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

Abstract

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 Reτ ≈ 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 kθ−1+α decay (with kθ the wavenumber in the azimuthal direction, and α ≈ 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−τα, and the integrated inner-scaled variance follows a defect power law of the type 〈u2〉+ = A − BRe−τα, 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.