Near-wall streamwise turbulence intensity as ${\text{Re}}_{\tau }\to \infty$

In this study, asymptotic scaling of near-wall streamwise turbulence intensity $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2$ ($u_{\tau }$ is the friction velocity) is theoretically explored. The three scalings previously proposed are first reviewed with their derivation and physical justification: (1) $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2\sim ln{\text{Re}}_{\tau }$ (${\text{Re}}_{\tau }$ is the friction velocity); (2) $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2\sim 1/U_{\infty }^{+}$ ($U_{\infty }^{+}$ is the inner-scaled freestream velocity in boundary layer); (3) $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2\sim {\text{Re}}_{\tau }^{-1/4}$. A new analysis is subsequently developed based on velocity spectrum, and two possible scenarios are identified based on the asymptotic behavior of the outer-scaling part of the near-wall velocity spectrum. In the former case, the outer-scaling part of the spectrum is assumed to reach a nonzero constant as ${\text{Re}}_{\tau }\to \infty$, and it results in the scaling of $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2\sim ln{\text{Re}}_{\tau }$, both physically and theoretically consistent with the classical attached eddy model. In the latter case, a sufficiently rapid decay of the outer-scaling part of the spectrum with Re is assumed due to the effect of viscosity, such that $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2<\infty$ for all ${\text{Re}}_{\tau }$. The following analysis yields $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2\sim 1/ln{\text{Re}}_{\tau }$, asymptotically consistent with the scaling of $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2\sim 1/U_{\infty }^{+}$. The scalings are further verified with the existing simulation and experimental data and those from a quasilinear approximation [Holford et al. J. Fluid Mech. 980, A12 (2024)], the spectra of which all appear to favor $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2\sim 1/ln{\text{Re}}_{\tau }$, although new datasets for ${\text{Re}}_{\tau }\gtrsim O\left({10}^4\right)$ would be necessary to conclude this issue.

Figure 1

Peak streamwise turbulence intensities from experiments (open symbols) and DNSs (filled symbols) and their scaling laws: - - - -, $A_S+B_{S}ln{\text{Re}}_{\tau }$, with $A_S=0.646$ and $B_S=3.54$ [20]; —, $A_M-B_M/U_{\infty }^{+}$, with $A_M=15.2$ and $B_M=174$, where the free stream velocity $U_{\infty }$ is given by $U_{\infty }^{+}=(1/\kappa )ln{\text{Re}}_{\tau }+6.5$, with $\kappa =0.384$ (note $A_M$ and $B_M$ here are different from those proposed in Refs. [21, 22], where an error was found in the description of $U_{\infty }^{+}$); – - – - –, $A_C+B_{C}ln{\text{Re}}_{\tau }$, with $A_C=11.5$ and $B_C=19.32$ [23]. The data from pipe, channel, and boundary layer are denoted by the orange, blue and green colors, respectively. Here, $\overline{(·)}$ indicates a time average and the superscript ${(·)}^{+}$ denotes the inner-scaling with $u_{\tau }$ and $\nu$.

Figure 2

Near-wall streamwise turbulence intensity in channel for ${\text{Re}}_{\tau }=544,1000,1995,5186$ [18]: (a) near-wall profile of $\overline{u^{\prime }{u}^{\prime }}\left(y^{+}\right)/u_{\tau }^2$; (b) values of $\overline{u^{\prime }{u}^{\prime }}\left(y^{+}\right)/u_{\tau }^2$ at $y^{+}=y_{\mathrm{peak}}^{+},25,30$ with $y_{\mathrm{peak}}^{+}\simeq 15$. In panel (b), the solid black lines indicate fits in the form of $Alog{\text{Re}}_{\tau }+B$, where $A$ and $B$ are determined by the data at two lowest Reynolds numbers.

Figure 3

Premultiplied streamwise wave number spectra of streamwise velocity at $y^{+}=15$ in panels (a), (b) channel [DNS from 18] and in panels (c), (d) boundary layer [experiment from Ref. 20]: (a), (c) inner scaling; (b), (d) outer scaling. In panels (a), (b), ${\text{Re}}_{\tau }=1000,1995,5186$, and, in panels (c), (d), ${\text{Re}}_{\tau }=6123,10100,19680$.

Figure 4

A schematic diagram of the premultiplied one-dimensional spectrum of near-wall streamwise velocity at a fixed inner-scaled wall-normal location ($y^{+}=c$).

Figure 5

Premultiplied streamwise wave number spectra of streamwise velocity at $y^{+}=20$ in channel (from a quasilinear approximation by Ref. [54]): (a) inner scaling; (b) outer scaling. Here, ${\text{Re}}_{\tau }=10000,20000,50000,100000$.

Figure 6

Scaling of streamwise turbulence intensity at $y^{+}=20$ for the quasilinear approximation of Ref. [54] (${\text{Re}}_{\tau }=5200,10000,20000,50000,100000$): $\left(\text{a}\right)ln{\text{Re}}_{\tau }$ scaling; $\left(\text{b}\right)1/ln{\text{Re}}_{\tau }$ scaling.

Figure 7

${\text{Re}}_{\tau }$ dependence of $B\left({\text{Re}}_{\tau }\right)$ in Eq. (11a): $\left(\text{a}\right)B\left({\text{Re}}_{\tau }\right)$ vs $\sigma (={(1/ln{\text{Re}}_{\tau })}^{-1})$; $\left(\text{b}\right)B\left(\sigma \right)$ vs ${\text{Re}}_{\tau }$. Here, data are from quasilinear approximation of Ref. [68] (orange) with $a=0.1$ and $b\simeq 1\times {10}^{-4}$ and experiment of Ref. [20] (green) with $a=0.2$ and $b\simeq 3\times {10}^{-4}$. Here, ${\text{Re}}_{\tau }=5200,10000,20000,50000,100000$ for the quasilinear approximation and ${\text{Re}}_{\tau }=6123,10100,19680$ for the experiment.

Figure 8

Peak streamwise turbulence intensities from experiments (open symbols) and DNSs (filled symbols) with the $1/ln{\text{Re}}_{\tau }$ scaling. The curve here is given by $13.8–40/ln{\text{Re}}_{\tau }$.