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Indicator functions of the streamwise normal-stress profiles (NSP), based on careful differentiation of some of the best direct numerical simulations (DNS) data from channel and pipe flows, over the range 550Reτ16 000, are examined to establish the existence and range in wall distances of either a logarithmic-trend segment or a 1/4-power region. For nine out of 15 cases of DNS data we examined where Reτ2000, the NSP did not contain either of the proposed trends. As Reτ exceeds around 2000 a 1/4-power, reflecting the “bounded-dissipation” predictions of Chen and Sreenivasan “Law of bounded dissipation and its consequences in turbulent wall flows,” J. Fluid Mech. 933, A20 (2022); “Reynolds number asymptotics of wall-turbulence fluctuations,” J. Fluid Mech. 976, A21 (2023) and data analysis of Monkewitz “Reynolds number scaling and inner-outer overlap of stream-wise Reynoldss stress in wall turbulence,” arXiv:2307.00612 (2023), develops near y+=1000 and expands with Reynolds numbers extending to 1000y+10 000 for Reτ around 15 000. This range of 1/4-power NSP corresponds to a range of outer-scaled Y between around 0.3 and 0.7. The computational database examined did not include the zero-pressure-gradient boundary layer experiments at higher Reynolds numbers where the logarithmic trend in the NSP has been previously reported around y+ of 1000 by Marusic et al. “Attached eddy model of wall turbulence,” Annu. Rev. Fluid Mech. 51, 49–74 (2019); “The logarithmic variance of streamwise velocity and conundrum in wall turbulence,” J. Fluid Mech. 933, A8 (2022) according to a “wall-scaled eddy model.”
Nagib et al. (Mon,) studied this question.