.: Some mathematical problems in the theory of the stability of parallel flows. J. Fluid Mech. 10, 430–438 (1961)CrossRefGoogle Scholar
  15. Manole D.M., Lage J.L., Nield D.A.: Convection induced by inclined thermal and solutal gradients, with horizontal mass flow, in a shallow horizontal layer of a porous medium. Int. J. Heat Mass Transf. 37, 2047–2057 (1994)CrossRefGoogle Scholar
  16. Nield D.A.: Convection in a porous medium with inclined temperature gradient. Int. J. Heat Mass Transf. 34, 87–92 (1991)CrossRefGoogle Scholar
  17. Nield D.A.: Convection in a porous medium with inclined temperature gradient: additional results. Int. J. Heat Mass Transf. 37, 3021–3025 (1994)CrossRefGoogle Scholar
  18. Nield D.A.: Convection in a porous medium with inclined temperature gradient and vertical throughflow. Int. J. Heat Mass Transf. 41, 241–243 (1998)CrossRefGoogle Scholar
  19. Nield D.A., Bejan A.: Convection in Porous Media. Springer, Berlin (2006)Google Scholar
  20. Nield D.A., Manole D.M., Lage J.L.: Convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. J. Fluid Mech. 257, 559–574 (1993)CrossRefGoogle Scholar
  21. Pearlstein A.J., Goussis D.A.: transformation of certain singular polynomial matrix eigenvalue problems. J. Comput. Phys. 78, 305–312 (1988)CrossRefGoogle Scholar
  22. Ruderman M.S., Brevdo L., Erdélyi R.: Kelvin–Helmholtz absolute and convective instabilities, and signalling in an inviscid fluid–viscous fluid configuration. Proc. R. Soc. Lond. A 460, 847–874 (2004)CrossRefGoogle Scholar
  23. Straughan B.: The Energy Method, Stability and Nonlinear Convection. Springer, Berlin (2004a)Google Scholar
  24. Straughan B.: Resonant penetrative convection. Proc. R. Soc. Lond. A 260, 2913–2927 (2004b)Google Scholar
  25. Straughan B., Walker D.W.: Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. J. Comput. Phys. 127, 128–141 (1996)CrossRefGoogle Scholar
  26. Weber J.E.: Convection in a porous medium with horizontal and vertical temperature gradients. Int. J. Heat Mass Transf. 17, 241–248 (1974)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institut de Mécanique des Fluides et des Solides - UMR 7507 ULP - CNRSUniversité de StrasbourgStrasbourgFrance
  2. 2.Department of Applied MathematicsUniversity of SheffieldSheffieldUK

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