Heat Transport in Horizontal and Inclined Convection

  • Olga ShishkinaEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 196)


We discuss three classical paradigmatic systems of thermally driven flows: Rayleigh–Bénard convection, where a fluid is confined between a heated bottom plate and a cooled top plate, horizontal convection, where the fluid is heated at one part of the bottom and cooled at some other part, and vertical convection, where the fluid is confined between two differently heated isothermal vertical plates. Rayleigh–Bénard and vertical convection can be also considered as limiting cases of so-called inclined convection. For these systems we study how the heat and momentum transport, which is represented by the Nusselt number and Reynolds number, scales with the main governing parameters of the system, which are the Rayleigh number and Prandtl number. We show that different boundary conditions generally lead to different scaling diagrams in the Prandtl–Rayleigh plane. For laminar vertical convection the scalings can be derived from the boundary layer equations, see Shishkina (Phys Rev E 93:051102, 2016, [8]). In the case of horizontal convection, the scalings can be derived from the analysis of the boundary-layer and bulk contributions of the kinetic and thermal dissipation rates, see Shishkina et al. (Geophys Res Lett 43:1219–1225, 2016, [5]). Here we summarize some previous results and discuss the applicability of the developed theory to global ocean circulation.



OS is grateful to G. Ahlers, E. Bodenschatz, E. Ching, S. Grossmann, X. He, S. Horn, D. Lohse and S. Weiss for useful discussions and acknowledges the financial support of the Deutsche Forschungsgemeinschaft (DFG) under Grant Sh405/4 – Heisenberg fellowship.


  1. 1.
    G. Ahlers, S. Grossmann, D. Lohse, Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537 (2009)CrossRefGoogle Scholar
  2. 2.
    F. Chillà, J. Schumacher, New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35, 58 (2012)CrossRefGoogle Scholar
  3. 3.
    G.O. Hughes, R.W. Griffiths, Horizontal convection. Ann. Rev. Fluid Mech. 40, 185–208 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    R.W. Griffiths, G.O. Hughes, B. Gayen, Horizontal convection dynamics: insights from transient adjustment. J. Fluid Mech. 726, 559–595 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    O. Shishkina, S. Grossmann, D. Lohse, Heat and momentum transport scalings in horizontal convection. Geophys. Res. Lett. 43, 1219–1225 (2016)CrossRefGoogle Scholar
  6. 6.
    O. Shishkina, S. Grossmann, D. Lohse, Prandtl-number dependences of the heat and momentum transport in horizontal convection. Proc. IUTAM 00, 000–000 (2016)Google Scholar
  7. 7.
    S. Ostrach, An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force. NACA report, vol. 1111 (1953)Google Scholar
  8. 8.
    O. Shishkina, Momentum and heat transport scalings in laminar vertical convection. Phys. Rev. E (R) 93, 051102 (2016)CrossRefGoogle Scholar
  9. 9.
    C.S. Ng, A. Ooi, D. Lohse, D. Chung, Vertical natural convection: application of the unifying theory of thermal convection. J. Fluid Mech. 764, 349–361 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Grossmann, D. Lohse, Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    S. Grossmann, D. Lohse, Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86, 3316–3319 (2001)CrossRefGoogle Scholar
  12. 12.
    P. Frick, R. Khalilov, I. Kolesnichenko, A. Mamykin, V. Pakholkov, A. Pavlinov, S.A. Rogozhkin, Turbulent convective heat transfer in a long cylinder with liquid sodium. Europhys. Lett. 109, 14002 (2015)CrossRefGoogle Scholar
  13. 13.
    O. Shishkina, S. Horn, Thermal convection in inclined cylindrical containers. J. Fluid Mech. 790, R3 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    O. Shishkina, S. Horn, S. Wagner, E.S.C. Ching, Thermal boundary layer equation for turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 114, 114302 (2015)CrossRefGoogle Scholar
  15. 15.
    H.T. Rossby, On thermal convection driven by non-uniform heating from below: an experimental study. Deep Sea Res. 12, 9–16 (1965)Google Scholar
  16. 16.
    J.H. Siggers, R.R. Kerswell, N.J. Balmforth, Bounds on horizontal convection. J. Fluid Mech. 517, 55–70 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    O. Shishkina, Mean flow structure in horizontal convection. J. Fluid Mech. 812, 525–540 (2017)Google Scholar
  18. 18.
    O. Shishkina, S. Wagner, Prandtl-number dependence of heat transport in laminar horizontal convection. Phys. Rev. Lett. 116, 024302 (2016)CrossRefGoogle Scholar
  19. 19.
    W. Munk, C. Wunsch, Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. 45, 1977–2010 (1998)CrossRefGoogle Scholar
  20. 20.
    L.D. Landau, E.M. Lifshitz, Fluid Mechanics. Course of Theoretical Physics, vol. 6, 2nd edn. (Butterworth-Heinemann, Oxford, 1987)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Max Planck Institute for Dynamics and Self-OrganizationGöttingenGermany

Personalised recommendations