Skip to main content Accessibility help

Direct numerical simulation of free convection over a heated plate

  • Juan Pedro Mellado (a1)


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.


Corresponding author

Email address for correspondence:


Hide All
Adrian, R. J. 1996 Variations of temperature and velocity fluctuations in turbulent thermal convection over horizontal surfaces. Intl J. Heat Mass Transfer 11, 23032310.
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.
Asaeda, T. & Watanabe, K. 1989 The mechanism of heat transport in thermal convection at high Rayleigh numbers. Phys. Fluids A 1 (5), 861867.
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, 152173.
Beljaars, A. C. M. 1994 The parametrization of surface fluxes in large-scale models under free convection. Q. J. R. Meteorol. Soc. 121, 255270.
Belmonte, A., Tilgner, A. & Libchaber, A. 1994 Temperature and velocity boundary layers in turbulent convection. Phys. Rev. E 50, 269279.
Businger, J. A. 1973 A note on free convection. Boundary-Layer Meteorol. 4, 323326.
Businger, J. A., Wyngaard, J. C., Izumi, Y. & Bradley, E. F. 1971 Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci. 28, 181189.
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, 130.
Deardorff, J. W. 1970 Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 12111213.
Deardorff, J. W. 1980 Cloud top entrainment instability. J. Atmos. Sci. 37, 131147.
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, 4164.
Eckhardt, B., Grossmann, S. & Lohse, D. 2000 Scaling global momentum transport in Taylor–Couette and pipe flow. Eur. Phys. J. B 18, 541544.
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Fluxes and energy dissipation in thermal convection and shear flows. Europhys. Lett. 78, 24001.
Emanuel, K. A. 1994 Atmospheric Convection. Oxford University Press.
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, 281295.
Fedorovich, E. & Shapiro, A. 2009 Turbulent natural convection along a vertical plane immersed in a stably stratified medium. J. Fluid Mech. 636, 4157.
Fernandes, R. L. J. & Adrian, R. J. 2002 Scaling of velocity and temperature fluctuations in turbulent thermal convection. Exp. Therm. Fluid Sci. 26, 355360.
Fernando, H. J. S. & Little, L. J. 1990 Molecular-diffusive effects in penetrative convection. Phys. Fluids A 2, 15921596.
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), 33383345.
Garratt, J. R. 1992 The Atmospheric Boundary Layer. Cambridge University Press.
Grachev, A. A., Fairall, C. W. & Bradley, E. F. 2000 Convective profile constants revisited. Boundary-Layer Meteorol. 94, 495515.
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying view. J. Fluid Mech. 407, 2756.
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.
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, 183205.
Jimenez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 2745.
Johnston, H. & Doering, C. R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.
Julien, K., Legg, S. & McWilliams, J. 1996 Rapidly rotating turbulent Rayleigh–Bénard convection. J. Fluid Mech. 322, 243273.
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, 311335.
Kraus, E. B. & Businger, J. A. 1994 Atmosphere–Ocean Interaction. Oxford University Press.
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), 31813193.
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335364.
Lui, S.-L. & Xia, K.-Q. 1998 Spatial structure of the thermal boundary layer in turbulent convection. Phys. Rev. E 57 (5), 54945503.
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.
Mellado, J. P. 2010 The evaporatively driven cloud-top mixing layer. J. Fluid Mech. 660, 132.
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, 380392.
Monin, A. S. & Yaglom, A. M. 2007 Statistical Fluid Mechanics, vol. I. Mechanics of Turbulence , Dover.
Obukhov, A. M. 1946 Turbulence in an atmosphere with a non-uniform temperature. Tr. Inst. Teo. Geofiz. Akad. Nauk. SSSR 1, 95115, in Russian. English translation: Boundary-Layer Meteorol. (1971) 2, 7–29.
Panosfky, H. A. & Dutton, J. A. 1984 Atmospheric Turbulence. Wiley.
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, 355361.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Prandtl, L. 1932 Meteorologische Anwendung der Strömungslehre. Beitr. Phys. Atmos. 19, 188202.
Priestley, C. H. B. 1954 Convection from a large horizontal surface. Austral. J. Phys. 7, 176201.
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, 231254.
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.
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.
Schmitz, S. & Tilgner, A. 2009 Heat transport in rotating convection without Ekman layers. Phys. Rev. E 80, 015305(R).
Schmitz, S. & Tilgner, A. 2010 Transitions in turbulent rotating Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn. 104, 481489.
Schumacher, J. 2008 Lagrangian dispersion and heat transport in convective turbulence. Phys. Rev. Lett. 100, 134502.
Schumacher, J. 2009 Lagrangian studies in convective turbulence. Phys. Rev. E 79, 056301.
Shishkina, O. & Wagner, C. 2006 Analysis of thermal dissipation rates in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 546, 5160.
Shishkina, O. & Wagner, C. 2008 Analysis of sheet-like thermal plumes in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 599, 383404.
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137168.
Sorbjan, Z. 1996 Comments on ‘A convective transport theory for surface fluxes’. J. Atmos. Sci. 54, 576578.
Stevens, B. 2005 Atmospheric moist convection. Annu. Rev. Earth Planet. Sci. 33, 605643.
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, 3143.
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, 495507.
Stull, R. B. 1988 An Introduction to Boundary Layer Meteorology. Kluwer Academic.
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, 23952415.
Tennekes, H. & Driedonks, A. G. M. 1981 Basic entrainment equations for the atmospheric boundary layer. Boundary-Layer Meteorol. 20, 515531.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.
Theerthan, S. A. & Arakeri, J. H. 2000 Planform structure and heat transfer in turbulent free convection over horizontal surfaces. Phys. Fluids 12 (4), 884894.
Townsend, A. A. 1959 Temperature fluctuations over a heated horizontal surface. J. Fluid Mech. 5, 209241.
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Verzicco, R. 2004 Effects of nonperfect thermal sources in turbulent thermal convection. Phys. Fluids 16, 19651979.
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, 203219.
Weidauer, T., Pauluis, O. & Schumacher, J. 2010 Cloud patterns and mixing properties in shallow moist Rayleigh–Bénard convection. New J. Phys. 12, 105002.
Willis, G. E. & Deardorff, J. W. 1974 A laboratory model of the unstable planetary boundary layer. J. Atmos. Sci. 31, 12971307.
Wyngaard, J. C. 2010 Turbulence in the Atmosphere. Cambridge University Press.
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, 11711182.
Zhou, Q., Sun, C. & Xia, K.-Q. 2007 Morphological evolution of thermal plumes in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 074501.
Zilitinkevich, S. S. 1991. Turbulent Penetrative Convection. Avebury Technical.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Direct numerical simulation of free convection over a heated plate

  • Juan Pedro Mellado (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.