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Modified Rayleigh–Bénard convection driven by long-wavelength heating from above and below

  • J. M. FloryanEmail author
  • M. Z. Hossain
  • Andrew P. Bassom
Original Article

Abstract

Classical Rayleigh–Bénard convection occurs when a horizontal fluid layer is heated sufficiently strongly from below. In more recent times, there has been increased interest in structured convection, when the heating that is applied is no longer uniform but rather spatially varying in some way. Here, we examine the effect of structured convection in a fluid layer when both the lower and upper boundaries are heated so that their temperatures fluctuate sinusoidally over a common long length scale offset by some phase \(\varOmega \). While this non-uniform heating can be shown to induce small fluid motions by way of a primary form of convection, some previous computations have shown that the layer is also susceptible to a stronger, secondary form of convection. When the heating is just sufficient to induce this secondary motion, the cells tend to conglomerate near the local hot spots where the underlying heating is at its most intense. The strength of the secondary convection falls off away from the hot spot on a length scale which is appreciably longer than the wavelength of the individual cells but also much shorter than the wavelength of the underlying applied heating. In this work, we derive a second-order amplitude equation that describes the secondary convection and discuss the important features of its solution. These are compared with some direct numerical simulations and are likely to apply for other situations in which long-scale heating gives rise to structured convection patterns.

Keywords

Distributed heating Secondary flow Long wavelength modes 

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Notes

Acknowledgements

The referees are thanked for their numerous helpful comments which led to significant improvements in the presentation and discussion of this work. The first author would like to acknowledge support through the NSERC of Canada and from an Erskine Fellowship at the University of Canterbury, New Zealand.

References

  1. 1.
    Rizwan, A.M., Dennis, L.Y.C., Liu, C.: A review on the generation, determination and mitigation of urban heat island. J. Environ. Sci. 20, 120–128 (2008)CrossRefGoogle Scholar
  2. 2.
    Finney, M.A., Cohen, J.D., McAllister, S.S., Jolly, W.M.: On the need for a theory of wildland fire spread. Int. J. Wildland Fire 22, 25–36 (2012)CrossRefGoogle Scholar
  3. 3.
    Hossain, M.Z., Floryan, J.M.: Drag reduction in a thermally modulated channel. J. Fluid Mech. 791, 122–153 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abtahi, A., Floryan, J.M.: Natural convection in corrugated slots. J. Fluid Mech. 815, 537–569 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hossain, M.Z., Floryan, J.M.: Natural convection under sub-critical conditions in the presence of heating non-uniformities. Int. J. Heat Mass Trans. 114, 8–19 (2017)CrossRefGoogle Scholar
  6. 6.
    Hossain, M.Z., Floryan, J.M.: Mixed convection in a periodically heated channel. J. Fluid Mech. 768, 51–90 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bénard, H.: Les tourbillons cellulaires dans une nappe liquide. Rev. Gén. Sci. Pure et Appl. 11, 1261–1271 (1900)Google Scholar
  8. 8.
    Rayleigh, J.W.S.: On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Philos. Mag. 32, 529–546 (1916)CrossRefGoogle Scholar
  9. 9.
    Freund, G., Pesch, W., Zimmermann, W.: Rayleigh-Bénard convection in the presence of spatial temperature modulations. J. Fluid Mech. 673, 318–348 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Weiss, S., Seiden, G., Bodenschatz, E.: Pattern formation in spatially forced thermal convection. New J. Phys. 14, 053010 (2012)CrossRefGoogle Scholar
  11. 11.
    McCoy, J.H., Brunner, W., Pesch, W., Bodenschatz, E.: Self-organization of topological defects due to applied constraints. Phys. Rev. Lett. 101, 254102 (2008)CrossRefGoogle Scholar
  12. 12.
    Shen, Y., Tong, P., Xia, K.Q.: Turbulent convection over rough surfaces. Phys. Rev. Lett. 76, 908–911 (1996)CrossRefGoogle Scholar
  13. 13.
    Wei, P., Chan, T.S., Ni, R., Zhao, X.Z., Xia, K.Q.: Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection. J. Fluid Mech. 740, 28–46 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Toppaladoddi, S., Succi, S., Wettlaufer, J.S.: Roughness as a route to the ultimate regime of thermal convection. Phys. Rev. Lett. 118, 074503 (2017)CrossRefGoogle Scholar
  15. 15.
    Du, Y.B., Tong, P.: Enhanced heat transport in turbulent convection over a rough surface. Phys. Rev. Lett. 81, 987–990 (1998)CrossRefGoogle Scholar
  16. 16.
    Wagner, S., Shishkina, O.: Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers. Phys. Fluids 25, 085110 (2013)CrossRefGoogle Scholar
  17. 17.
    Chong, K.L., Huang, S.D., Kaczorowski, M., Xia, K.Q.: Condensation of coherent structures in turbulent flows. Phys. Rev. Lett. 115, 264503 (2015)CrossRefGoogle Scholar
  18. 18.
    Huang, S.D., Kaczorowski, M., Ni, R., Xia, K.Q.: Confinement-induced heat-transport enhancement in turbulent thermal convection. Phys. Rev. Lett. 111, 104501 (2013)CrossRefGoogle Scholar
  19. 19.
    Bizon, C., Werne, J., Predtechensky, A.A., Julien, K., McCormick, W.D., Swift, J.B., Swinney, H.L.: Plume dynamics in quasi-2D turbulent convection. Chaos 7, 107–124 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Bao, Y., Chen, J., Liu, B.F., She, Z.S., Zhang, J., Zhou, Q.: Enhanced heat transport in partitioned thermal convection. J. Fluid Mech. 784(R5), 1–11 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Abtahi, A., Floryan, J.M.: Natural convection and thermal drift. J. Fluid Mech. 826, 553–582 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Walton, I.C.: The effects of slow spatial variations on Bénard convection. Quart. Jl. Mech. Appl. Math. 35, 33–48 (1982)CrossRefGoogle Scholar
  23. 23.
    Hossain, M.Z., Floryan, J.M.: Instabilities of natural convection in a periodically heated layer. J. Fluid Mech. 733, 33–67 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Asgarian, A., Hossain, M.Z., Floryan, J.M.: Rayleigh–Bénard convection driven by a long wavelength heating. Theor. Comput. Fluid Dyn. 30, 313–337 (2016)CrossRefGoogle Scholar
  25. 25.
    Hossain, M.Z., Floryan, D., Floryan, J.M.: Drag reduction due to spatial thermal modulations. J. Fluid Mech. 713, 398–419 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)zbMATHGoogle Scholar
  27. 27.
    Drazin, P.G., Reid, W.H.: Hydrodynamic Stability, second edn. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • J. M. Floryan
    • 1
    Email author
  • M. Z. Hossain
    • 1
  • Andrew P. Bassom
    • 2
  1. 1.Department of Mechanical and Materials EngineeringUniversity of Western OntarioLondonCanada
  2. 2.School of Natural SciencesUniversity of TasmaniaHobartAustralia

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