On the streamwise velocity variance in the near-wall region of turbulent flows

We study the behaviour of the streamwise velocity variance in turbulent wall-bounded flows using a direct numerical simulation (DNS) database of pipe flow up to friction Reynolds number {Re} 12000. The analysis of the spanwise spectra in the viscous near-wall region strongly hints to the presence of an overlap layer between the inner- and the outer-scaled spectral ranges, featuring a k ^-1+ decay (with k the wavenumber in the azimuthal direction, and 0. 18), hence shallower than suggested by the classical formulation of the attached-eddy model. The key implication is that the contribution to the streamwise velocity variance (u²) from the largest scales of motion (superstructures) slowly declines as {Re} ^-, and the integrated inner-scaled variance follows a defect power law of the type u² ^+ = A - B \, {Re} ^-, with constants A and B depending on y^+. The DNS data very well support this behaviour, which implies that strict wall scaling is restored in the infinite-Reynolds-number limit. The extrapolated limit distribution of the streamwise velocity variance features a buffer-layer peak value of u² ^+ 12. 1, and an additional outer peak with larger magnitude. The analysis of the velocity spectra also suggests a similar behaviour of the dissipation rate of the streamwise velocity variance at the wall, which is expected to attain a limiting value of approximately 0. 28, hence slightly exceeding the value 0. 25 which was assumed in previous analyses (Chen & Sreenivasan, J. Fluid Mech. , vol. 908, 2021, R3). We have found evidence suggesting that the reduced near-wall influence of wall-attached eddies is likely linked to the formation of underlying turbulent Stokes layers.