New fundamental developments in wall-bounded turbulence*

With a fresh look at matched asymptotic approximations, we develop new directions in our understanding of overlap regions in wall bounded flows and employ the best available DNS data in channel and pipe flows, as well as experimental data in boundary layer and pipe flows. For the mean velocity profiles, a logarithmic plus linear overlap is found to be superior to the classical pure logarithmic form, which confirms the non-universality of the overlap parameters including the Kármán coefficient, and their dependence on flow geometry and pressure gradient. The self-similarity of the logarithmic plus linear overlap is demonstrated using several data sets, including those from the Superpipe, while improving on the pure logarithmic overlap because of the wider overlap region for the same Reynolds numbers. For the streamwise normal stress, the one-quarter power trend, based on bounded dissipation, is confirmed for channels, pipes and zero-pressure-gradient (ZPG) boundary layers to represent the data better and over a wider part of the flow, than the commonly used inverse logarithmic representation developed by the wall-scaled eddy model. For ZPG boundary layers and Re_τ ≥ 10,000, where Re_τ is the friction Reynolds number, the inverse logarithmic trend is found only in the limited range 0.05 ≤ y/δ ≤ 0.15, compared to a range of 0.02 ≤ y/δ ≤ 0.38 for the one-quarter power trend. In the Superpipe data with Re_τ ≥ 10,000, the inverse logarithmic trend is found in the limited range 0.03 ≤ y/R ≤ 0.18, compared to a range of 0.04 ≤ y/R ≤ 0.5 for the one-quarter power trend. For ZPG boundary layers, a simple intermittency correction extends the range for the one-quarter trend up to y/δ around 0.5, as well. Finally, a somewhat larger exponent for the power law of streamwise normal stress equal to 0.28, instead of 0.25, is found to slightly extend its range of validity.