Stochastic modeling of transient Ekman flow at arbitrary Reynolds number driven by horizontal bottom wall oscillation

Key challenges in the modeling and simulation of Ekman boundary layers (EBLs) in the atmosphere and in the ocean are related to the interplay of Coriolis, viscous, and inertial forces that act on all relevant scales of the turbulent flow. The interplay of these forces can lead to nonuniversal flow properties such as intermittency, boundary-layer resonance, and ‘eruption’ of the surface layer thickness, which can locally excite waves, mean flows, and turbulence (e.g., [1,2,3]). Advanced wall models should sufficiently capture the dynamical complexity in order to improve upon parameterization schemes that describe the mean effects but do not directly resolve the small-scale processes (e.g. [4]).

The difficulties associated with such a cross-regime modeling of EBL flows is addressed in the present study by utilizing the stochastic one-dimensional turbulence (ODT) model formulated for atmospheric boundary layers [5] using an adaptive grid [6]. ODT is as self-contained flow model that aims to resolve all relevant flow scales but only for a single vertical column. Turbulent eddy events are sampled from a stochastic process and punctuate the deterministic flow evolution. The stand-alone model resolves vertical viscous and nongeostrophic horizontal Coriolis forces along the wall-normal one-dimensional domain.

Here, we investigate a periodically forced EBL flow over a smooth and flat surface. The configuration aims to model the effects of wind-shear and tidal forcing in the ocean and coastal environment, respectively, and is mathematically related also to alternating up- and down-slope winds in the atmosphere. In this study, the local wall-shear stress is the result of the forcing surplus laminar Ekman dynamics and turbulent mixing. We show that ODT is able to capture the regime transition from the laminar to an intermittently turbulent flow with fixed model parameters when the Reynolds number is increased. ODT furthermore captures the boundary-layer resonance and the dependence of the turbulence intensity on the wall oscillation phase. Preliminary model predictions are in agreement with theory, direct numerical simulations, and reference measurements. It is concluded that ODT has reasonable capabilities for regime-independent fluctuation modeling yielding improvements of the bulk–surface coupling by capturing transient small-scale processes.

References

[1] M. Klein, T. Seelig, M. V. Kurgansky, A. Ghasemi, I. D. Borcia, A. Will, E. Schaller, C. Egbers, U. Harlander (2014). J. Fluid Mech., 751:255–297.

[2] A. Ghasemi, M. Klein, A. Will, U. Harlander (2018). J. Fluid Mech., 853:111–149.

[3] M. Vincze, N. Fenyvesi, M. Klein, J. Sommeria, S. Viboud, Y. Ashkenazy (2019). EPL, 125:44001.

[4] M. Klein, H. Schmidt, D. O. Lignell (2022). Int. J. Heat Fluid Flow, 93:108889.

[5] A. R. Kerstein, S. Wunsch (2006). Boundary-Layer Meteorol., 118:325–356.

[6] D. O. Lignell, A. R. Kerstein, G. Sun, E. I. Monson (2013). Theor. Comp. Fluid Dyn., 27(3):273–295.