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

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## 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.

## Keywords

Convective boundary layer Ground boundary conditions Penetrative convection## Notes

### Acknowledgments

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

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