A method for evaluating relations of turbulent normal stresses by experimental data over a wide range of Reynolds numbers

Recently, Nagib et al. ( Phys. Fluids, vol. 36, no. 7, 2024, 075145) used indicator functions of streamwise normal stress profiles to identify the valid wall-distance and Reynolds number ranges for two models in direct numerical sumulation (DNS) of channel and pipe flows. Since such functions are challenging to construct from experimental data, we propose a simpler, more robust method better suited to experiments. Applied to the two leading models – logarithmic and power-law – for normal stresses in the ‘fitting region’ of wall-bounded flows, this method is tested on prominent experimental data sets in zero-pressure-gradient (ZPG) boundary layers and pipe flows across a wide Reynolds number range ( Re_τ ). Valid regions for the models appear only for Re_τ 10\,000 , with a lower bound y⁺ᵢₙ ~ (Re_τ )0.5 and y⁺ᵢₙ 400 . The upper bound is a fixed fraction of the boundary layer thickness or pipe radius, independent of Re_τ . The power-law model is found to hold over a broader range, up to Y ≈ 0.4 in ZPG and Y ≈ 0.5 in pipe flows, compared with the logarithmic trend, which is formulated to be coincident with the classical logarithmic region for the mean flow ( Y 0.15 ). A slightly higher exponent ( $0.28$ ) than that of Chen & Sreenivasan ( J. Fluid Mech. vol. 933, 2022, A20; J. Fluid Mech. vol. 976, 2023, A21) extends the power-law model’s validity and correcting for outer intermittency in ZPG flows further broadens it. Projections to the near-wall region of both models yield nearly identical predictions of near-wall peak stress across the highest available Re_τ . These findings, alongside results from Monkewitz & Nagib ( J. Fluid Mech. vol. 967, 2023, A15) and Baxerras et al. ( J. Fluid Mech. vol. 987, 2024, R8), highlight the importance of nonlinear eddy growth and residual viscous effects in wall-bounded flow modelling, informing potential refinements to the logarithmic model, such as those proposed by Deshpande et al. ( J. Fluid Mech. vol. 914, 2021, A5).