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Direct numerical simulations have been carried out for a fully developed turbulent channel flow with a smooth upper wall and a lower wall consisting of square bars separated by a rectangular cavity. A wide range of w/k, the cavity width to roughness height ratio, was considered. For w/k > 7, recirculation zones occur immediately upstream and downstream of each element while mean streamlines and spatial distributions of the skin frictional drag indicate that each element is virtually isolated. The maximum form drag occurs at w/k = 7 and coincides with the minimum skin frictional drag. The dependence on w/k of the Clauser roughness function reflects that of the form drag.

We study passive scalars in turbulent plane channels at computationally high Reynolds number, thus allowing us to observe previously unnoticed effects. The mean scalar profiles are found to obey a generalized logarithmic law which includes a linear correction term in the whole lower half-channel, and they follow a universal parabolic defect profile in the core region. This is consistent with recent findings regarding the mean velocity profiles in channel flow. The scalar variances also exhibit a near universal parabolic distribution in the core flow and hints of a sizeable log layer, unlike the velocity variances. The energy spectra highlight the formation of large scalar-bearing eddies with size proportional to the channel height which are caused by a local production excess over dissipation, and which are clearly visible in the flow visualizations. Close correspondence of the momentum and scalar eddies is observed, with the main difference being that the latter tend to form sharper gradients, which translates into higher scalar dissipation. Another notable Reynolds number effect is the decreased correlation of the passive scalar field with the vertical velocity field, which is traced to the reduced effectiveness of ejection events.

The high-Reynolds-number behaviour of the canonical incompressible turbulent channel flow is investigated through large-scale direct numerical simulation (DNS). A Reynolds number is achieved (Re τ = h/δ v ≈ 4000, where h is the channel half-height, and δ v is the viscous length scale) at which theory predicts the onset of phenomena typical of the asymptotic Reynolds number regime, namely a sensible layer with logarithmic variation of the mean velocity profile, and Kolmogorov scaling of the velocity spectra. Although higher Reynolds numbers can be achieved in experiments, the main advantage of the present DNS study is access to the full three-dimensional flow field. Consistent with refined overlap arguments (results suggest that the mean velocity profile never achieves a truly logarithmic profile, and the logarithmic diagnostic function instead exhibits a linear variation in the outer layer whose slope decreases with the Reynolds number. The extrapolated value of the von Kármán constant is k ≈ 0.41. A near logarithmic layer is observed in the spanwise velocity variance, as predicted by Townsend's attached eddy hypothesis, whereas the streamwise variance seems to exhibit a shoulder, perhaps being still affected by low-Reynolds-number effects. Comparison with previous DNS data at lower Reynolds number suggests enhancement of the imprinting effect of outer-layer eddies onto the near-wall region. This mechanisms is associated with excess turbulence kinetic energy production in the outer layer, and it reflects in flow visualizations and in the streamwise velocity spectra, which exhibit sharp peaks in the outer layer. Associated with the outer energy production site, we find evidence of a Kolmogorov-like inertial range, limited to the spanwise spectral density of u, whereas power laws with different exponents are found for the other spectra. Finally, arguments are given to explain the 'odd' scaling of the streamwise velocity variances, based on the analysis of the kinetic energy production term.