Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-29T04:38:22.883Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection

Published online by Cambridge University Press:  03 February 2017

Yang Zhang ,
Quan Zhou  and
Chao Sun
Yang Zhang
Affiliation:
Shanghai Institute of Applied Mathematics and Mechanics, and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
Quan Zhou*
Affiliation:
Shanghai Institute of Applied Mathematics and Mechanics, and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
Chao Sun
Affiliation:
Center for Combustion Energy and Department of Thermal Engineering, Tsinghua University, 100084 Beijing, China
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the statistical properties of the kinetic $\unicode[STIX]{x1D700}_{u}$ and thermal $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}$ energy dissipation rates in two-dimensional (2-D) turbulent Rayleigh–Bénard (RB) convection. Direct numerical simulations were carried out in a box with unit aspect ratio in the Rayleigh number range $10^{6}\leqslant Ra\leqslant 10^{10}$ for Prandtl numbers $Pr=0.7$ and 5.3. The probability density functions (PDFs) of both dissipation rates are found to deviate significantly from a log-normal distribution. The PDF tails can be well described by a stretched exponential function, and become broader for higher Rayleigh number and lower Prandtl number, indicating an increasing degree of small-scale intermittency with increasing Reynolds number. Our results show that the ensemble averages $\langle \unicode[STIX]{x1D700}_{u}\rangle _{V,t}$ and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ scale as $Ra^{-0.18\sim -0.20}$, which is in excellent agreement with the scaling estimated from the two global exact relations for the dissipation rates. By separating the bulk and boundary-layer contributions to the total dissipations, our results further reveal that $\langle \unicode[STIX]{x1D700}_{u}\rangle _{V,t}$ and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ are both dominated by the boundary layers, corresponding to regimes $I_{l}$ and $I_{u}$ in the Grossmann–Lohse (GL) theory (J. Fluid Mech., vol. 407, 2000, pp. 27–56). To include the effects of thermal plumes, the plume–background partition is also considered and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ is found to be plume dominated. Moreover, the boundary-layer/plume contributions scale as those predicted by the GL theory, while the deviations from the GL predictions are observed for the bulk/background contributions. The possible reasons for the deviations are discussed.

JFM classification

Convection: Convection Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 814 , 10 March 2017 , pp. 165 - 184
Check for updates
Copyright
© 2017 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, 503–537.Google Scholar
Ashkenazi, S. & Steinberg, V. 1999 Spectra and statistics of velocity and temperature fluctuations in turbulent convection. Phys. Rev. Lett. 83, 4760.CrossRefGoogle Scholar
Brown, E., Funfschilling, D. & Ahlers, G. 2007 Anomalous Reynolds-number scaling in turbulent Rayleigh–Bénard convection. J. Stat. Mech. 10, P10005.Google Scholar
Calzavarini, E., Lohse, D., Toschi, F. & Tripiccione, R. 2005 Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard convection. Phys. Fluids 17, 055107.Google Scholar
Celani, A., Matsumoto, T., Mazzino, A. & Vergassola, M. 2002 Scaling and universality in turbulent convection. Phys. Rev. Lett. 88, 054503.Google Scholar
Chandra, M. & Verma, M. K. 2013 Flow reversals in turbulent convection via vortex reconnections. Phys. Rev. Lett. 110, 114503.CrossRefGoogle ScholarPubMed
Chertkov, M., Falkovich, G. & Kolokolov, I. 1998 Intermittent dissipation of a passive scalar in turbulence. Phys. Rev. Lett. 80, 2121–2124.Google Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35, 58.Google Scholar
Emran, M. S. & Schumacher, J. 2008 Fine-scale statistics of temperature and its derivatives in convective turbulence. J. Fluid Mech. 611, 13–34.Google Scholar
Emran, M. S. & Schumacher, J. 2012 Conditional statistics of thermal dissipation rate in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35, 108.Google ScholarPubMed
Ferchichi, M. & Tavoularis, S. 2002 Scalar probability density function and fine structure in uniformly sheared turbulence. J. Fluid Mech. 461, 155–182.Google Scholar
Gamba, A. & Kolokolov, I. 1999 Dissipation statistics of a passive scalar in a multidimensional smooth flow. J. Stat. Phys. 94, 759–777.Google Scholar
Gastine, T., Wicht, J. & Aurnou, J. M. 2015 Turbulent Rayleigh–Bénard convection in spherical shells. J. Fluid Mech. 778, 721–764.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.Google Scholar
Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16, 4462.Google Scholar
Grötzbach, G. 1983 Spatial resolution requirements for direct numerical simulation of the Rayleigh–Bénard convection. J. Comput. Phys. 49, 241–269.CrossRefGoogle Scholar
He, X.-Z., Ching, E. S. C. & Tong, P. 2011 Locally averaged thermal dissipation rate in turbulent thermal convection: a decomposition into contributions from different temperature gradient components. Phys. Fluids 23, 025106.CrossRefGoogle Scholar
He, X.-Z. & Tong, P. 2009 Measurements of the thermal dissipation field in turbulent Rayleigh–Bénard convection. Phys. Rev. E 79, 026306.Google Scholar
He, X.-Z., Tong, P. & Ching, E. S. C. 2010 Statistics of the locally averaged thermal dissipation rate in turbulent Rayleigh–Bénard convection. J. Turbul. 11 (35), 1–10.CrossRefGoogle Scholar
He, X.-Z., Tong, P. & Xia, K.-Q. 2007 Measured thermal dissipation field in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 144501.CrossRefGoogle ScholarPubMed
Huang, S.-D., Kaczorowski, M., Ni, R. & Xia, K.-Q. 2013 Confinement-induced heat-transport enhancement in turbulent thermal convection. Phys. Rev. Lett. 111, 104501.Google Scholar
Huang, Y.-X. & Zhou, Q. 2013 Counter-gradient heat transport in two-dimensional turbulent Rayleigh–Bénard convection. J. Fluid Mech. 737, R3.Google Scholar
Johnston, H. & Doering, C. R. 2009 Comparison of temperature thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.CrossRefGoogle ScholarPubMed
Kaczorowski, M. & Wagner, C. 2009 Analysis of the thermal plumes in turbulent Rayleigh–Bénard convection based on well-resolved numerical simulations. J. Fluid Mech. 618, 89–112.CrossRefGoogle Scholar
Kadanoff, L. P. 2001 Turbulent heat flow: structures and scaling. Phys. Today 54 (8), 34–39.CrossRefGoogle Scholar
Kerr, R. 1996 Rayleigh number scaling in numerical convection. J. Fluid Mech. 310, 139–179.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.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, 066306.Google ScholarPubMed
Liu, J.-G., Wang, C. & Johnston, H. 2003 A fourth order scheme for incompressible Boussinesq equations. J. Sci. Comput. 18, 253–285.Google Scholar
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.CrossRefGoogle Scholar
Ng, C. S., Ooi, A., Lohse, D. & Chung, D. 2015 Vertical natural convection: application of the unifying theory of thermal convection. J. Fluid Mech. 764, 349–361.Google Scholar
Ni, R., Huang, S.-D. & Xia, K.-Q. 2011 Local energy dissipation rate balances local heat flux in the center of turbulent thermal convection. Phys. Rev. Lett. 107, 174503.Google Scholar
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2001 The wind in confined thermal convection. J. Fluid Mech. 449, 169–178.CrossRefGoogle Scholar
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.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, 114502.Google Scholar
Petschel, K., Stellmach, S., Wilczek, M., Lülff, J. & Hansen, U. 2015 Kinetic energy transport in Rayleigh–Bénard convection. J. Fluid Mech. 773, 395–417.Google Scholar
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, 177–194.Google Scholar
van der Poel, E. P., Stevens, R. J. A. M., Sugiyama, K. & Lohse, D. 2012 Flow states in two-dimensional Rayleigh–Bénard convection as a function of aspect-ratio and Rayleigh number. Phys. Fluids 24, 085104.Google Scholar
van der Poel, E. P., Verzicco, R., Grossmann, S. & Lohse, D. 2015 Plume emission statistics in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 772, 5–15.CrossRefGoogle Scholar
du Puits, R., Resagk, C. & Thess, A. 2007 Mean velocity profile in confined turbulent convection. Phys. Rev. Lett. 99, 234504.Google Scholar
du Puits, R., Resagk, C. & Thess, A. 2010 Measurements of the instantaneous local heat flux in turbulent Rayleigh–Bénard convection. New J. Phys. 12, 075023.Google Scholar
Qiu, X., Liu, Y.-L. & Zhou, Q. 2014 Local dissipation scales in two-dimensional Rayleigh–Taylor turbulence. Phys. Rev. E 90, 043012.Google Scholar
Qiu, X.-L. & Tong, P. 2001 Large-scale velocity structures in turbulent thermal convection. Phys. Rev. Lett. 64, 036304.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
Schumacher, J. & Sreenivasan, K. R. 2005 Statistics and geometry of passive scalars in turbulence. Phys. Fluids 17, 125107.CrossRefGoogle Scholar
Shishkina, O., Horn, S., Wagner, S. & Ching, E. S. C. 2015 Thermal boundary layer equation for turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 114, 114302.Google 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, 075022.CrossRefGoogle Scholar
Shishkina, O. & Wagner, C. 2006 Analysis of thermal dissipation rates in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 546, 51–60.Google Scholar
Shishkina, O. & Wagner, C. 2007 Local heat fluxes in turbulent Rayleigh–Bénard convection. Phys. Fluids 19, 085107.Google Scholar
Shishkina, O. & Wagner, C. 2008 Analysis of sheet-like thermal plumes in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 599, 383–404.Google Scholar
Silano, G., Sreenivasan, K. R. & Verzicco, R. 2010 Numerical simulations of Rayleigh–Bénard convection for Prandtl numbers between 10-1 and 104 and Rayleigh numbers between 105 and 109 . J. Fluid Mech. 662, 409–446.Google Scholar
Stevens, R. J. A. M., Lohse, D. & Verzicco, R. 2011 Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid Mech. 688, 31–43.Google Scholar
Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2010 Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection. J. Fluid Mech. 643, 495–507.Google Scholar
Sugiyama, K., Calzavarini, E., Grossmann, S. & Lohse, D. 2009 Flow organization in two-dimensional non-Oberbeck–Boussinesq Rayleigh–Bénard convection in water. J. Fluid Mech. 637, 105–135.CrossRefGoogle Scholar
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, 034503.Google Scholar
Sun, C., Ren, L.-Y., Song, H. & Xia, K.-Q. 2005 Heat transport by turbulent Rayleigh–Bénard convection in 1 m diameter cylindrical cells of widely varying aspect ratio. J. Fluid Mech. 542, 165–174.Google Scholar
Sun, C. & Xia, K.-Q. 2005 Scaling of the Reynolds number in turbulent thermal convection. Phys. Rev. E 72, 067302.Google Scholar
Sun, C. & Zhou, Q. 2014 Experimental techniques for turbulent Taylor–Couette flow and Rayleigh–Bénard convection. Nonlinearity 27, R89–R121.Google Scholar
Verzicco, R. 2003 Turbulent thermal convection in a closed domain: viscous boundary layer and mean flow effects. Eur. Phys. J. B 35, 133–141.Google 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. & Sreenivasan, K. R. 2008 A comparison of turbulent thermal convection between conditions of constant temperature and constant heat flux. J. Fluid Mech. 595, 203–219.Google Scholar
Wagner, S., Shishkina, O. & Wagner, C. 2012 Boundary layers and wind in cylindrical Rayleigh–Bénard cells. J. Fluid Mech. 697, 336–366.Google Scholar
Whitehead, J. P. & Doering, C. R. 2011 Ultimate state of two-dimensional Rayleigh–Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106, 244501.Google Scholar
Zhou, Q. 2013 Temporal evolution and scaling of mixing in two-dimensional Rayleigh–Taylor turbulence. Phys. Fluids 25, 085107.Google Scholar
Zhou, Q., Huang, Y.-X., Lu, Z.-M., Liu, Y.-L. & Ni, R. 2016 Scale-to-scale energy and enstrophy transport in two-dimensional Rayleigh–Taylor turbulence. J. Fluid Mech. 786, 294–308.CrossRefGoogle Scholar
Zhou, Q. & Jiang, L.-F. 2016 Kinetic and thermal energy dissipation rates in two-dimensional Rayleigh–Taylor turbulence. Phys. Fluids 28, 045109.Google Scholar
Zhou, Q., Liu, B.-F., Li, C.-M. & Zhong, B.-C. 2012 Aspect ratio dependence of heat transport by turbulent Rayleigh–Bénard convection in rectangular cells. J. Fluid Mech. 710, 260–276.Google 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, 125104.CrossRefGoogle Scholar
Zhou, Q. & Xia, K.-Q. 2010 Measured instantaneous viscous boundary layer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 104, 104301.Google Scholar
Zhou, Q. & Xia, K.-Q. 2013 Thermal boundary layer structure in turbulent Rayleigh–Bénard convection in a rectangular cell. J. Fluid Mech. 721, 199–224.Google Scholar