Logarithmic scaling of turbulence in smooth- and rough-wall pipe flow

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Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number (2018, background): “Several experimental studies have found that 〈u′2〉+ at very high Reτ (say Reτ > 20 000) decreases logarithmically with y for y+ > 3Re1/2τ and y/δ < 0.15 (Hultmark et al. 2013; Marusic et al. 2013).”
Turbulent boundary layer statistics at very high Reynolds number (2015, methodology): “For example, examinations of pipe flow have reported mean flow data at values of Reτ as high as 530 000 (Zagarola & Smits 1998; McKeon et al. 2004), where Reτ is the friction Reynolds number, with turbulence data at values up to 98 000 (Hultmark et al. 2012, 2013; Rosenberg et al. 2013).”
Direct numerical simulation of turbulent channel flow up to $\mathit{Re}_{{\it\tau}}\approx 5200$ (2014, background): “In contrast, Hultmark et al. (2012, 2013) observed a logarithmic region in 〈u′2〉+ over the range 800< y+ < 0.15Reτ in pipe flow with Reτ > 2× 104.”