The celebrated wall-scaling works for many statistical averages in turbulent flows near smooth walls, but the streamwise velocity fluctuation, owing to the natural constraint of boundedness on the dissipation rate at the wall. This new formula agrees well with the existing data and, in contrast to the logarithmic growth, supports the classical wall-scaling for turbulent intensity at asymptotically high Reynolds numbers.

The effect of Reynolds number (Re_τ) on drag reduction using spanwise wall oscillation is studied through direct numerical simulation of incompressible turbulent channel flows with Re_τ ranging from 200 to 2000. For the nondimensional oscillation period T⁺ = 100 with maximum velocity amplitude A⁺ = 12, the drag reduction ($\mathcal{D}\mathcal{R}$ ) decreases from 35.3% ± 0.5% at Re_τ = 200 to 22.3% ± 0.7% at Re_τ = 2000. The oscillation frequency ω⁺ for maximum $\mathcal{D}\mathcal{R}$ slightly increases with Re_τ, i.e., from ω⁺ ≈ 0.06 at Re_τ = 200 to 0.08 at Re_τ = 2000, with ${\mathcal{D}\mathcal{R}}_{max}=23.2\%\pm 0.6\%$ . These results show that $\mathcal{D}\mathcal{R}$ progressively decreases with increasing Re_τ. Turbulent statistics and coherent structures are examined to explain the degradation of drag control effectiveness at high Re_τ. Fukagata, Iwamoto, and Kasagi analysis in combination with the spanwise wavenumber spectrum of Reynolds stresses reveals that the decreased drag reduction at higher Re_τ is due to the weakened effectiveness in suppressing the near-wall large-scale turbulence, whose contribution continuously increases due to the enhanced modulation and penetration effect of the large-scale and very large-scale motions in the log and outer regions. Both the power-law model ($DR\propto R{e}_{\tau }^{-\gamma }$ ) and the log-law model [DR = f(Re_τ, ΔB), where ΔB is the vertical shift of the log-law intercept under control] are examined here by comparing them with our simulation data, from these two models we predict more than 10% drag reduction at very high Reynolds numbers, say, Re_τ = 10⁵.

Well-resolved direct numerical simulations of turbulent open channel flows (OCFs) are performed for friction Reynolds numbers up to $Re_\tau =2000$ . Various turbulent statistics are documented and compared with the closed channel flows (CCFs). As expected, the mean velocity profiles of the OCFs match well with the CCFs in the near-wall region but diverge notably in the outer region. Interestingly, a logarithmic layer with Kárman constant $\kappa =0.363$ occurs for OCF at $Re_\tau =2000$ , distinctly different from CCF. Except for a very thin layer near the free surface, most of the velocity and vorticity variances match between OCFs and CCFs. The one-dimensional energy spectra reveal that the very-large-scale motions (VLSMs) with streamwise wavelength $\lambda _x>3 h$ or spanwise wavelength $\lambda _z>0.5 h$ contribute the most to turbulence intensity and Reynolds shear stress in the overlap and outer layers (where h is the water depth). Furthermore, the VLSMs in OCFs are stronger than those in CCFs, resulting in a slightly higher streamwise velocity variance in the former. Due to the footprint effect, these structures also have significant contributions to the mean wall shear stress, and the difference between OCF and CCF enlarges with increasing $Re_\tau$ . In summary, the free surface in OCFs plays an essential role in various flow phenomena, including the formation of stronger VLSMs and turbulent kinetic energy redistribution.

In a recent article, Canton et al. [J. Canton et al., Phys. Rev. Fluids 1, 081501(R) (2016)] reported significant drag reduction in turbulent channel flow by using large-scale, near-wall streamwise swirls following the control strategy of Schoppa and Hussain [W. Schoppa and F. Hussain, Phys. Fluids 10, 1049 (1998)] for low Reynolds numbers only, but found no drag reduction at high friction Reynolds numbers $\left({\text{Re}}_{\tau }=550\right)$. Here we show that the lack of drag reduction at high $\text{Re}$ observed by Canton et al. is remedied by the proper choice of the large-scale control flow. In this study, we apply near-wall opposed wall-jet forcing to achieve drag reduction at the same (high) Reynolds number where Canton et al. found no drag reduction. The steady excitation is characterized by three control parameters, namely, the wall-jet-forcing amplitude $A^{+}$, the spanwise spacing ${\mathrm{\Lambda }}^{+}$, and the wall jet height $y_c^{+}$ $(+$ indicates viscous scaling); the primary difference between Schoppa and Hussain's work (also that of Canton et al.) and this Rapid Communication is the emphasis on the explicit choice of $y_c^{+}$ here. We show as an example that with a choice of $A^{+}\approx 0.015,{\mathrm{\Lambda }}^{+}\approx 1200$, and $y_c^{+}\approx 30$ the flow control definitely suppresses the wall shear stress at a series of Reynolds numbers, namely, $19\%,14\%$, and $12\%$ drag reductions at ${\text{Re}}_{\tau }=180$, 395, and 550, respectively. Further study should explore optimization of these parameter values.

The dominant paradigm in turbulent wall flows is that the mean velocity near the wall, when scaled on wall variables, is independent of the friction Reynolds number $Re_\tau$ . This paradigm faces challenges when applied to fluctuations but has received serious attention only recently. Here, by extending our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, p. R3) we present a promising perspective, and support it with data, that fluctuations displaying non-zero wall values, or near-wall peaks, are bounded for large values of $Re_\tau$ , owing to the natural constraint that the dissipation rate is bounded. Specifically, $\varPhi _\infty - \varPhi = C_\varPhi \,Re_\tau ^{-1/4},$ where $\varPhi$ represents the maximum value of any of the following quantities: energy dissipation rate, turbulent diffusion, fluctuations of pressure, streamwise and spanwise velocities, squares of vorticity components, and the wall values of pressure and shear stresses; the subscript $\infty$ denotes the bounded asymptotic value of $\varPhi$ , and the coefficient $C_\varPhi$ depends on $\varPhi$ but not on $Re_\tau$ . Moreover, there exists a scaling law for the maximum value in the wall-normal direction of high-order moments, of the form $\langle \varphi ^{2q}\rangle ^{{1}/{q}}_{max}= \alpha _q-\beta _q\,Re^{-1/4}_\tau$ , where $\varphi$ represents the streamwise or spanwise velocity fluctuation, and $\alpha _q$ and $\beta _q$ are independent of $Re_\tau$ . Excellent agreement with available data is observed. A stochastic process for which the random variable has the form just mentioned, referred to here as the ‘linear $q$ -norm Gaussian’, is proposed to explain the observed linear dependence of $\alpha _q$ on $q$ .

As a counterpart of energy cascade, turbulent momentum cascade (TMC) in the wall-normal direction is important for understanding wall turbulence. Here, we report an analytic prediction of non-universal Reynolds number ( $Re_{\unicode[STIX]{x1D70F}}$ ) scaling transition of the maximum TMC located at $y_{p}$ . We show that in viscous units, $y_{p}^{+}$ (and $1+\overline{u^{\prime }v^{\prime }}_{p}^{+}$ ) displays a scaling transition from $Re_{\unicode[STIX]{x1D70F}}^{3/7}$ ( $Re_{\unicode[STIX]{x1D70F}}^{-6/7}$ ) to $Re_{\unicode[STIX]{x1D70F}}^{3/5}$ ( $Re_{\unicode[STIX]{x1D70F}}^{-3/5}$ ) in turbulent boundary layer, in sharp contrast to that from $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ ( $Re_{\unicode[STIX]{x1D70F}}^{-2/3}$ ) to $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ ( $Re_{\unicode[STIX]{x1D70F}}^{-1/2}$ ) in a channel/pipe, countering the prevailing view of a single universal near-wall scaling. This scaling transition reflects different near-wall motions in the buffer layer for small $Re_{\unicode[STIX]{x1D70F}}$ and log layer for large $Re_{\unicode[STIX]{x1D70F}}$ , with the non-universality being ascribed to the presence/absence of mean wall-normal velocity $V$ . Our predictions are validated by a large set of data, and a probable flow state with a full coupling between momentum and energy cascades beyond a critical $Re_{\unicode[STIX]{x1D70F}}$ is envisaged.