One-point statistics are presented for new direct simulations of the zero-pressuregradient turbulent boundary layer in the range Re θ = 2780-6680, matching channels and pipes at δ + ≈ 1000-2000. For tripped boundary layers, it is found that the eddyturnover length is a better criterion than the Reynolds number for the recovery of the largest flow scales after an artificial inflow. Beyond that limit, the integral parameters, mean velocities, Reynolds stresses, and pressure fluctuations of the new simulations agree very well with the available numerical and experimental data, but show clear differences with internal flows when expressed in wall units at the same wall distance and Reynolds number. Those differences are largest in the outer layer, independent of the Reynolds number, and apply to the three velocity components. The logarithmic increase with the Reynolds number of the maximum of the streamwise velocity and pressure fluctuations is confirmed to apply to experimental and numerical internal and external flows. The new simulations also extend to a wider range of Reynolds numbers, and to more than a decade in wall distance, the evidence for logarithmic intensity profiles of the spanwise velocity and of the pressure intensities. Streamwise velocity fluctuations appear to require higher Reynolds numbers to develop a clear logarithmic profile, but it is argued that the comparison of the available near-wall data with fluctuation profiles experimentally obtained by other groups at higher Reynolds numbers can only be explained by assuming the existence of a mesolayer for the fluctuations. The statistics of the new simulation are available in our website.

Two-point statistics are presented for a new direct simulation of the zero-pressuregradient turbulent boundary layer in the range Re θ = 2780-6680, and compared with channels in the same range of Reynolds numbers, δ + ≈ 1000-2000. Threedimensional spatial correlations are investigated in very long domains to educe the average structure of the velocity and pressure fluctuations. The streamwise velocity component is found to be coherent over longer distances in channels than in boundary layers, especially in the direction of the flow. For weakly correlated structures, the maximum streamwise length is O(7δ) for boundary layers and O(18δ) for channels, attained at the logarithmic and outer regions, respectively. The corresponding lengths for the spanwise and wall-normal velocities and for the pressure are shorter, O(δ-2δ). The correlations are shown to be inclined to the wall at angles that depend on the distance from the wall, on the variable being considered, and on the correlation level used to define them. All these features change little between the two types of flows. Most the above features are also approximately independent of the Reynolds number, except for the pressure, and for the streamwise velocity structures in the channel. Further insight into the flow is provided by correlations conditioned on the intensity of the perturbations at the reference point, or on their sign. The statistics of the new simulation are available in our website.

The statistical properties are presented for the direct numerical simulation (DNS) of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) at the verge of separation. The APG TBL has a momentum thickness based Reynolds number range from Re δ2 = 570 to 13800, with a self-similar region from Re δ2 = 10000 to 12300. Within this domain the average non-dimensional pressure gradient parameter β = 39, where for a unit density β = δ 1 P ′ e /τ w , with δ 1 the displacement thickness, τ w the mean shear stress at the wall, and P ′ e the farfield pressure gradient. This flow is compared to previous zero pressure gradient (ZPG) and mild APG TBL (β = 1) results of similar Reynolds number. All flows are generated via the DNS of a TBL on a flat surface with farfield boundary conditions tailored to apply the desired pressure gradient. The conditions for self-similarity, and the appropriate length and velocity scales are derived. The mean and Reynolds stress profiles are shown to collapse when non-dimensionalised on the basis of these length and velocity scales. As the pressure gradient increases, the extent of the wake region in the mean streamwise velocity profiles increases, whilst the extent of the log-layer and viscous sub-layer decreases. The Reynolds stress, production and dissipation profiles of the APG TBL cases exhibit a second outer peak, which becomes more pronounced and more spatially localised with increasing pressure gradient. This outer peak is located at the point of inflection of the mean velocity profiles, and is suggestive of the presence of a shear flow instability. The maximum streamwise velocity variance is located at a wall normal position of δ 1 of spanwise wavelength of 2δ 1 . In summary as the pressure gradient increases the flow has properties less like a ZPG TBL and more akin to a free shear layer.

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The statistical scaling properties of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) are presented. The intended flow is generated using the direct numerical simulation (DNS) TBL code of Simens et al. (2009) and Borrell et al. (2013), with a modified farfield boundary condition (BC). The conditions for self-similarity and appropriate scaling are derived, with mean and Reynolds stress profiles presented using this scaling. The APG and ZPG DNS are also compared under the classical viscous scaling.