On the wall-normal velocity variance in canonical wall-bounded turbulence

The variance and spectra of wall-normal velocities are investigated for direct numerical simulations of turbulent flow in a channel, pipe, and zero-pressure-gradient boundary layer across a decade of friction Reynolds numbers. Spectra along the spanwise wavenumber have a pronounced peak well described by the turbulent dissipation rate and the local shear stress throughout the bottom half of the boundary layer. Deviations in the local stress from the surface shear velocity $U_τ$ account for almost all of the differences in wall-normal velocity variance observed across different canonical flows, including for plane Couette flow. The dependence on the local stress is attributed to the fact that wall-normal motions are predominately `active' per Townsend's attached eddy hypothesis and directly contribute to the local shear stress, noting this hypothesis assumes simplified ideal conditions with constant turbulent shear stress. A semi-empirical fit applied to the Reynolds number dependence of the variance matches the simulations across the lower half of the boundary layer and aligns with observed values in the literature. The fit extrapolates to a value between 1.45 and 1.65 times the local shear stress in the high-Reynolds-number limit, consistent with previous predictions relative to $U_τ$ including for the vertical velocity in the near-neutral atmospheric boundary layer. However, universality in the exact proportional constant is precluded by small discrepancies in the variances corresponding to dissimilarity in the low-wavenumber contributions across different flow configurations and wall-normal positions. We speculate the dissimilarity is due to relatively weak `inactive' wall-normal motions that are excluded from Townsend's original hypothesis.