## Abstract

We performed a numerical simulation of penetrative convection of an inversion-topped weakly stratified atmospheric boundary layer over urban terrain with a strong localized source of heat and moisture. With some simplifications, the case mimics the real environment of the Krasnoyarsk region in Russia where the non-freezing river Yenisei acts as a thermal and humidity source during winter, generating an undulating fog pattern along the river accompanied with scattered ‘steam devils’. An idealized full diurnal cycle was simulated using an unsteady Reynolds-averaged Navier–Stokes (RANS) three-equation algebraic flux model and the novel buoyancy-accounting functions for treating the ground boundary conditions. The results show a significant effect of the river on the net temperature and moisture distribution. The localized heat and moisture source leads to strong horizontal convection and marked non-uniformity of humidity concentration in the air. An interplay of several distinct large-scale vortex systems leads to a wavy pattern of moisture plumes over the river. The simulations deal with rare natural phenomena and show the capability of the RANS turbulence closure to capture the main features of flow and scalar fields on an affordable, relatively coarse, computational grid.

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## Notes

- 1.
- 2.
The approach here followed was earlier labelled as “T-RANS” (transient, time-resolved RANS, Kenjereš and Hanjalić 1999) to indicate its triple-decomposition framework and resolving the large-scale, deterministic motion. It can also be interpreted as very-large eddy simulations in which the constitutive relations play the role of the “subscale” model in the spirit of LES, but not related to the grid-cell size.

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## Acknowledgements

This work was supported by Russian Science Foundation (Grant 114-29-00203). The authors thank Dr. Saša Kenjereš from TU Delft, Nl, for invaluable help and advice to MH in mastering the *BuoyFlow* CFD code.

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## Appendix: Turbulence Model and Ground Boundary Conditions

### Appendix: Turbulence Model and Ground Boundary Conditions

The equation set 1–3 is closed by solving the transport equations for the scalar quantities

where

with \(\phi \) standing for a scalar (*k*, \(\varepsilon \), \(\overline{\theta ^{2}} )\), \(\sigma _{\phi } \) for the corresponding turbulent Prandtl-Schmidt number, and the coefficients in the \(\varepsilon \) equation take the common values \(C_{\varepsilon 1} =1.44\) and \(C_{\varepsilon 2} =1.92\).

The model (with inclusion of the molecular terms and the appropriate low-Re-number modifications) was shown earlier to reproduce well the Rayleigh–Bénard convection with integration up to the walls using an adequately refined/clustered grid in the wall-normal direction (Kenjereš and Hanjalić 2006). In the present work we employed modified ground (“wall”) functions approach suitable for high *Ra* buoyancy-dominated flows, (Hanjalić and Hrebtov 2016), which provide boundary conditions for all mean and turbulence variables at the ground-adjacent grid nodes, well outside the molecular layer.

For the vapour concentration *H* one can use the same approach as for the temperature by specifying the vapour surface flux \(\dot{m}_s^{\prime \prime } =D_s^{{ eff}} \left( {H_P -H_s } \right) /z_P\), where \(D_s^{{ eff}} \) is the effective wall mass diffusivity. However in view of humidity being treated as a passive scalar, we assume that the absolute humidity (vapour concentration) at the ground-nearest grid node can be considered as equal to that at the water surface corresponding to the saturation pressure, for which we employed the following approximation (World Meteorological Organization)

where

is the saturation pressure, resulting in the value of absolute humidity at point P

The model and the ground functions were re-tested in Rayleigh–Bénard convection and in the penetrative convection of a mixed layer heated from below (Deardorff et al. 1969) using one-dimensional \((1 \times 1 \times N_{z})\) and three-dimensional (\(60 \times 60 \times N_{z}\) and \(10 \times 10 \times N_{z)}\) fine and coarse grids, the former with \(N_{z} =100\) and the latter with \(N_{z} = 20\) uniformly spaced grid nodes in the vertical direction (the latter chosen deliberately to be extremely coarse), both showing very good agreement with the reference DNS/LES and experimental data. The tests show that the model and the applied ground functions are capable of predicting adequately the mean temperature and the vertical turbulent heat-flux evolution, as well as other properties, on very coarse meshes even in the limit of zero convection, when the second moments are provided solely by the model without any resolved contribution (Hanjalić and Hrebtov 2016).

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Hrebtov, M., Hanjalić, K. Numerical Study of Winter Diurnal Convection Over the City of Krasnoyarsk: Effects of Non-freezing River, Undulating Fog and Steam Devils.
*Boundary-Layer Meteorol* **163, **469–495 (2017). https://doi.org/10.1007/s10546-016-0231-0

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### Keywords

- Atmospheric boundary layer
- Diurnal convection
- Steam devils
- Undulating fog pattern
- Reynolds-averaged Navier–Stokes modelling