Transport in Porous Media

, Volume 122, Issue 1, pp 97–124 | Cite as

The Onset of Double-Diffusive Convection in a Superposed Fluid and Porous Layer Under High-Frequency and Small-Amplitude Vibrations

  • T. P. LyubimovaEmail author
  • E. A. Kolchanova


We numerically simulate the initiation of an average convective flow in a system composed of a horizontal binary fluid layer overlying a homogeneous porous layer saturated with the same fluid under gravitational field and vibration. In the layers, fixed equilibrium temperature and concentration gradients are set. The layers execute high-frequency oscillations in the vertical direction. The vibration period is small compared with characteristic timescales of the problem. The averaging method is applied to obtain vibrational convection equations. Using for computation the shooting method, a numerical investigation is carried out for an aqueous ammonium chloride solution and packed glass spheres saturated with the solution. The instability threshold is determined under two heating conditions—on heating from below and from above. When the solution is heated from below, the instability character changes abruptly with increasing solutal Rayleigh number, i.e., there is a jump-wise transition from the most dangerous shortwave perturbations localized in the fluid layer to the long-wave perturbations covering both layers. The perturbation wavelength increases by almost 10 times. Vibrations significantly stabilize the fluid equilibrium state and lead to an increase in the wavelength of its perturbations. When the fluid with the stabilizing concentration gradient is heated from below, convection can occur not only in a monotonous manner but also in an oscillatory manner. The frequency of critical oscillatory perturbations decreases by 10 times, when the long-wave instability replaces the shortwave instability. When the fluid is heated from above, only stationary convection is excited over the entire range of the examined parameters. A lower monotonic instability level is associated with the development of perturbations with longer wavelength even at a relatively large fluid layer thickness. Vibrations speed up the stationary convection onset and lead to a decrease in the wavelength of most dangerous perturbations of the motionless equilibrium state. In this case, high enough amplitudes of vibration are needed for a remarkable change in the stability threshold. The results of numerical simulation show good agreement with the data of earlier works in the limiting case of zero fluid layer thickness.


Double-diffusive convection Porous medium Superposed fluid and porous layers Binary fluid High-frequency vibrations Theoretical relationships bimodal neutral curves 



Funding was provided by President of Russian Federation, Program of the support of Leading Scientific Schools of Russian Federation (Grant No.NSh-9176.2016.1).


  1. Bardan, G., Mojtabi, A.: On the Horton–Rogers–Lapwood convective instability with vertical vibration: onset of convection. Phys. Fluids 12(11), 2723–2731 (2000)CrossRefGoogle Scholar
  2. Bardan, G., Knobloch, E., Mojtabi, A., Khallouf, H.: Natural doubly diffusive convection with vibration. Fluid Dyn. Res. 28, 159–187 (2001)CrossRefGoogle Scholar
  3. Bardan, G., Razi, Y.P., Mojtabi, A.: Comments on the mean flow averaged model. Phys. Fluids 16(12), 4535 (2004)CrossRefGoogle Scholar
  4. Bejan, A.: Convection Heat Transfer. Wiley, New York (2013)CrossRefGoogle Scholar
  5. Carman, P.C.: Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15, S32–S48 (1937)Google Scholar
  6. Charrier-Mojtabi, M.C., Razi, Y.P., Maliwan, K., Mojtabi, A.: Influence of vibration on Soret-driven convection in porous media. Numer. Heat Transf. 46, 981–993 (2004)CrossRefGoogle Scholar
  7. Charrier-Mojtabi, M.C., Razi, Y.P., Maliwan, K., Mojtabi, A.: Effect of vibration on the onset of double-diffusive convection in porous media. In: Ingham, D.B., Pop, I. (eds.) Transport Phenomena in Porous Media III, pp. 261–286. Elsevier, Oxford (2005)CrossRefGoogle Scholar
  8. Chen, F., Chen, C.F.: Onset of finger convection in a horizontal porous layer underlying a fluid layer. J. Heat Transf. 110(2), 403–409 (1988)CrossRefGoogle Scholar
  9. Chen, F., Chen, C.F.: Experimental investigation of convective stability in a superposed fluid and porous layer when heated from below. J. Fluid Mech. 207, 311–321 (1989)CrossRefGoogle Scholar
  10. Chen, C.F., Chen, F.: Experimental study of directional solidification of aqueous ammonium chloride solution. J. Fluid Mech. 227, 567–586 (1991)CrossRefGoogle Scholar
  11. Chen, F., Lu, J.W., Yang, T.L.: Convective instability in ammonium chloride solution directionally solidified from below. J. Fluid Mech. 216, 163–187 (1994)CrossRefGoogle Scholar
  12. Fand, R.M., Kim, B.Y.K., Lam, A.C.C., Phan, R.T.: Resistance to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres. J. Fluids Eng. 109, 268–273 (1987)CrossRefGoogle Scholar
  13. Gershuni, G.Z., Zhukovitskii, E.M.: Convective Stability of Incompressible Fluids, p. 392. Nauka, Moscow (1972)Google Scholar
  14. Gershuni, G.Z., Lyubimov, D.V.: Thermal Vibrational Convection, p. 358. Wiley, New York (1998)Google Scholar
  15. Glukhov, A.F., Putin, G.F.: Experimental investigation of convective structures in a fluid-saturated porous medium in the vicinity of the instability threshold of the mechanical equilibrium. Hydrodynamics 12, 104–119 (1999). (in Russian) Google Scholar
  16. Govender, S.: Stability of convection in a gravity modulated porous layer heated from below. Transp. Porous Med. 57, 113–123 (2004)CrossRefGoogle Scholar
  17. Govender, S.: Destabilizing a fluid saturated gravity modulated porous layer heated from above. Transp. Porous Med. 59, 215–225 (2005a)CrossRefGoogle Scholar
  18. Govender, S.: Linear stability and convection in a gravity modulated porous layer heated from below: transition from synchronous to subharmonic oscillations. Transp. Porous Med. 59, 227–238 (2005b)CrossRefGoogle Scholar
  19. Govender, S.: Natural convection in gravity-modulated porous layers. In: Vadasz, P. (ed.) Emerging Topics in Heat and Mass Transfer in Porous Media, pp. 133–148. Springer, New York (2008)CrossRefGoogle Scholar
  20. Hirata, S.C., Goyeau, B., Gobin, D.: Stability of thermosolutal natural convection in superposed fluid and porous layers. Transp. Porous Med. 78, 525–536 (2009)CrossRefGoogle Scholar
  21. Huppert, H.E., Turner, J.S.: Double-diffusive convection. J. Fluid Mech. 106, 299–329 (1981)CrossRefGoogle Scholar
  22. Jounet, A., Bardan, G.: Onset of thermohaline convection in a rectangular porous cavity in the presence of vertical vibration. Phys. Fluids 13, 3234–3246 (2001)CrossRefGoogle Scholar
  23. Katto, Y., Matsuoka, T.: Criterion for onset of convective flow in a fluid in a porous medium. Int. J. Heat Mass Transf. 10, 297–309 (1967)CrossRefGoogle Scholar
  24. Kolchanova, E.A., Lyubimov, D.V., Lyubimova, T.P.: Influence of effective medium permeability on stability of a two-layer system pure fluid porous medium under high-frequency vibrations. Comput. Contin. Mech. 5(2), 225–232 (2012). (in Russian) CrossRefGoogle Scholar
  25. Kolchanova, E., Lyubimov, D., Lyubimova, T.: The onset and nonlinear regimes of convection in a two-layer system of fluid and porous medium saturated by the fluid. Transp. Porous Med. 97, 25–42 (2013)CrossRefGoogle Scholar
  26. Lobov, N.I., Lyubimov, D.V., Lyubimova, T.P.: Numerical Methods of Solving the Problems in the Theory of Hydrodynamic Stability: A Textbook, p. 101. PSU Publishers, Perm (2004). (in Russian) Google Scholar
  27. Lyubimov, D.V., Muratov, I.D.: On convective instability in a multilayer system. Hydrodynamics 10, 38–46 (1977). (in Russian) Google Scholar
  28. Lyubimov, D.V., Lyubimova, T.P., Muratov, I.D.: Competition between long-wave and short-wave instability in a three-layer system. Hydrodynamics 13, 121–127 (2002). (in Russian) Google Scholar
  29. Lyubimov, D.V., Lyubimova, T.P., Muratov I.D.: Numerical study of the onset of convection in a horizontal fluid layer confined between two porous layers. In: Proceedings of International Conference on “Advanced Problems in Thermal Convection”, pp. 105–109 (2004a)Google Scholar
  30. Lyubimov, D.V., Lyubimova, T.P., Muratov, I.D.: The effect of vibrations on the excitation of convection in a two-layer system consisting of porous medium and homogeneous liquid. Hydrodynamics 14, 148–159 (2004b). (in Russian) Google Scholar
  31. Lyubimov, D.V., Lyubimova, T.P., Muratov, I.D., Shishkina, E.A.: The influence of vibrations on the onset of convection in the system of a horizontal layer of pure liquid and a layer of porous medium saturated with liquid. Fluid Dyn. 5, 132–143 (2008)Google Scholar
  32. Lyubimov, D., Kolchanova, E., Lyubimova, T.: Vibration effect on the nonlinear regimes of thermal convection in a two-layer system of fluid and saturated porous medium. Transp. Porous Med. 106, 237–257 (2015)CrossRefGoogle Scholar
  33. Maryshev, B., Lyubimova, T., Lyubimov, D.: Two-dimensional thermal convection in porous enclosure subjected to the horizontal seepage and gravity modulation. Phys. Fluids 25, 084105 (2013)CrossRefGoogle Scholar
  34. Nield, D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4, 553–560 (1968)CrossRefGoogle Scholar
  35. Nield, D.A., Bejan, A.: Convection in Porous Media, p. 778. Springer, New York (2013)CrossRefGoogle Scholar
  36. Peppin, S.L., Huppert, H.E., Worster, M.G.: Steady-state solidification of aqueous ammonium chloride. J. Fluid Mech. 599, 465–476 (2008)CrossRefGoogle Scholar
  37. Prasad, V.: Flow instabilities and heat transfer in fluid overlying horizontal porous layers. Exp. Therm. Fluid Sci. 6, 135–146 (1993)CrossRefGoogle Scholar
  38. Razi, Y.P., Charrier-Mojtabi, M.C., Mojtabi, A.: Thermal vibration convection in a porous medium saturated by a pure or binary fluid. In: Vadasz, P. (ed.) Emerging Topics in Heat and Mass Transfer in Porous Media, pp. 149–179. Springer, New York (2008)Google Scholar
  39. Razi, Y.P., Mojtabi, I., Charrier-Mojtabi, M.C.: A summary of new predictive high frequency thermo-vibrational modes in porous media. Transp. Porous Med. 77, 207–208 (2009)CrossRefGoogle Scholar
  40. Rrazi, Y., Maliwan, K., Mojtabi, A.: Two different approaches for studying the stability of the Horton–Rogers–Lapwood problem under the effect of vertical vibration. In: Proceedings of The First International Conference in Applications of Porous Media, pp. 479–488, Tunisia (2002)Google Scholar
  41. Rrazi, Y., Maliwan, K., Charrier-Mojtabi, M., Mojtabi, A.: The influence of mechanical vibrations on buoyancy induced convection in porous media. In: Vafai, K. (ed.) Handbook of Porous Media, pp. 321–370. Taylor and Francis Group, New York (2005)Google Scholar
  42. Tait, S., Jaupart, C.: Compositional convection in a reactive crystalline mush and melt differentiation. J. Geophys. Res. 97, 6735–6756 (1992)CrossRefGoogle Scholar
  43. Worster, G.: Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359 (1991)CrossRefGoogle Scholar
  44. Worster, G.: Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237, 649–669 (1992)CrossRefGoogle Scholar
  45. Zen’kovskaya, S.M.: The effect of high-frequency vibrations on filtration convection. AMTP 33(5), 83–88 (1992)Google Scholar
  46. Zen’kovskaya, S.M., Simonenko, I.B.: Effect of high frequency vibration on convection initiation. Fluid Dyn. 1(5), 35–37 (1966)CrossRefGoogle Scholar
  47. Zen’kovskaya, S.M., Rogovenko, T.N.: Filtration convection in high-frequency vibration field. AMTP 40(3), 22–29 (1999)Google Scholar
  48. Zhao, P., Chen, C.F.: Stability analysis of double-diffusive convection in superposed fluid and porous layers using a one-equation model. Int. J. Heat Mass Transf. 44(24), 4625–4633 (2001)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Continuous Media Mechanics UB RASPermRussia
  2. 2.Perm State UniversityPermRussia
  3. 3.Perm National Research Polytechnic UniversityPermRussia

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