Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-20T16:18:35.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Fine-scale statistics of temperature and its derivatives in convective turbulence

Published online by Cambridge University Press:  25 September 2008

M. S. EMRAN  and
J. SCHUMACHER
M. S. EMRAN
Affiliation:
Department of Mechanical Engineering, Technische Universität Ilmenau, D-98684 Ilmenau, Germany
J. SCHUMACHER
Affiliation:
Department of Mechanical Engineering, Technische Universität Ilmenau, D-98684 Ilmenau, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the fine-scale statistics of temperature and its derivatives in turbulent Rayleigh–Bénard convection. Direct numerical simulations are carried out in a cylindrical cell with unit aspect ratio filled with a fluid with Prandtl number equal to 0.7 for Rayleigh numbers between 107 and 109. The probability density function of the temperature or its fluctuations is found to be always non-Gaussian. The asymmetry and strength of deviations from the Gaussian distribution are quantified as a function of the cell height. The deviations of the temperature fluctuations from the local isotropy, as measured by the skewness of the vertical derivative of the temperature fluctuations, decrease in the bulk, but increase in the thermal boundary layer for growing Rayleigh number, respectively. Similarly to the passive scalar mixing, the probability density function of the thermal dissipation rate deviates significantly from a log-normal distribution. The distribution is fitted well by a stretched exponential form. The tails become more extended with increasing Rayleigh number which displays an increasing degree of small-scale intermittency of the thermal dissipation field for both the bulk and the thermal boundary layer. We find that the thermal dissipation rate due to the temperature fluctuations is not only dominant in the bulk of the convection cell, but also yields a significant contribution to the total thermal dissipation in the thermal boundary layer. This is in contrast to the ansatz used in scaling theories and can explain the differences in the scaling of the total thermal dissipation rate with respect to the Rayleigh number.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 611 , 25 September 2008 , pp. 13 - 34
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Belmonte, A. & Libchaber, A. 1996 Thermal signature of plumes in turbulent convection: the skewness of the derivative. Phys. Rev. E 53, 4893–4898.Google ScholarPubMed
Brown, E. & Ahlers, G. 2007 Temperature gradients, and search for non-Boussinesq effects, in the interior of turbulent Rayleigh–Bénard convection. Europhys. Lett. 80, 14001.CrossRefGoogle Scholar
Brown, E., Funfschilling, D. & Ahlers, G. 2007 Anomalous Reynolds-number scaling in turbulent Rayleigh–Bénard convection. J. Stat. Mech. P10005.CrossRefGoogle 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
Castaing, B., Gagne, Y. & Hopfinger, E. J. 1990 Velocity probability density functions of high Reynolds number turbulence. Physica D 46, 177–200.Google Scholar
Chertkov, M., Falkovich, G. & Kolokolov, I. 1998 Intermittent dissipation of a passive scalar in turbulence. Phys. Rev. Lett. 80, 2121–2124.CrossRefGoogle Scholar
Chevillard, L., Castaing, B., Lévêque, E. & Arneodo, A. 2006 Unified multifractal description of velocity increments statistics in turbulence: Intermittency and skewness. Physica D 218, 77–82.Google Scholar
Ching, E. S. C. 1991 Probabilities for temperature differences in Rayleigh–Bénard convection. Phys. Rev. A 44, 3622–3628.CrossRefGoogle ScholarPubMed
Ching, E. S. C. 1993 Probability densities of turbulent temperature fluctuations. Phys. Rev. Lett. 70, 283–286.CrossRefGoogle ScholarPubMed
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37, 329–356.CrossRefGoogle Scholar
Donzis, D. A., Sreenivasan, K. R. & Yeung, P. K. 2005 Scalar dissipation rate and dissipative anomaly in isotropic turbulence. J. Fluid Mech. 532, 199–216.CrossRefGoogle Scholar
Ferchichi, M. & Tavoularis, S. 2002 Scalar probability density function and fine structure in uniformly sheared turbulence. J. Fluid Mech. 461, 155–182.CrossRefGoogle Scholar
Funfschilling, D., Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Heat transport by turbulent Rayleigh–Bénard convection in cylindrical samples with aspect ratio one and larger. J. Fluid Mech. 536, 145–154.CrossRefGoogle Scholar
Gamba, A. & Kolokolov, I. 1999 Dissipation statistics of a passive scalar in a multi-dimensional smooth flow. J. Stat. Phys. 94, 759–777.CrossRefGoogle Scholar
Gollub, J. P., Clarke, J., Gharib, M., Lane, B. & Mesquita, O. N. 1991 Fluctuations and transport in a stirred fluid with a mean gradient. Phys. Rev. Lett. 67, 3507–3510.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.Google ScholarPubMed
Grötzbach, G. 1983 Spatial resolution requirements for direct numerical simulation of the Rayleigh–Bénard convection. J. Comput. Phys. 49, 241–269.CrossRefGoogle Scholar
Gylfason, A. & Warhaft, Z. 2004 On higher order passive scalar structure functions in grid turbulence. Phys. Fluids 16, 4012–4019.CrossRefGoogle Scholar
Hartlep, T., Tilgner, A. & Busse, F. H. 2005 Transition to turbulent convection in a fluid layer heated from below at moderate aspect ratio. J. Fluid Mech. 544, 309–322.CrossRefGoogle Scholar
He, X., Tong, P. & Xia, K.-Q. 2007 Measured thermal dissipation field in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 144501.CrossRefGoogle ScholarPubMed
Heslot, F., Castaing, B. & Libchaber, A. 1987 Transitions to turbulence in helium gas. Phys. Rev. A 36, 5870–5873.CrossRefGoogle ScholarPubMed
Jayesh, & Warhaft, Z. 1991 Probability distribution of a passive scalar in grid-generated turbulence. Phys. Rev. Lett. 67, 3503–3506.CrossRefGoogle ScholarPubMed
Jayesh, & Warhaft, Z. 1992 Probability distribution, conditional dissipation and transport of passive temperature fluctuations in grid-generated turbulence. Phys. Fluids A 4, 2292–2307.CrossRefGoogle Scholar
Kaczorowski, M. & Wagner, C. 2007 Direct numerical simulation of turbulent convection in a rectangular Rayleigh–Bénard cell. Proc. Fifth International Symp. on Turbulence and Shear Flow Phenomena, Garching, 2007, vol. 2, pp. 499–504.Google Scholar
Kadanoff, L. P. 2001 Turbulent heat flow: structures and scaling. Phys. Today 54, 34–99.CrossRefGoogle Scholar
Kerr, R. M. 1996 Rayleigh number scaling in numerical convection. J. Fluid Mech. 310, 139–179.CrossRefGoogle Scholar
Kushnir, D., Schumacher, J. & Brandt, A. 2006 Geometry of intensive scalar dissipation events in turbulence. Phys. Rev. Lett. 97, 124502.CrossRefGoogle ScholarPubMed
Lumley, J. L. 1967 Similarity and the turbulent energy spectrum. Phys. Fluids 10, 855–858.CrossRefGoogle Scholar
Mydlarski, L. & Warhaft, Z. 1998 Passive scalar statistics in high-Péclet-number grid turbulence. J. Fluid Mech. 358, 135–175.CrossRefGoogle Scholar
Niemela, J. J. & Sreenivasan, K. R. 2003 Turbulent confined convection. J. Fluid Mech. 481, 355–384.CrossRefGoogle Scholar
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donelly, R. J. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840.CrossRefGoogle ScholarPubMed
Overholt, M. R. & Pope, S. B. 1996 Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence. Phys. Fluids 8, 3128–3148.CrossRefGoogle Scholar
du Puits, R., Resagk, C., Tilgner, A., Busse, F. H. & Thess, A. 2007 a Structure of thermal boundary layers in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 572, 231–254.CrossRefGoogle Scholar
du Puits, R., Resagk, C. & Thess, A. 2007 b Breakdown of wind in turbulent thermal convection. Phys. Rev. E 75, 016302.Google ScholarPubMed
Pumir, A. 1996 Turbulence in homogeneous shear flows. Phys. Fluids 8, 3112–3127.CrossRefGoogle Scholar
Pumir, A., Shraiman, B. I. & Siggia, E. D. 1991 Exponential tails and random advection. Phys. Rev. Lett. 23, 2984–2987.CrossRefGoogle Scholar
Schumacher, J. 2008 Lagrangian dispersion and heat transport in convective turbulence. Phys. Rev. Lett. 100, 134502.CrossRefGoogle ScholarPubMed
Schumacher, J. & Sreenivasan, K. R. 2003 Geometric features of the mixing of passive scalars at high Schmidt numbers. Phys. Rev. Lett. 91, 174501.CrossRefGoogle ScholarPubMed
Schumacher, J. & Sreenivasan, K. R. 2005 Statistics and geometry of passive scalars in turbulence. Phys. Fluids 17, 125107.CrossRefGoogle Scholar
Schumacher, J., Sreenivasan, K. R. & Yeung, P. K. 2005 Very fine structures in scalar mixing. J. Fluid Mech. 531, 113–122.CrossRefGoogle Scholar
Shishkina, O. & Wagner, C. 2006 Analysis of thermal dissipation rates in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 546, 51–60.CrossRefGoogle Scholar
Shishkina, O. & Wagner, C. 2007 Local heat flux in turbulent Rayleigh–Bénard convection. Phys. Fluids 19, 085107.CrossRefGoogle Scholar
Shraiman, B. I. & Siggia, E. D. 2000 Scalar turbulence. Nature 405, 639–646.CrossRefGoogle ScholarPubMed
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137–68.CrossRefGoogle Scholar
Verzicco, R. & Camussi, R. 2003 Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J. Fluid Mech. 477, 19–49.CrossRefGoogle Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402–414.CrossRefGoogle Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.CrossRefGoogle Scholar
Warhaft, Z. 2002 Turbulence in nature and laboratory. Proc. Natl Acad. Sci. 99, 2481–2486.CrossRefGoogle Scholar
Watanabe, T. & Gotoh, T. 2004 Statistics of a passive scalar in homogeneous turbulence. New J. Phys. 6, 40.CrossRefGoogle Scholar
Xi, H.-D., Lam, S. & Xia, K.-Q. 2004 From laminar plumes to organized flows: the onset of large-scale circulation in turbulent thermal convection. J. Fluid Mech. 503, 47–56.CrossRefGoogle Scholar
Yakhot, V. 1989 Probability distributions in high-Rayleigh-number Bénard convection. Phys. Rev. Lett. 63, 1965–1967.CrossRefGoogle ScholarPubMed
Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2005 High-Reynolds-number simulation of turbulent mixing. Phys. Fluids 17 081703.CrossRefGoogle Scholar
Zhou, S.-Q. & Xia, K.-Q. 2002 Plume statistics in thermal turbulence: mixing of an active scalar. Phys. Rev. Lett. 89, 184502.CrossRefGoogle ScholarPubMed
Zhou, Q., Sun, C. & Xia, K.-Q. 2007 Morphological evolution of thermal plumes in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 074501.CrossRefGoogle ScholarPubMed