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Direct numerical simulation of free convection over a heated plate

Published online by Cambridge University Press:  08 October 2012

Juan Pedro Mellado
Juan Pedro Mellado*
Affiliation:
Max Planck Institute for Meteorology, Bundesstrasse 53, 20146 Hamburg, Germany
*
†Email address for correspondence: [email protected]
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Abstract

Direct numerical simulations of free convection over a smooth, heated plate are used to investigate unbounded, unsteady turbulent convection. Four different boundary conditions are considered: free-slip or no-slip walls, and constant buoyancy or constant buoyancy flux. It is first shown that, after the initial transient, the vertical structure agrees with observations in the atmospheric boundary layer and predictions from classical similarity theory. A quasi-steady inner layer and a self-preserving outer layer are clearly distinguished, with an overlap region between them of constant turbulent buoyancy flux. The extension of the overlap region reached in our simulations is more than 100 wall units $ \mathop{ ({\kappa }^{3} / {B}_{s} )}\nolimits ^{1/ 4} $, where ${B}_{s} $ is the surface buoyancy flux and $\kappa $ the corresponding molecular diffusivity (the Prandtl number is one). The buoyancy fluctuation inside the overlap region already exhibits the $\ensuremath{-} 1/ 3$ power-law scaling with height for the four types of boundary conditions, as expected in the local, free-convection regime. However, the mean buoyancy gradient and the vertical velocity fluctuation are still evolving toward the corresponding power laws predicted by the similarity theory. The second major result is that the relation between the Nusselt and Rayleigh numbers agrees with that reported in Rayleigh–Bénard convection when the heated plate is interpreted as half a convection cell. The range of Rayleigh numbers covered in the simulations is then $5\ensuremath{\times} 1{0}^{7} \text{{\ndash}} 1{0}^{9} $. Further analogies between the two problems indicate that knowledge can be transferred between steady Rayleigh–Bénard and unsteady convection. Last, we find that the inner scaling based on $\{ {B}_{s} , \hspace{0.167em} \kappa \} $ reduces the effect of the boundary conditions to, mainly, the diffusive wall layer, the first 10 wall units. There, near the plate, free-slip conditions allow stronger mixing than no-slip ones, which results in 30 % less buoyancy difference between the surface and the overlap region and 30–40 % thinner diffusive sublayers. However, this local effect also entails one global, substantial effect: with an imposed buoyancy, free-slip systems develop a surface flux 60 % higher than that obtained with no-slip walls, which implies more intense turbulent fluctuations across the whole boundary layer and a faster growth.

JFM classification

Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 712 , 10 December 2012 , pp. 418 - 450
Copyright
©2012 Cambridge University Press

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References

Adrian, R. J. 1996 Variations of temperature and velocity fluctuations in turbulent thermal convection over horizontal surfaces. Intl J. Heat Mass Transfer 11, 2303–2310.CrossRefGoogle Scholar
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.CrossRefGoogle Scholar
Asaeda, T. & Watanabe, K. 1989 The mechanism of heat transport in thermal convection at high Rayleigh numbers. Phys. Fluids A 1 (5), 861–867.CrossRefGoogle Scholar
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. 665, 152–173.CrossRefGoogle Scholar
Beljaars, A. C. M. 1994 The parametrization of surface fluxes in large-scale models under free convection. Q. J. R. Meteorol. Soc. 121, 255–270.Google Scholar
Belmonte, A., Tilgner, A. & Libchaber, A. 1994 Temperature and velocity boundary layers in turbulent convection. Phys. Rev. E 50, 269–279.CrossRefGoogle ScholarPubMed
Businger, J. A. 1973 A note on free convection. Boundary-Layer Meteorol. 4, 323–326.CrossRefGoogle Scholar
Businger, J. A., Wyngaard, J. C., Izumi, Y. & Bradley, E. F. 1971 Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci. 28, 181–189.2.0.CO;2>CrossRefGoogle Scholar
Castaign, 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.Google Scholar
Deardorff, J. W. 1970 Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 1211–1213.2.0.CO;2>CrossRefGoogle Scholar
Deardorff, J. W. 1980 Cloud top entrainment instability. J. Atmos. Sci. 37, 131–147.2.0.CO;2>CrossRefGoogle Scholar
Deardorff, J. W., Willis, G. E. & Stockton, B. H. 1980 Laboratory studies of the entrainment zone of a convectively mixed layer. J. Fluid Mech. 100, 41–64.CrossRefGoogle Scholar
Eckhardt, B., Grossmann, S. & Lohse, D. 2000 Scaling global momentum transport in Taylor–Couette and pipe flow. Eur. Phys. J. B 18, 541–544.CrossRefGoogle Scholar
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Fluxes and energy dissipation in thermal convection and shear flows. Europhys. Lett. 78, 24001.CrossRefGoogle Scholar
Emanuel, K. A. 1994 Atmospheric Convection. Oxford University Press.CrossRefGoogle Scholar
Fedorovich, E., Conzemius, R. & Mironov, D. 2004 Convective entrainment into a shear-free linearly stratified atmosphere: bulk models reevaluated through large-eddy simulation. J. Atmos. Sci. 61, 281–295.2.0.CO;2>CrossRefGoogle Scholar
Fedorovich, E. & Shapiro, A. 2009 Turbulent natural convection along a vertical plane immersed in a stably stratified medium. J. Fluid Mech. 636, 41–57.CrossRefGoogle Scholar
Fernandes, R. L. J. & Adrian, R. J. 2002 Scaling of velocity and temperature fluctuations in turbulent thermal convection. Exp. Therm. Fluid Sci. 26, 355–360.CrossRefGoogle Scholar
Fernando, H. J. S. & Little, L. J. 1990 Molecular-diffusive effects in penetrative convection. Phys. Fluids A 2, 1592–1596.CrossRefGoogle Scholar
Flack, K. A., Saylor, J. R. & Smith, G. B. 2001 Near-surface turbulence for evaporative convection at an air/water interface. Phys. Fluids 13 (11), 3338–3345.CrossRefGoogle Scholar
Garratt, J. R. 1992 The Atmospheric Boundary Layer. Cambridge University Press.Google Scholar
Grachev, A. A., Fairall, C. W. & Bradley, E. F. 2000 Convective profile constants revisited. Boundary-Layer Meteorol. 94, 495–515.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying view. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. 2012 Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 024502.CrossRefGoogle Scholar
Hunt, J. C. R., Vrieling, A. J., Nieuwstadt, F. T. M. & Fernando, H. J. S. 2003 The influence of the thermal diffusivity of the lower boundary on eddy motion in convection. J. Fluid Mech. 491, 183–205.CrossRefGoogle Scholar
Jimenez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 27–45.CrossRefGoogle Scholar
Johnston, H. & Doering, C. R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.CrossRefGoogle ScholarPubMed
Julien, K., Legg, S. & McWilliams, J. 1996 Rapidly rotating turbulent Rayleigh–Bénard convection. J. Fluid Mech. 322, 243–273.CrossRefGoogle Scholar
Katsaros, K. B., Liu, W. T., Businger, J. A. & Tillman, J. E. 1977 Heat transport and thermal structure in the interfacial boundary layer measured in an open tank of water in turbulent free convection. J. Fluid Mech. 83, 311–335.CrossRefGoogle Scholar
Kraus, E. B. & Businger, J. A. 1994 Atmosphere–Ocean Interaction. Oxford University Press.Google Scholar
Leighton, R. I., Smith, G. B. & Handler, R. A. 2003 Direct numerical simulation of free convection beneath an air–water interface at low Rayleigh numbers. Phys. Fluids 15 (10), 3181–3193.CrossRefGoogle Scholar
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.CrossRefGoogle Scholar
Lui, S.-L. & Xia, K.-Q. 1998 Spatial structure of the thermal boundary layer in turbulent convection. Phys. Rev. E 57 (5), 5494–5503.CrossRefGoogle Scholar
Maystrenko, A., Resagk, C. & Thess, A. 2007 Structure of the thermal boundary layer for turbulent Rayleigh–Bénard convection of air in a long rectangular enclosure. Phys. Rev. E 75, 066303.CrossRefGoogle Scholar
Mellado, J. P. 2010 The evaporatively driven cloud-top mixing layer. J. Fluid Mech. 660, 1–32.CrossRefGoogle Scholar
Mellado, J. P. & Ansorge, C. 2012 Factorization of the Fourier transform of the pressure-Poisson equation using finite differences in colocated grids. Z. Angew. Math. Mech. 92, 380–392.CrossRefGoogle Scholar
Monin, A. S. & Yaglom, A. M. 2007 Statistical Fluid Mechanics, vol. I. Mechanics of Turbulence , Dover.Google Scholar
Obukhov, A. M. 1946 Turbulence in an atmosphere with a non-uniform temperature. Tr. Inst. Teo. Geofiz. Akad. Nauk. SSSR 1, 95–115, in Russian. English translation: Boundary-Layer Meteorol. (1971) 2, 7–29.Google Scholar
Panosfky, H. A. & Dutton, J. A. 1984 Atmospheric Turbulence. Wiley.Google Scholar
Panofsky, H. A., Tennekes, H., Lenschow, D. H. & Wyngaard, J. C. 1977 The characteristics of turbulent velocity components in the surface layer under convective conditions. Boundary-Layer Meteorol. 11, 355–361.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Prandtl, L. 1932 Meteorologische Anwendung der Strömungslehre. Beitr. Phys. Atmos. 19, 188–202.Google Scholar
Priestley, C. H. B. 1954 Convection from a large horizontal surface. Austral. J. Phys. 7, 176–201.CrossRefGoogle Scholar
du Puits, R., Resagk, C., Tilgner, A., Busse, F. H. & Thess, A. 2007 Structure of thermal boundary layers in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 572, 231–254.CrossRefGoogle Scholar
Reeuwijk, M., Jonker, H. J. J. & Hanjalić, K. 2008a Wind and boundary layers in Rayleigh–Bénard convection. Part 1. Analysis and modeling. Phys. Rev. E 77, 036311.Google Scholar
Reeuwijk, M., Jonker, H. J. J. & Hanjalić, K. 2008b Wind and boundary layers in Rayleigh–Bénard convection. Part 2. Boundary layer character and scaling. Phys. Rev. E 77, 036312.Google Scholar
Schmitz, S. & Tilgner, A. 2009 Heat transport in rotating convection without Ekman layers. Phys. Rev. E 80, 015305(R).CrossRefGoogle ScholarPubMed
Schmitz, S. & Tilgner, A. 2010 Transitions in turbulent rotating Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn. 104, 481–489.CrossRefGoogle Scholar
Schumacher, J. 2008 Lagrangian dispersion and heat transport in convective turbulence. Phys. Rev. Lett. 100, 134502.CrossRefGoogle ScholarPubMed
Schumacher, J. 2009 Lagrangian studies in convective turbulence. Phys. Rev. E 79, 056301.CrossRefGoogle ScholarPubMed
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. 2008 Analysis of sheet-like thermal plumes in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 599, 383–404.CrossRefGoogle Scholar
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137–168.CrossRefGoogle Scholar
Sorbjan, Z. 1996 Comments on ‘A convective transport theory for surface fluxes’. J. Atmos. Sci. 54, 576–578.2.0.CO;2>CrossRefGoogle Scholar
Stevens, B. 2005 Atmospheric moist convection. Annu. Rev. Earth Planet. Sci. 33, 605–643.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Stull, R. B. 1988 An Introduction to Boundary Layer Meteorology. Kluwer Academic.CrossRefGoogle Scholar
Sullivan, P. P. & Patton, E. G. 2011 The effect of mesh resolution on convective boundary layer statistics and structures generated by large-eddy simulations. J. Atmos. Sci. 68, 2395–2415.CrossRefGoogle Scholar
Tennekes, H. & Driedonks, A. G. M. 1981 Basic entrainment equations for the atmospheric boundary layer. Boundary-Layer Meteorol. 20, 515–531.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Theerthan, S. A. & Arakeri, J. H. 2000 Planform structure and heat transfer in turbulent free convection over horizontal surfaces. Phys. Fluids 12 (4), 884–894.CrossRefGoogle Scholar
Townsend, A. A. 1959 Temperature fluctuations over a heated horizontal surface. J. Fluid Mech. 5, 209–241.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Verzicco, R. 2004 Effects of nonperfect thermal sources in turbulent thermal convection. Phys. Fluids 16, 1965–1979.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.CrossRefGoogle Scholar
Weidauer, T., Pauluis, O. & Schumacher, J. 2010 Cloud patterns and mixing properties in shallow moist Rayleigh–Bénard convection. New J. Phys. 12, 105002.CrossRefGoogle Scholar
Willis, G. E. & Deardorff, J. W. 1974 A laboratory model of the unstable planetary boundary layer. J. Atmos. Sci. 31, 1297–1307.2.0.CO;2>CrossRefGoogle Scholar
Wyngaard, J. C. 2010 Turbulence in the Atmosphere. Cambridge University Press.CrossRefGoogle Scholar
Wyngaard, J. C., Coté, O. R. & Izumi, Y. 1971 Local free convection, similarity, and the budget of shear stress and heat flux. J. Atmos. Sci. 28, 1171–1182.2.0.CO;2>CrossRefGoogle Scholar
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
Zilitinkevich, S. S. 1991. Turbulent Penetrative Convection. Avebury Technical.Google Scholar