Skip to main content
SpringerLink
Log in

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

  • Research Article
  • Published:
Boundary-Layer Meteorology Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. The notion of unsteady Reynold-averaged Navier–Stokes (URANS) approach commonly implies the straightforward application of a RANS model in time-dependent mode to solve flows that are unsteady in the mean, and when the externally imposed time scale is much greater than the characteristic turbulence scale, justifying the use of the time-averaged equations. The URANS concept can also be applied to flows that can be regarded as steady in the mean, but inherently unsteady due to strong internal instabilities that generate quasi-periodic, deterministic turbulence structures, such as vortex shedding or convective cells in flows dominated by buoyancy or rotation. The T-RANS label is used for the latter to distinguish it from the common URANS applied to flows subjected to external mean flow unsteadiness.

  2. For convenience, in the following we deleted braces \(\langle {}\rangle \) for the resolved properties, as the analysis is applicable to general steady or unsteady RANS methods.

  3. Admittedly, using the well-resolved LES or DNS results would have been more appropriate for this exercise. However, detailed field data for the two cases considered (ensemble averaged wind velocity and temperature, friction velocity and heat flux along the wall, the reference near-wall point to define the characteristic turbulence velocity scale) are not readily available and would require laborious processing of the raw data. As the present analysis is more of a qualitative nature (collapsing of curves) and in view of the earlier satisfactory reproduction of LES by T-RANS, we opted for using the current dedicated T-RANS solutions.

References

  • Adrian RJ, Ferreira RTDS, Boberg T (1986) Turbulent thermal convection in wide horizontal fluid layers. Exp Fluids 4:121–141

    Google Scholar 

  • Belmonte A, Tilgner A, Libchaber A (1994) Temperature and velocity boundary layers in turbulent convection. Phys Rev E 50:269–281

    Article  Google Scholar 

  • 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 35:43–51

    Article  Google Scholar 

  • Craft TJ, Gerasimov AV, Iacovides H, Launder BE (2002) Progress in the generalization of wall-function treatments. Int J Heat Fluid Flow 23:148–160

    Article  Google Scholar 

  • Deardorff JW, Willis GE, Lilly DK (1969) Laboratory investigation of non-steady penetrative convection. J Fluid Mech 35(1):7–31

    Article  Google Scholar 

  • 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 61:281–295

    Article  Google Scholar 

  • Gibson MM, Launder BE (1976) Ground effects on pressure fluctuations in the atmospheric boundary layer. J Fluid Mech 86:491–511

    Article  Google Scholar 

  • Hanjalić K (2002) One-point closure models for buoyancy-driven turbulent flows. Annu Rev Fluid Mech 34:321–347

    Article  Google Scholar 

  • Hanjalić K, Launder BE (2011) Modelling turbulence in engineering and the environment. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Kenjereš S, Hanjalić K (2009) Tackling turbulent flow by TRANS, Fluid Dyn Res 41:012201, 32 pp

  • Kenjereš S, Hanjalić K, (2002a) Numerical insight into flow structure in ultraturbulent thermal convection. Phys Rev E 66 (3): 1–5. Art.No.036307

  • Kenjereš S, Hanjalić K (1999) Transient analysis of Rayleigh-Bénard convection with a RANS model. Int J Heat and Fluid Flow 20(3):329–340

    Article  Google Scholar 

  • 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 3:1–25

    Article  Google Scholar 

  • 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):800–810

    Article  Google Scholar 

  • 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)

  • Mellor GL, Yamada T (1974) A hierarchy of turbulence closure models for planetary boundary layers. J Atmos Sci 31:1791–1806

    Article  Google Scholar 

  • 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):1–29

    Google Scholar 

  • Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge, UK, 771 pp

  • Pope SB (2004) Ten questions concerning the LES. New J Phys 6:35

    Article  Google Scholar 

  • Popovac M, Hanjalić K (2007) Compound wall treatment for RANS computation of complex turbulent flows and heat transfer. Flow Turbul Combust 78:177–202

    Article  Google Scholar 

  • Townsend AA (1980) The structure of turbulent shear flow, 2nd edn. Cambridge University Press, Cambridge, UK, 442 pp

  • van Reeuwijk M, Jonker HJJ, Hanjalić K (2005) Identification of the wind in Rayleigh–Bénard convection. Phys Fluids 17, 051704, 1–4

  • 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)

  • Xin YB, Xia KQ (1997) Boundary layer length scales in convective turbulence. Phys Rev E 56:3010–3015

    Article  Google Scholar 

  • Yamada T, Mellor G (1975) A simulation of the Wangara atmospheric boundary layer data. J Atmos Sci 32:2309–2329

    Article  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Novosibirsk State University, Pirogova st., 2, Novosibirsk, Russian Federation, 630090

    K. Hanjalić & M. Hrebtov

  2. Institute of Thermophysics of SB RAS, Ak. Lavrentieva ave. 1, Novosibirsk, Russian Federation, 630090

    M. Hrebtov

  3. Delft University of Technology, ChemE, Julianalaan 136, 2628BL, Delft, The Netherlands

    K. Hanjalić

Authors
  1. K. Hanjalić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Hrebtov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Hrebtov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hanjalić, K., Hrebtov, M. Ground Boundary Conditions for Thermal Convection Over Horizontal Surfaces at High Rayleigh Numbers. Boundary-Layer Meteorol 160, 41–61 (2016). https://doi.org/10.1007/s10546-016-0135-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10546-016-0135-z

Keywords

Search

Navigation