The logarithmic variance of streamwise velocity and conundrum in wall turbulence

The logarithmic dependence of streamwise turbulence intensity has been observed repeatedly in recent experimental and direct numerical simulation data. However, its spectral counterpart, a well-developed $k^{-1}$ spectrum ( $k$ is the spatial wavenumber wall-parallel direction), not convincingly from same In present study, we revisit spectrum-based attached eddy model Perry co-workers, who proposed emergence inviscid limit, for small but finite $z/\delta$ Reynolds numbers $z$ wall-normal coordinate, $\delta$ outer length scale). upper layer (or inertial sublayer), reexamination reveals that must vary with location at order , consistent early observation argued ‘incomplete similarity’. subsequently calculated, demonstrating existence necessary condition approximate – more general premultiplied power-spectral $O(1)$ $O(1/\delta ) < k O(1/z)$ . Furthermore, it shown Townsend–Perry constant be weakly dependent on number. Finally, analysis semi-empirically extended to lower mesolayer), near-wall correction proposed. All predictions related analyses/assumptions are validated high-fidelity data (Samie et al. J. Fluid Mech. vol. 851, 2018, pp. 391–415).