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

TL;DR

This work tests Townsend's Attached Eddy Hypothesis by examining the spanwise spectrum of wall-normal velocity in DNS of ZPG TBL, channel, and pipe flows across a broad range of Reynolds numbers. By fitting a generalized von Kármán spectrum to $E_{ww}(k_y)$ and introducing local scaling based on the dissipation-driven length $\ell_\epsilon$ and local shear $u_{\tau z}$, the authors show that the peak energy and its location scale with these local quantities, extending validity from the near-wall region into the outer layer. A semi-empirical variance relation emerges: $\frac{\overline{w'^2}}{u_{\tau z}^2} \approx 1.55\pm0.1 - 1.5 C_w Re_\epsilon^{-2/3} - 18 Re_\epsilon^{-2}$, predicting a high-Re limit $B_3\approx 1.55\pm0.1$ when extrapolated, but acknowledging departures due to inactive motions that vary with wall distance and flow type. The results highlight that wall-normal variance is governed by local, not global, scaling and provide a framework that connects AEH to finite-Reynolds-number effects and to atmospheric boundary-layer behavior, with implications for cross-configuration turbulence modeling.

Abstract

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 (ZPG) boundary layer, across a decade of wall friction Reynolds numbers. Spectra along the spanwise wavenumber have a pronounced peak at intermediate wavenumbers that is proportional to the turbulent kinetic energy dissipation rate and the cube of the local shear stress throughout the bottom half of the boundary layer for all flow cases. The observed scaling with the local stress rather than the surface shear velocity $U_τ$ accounts for differences in the ZPG wall-normal variance seen in previous studies. The scaling 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, while noting this hypothesis does not account for the linear decay of the total stress in enclosed flows. The Reynolds number dependence of the variance is determined from the scale separation between the spectrum peak and the dissipative cutoff, where the resulting semi-empirical fit aligns with variance values in the literature. At the high-Reynolds-number limit, the spectral peak leads to a variance between 1.45 to 1.65 times the local shear stress. This range is 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 relatively minor contributions from the `inactive' motions at low wavenumbers, which vary with wall-normal position and for different flow configurations.

Figures (10)

• Figure 1: Profiles of the wall-normal velocity variance $\overline{{w^\prime}^2}$ normalized by the surface shear velocity $U_{\tau}$: (a) wall-normal position $z$ in viscous units $z^+=z U_\tau / \nu$; (b) $z$ in outer units relative to the boundary layer thickness $\delta$. In all figures, the data are from direct numerical simulations (DNS) Sillero2013Lee2015Yao2023, with color indicating the flow case and shade corresponding to the friction Reynolds numbers $Re_\tau = \delta U_\tau /\nu$ given in Table \ref{['tbl']}.
• Figure 2: Example wall-normal velocity spectra $E_{ww}$ as a function of spanwise wavenumber $k_y$ for $z/\delta =$ 0.1, where the dashed lines are fits using the model von Kármán spectrum in Eq. \ref{['eq:vks']}. (a) for the $Re_\tau=$ 5 200 channel flow case Lee2015 to demonstrate the wavenumber $k_{peak}$ and energy density $E_{peak}$ of the spectrum peak, the low-wavenumber plateau $E_{k_y \to 0}$, and the contribution of large-scale motions $\overline{{w^\prime}^2}_{l} = \int_0^{k_{peak}} E_{ww} dk_y$ and small-scale motions $\overline{{w^\prime}^2}_{s} = \int_{k_{peak}}^\infty E_{ww} dk_y$ to the wall-normal variance. (b) for all cases in Table \ref{['tbl']} to demonstrate the model spectrum fit.
• Figure 3: Profiles of the dissipation-based length scale $\ell_\epsilon$ defined in Eq. \ref{['eq:ell']}: (a) $\ell_\epsilon$ in viscous units; (b) $\ell_\epsilon$ in outer units; (c) the Reynolds number $Re_\epsilon = \ell_\epsilon / \eta$ representing separation between length scales. The dashed lines in each panel all correspond to $\kappa z$.
• Figure 4: Comparison of $E_{ww}(k_y)$ at fixed $z$ positions in viscous and outer units: (a) using traditional wall-scaling parameters $U_{\tau}^2$ and $z$; (b) using local-in-$z$ parameters with shear velocity $u_{\tau z}^2 = -\overline{u^\prime w^\prime} + \nu \partial U / \partial z$ and $\ell_\epsilon$. The inset plot in (b) shows the full spectrum with dissipative scales for reference.
• Figure 5: Profiles for the properties of $E_{ww}(k_y)$ identified in Fig. \ref{['fig:example']}(a). Columns correspond to the wavenumber $k_{peak}$ of the spectrum peak (a,d), the amplitude $E_{peak}$ of the peak (b,e), and the low-wavenumber plateau $E_{k_y \to 0}$ (c,f). Rows correspond to normalization with wall-scaling parameters (a,b,c) and local-in-$z$ parameters (d,e,f). The peak properties are detected directly from the spectra, and the low-wavenumber plateau is inferred from the fitted model von Kármán spectrum.