Skip to main content

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

  • Johannes Lülff (a1), Michael Wilczek (a2), Richard J. A. M. Stevens (a3) (a4), Rudolf Friedrich (a1) and Detlef Lohse (a2) (a4)...

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.

Corresponding author
Email address for correspondence:
Hide All


Hide All
Ahlers G., Grossmann S. & Lohse D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503537.
Ahlers G., He X., Funfschilling D. & Bodenschatz E. 2012 Heat transport by turbulent Rayleigh–Bénard convection for inline-graphic $Pr\simeq 0.8$ and inline-graphic $3\times 10^{12}\lesssim Ra\lesssim 10^{15}$ : aspect ratio inline-graphic ${\it\Gamma}=0.5$ . New J. Phys. 14 (10), 103012.
Angot P., Bruneau C.-H. & Fabrie P. 1999 A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81 (4), 497520.
Bailon-Cuba J., Emran M. S. & Schumacher J. 2010 Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection. J. Fluid Mech. 655, 152173.
Boussinesq J. 1903 Théorie Analytique de la Chaleur. Gauthier-Villars.
Calzavarini E., Lohse D., Toschi F. & Tripiccione R. 2005 Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence. Phys. Fluids 17 (5), 055107.
Chillà F. & Schumacher J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. J. Phys. E 35, 125.
Ching E. S. C. 1993 Probability densities of turbulent temperature fluctuations. Phys. Rev. Lett. 70 (3), 283286.
Ching E. S. C., Guo H., Shang X.-D., Tong P. & Xia K.-Q. 2004 Extraction of plumes in turbulent thermal convection. Phys. Rev. Lett. 93, 124501.
Courant R. & Hilbert D. 1962 Methods of Mathematical Physics, Vol. II. Wiley.
Friedrich R., Daitche A., Kamps O., Lülff J., Voßkuhle M. & Wilczek M. 2012 The Lundgren–Monin–Novikov hierarchy: kinetic equations for turbulence. C. R. Phys. 13 (9–10), 929953.
Gasteuil Y., Shew W. L., Gibert M., Chillà F., Castaing B. & Pinton J.-F. 2007 Lagrangian temperature, velocity, and local heat flux measurement in Rayleigh–Bénard convection. Phys. Rev. Lett. 99 (23), 234302.
Grossmann S. & Lohse D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.
Grossmann S. & Lohse D. 2001 Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86 (15), 33163319.
Grossmann S. & Lohse D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23 (4), 045108.
Grossmann S. & Lohse D. 2012 Logarithmic temperature profiles in the ultimate regime of thermal convection. Phys. Fluids 24 (12), 125103.
Keetels G. H., D’Ortona U., Kramer W., Clercx H. J. H., Schneider K. & van Heijst G. J. F. 2007 Fourier spectral and wavelet solvers for the incompressible Navier–Stokes equations with volume-penalization: Convergence of a dipole-wall collision. J. Comput. Phys. 227 (2), 919945.
Lohse D. & Toschi F. 2003 Ultimate state of thermal convection. Phys. Rev. Lett. 90, 034502.
Lohse D. & Xia K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42 (1), 335364.
Lülff J., Wilczek M. & Friedrich R. 2011 Temperature statistics in turbulent Rayleigh–Bénard convection. New J. Phys. 13 (1), 015002.
Lundgren T. S. 1967 Distribution functions in the statistical theory of turbulence. Phys. Fluids 10 (5), 969975.
Monin A. S. 1967 Equations of turbulent motion. Prikl. Mat. Mekh. 31 (6), 10571068.
Novikov E. A. 1968 Kinetic equations for a vortex field. Sov. Phys. Dokl. 12 (11), 10061008.
Oberbeck A. 1879 Über die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem. 7, 271292.
Petschel K., Stellmach S., Wilczek M., Lülff J. & Hansen U. 2013 Dissipation layers in Rayleigh–Bénard convection: a unifying view. Phys. Rev. Lett. 110, 114502.
Petschel K., Wilczek M., Breuer M., Friedrich R. & Hansen U. 2011 Statistical analysis of global wind dynamics in vigorous Rayleigh–Bénard convection. Phys. Rev. E 84 (2), 026309.
van der Poel E. P., Ostilla-Mónico R., Verzicco R. & Lohse D. 2014 Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 90, 013017.
van der Poel E. P., Stevens R. J. A. M. & Lohse D. 2011 Connecting flow structures and heat flux in turbulent Rayleigh–Bénard convection. Phys. Rev. E 84 (4), 045303.
van der Poel E. P., Stevens R. J. A. M. & Lohse D. 2013 Comparison between two- and three-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 736, 177194.
Pope S. B. 1984 Calculations of a plane turbulent jet. AIAA J. 22 (7), 896904.
Pope S. B. 1985 PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119192.
Pope S. B. 2000 Turbulent Flows. Cambridge University Press.
Sarra S. A. 2003 The method of characteristics with applications to conservation laws. J. Onl. Math. Appl. 3, 16.
Schmalzl J., Breuer M., Wessling S. & Hansen U. 2004 On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 67 (3), 390396.
Schneider K. 2005 Numerical simulation of the transient flow behaviour in chemical reactors using a penalisation method. Comput. Fluids 34 (10), 12231238.
Schumacher J. 2009 Lagrangian studies in convective turbulence. Phys. Rev. E 79 (5), 056301.
Shang X.-D., Tong P. & Xia K.-Q. 2008 Scaling of the local convective heat flux in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 100, 244503.
Shishkina O., Stevens R. J. A. M., Grossmann S. & Lohse D. 2010 Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New J. Phys. 12 (7), 075022.
Stevens R. J. A. M., van der Poel E. P., Grossmann S. & Lohse D. 2013 The unifying theory of scaling in thermal convection: The updated prefactors. J. Fluid Mech. 730, 295308.
Sugiyama K., Ni R., Stevens R. J. A. M., Chan T. S., Zhou S.-Q., Xi H.-D., Sun C., Grossmann S., Xia K.-Q. & Lohse D. 2010 Flow reversals in thermally driven turbulence. Phys. Rev. Lett. 105 (3), 034503.
Verzicco R. & Camussi R. 2003 Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J. Fluid Mech. 477, 1949.
Wilczek M., Daitche A. & Friedrich R. 2011 On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity. J. Fluid Mech. 676, 191217.
Wilczek M. & Friedrich R. 2009 Dynamical origins for non-Gaussian vorticity distributions in turbulent flows. Phys. Rev. E 80 (1), 016316.
Yakhot V. 1989 Probability distributions in high-Rayleigh number Bénard convection. Phys. Rev. Lett. 63 (18), 19651967.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 5
Total number of PDF views: 74 *
Loading metrics...

Abstract views

Total abstract views: 174 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th November 2017. This data will be updated every 24 hours.