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We discuss the Onsager theory of wall-bounded turbulence, analysing the momentum dissipation anomaly hypothesized by Taylor. Turbulent drag laws observed with both smooth and rough walls imply ultraviolet divergences of velocity gradients. These are eliminated by a coarse-graining operation, filtering out small-scale eddies and windowing out near-wall eddies, thus introducing two arbitrary regularization length-scales. The regularized equations for resolved eddies correspond to the weak formulation of the Navier–Stokes equation and contain, in addition to the usual turbulent stress, also an inertial drag force modelling momentum exchange with unresolved near-wall eddies. Using an Onsager-type argument based on the principle of renormalization group invariance, we derive an upper bound on wall friction by a function of Reynolds number determined by the modulus of continuity of the velocity at the wall. Our main result is a deterministic version of Prandtl’s relation between the Blasius − 1 / 4 drag law and the 1/7 power-law profile of the mean streamwise velocity. At higher Reynolds, the von Kármán–Prandtl drag law requires instead a slow logarithmic approach of velocity to zero at the wall. We discuss briefly also the large-eddy simulation of wall-bounded flows and use of iterative renormalization group methods to establish universal statistics in the inertial sublayer. This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’. 

Aiming to study the rough-wall turbulent boundary layer structure over differently arranged roughness elements, an experimental study was conducted on flows with regular and random roughness. Varying planform densities of truncated cone roughness elements in a square staggered pattern were investigated. The same planform densities were also investigated in random arrangements. Velocity statistics were measured via two-component laser Doppler velocimetry and stereoscopic particle image velocimetry. Friction velocity, thickness, roughness length and zero-plane displacement, determined from spatially averaged flow statistics, showed only minor differences between the regular and random arrangements at the same density. Recent a priori morphometric and statistical drag prediction methods were evaluated against experimentally determined roughness length. Observed differences between regular and random surface flow parameters were due to the presence of secondary flows which manifest as high-momentum pathways and low-momentum pathways in the streamwise velocity. Contrary to expectation, these secondary flows were present over the random surfaces and not discernible over the regular surfaces. Previously identified streamwise-coherent spanwise roughness heterogeneity does not seem to be present, suggesting that such roughness heterogeneity is not necessary to sustain secondary flows. Evidence suggests that the observed secondary flows were initiated at the front edge of the roughness and sustained over irregular roughness. Due to the secondary flows, local turbulent boundary layer profiles do not scale with local wall shear stress but appear to scale with local turbulent shear stress above the roughness canopy. Additionally, quadrant analysis shows distinct changes in the populations of ejection and sweep events. 

Motivated by the need for accurate determination of wall shear stress from profile measurements in turbulent boundary layer flows, the total shear stress balance is analysed and reformulated using several well-established semi-empirical relations. The analysis highlights the significant effect that small pressure gradients can have on parameters deduced from data even in nominally zero pressure gradient boundary layers. Using the comprehensive shear stress balance together with the log-law equation, it is shown that friction velocity, roughness length and zero-plane displacement can be determined with only velocity and turbulent shear stress profile measurements at a single streamwise location for nominally zero pressure gradient turbulent boundary layers. Application of the proposed analysis to turbulent smooth- and rough-wall experimental data shows that the friction velocity is determined with accuracy comparable to force balances (approximately 1 %–4 %). Additionally, application to boundary layer data from previous studies provides clear evidence that the often cited discrepancy between directly measured friction velocities (e.g. using force balances) and those derived from traditional total shear stress methods is likely due to the small favourable pressure gradient imposed by a fixed cross-section facility. The proposed comprehensive shear stress analysis can account for these small pressure gradients and allows more accurate boundary layer wall shear stress or friction velocity determination using commonly available mean velocity and shear stress profile data from a single streamwise location. 

A displacement thickness based inflow generation method, for simulation of a developing turbulent boundary layer, is proposed. Following existing rescaling/recycling methods, velocities from a plane sufficiently downstream of the inlet are recycled back and used as the inflow after re-scaling based on inner and outer length-scales. The inner length-scale is based on the viscous length-scale (for smooth walls) or surface specific scales (for rough walls). Prior recycling methods for smooth and rough boundary layers typically use d99 as the outer length-scale. Since d99 is a threshold based quantity, it is strongly dependent on the mean velocity profile and can have large undesired fluctuations, particularly if the profile shape is atypical or unsteady. Here, we propose the use of profile integrated quantities such as the displacement thickness (d1) to obtain a ‘surrogate’ for d99 in order to mitigate the adverse effects of having to determine the outer scale from a point-wise measurement of the mean velocity profile. The outer length- scale at the downstream plane is determined based on the local displacement thickness and higher-order moments of the integrated velocity profile. The inlet displacement thick- ness is fixed at a desired value and the outer length-scale at the inlet is determined through an iterative method. The use of high-order moments of the velocity profile is tested a- priori on DNS data for a developing boundary layer. Also, an initial application to LES over a surface with roughness elements is presented.