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Turbulent Rayleigh–Bénard convection described by projected dynamics in phase space

Published online by Cambridge University Press:  21 September 2015

Johannes Lülff*
Affiliation:
Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany
Michael Wilczek
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, D-37077 Göttingen, Germany
Richard J. A. M. Stevens
Affiliation:
Department of Mechanical Engineering and Center for Environmental and Applied Fluid Mechanics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA Physics of Fluids Group, Faculty of Science and Technology, Mesa+ Research Institute, and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands
Rudolf Friedrich
Affiliation:
Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany
Detlef Lohse
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, D-37077 Göttingen, Germany Physics of Fluids Group, Faculty of Science and Technology, Mesa+ Research Institute, and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands
*
Email address for correspondence: johannes.luelff@uni-muenster.de

Abstract

Rayleigh–Bénard convection, i.e. the flow of a fluid between two parallel plates that is driven by a temperature gradient, is an idealised set-up to study thermal convection. Of special interest are the statistics of the turbulent temperature field, which we are investigating and comparing for three different geometries, namely convection with periodic horizontal boundary conditions in three and two dimensions as well as convection in a cylindrical vessel, in order to determine the similarities and differences. To this end, we derive an exact evolution equation for the temperature probability density function. Unclosed terms are expressed as conditional averages of velocities and heat diffusion, which are estimated from direct numerical simulations. This framework lets us identify the average behaviour of a fluid particle by revealing the mean evolution of a fluid with different temperatures in different parts of the convection cell. We connect the statistics to the dynamics of Rayleigh–Bénard convection, giving deeper insights into the temperature statistics and transport mechanisms. We find that the average behaviour is described by closed cycles in phase space that reconstruct the typical Rayleigh–Bénard cycle of fluid heating up at the bottom, rising up to the top plate, cooling down and falling again. The detailed behaviour shows subtle differences between the three cases.

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Papers
Copyright
© 2015 Cambridge University Press 

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Footnotes

Deceased.

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