Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-30T15:38:14.464Z Has data issue: false hasContentIssue false
  • English
  • Français

Kinetic energy transport in Rayleigh–Bénard convection

Published online by Cambridge University Press:  22 May 2015

K. Petschel ,
S. Stellmach ,
M. Wilczek ,
J. Lülff  and
U. Hansen
K. Petschel*
Affiliation:
Institut für Geophysik, Westfälische Wilhelms-Universität, Münster, D-48149, Germany
S. Stellmach
Affiliation:
Institut für Geophysik, Westfälische Wilhelms-Universität, Münster, D-48149, Germany
M. Wilczek
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, Göttingen, D-37077, Germany
J. Lülff
Affiliation:
Institut für Theoretische Physik, Westfälische Wilhelms-Universität, Münster, D-48149, Germany
U. Hansen
Affiliation:
Institut für Geophysik, Westfälische Wilhelms-Universität, Münster, D-48149, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The kinetic energy balance in Rayleigh–Bénard convection is investigated by means of direct numerical simulations for the Prandtl number range $0.01\leqslant \mathit{Pr}\leqslant 150$ and for fixed Rayleigh number $\mathit{Ra}=5\times 10^{6}$. The kinetic energy balance is divided into a dissipation, a production and a flux term. We discuss the profiles of all the terms and find that the different contributions to the energy balance can be spatially separated into regions where kinetic energy is produced and where kinetic energy is dissipated. By analysing the Prandtl number dependence of the kinetic energy balance, we show that the height dependence of the mean viscous dissipation is closely related to the flux of kinetic energy. We show that the flux of kinetic energy can be divided into four additive contributions, each representing a different elementary physical process (advection, buoyancy, normal viscous stresses and viscous shear stresses). The behaviour of these individual flux contributions is found to be surprisingly rich and exhibits a pronounced Prandtl number dependence. Different flux contributions dominate the kinetic energy transport at different depths, such that a comprehensive discussion requires a decomposition of the domain into a considerable number of sublayers. On a less detailed level, our results reveal that advective kinetic energy fluxes play a key role in balancing the near-wall dissipation at low Prandtl number, whereas normal viscous stresses are particularly important at high Prandtl number. Finally, our work reveals that classical velocity boundary layers are deeply connected to the kinetic energy transport, but fail to correctly represent regions of enhanced viscous dissipation.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 773 , 25 June 2015 , pp. 395 - 417
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503–537.Google Scholar
Breuer, M., Wessling, S., Schmalzl, J. & Hansen, U. 2004 Effect of inertia in Rayleigh–Bénard convection. Phys. Rev. E 69 (2), 026302.Google Scholar
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in Rayleigh–Bénard convection. J. Fluid Mech. 204, 1–30.CrossRefGoogle Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Euro. Phys. J. E 35 (7), 1–25.Google Scholar
Deardorff, J. W. & Willis, G. E. 1967 Investigation of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28 (04), 675–704.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.Google Scholar
Grötzbach, G. 1983 Spatial resolution requirements for direct numerical simulation of the Rayleigh–Bénard convection. J. Comput. Phys. 49 (2), 241–264.Google Scholar
Kerr, R. M. 2001 Energy budget in Rayleigh–Bénard convection. Phys. Rev. Lett. 87 (24), 244502.Google Scholar
Kerr, R. M. & Herring, J. R. 2000 Prandtl number dependence of Nusselt number in direct numerical simulations. J. Fluid Mech. 419, 325–344.Google Scholar
Lam, S., Shang, X.-D., Zhou, S.-Q. & Xia, K.-Q. 2002 Prandtl number dependence of the viscous boundary layer and the Reynolds numbers in Rayleigh–Bénard convection. Phys. Rev. E 65 (6), 066306.CrossRefGoogle ScholarPubMed
Li, L., Shi, N., du Puits, R., Resagk, C., Schumacher, J. & Thess, A. 2012 Boundary layer analysis in turbulent Rayleigh–Bénard convection in air: experiment versus simulation. Phys. Rev. E 86 (2), 026315.Google Scholar
Petschel, K., Stellmach, S., Wilczek, M., Lülff, J. & Hansen, U. 2013 Dissipation layers in Rayleigh–Bénard convection: a unifying view. Phys. Rev. Lett. 110 (11), 114502.Google Scholar
du Puits, R., Resagk, C. & Thess, A. 2007 Mean velocity profile in confined turbulent convection. Phys. Rev. Lett. 99 (23), 234504.Google Scholar
Scheel, J. D., Kim, E. & White, K. R. 2012 Thermal and viscous boundary layers in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 711, 281–305.Google Scholar
Scheel, J. D. & Schumacher, J. 2014 Local boundary layer scales in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 758, 344–373.Google Scholar
Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory, 8th edn. Springer.CrossRefGoogle Scholar
Shi, N., Emran, M. S. & Schumacher, J. 2012 Boundary layer structure in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 706, 5–33.CrossRefGoogle Scholar
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.Google Scholar
Stellmach, S. & Hansen, U. 2008 An efficient spectral method for the simulation of dynamos in Cartesian geometry and its implementation on massively parallel computers. Geochem. Geophys. Geosyst. 9 (5), Q05003.CrossRefGoogle Scholar
Tilgner, A., Belmonte, A. & Libchaber, A. 1993 Temperature and velocity profiles of turbulent convection in water. Phys. Rev. E 47 (4), R2253–R2256.CrossRefGoogle ScholarPubMed
Wagner, S., Shishkina, O. & Wagner, C. 2012 Boundary layers and wind in cylindrical Rayleigh–Bénard cells. J. Fluid Mech. 697, 336–366.Google Scholar
Wallace, J. M. 2009 Twenty years of experimental and direct numerical simulation access to the velocity gradient tensor: what have we learned about turbulence? Phys. Fluids 21 (2), 021301.CrossRefGoogle Scholar
Zhou, Q., Sugiyama, K., Stevens, R. J. A. M., Grossmann, S., Lohse, D. & Xia, K.-Q. 2011 Horizontal structures of velocity and temperature boundary layers in two-dimensional numerical turbulent Rayleigh–Bénard convection. Phys. Fluids 23 (12), 125104,1–15.CrossRefGoogle Scholar
Zhou, Q. & Xia, K.-Q. 2010 Measured instantaneous viscous boundary layer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 104 (10), 104301.CrossRefGoogle ScholarPubMed