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Mean temperature profiles in turbulent thermal convection

Olga Shishkina*

Susanne Horn†

Mohammad S. Emran‡

Emily S. C. Ching§

  • Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany

  • Earth, Planetary, and Space Sciences, University of California, Los Angeles, California 900 95, USA

  • Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany

  • Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong

  • *Olga.Shishkina@ds.mpg.de
  • †SusanneHorn@ucla.edu
  • ‡Mohammad.Emran@ds.mpg.de
  • §ching@phy.cuhk.edu.hk
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Phys. Rev. Fluids 2, 113502 – Published 15 November, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.113502

Abstract

To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al. [Phys. Rev. Lett. 114, 114302 (2015)], the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated, respectively, as a cubic and a linear function of the distance from the plate. In the present work, we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from 107 to 109.

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Phys. Rev. Fluids 2, 113502– Published 15 November, 2017
  • Received 18 August 2017

©2017 American Physical Society

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