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A Turbulence Closure Study of the Flow and Thermal Fields in the Ekman Layer

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Abstract

We assess the performance of turbulence closures of varying degrees of sophistication in the prediction of the mean flow and the thermal fields in a neutrally-stratified Ekman layer. The Reynolds stresses that appear in the Reynolds-averaged momentum equations are determined using both eddy-viscosity and complete differential Reynolds-stress-transport closures. The results unexpectedly show that the assumption of an isotropic eddy viscosity inherent in eddy-viscosity closures does not preclude the attainment of accurate predictions in this flow. Regarding the Reynolds-stress transport closure, two alternative strategies are examined: one in which a high turbulence–Reynolds–number model is used in conjunction with a wall function to bridge over the viscous sublayer and the other in which a low turbulence–Reynolds-number model is used to carry out the computations through this layer directly to the surface. It is found that the wall-function approach, based on the assumption of the applicability of the universal logarithmic law-of-the-wall, yields predictions that are on par with the computationally more demanding alternative. Regarding the thermal field, the unknown turbulent heat fluxes are modelled (i) using the conventional Fourier’s law with a constant turbulent Prandtl number of 0.85, (ii) by using an alternative algebraic closure that includes dependence on the gradients of mean velocities and on rotation, and (iii) by using a differential scalar-flux transport model. The outcome of these computations does not support the use of Fourier’s law in this flow.

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Acknowledgements

Lukas Braun gratefully acknowledges the financial support provided by the Studienstiftung des deutschen Volkes that facilitated this research at UC Davis.

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Authors and Affiliations

  1. Institut für Thermodynamik der Luft- und Raumfahrt, Universität Stuttgart, Stuttgart, Germany

    Lukas Braun & Bernhard Weigand

  2. Department of Civil and Environmental Engineering, University of California - Davis, Davis, CA, 95616, USA

    Bassam A. Younis

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  1. Lukas Braun
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  2. Bassam A. Younis
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  3. Bernhard Weigand
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Correspondence to Bassam A. Younis.

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Appendix: Comparisons with the DNS of Deusebio et al. (2014)

Appendix: Comparisons with the DNS of Deusebio et al. (2014)

A reviewer drew our attention to the DNS results of Deusebio et al. (2014) for the mean temperature and vertical heat flux in a neutrally-stratified Ekman layer. These results are presented in Fig. 16 where they are compared with the present predictions. In the viscous sub-layer (\(z^+<8\)), the correspondence between the DNS results for mean temperature and the predictions of the differential and the non-linear flux models is quite close. However, differences appear further away from the surface. There, the pronounced change in the slope of the temperature profile exhibited by the DNS is not reproduced in the models’ predictions. Concerning the vertical turbulent heat flux, significant differences between the present results and the DNS are apparent. We are at a loss to explain the observed differences in the profiles shape, especially in the outer region of the boundary layer where the DNS results show an extensive region of constant heat flux. We are however encouraged to see that the two models yield almost identical results even though they differ in so many ways (e.g. algebraic vs. differential), and share no assumptions in their formulation.

Fig. 16
figure 16

DNS and present predictions of mean temperature (top) and vertical heat flux (\(Re = 1600\)). \(\bigcirc \) DNS of Deusebio et al. (2014); differential transport model; non-linear model

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Braun, L., Younis, B.A. & Weigand, B. A Turbulence Closure Study of the Flow and Thermal Fields in the Ekman Layer. Boundary-Layer Meteorol 175, 25–55 (2020). https://doi.org/10.1007/s10546-019-00495-8

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