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 ${{Re}}_{\tau } \approx 12000$ . 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 $k_{\theta }^{-1+\alpha }$ decay (with $k_{\theta }$ the wavenumber in the azimuthal direction, and $\alpha \approx 0.18$ ), 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 $(\langle{u}^2\rangle)$ from the largest scales of motion (superstructures) slowly declines as ${{Re}}_{\tau }^{-\alpha }$ , and the integrated inner-scaled variance follows a defect power law of the type $\langle u^2 \rangle ^+ = A - B \, {{Re}}_{\tau }^{-\alpha }$ , 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 $\langle u^2 \rangle ^+ \approx 12.1$ , 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.

Topics & Concepts