Boundary-Layer Meteorology

, Volume 160, Issue 1, pp 41–61 | Cite as

Ground Boundary Conditions for Thermal Convection Over Horizontal Surfaces at High Rayleigh Numbers

Research Article


We present “wall functions” for treating the ground boundary conditions in the computation of thermal convection over horizontal surfaces at high Rayleigh numbers using coarse numerical grids. The functions are formulated for an algebraic-flux model closed by transport equations for the turbulence kinetic energy, its dissipation rate and scalar variance, but could also be applied to other turbulence models. The three-equation algebraic-flux model, solved in a T-RANS mode (“Transient” Reynolds-averaged Navier–Stokes, based on triple decomposition), was shown earlier to reproduce well a number of generic buoyancy-driven flows over heated surfaces, albeit by integrating equations up to the wall. Here we show that by using a set of wall functions satisfactory results are found for the ensemble-averaged properties even on a very coarse computational grid. This is illustrated by the computations of the time evolution of a penetrative mixed layer and Rayleigh–Bénard (open-ended, 4:4:1 domain) convection, using \(10 \times 10 \times 100\) and \(10 \times 10 \times 20\) grids, compared also with finer grids (e.g. \(60 \times 60 \times 100\)), as well as with one-dimensional treatment using \(1 \times 1 \times 100\) and \(1 \times 1 \times 20\) nodes. The approach is deemed functional for simulations of a convective boundary layer and mesoscale atmospheric flows, and pollutant transport over realistic complex hilly terrain with heat islands, urban and natural canopies, for diurnal cycles, or subjected to other time and space variations in ground conditions and stratification.


Convective boundary layer Ground boundary conditions  Penetrative convection 



The work is supported by the Russian Science Fund, Grant No 14-29-00203_2014-16.


  1. Adrian RJ, Ferreira RTDS, Boberg T (1986) Turbulent thermal convection in wide horizontal fluid layers. Exp Fluids 4:121–141Google Scholar
  2. Belmonte A, Tilgner A, Libchaber A (1994) Temperature and velocity boundary layers in turbulent convection. Phys Rev E 50:269–281CrossRefGoogle Scholar
  3. Chung MK, Yun HC, Adrian RJ (1992) Scale analysis and wall-layer model for the temperature profile in a turbulent thermal convection. Int J Heat Mass Transfer 35:43–51CrossRefGoogle Scholar
  4. Craft TJ, Gerasimov AV, Iacovides H, Launder BE (2002) Progress in the generalization of wall-function treatments. Int J Heat Fluid Flow 23:148–160CrossRefGoogle Scholar
  5. Deardorff JW, Willis GE, Lilly DK (1969) Laboratory investigation of non-steady penetrative convection. J Fluid Mech 35(1):7–31CrossRefGoogle Scholar
  6. Fedorovich E, Conzemius R, Mironov D (2004) Convective entrainment into a shear-free linearly stratified atmosphere: bulk models re-evaluated through large-eddy simulation. J Atmos Sci 61:281–295CrossRefGoogle Scholar
  7. Gibson MM, Launder BE (1976) Ground effects on pressure fluctuations in the atmospheric boundary layer. J Fluid Mech 86:491–511CrossRefGoogle Scholar
  8. Hanjalić K (2002) One-point closure models for buoyancy-driven turbulent flows. Annu Rev Fluid Mech 34:321–347CrossRefGoogle Scholar
  9. Hanjalić K, Launder BE (2011) Modelling turbulence in engineering and the environment. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  10. Kenjereš S, Hanjalić K (2009) Tackling turbulent flow by TRANS, Fluid Dyn Res 41:012201, 32 ppGoogle Scholar
  11. Kenjereš S, Hanjalić K, (2002a) Numerical insight into flow structure in ultraturbulent thermal convection. Phys Rev E 66 (3): 1–5. Art.No.036307Google Scholar
  12. Kenjereš S, Hanjalić K (1999) Transient analysis of Rayleigh-Bénard convection with a RANS model. Int J Heat and Fluid Flow 20(3):329–340CrossRefGoogle Scholar
  13. Kenjereš S, Hanjalić K (2002b) Combined effects of terrain orography and thermal stratification on pollutant dispersion in a town valley: a T-RANS simulation. J Turbul 3:1–25CrossRefGoogle Scholar
  14. Kenjereš S, Hanjalić K (2006) LES, T-RANS and hybrid simulations of thermal convection at high Ra numbers. Int J Heat Fluid Flow 27(5):800–810CrossRefGoogle Scholar
  15. Kenjereš S, Hanjalić K (2016) Near-wall scaling in turbulent thermal convection over horizontal surfaces in infinite and finite enclosures. Int J Heat Fluid Flow (submitted)Google Scholar
  16. Mellor GL, Yamada T (1974) A hierarchy of turbulence closure models for planetary boundary layers. J Atmos Sci 31:1791–1806CrossRefGoogle Scholar
  17. Peng Sh-H, Hanjalić K, Davidson L (2006) Large-eddy simulation and deduced scaling analysis of Rayleigh–Bénard convection up to Ra = 10\(^{9}\). J Turbul 7(66):1–29Google Scholar
  18. Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge, UK, 771 ppGoogle Scholar
  19. Pope SB (2004) Ten questions concerning the LES. New J Phys 6:35CrossRefGoogle Scholar
  20. Popovac M, Hanjalić K (2007) Compound wall treatment for RANS computation of complex turbulent flows and heat transfer. Flow Turbul Combust 78:177–202CrossRefGoogle Scholar
  21. Townsend AA (1980) The structure of turbulent shear flow, 2nd edn. Cambridge University Press, Cambridge, UK, 442 ppGoogle Scholar
  22. van Reeuwijk M, Jonker HJJ, Hanjalić K (2005) Identification of the wind in Rayleigh–Bénard convection. Phys Fluids 17, 051704, 1–4Google Scholar
  23. van Reeuwijk M, Jonker HJJ, Hanjalić K (2008) Wind and boundary layers in Rayleigh-Bénard convection. Phys Rev E 77, Pt 1: 036311 (1–15), Pt2: 036312(1–10)Google Scholar
  24. Xin YB, Xia KQ (1997) Boundary layer length scales in convective turbulence. Phys Rev E 56:3010–3015CrossRefGoogle Scholar
  25. Yamada T, Mellor G (1975) A simulation of the Wangara atmospheric boundary layer data. J Atmos Sci 32:2309–2329CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussian Federation
  2. 2.Institute of Thermophysics of SB RASNovosibirskRussian Federation
  3. 3.Delft University of TechnologyDelftThe Netherlands

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