Direct numerical simulation of the Ekman layer: a step in Reynolds number, and cautious support for a log law with a shifted origin

Abstract

Results at Ekman Reynolds numbers Re ranging from 1000 to 2828 expand the DNS contribution to the theory of wall-bounded turbulence. An established spectral method is used, with rules for domain size and grid resolution at each Reynolds number derived from the theory. The Re increase is made possible by better computers and by optimizing the grid in relation to the wall shear-stress direction. The boundary-layer thickness in wall units ?+ varies here by a factor of about 5.3, and reaches values near 5,000, or 22 times the minimum at which turbulence has been sustained. An equivalent channel Reynolds number, based on the pressure gradient in wall units, would reach about Re? = 1250. The principal goal of the analysis, the impartial identification of a log law, is summarized in the local ‘Karman Measure’ d(ln z+)/dU+. The outcome differs from that for Hoyas & Jim´enez and for Hu, Morfey & Sandham in channel-flow DNS at similar Reynolds numbers, for reasons unknown: here, the law of the wall is gradually established up to a z+ around 400, with little statistical scatter. To leading order, it is consistent with the experiments of ¨Osterlund et al. in boundary layers. With the traditional expression, a logarithmic law is not present, in that the Karman Measure drifts from about 0.41 at z+ ? 70 to the 0.37-0.38 range for z+ ? 500, with Re = 2828. However, if a virtual origin is introduced with a shift of a+ = 7.5 wall units, the data support a long logarithmic layer with ? = 0.38 a good fit to d(ln[z++a+])/dU+. A determination of the Karman constant from the variation of the skin-friction coefficients with Reynolds numbers also yields values near 0.38. The uncertainty is about ±0.01. These values are close to the boundary-layer experiments, but well below the accepted range of [0.40,0.41] and the experimental pipe-flow results near 0.42. The virtual-origin concept is also controversial, although non-essential at transportation or atmospheric Reynolds numbers. Yet, this series may reflect some success in verifying the law of the wall and investigating the logarithmic law by DNS, redundantly and with tools more impartial than the visual fit of a straight line to a velocity profile.

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