2. A hypothesis for $\overline {{u_iu_j}}$, $\overline {{u_iu_i}}$ and $\overline {{\theta \theta }}$

The transport equations for the turbulent kinetic energy $\overline {{u_i}{u_i}}$ and the scalar variance $\overline {{\theta }{\theta }}$ in a channel are given by (e.g. Pope Reference Pope2000; Abe, Antonia & Kawamura Reference Abe, Antonia and Kawamura2009; Alcántara-Ávila et al. Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021)

(2.1)$$\begin{gather} \underbrace{- \overline{{u_1}{u_2}}\,\frac{{\partial {{\bar U}_1}}}{{\partial {x_2}}}}_{{P_U}} \underbrace{{}- \frac{1}{2}\,\frac{{\partial \overline{u_i^2{u_2}} }}{{\partial {x_2}}}}_T \underbrace{{}- \frac{1}{2}\,\frac{{\partial \overline{{u_2}p} }}{{\partial {x_2}}}}_P + \underbrace{\frac{\nu }{2}\,\frac{{{\partial ^2}\overline{{u_i}{u_i}} }}{{\partial x_2^2}}}_{{D_\nu }} \underbrace {{}- \bar \varepsilon }_D = 0, \end{gather}$$

(2.2)$$\begin{gather}\underbrace{ - \overline{\theta {u_2}}\,\frac{{\partial \varTheta }}{{\partial {x_2}}}}_{{P_\varTheta }}\underbrace{{}- \frac{1}{2}\,\frac{{\partial \overline{{\theta ^2}{u_2}} }}{{\partial {x_2}}}}_{{T_\theta }} + \underbrace {\frac{{\gamma} }{2}\,\frac{{{\partial ^2}\overline{{\theta ^2}} }}{{\partial x_2^2}}}_{{D_{\gamma} }}\underbrace {{}- {{\bar \varepsilon }_\theta }}_{{D_\theta }} = 0, \end{gather}$$

where ${\bar U}_1$ is the mean velocity in the streamwise ($x_1$) direction, and $\varTheta$ is the mean scalar. In (2.1), the terms $P_U$, $T$, $P$, $D_\nu$ and $D$ are the production, turbulent diffusion, pressure diffusion, molecular diffusion and dissipation rate, respectively. Similarly, in (2.2), terms $P_\varTheta$, $T_\theta$, $D_{\gamma }$ and $D_\theta$ are the production, turbulent diffusion, molecular diffusion and dissipation, respectively. Except for the pressure term ($P$) in (2.1), which is negligible near the wall and in the outer flow region, the two equations are analogous. This analogy has been tested extensively (e.g. Fulachier & Dumas Reference Fulachier and Dumas1976; Fulachier & Antonia Reference Fulachier and Antonia1984; Antonia, Abe & Kawamura Reference Antonia, Abe and Kawamura2009) in both wall and free shear flows.

All terms in (2.1) and (2.2), after normalizing by wall parameters, are shown in figure 1(a,b), respectively. Close to the wall, both $\bar \varepsilon$ and ${\bar \varepsilon _\theta }$ are approximately balanced by the molecular diffusion. In particular, ${\bar \varepsilon _\theta }$ is very nearly equal to ${({{\gamma } }/{2})({{{\partial ^2}\overline {{\theta ^2}} }}/{{\partial x_2^2}})}$ in the region $x_2^+\lesssim 2$ at $Re_\tau =500\unicode{x2013}5000$ and $Pr=\nu /{\gamma }=0.71$ (figure 1b). However, ${\bar \varepsilon }$ is approximately equal to ${({\nu }/{2})({{{\partial ^2}\overline {{u_i}{u_i}} }}/{{\partial x_2^2}})}$ only in the region $x_2^+\lesssim 0.3$ at $Re_\tau =550\unicode{x2013}10^4$ (figure 1a). The pressure diffusion term ($P$) slowly increases away from the wall up to a maximum value near $x_2^+\approx 2$, and the magnitude of the ratio of this term to ${\bar \varepsilon }$ is 2.7 % at $x_2^+\approx 0.3$ and $Re_\tau =10^4$. Therefore, in these near wall regions, (2.1) and (2.2) reduce to

(2.3a,b)\begin{equation} \bar \varepsilon^+= \frac{1}{2}\,\frac{{{\partial ^2}\overline{{u_i}{u_i}}^+ }}{{\partial x_2^{{+}2}}},\quad {\bar \varepsilon _\theta^+ } = \frac{1}{2\,Pr}\, \frac{{{\partial ^2}\overline{{\theta \theta}}^+ }}{{\partial x_2^{{+}2}}}, \end{equation}

when the conventional scaling is used. It is evident that both sides of these two equations depend on $Re_\tau$. However, after normalizing (2.3a,b) by $\bar \varepsilon$, ${\bar \varepsilon _\theta }$ and $\nu$, we obtain

(2.4a,b)\begin{equation} \frac{{{\partial ^2}\overline{{u_i}{u_i}}^* }}{{\partial x_2^{*2}}}=2,\quad \frac{{{\partial ^2}\overline{{\theta \theta}}^* }}{{\partial x_2^{*2}}}=2\,Pr, \end{equation}

independently of $Re_\tau$. Equations (2.3a,b) and (2.4a,b) indicate that the scaling based on $\bar \varepsilon$, ${\bar \varepsilon _\theta }$ and $\nu$ is superior to that based on wall parameters. A similar analysis, based on the transport equations for $\overline {u_iu_i}$ in the range $40 \leq x_2^+ \leq 200$, leads to the same conclusion; see Appendix B. Note that $\bar \varepsilon ^+$ and ${\bar \varepsilon _\theta ^+ }$ do not collapse and their magnitudes are large near the wall (figure 1); they arise because of the presence of the wall and, concomitantly, the presence of the mean shear. Therefore, it seems natural to use the local $\bar \varepsilon$ and ${\bar \varepsilon _\theta }$, instead of $u_\tau$ and $\theta _\tau$, to describe the wall turbulence statistics; whilst $u_\tau$ and $\theta _\tau$ are local (wall) quantities, they also balance the momentum flux and heat flux, respectively, across the flow. On the other hand, $\bar \varepsilon$ and ${\bar \varepsilon _\theta }$ are quantities that are important in describing the dynamics at every location in the flow via (2.1) and (2.2). For example, $\bar \varepsilon$ and ${\bar \varepsilon _\theta }$ are approximately balanced by the molecular diffusion as $x_2^+\to 0$, while they are approximately balanced by the production and turbulent diffusion at $x_2^+=5$ (figure 1). The relationship between the present scaling and the wall scaling will be discussed in some detail in § 4.

Figure 1. Distributions of terms in (a) (2.1) and (b) (2.2), normalized by wall parameters, in a channel at $Pr=0.71$. Data of Lee & Moser (Reference Lee and Moser2015, Reference Lee and Moser2019), Alcántara-Ávila et al. (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021) and Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022).

Equations (2.4a,b) further indicate that the possible scaling parameters for $\overline {{u_i}{u_i}}$ and $\overline {{\theta }{\theta }}$ are $\bar \varepsilon$, ${\bar \varepsilon _\theta }$ and $\nu$; since ${\gamma }$ at any equation can be written as ${\gamma } \equiv \nu /Pr$ (see, for example, (2.4a,b)), for simplicity, we consider only $\bar \varepsilon$, ${\bar \varepsilon _\theta }$ and $\nu$ as the scaling parameters. In this situation, the distance from the wall should be normalized as

(2.5)\begin{equation} x_2^* = \frac{{{x_2}}}{\eta } = \frac{{{x_2}{u_K}}}{\nu }, \end{equation}

where $x_2^*$ is a local Reynolds number so that its magnitude can be expected to determine the relative importance of viscous effects. A more general definition of the local turbulent Reynolds number is

(2.6)\begin{equation} Re_{\lambda} = \frac{{{{\overline{{u_iu_i}} }^{1/2}}}}{{{3^{1/2}}}}\, \frac{{{\lambda }}}{\nu }=\sqrt {\frac{5}{3}}\,{\overline{{u_i}{u_i}} ^*}, \end{equation}

where $\lambda$ is given by

(2.7)\begin{equation} {\lambda } = {(5\nu )^{1/2}}\,\frac{{{{\overline{{u_iu_i}} }^{1/2}}}}{{{{\bar \varepsilon }^{1/2}}}}. \end{equation}

The Taylor microscale Reynolds number $Re_\lambda$ is defined via (2.6), and the Taylor microscale $\lambda$ is defined via (2.7). Similarly, a more general definition of the local turbulent Péclet number is

(2.8)\begin{equation} Pe_{\lambda_\theta} = \frac{{{{\overline{{u_iu_i}} }^{1/2}}{\lambda _\theta }}}{{\gamma} } = {(3\,Pr\,{\overline{{u_i}{u_i}} ^*}\,{\overline{\theta \theta } ^*})^{1/2}}, \end{equation}

where

(2.9)\begin{equation} \lambda_\theta = {(3{\gamma} )^{1/2}}\,\frac{{{{\overline{\theta \theta } }^{1/2}}}}{{\bar \varepsilon _\theta ^{1/2}}} \end{equation}

is the Corrsin microscale. (Fulachier & Antonia (Reference Fulachier and Antonia1983) suggested that the use of $u_i$, instead of $u_1$, yields a more meaningful comparison between turbulent Reynolds and Péclet numbers.) After integrating (2.4a,b) twice with respect to $x_2^*$, we obtain

(2.10a,b)\begin{equation} \overline{{u_i}{u_i}}^* ={{ x_2^{*2}}},\quad \overline{{\theta}{\theta}}^* =Pr\,x_2^{*2}. \end{equation}

Equations (2.10a,b) are tenable as $x_2^*\to 0$ in a channel flow and a boundary layer (figure 14 of Tang & Antonia Reference Tang and Antonia2022).

Substituting (2.10a,b) into (2.6) and (2.8), we obtain

(2.11a,b)\begin{equation} R{e_\lambda } = \sqrt {\frac{5}{3}}\,x_2^{*2}, \quad P{e_{{\lambda _\theta }}} = \sqrt 3\,Pr\,x_2^{*2}. \end{equation}

Equations (2.11a,b) indicate that both $Re_{\lambda }$ and $Pe_{\lambda _\theta }$ should be independent of $Re_\tau$ as $x_2^*\to 0$ (at a given $Pr$). On the other hand, in the region sufficiently far away from the wall where the effect of the wall on the flow is negligible (e.g. the central region of the channel or pipe), both $Re_{\lambda }$ and $Pe_{\lambda _\theta }$ should go to infinity when $Re_\tau$ and $Pr\,Re_\tau$ go to infinity; hereafter, this region will be described as $x_2^*\to \delta ^*$. It is expected that both $Re_{\lambda }$ and $Pe_{\lambda _\theta }$ cannot increase to infinity abruptly at small and moderate $x_2^*$. Consequently, there should be a near-wall region over which $\overline {{u_i}{u_i}}$ and $\overline {{\theta }{\theta }}$, after normalization by $\bar \varepsilon$, ${\bar \varepsilon _\theta }$ and $\nu$, increase with $x_2^*$ at sufficiently high $Re_\tau$ and $Pr\,Re_\tau$; this increase should not depend on $Re_\tau$ at a given $Pr$. Further, at a sufficiently, though not infinitely, large distance from the wall ($0\ll x_2^* \ll \delta ^*$) where the effect of $\nu$ is negligible, $\overline {u_iu_i}$ and $\overline {\theta \theta }$ should be determined by $\bar \varepsilon$ and $\bar \varepsilon _\theta$. Dimensional analysis leads to $\overline {u_iu_i}^*=C_{q} x_2^{*2/3}$ and $\overline {\theta \theta }^*=C_{\theta }x_2^{*2/3}$, over which an inertial-convective range is established; the details can be found in Appendix A. Here, $C_{q}$ and $C_{\theta }$ are constants. Finally, at large $Pr$, there should be a range between the inertial-convective range (at moderate $x_2^*$) and the viscous-diffusive range (at small $x_2^*$) over which the effect of viscosity, instead of diffusivity, is important. The previous arguments are best summarized by the following hypothesis.

In wall turbulence at sufficiently high Reynolds and Péclet numbers, velocity and passive scalar fluctuation statistics, such as $\overline {{u_i}{u_i}}$, $\overline {u_iu_j}$ and $\overline {\theta \theta }$, do not depend on the Reynolds and Péclet numbers when the normalization uses $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$. Consequently, they are independent of the Reynolds and Péclet numbers. In other words, they depend only on the distance from the wall. Further, at a sufficiently large distance from the wall where the effect of $\nu$ is negligible, $\overline {u_iu_i}$, $\overline {u_iu_j}$ and $\overline {\theta \theta }$ are determined by $\bar \varepsilon$ and $\bar \varepsilon _\theta$. Finally, there is a viscous-convective range for $\overline {\theta \theta }$ between the viscous-diffusion and inertial-convective ranges over which the effect of $\nu$, instead of ${\gamma }$, is important when the Prandtl number is sufficiently large. The larger the Reynolds and Péclet numbers, the larger the distance from the wall over which the above hypothesis applies.

Whilst it is tempting to hypothesize that in the wall region, $\overline {u_iu_i}^*$, $\overline {u_iu_j}^*$ and $\overline {\theta \theta }^*$ should not depend significantly on the flow type (especially when the outer boundary conditions are similar, such as for the pipe and the channel), we refrain from doing so at this stage. We will, however, test the above hypothesis in all three canonical flows, i.e. the channel, boundary layer and pipe.

Mathematically, the hypothesis predicts that in wall turbulence, the behaviour of $\overline {u_iu_j}$, $\overline {u_iu_i}$ and $\overline {\theta \theta }$ can be expressed as

(2.12a–c)\begin{equation} \overline{u_iu_j}^*= f_{ij}(x_2^*), \quad \overline{u_iu_i}^*= f_{q}(x_2^*), \quad \overline{\theta \theta}^*= f_{\theta }(x_2^*), \end{equation}

where the asterisk denotes normalization by $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ (or equivalently, ${\theta _B}$, ${u_K}$ and $\eta$), and the functions $f_{ij}$, $f_{q}$ and $f_{\theta }$ are independent of the Reynolds and Péclet numbers once appropriate values of the Reynolds and Péclet numbers are reached. A conceptual sketch of the predictions for $|\overline {u_iu_j}^*|$, $\overline {u_iu_i}^*$ and $\overline {\theta \theta }^*$ at $Pr\approx 1$ is shown in figure 2 using log–log scales. Also shown in this figure is a sketch for $\overline {\theta \theta }^*$ at large $Pr$. At small $x_2^*$, there should be a viscous-diffusive dominated range for $\overline {u_iu_j}^*$, $\overline {u_iu_i}^*$ and $\overline {\theta \theta }^*$. In the inertial-convective range $0\ll x_2^*\ll \delta ^*$,

(2.13a–c)\begin{equation} \overline{u_iu_j}^*= C_{ij} x_2^{*2/3},\quad \overline{u_iu_i}^*= C_{q} x_2^{*2/3},\quad \overline{\theta\theta}^*= C_{\theta } x_2^{*2/3}, \end{equation}

where $C_{ij}$ is constant; the detailed derivations in the context of $\overline {u_iu_i}^*$ and $\overline {\theta \theta }^*$ can be found in Appendix A. Finally, at large $Pr$, there exists a viscous-convective range for $\overline {\theta \theta }$ between the inertial-convective range and the viscous-diffusive range. We assume that $\overline {\theta \theta }/\bar \varepsilon _\theta$ scales with the Kolmogorov time scale $(\nu /\bar \varepsilon )^{1/2}$, which leads to

(2.14)\begin{equation} \overline{\theta\theta}^*= const. \end{equation}

We will test (2.12a–c)–(2.14) in § 3.

Figure 2. Predictions for $|\overline {u_iu_j}^*|$, $\overline {u_iu_i}^*$ and $\overline {\theta \theta }^*$ at $Pr\approx 1$ on log–log scales (red curve). Also shown, based on the present hypothesis, is the prediction for $\overline {\theta \theta }^*$ at large $Pr$ (black curve).

3. Test of the hypothesis

Figure 3 shows the Kolmogorov-normalized turbulent kinetic energies ${\overline {{u_i}{u_i}} ^*}$ in the channel, boundary layer and pipe for $Re_\tau =445\unicode{x2013}10^4$. They are plotted using the wall-parameter-normalized ${\overline {{u_i}{u_i}} ^+}$ in these three flows (Simens et al. Reference Simens, Jiménez, Hoyas and Mizuno2009; Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Borrell, Sillero & Jiménez Reference Borrell, Sillero and Jiménez2013; Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2013; Lee & Moser Reference Lee and Moser2015, Reference Lee and Moser2019; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022). Specifically, the horizontal axis $x_2^*$ is obtained via $x_2^*=x_2^+/\eta ^+=x_2^+\bar \varepsilon ^{+1/4}$, and the vertical axis ${\overline {{u_i}{u_i}} ^*}$ is obtained via ${\overline {{u_i}{u_i}} ^*} = {\overline {{u_i}{u_i}} ^ + }/u_K^{ + 2} = {\overline {{u_i}{u_i}} ^ + }/{{\bar \varepsilon }^{ + 1/2}}$. For clarity, we show in a separate figure (figure 4) the distributions of Kaneda & Yamamoto (Reference Kaneda and Yamamoto2021) in the channel flow over a relatively large range of $Re_\tau$ ($=500\unicode{x2013}8000$); the distribution at $Re_\tau =2000$ for the channel flow in figure 3 is added in figure 4 for reference. It can be seen from figures 3 and 4 that ${\overline {{u_i}{u_i}} ^*}$ collapses nearly perfectly for $x_2^*\lesssim 1$ in all wall flows over a wide range of $Re_\tau$ ($=445\unicode{x2013}10^4$). The larger the Reynolds number, the wider the $x_2^*$ range over which ${\overline {{u_i}{u_i}} ^*}$ is approximately independent of the type of flow as well as of $Re_\tau$; this can be seen more clearly in the linear–log scales (see figures 3(b) and 4(b)). In particular, the collapse of the ${\overline {{u_i}{u_i}} ^*}$ distribution at $Re_\tau =5200$ with the distribution at $Re_\tau =10^4$ can extend to $x_2^*\approx 4$. This collapse can also be seen from figure 5, which shows the variation of ${\overline {{u_i}{u_i}} ^*}$ at two typical locations: $x_2^*=4$ and $9$, on $Re_\tau$ in all wall flows. There is a clear $Re_\tau$ dependence of ${\overline {{u_i}{u_i}} ^*}$ at $x_2^*=9$ when $Re_\tau$ is small to moderate (<5200); this dependence becomes very small, if not negligible, when $Re_\tau >5200$. In contrast, ${\overline {{u_i}{u_i}} ^*}$ at $x_2^*=4$ increases slightly as $Re_\tau$ increases, eventually becoming constant for $Re_\tau \gtrsim 2000$. This confirms that the approximate independence of ${\overline {{u_i}{u_i}} ^*}$ on the flow and $Re_\tau$ in figures 3 and 4 extends to $x_2^*\approx 4$ when $Re_\tau \gtrsim 2000$. It is worth mentioning that the variation of $x_2^*$ with $x_2^+$ in a channel for $Re_\tau = 550\unicode{x2013}10^4$ has been presented and discussed by Tang & Antonia (Reference Tang and Antonia2022) (see their figure 2). At $Re_\tau =10^4$, $x_2^*=4$ and 9 correspond to $x_2^+=6.3$ and 14.8, respectively. For reference, the corresponding $x_2^+$ values, with an increment of 0.5 decade, at $Re_\tau =10^4$ are added onto the upper abscissa of figure 3.

Figure 3. (a) Kolmogorov-normalized turbulent kinetic energies ${\overline {{u_i}{u_i}} ^*}$ in a channel (Lee & Moser Reference Lee and Moser2015, Reference Lee and Moser2019; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022), boundary layer (Simens et al. Reference Simens, Jiménez, Hoyas and Mizuno2009; Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Borrell et al. Reference Borrell, Sillero and Jiménez2013; Sillero et al. Reference Sillero, Jiménez and Moser2013) and pipe (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). They are plotted using wall-parameter normalization in those three flows, using the relations $x_2^*=x_2^+/\eta ^+=x_2^+\bar \varepsilon ^{+1/4}$ and ${\overline {{u_i}{u_i}} ^*} = {\overline {{u_i}{u_i}} ^ + }/u_K^{ + 2} = {\overline {{u_i}{u_i}} ^ + }/{{\bar \varepsilon }^{ + 1/2}}$. The green dotted line indicates $\sim x_2^{*2/3}$. Note that the data close to and beyond the edge of the boundary layer ($x_2/\delta >$0.8) are not shown since they are affected by the intermittency associated with the turbulent/potential flow interface. (b) Corresponding distributions on linear–log scales. For reference, the corresponding $x_2^+$ values, with increment 0.5 decade, at $Re_\tau =10^4$ are added on the upper abscissa. Note that the variation of $x_2^+$ with $x_2^*$ is nonlinear; see figure 2 of Tang & Antonia (Reference Tang and Antonia2022).

Figure 4. (a) Kolmogorov-normalized ${\overline {{u_i}{u_i}} ^*}$ in a channel. They are plotted using the wall-parameter-normalized ${\overline {{u_i}{u_i}} ^+}$ of Kaneda & Yamamoto (Reference Kaneda and Yamamoto2021). For reference, also shown in this figure (cyan curve) is the distribution of Lee & Moser (Reference Lee and Moser2015, Reference Lee and Moser2019) at $Re_\tau =2000$ in figure 3. (b) Corresponding distributions on linear–log scales.

Figure 5. Dependence of ${\overline {{u_i}{u_i}} ^*}$ on $Re_\tau$ at $x_2^*=9$ and $4$, corresponding to the data in figures 3 and 4. Pipe: $\triangledown$, green (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). Channel: $\circ$, blue (Lee & Moser Reference Lee and Moser2015, Reference Lee and Moser2019; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022); $\triangledown$, magenta (Kaneda & Yamamoto Reference Kaneda and Yamamoto2021). Boundary layer: $\circ$, red (Simens et al. Reference Simens, Jiménez, Hoyas and Mizuno2009; Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Borrell et al. Reference Borrell, Sillero and Jiménez2013; Sillero et al. Reference Sillero, Jiménez and Moser2013).

Figure 6 shows the local slope of ${\overline {{u_i}{u_i}} ^*}$, i.e. $LS_k(x_2^*)= {\rm d}\log {{\overline {{u_i}{u_i}} ^*}}(x_2^*)/{\rm d}\log x_2^*$ in the channel flow. For clarity, only the channel data in figure 3 are shown in this figure. The collapse of $LS_k(x_2^*)$ at small $x_2^*$ is not surprising because of the collapse of ${\overline {{u_i}{u_i}} ^*}$ at small $x_2^*$. In particular, $LS_k(x_2^*) \to 2$ as $x_2^*\to 0$; see figure 6 or (2.10a,b). At the moderate $x_2^*$, say $x_2^*\sim 30$, although ${\overline {{u_i}{u_i}} ^*}$ does not collapse (figures 3 and 4), its slope increases with $Re_\tau$, gradually approaching the $2/3$ power-law scaling. We can see this behaviour more clearly in figure 6, where the local slope $LS_k(x_2^*)$ increases systematically with $Re_\tau$, and the $2/3$ power-law scaling is approached by the distribution at $Re_\tau =10^4$ over the range $30 \lesssim x_2^*\lesssim 70$. In this $x_2^*$ range at $Re_\tau =10^4$, $\overline {u_iu_i}^*\approx C_{q}x_2^{*2/3}=5.807x_2^{*2/3}$, although it is not clear whether the prefactor ($C_{q}$) has become $Re_\tau$-independent; higher $Re_\tau$ data are needed. In contrast, the variation of the wall-parameter-normalized $\overline {u_iu_i}^+$ with $Re_\tau$ is systematic at all $x_2^+$ in the channel, boundary layer and pipe (figure 7).

Figure 6. Local slope of ${\overline {{u_i}{u_i}} ^*}$, i.e. $LS_k(x_2^*)= {\rm d}\log {{\overline {{u_i}{u_i}} ^*}}(x_2^*) /{\rm d}\log x_2^*$, corresponding to the channel data in figure 3. The green dotted line corresponds to the value $2/3$.

Figure 7. Wall-parameter-normalized $\overline {u_iu_i}^+$ in a channel, boundary layer and pipe for the same data as in figure 3. The inset zooms in on the range $x_2^+ = 1\unicode{x2013}4$.

We next focus on the behaviour of the Reynolds stresses ${\overline {{u_i}{u_j}} ^*}$. Note that in the region $x_2\to 0$ (e.g. Pope Reference Pope2000),

(3.1)\begin{equation} {{\bar \varepsilon }_{ij}} = \nu\,\frac{{{\partial ^2}\overline{{u_i}{u_j}} }}{{\partial x_2^2}}, \end{equation}

where ${{\bar \varepsilon }_{ij}}$ is the dissipation rate tensor, which is given by

(3.2)\begin{equation} {\bar \varepsilon _{ij}} = 2\nu \overline{\left( {\frac{{\partial {u_i}}}{{\partial {x_k}}}\, \frac{{\partial {u_j}}}{{\partial {x_k}}}} \right)}. \end{equation}

After normalizing by the Kolmogorov scales, (3.1) can be rewritten as

(3.3)\begin{equation} {{\bar \varepsilon }^*_{ij}} = \frac{{{\partial ^2}\overline{{u_i}{u_j}}^* }}{{\partial x_2^{*2}}}. \end{equation}

Since $u_2\sim x_2^2$ as $x_2\to 0$ (e.g. Pope Reference Pope2000), ${{\bar \varepsilon }^*_{22}}$ and ${{\bar \varepsilon }^*_{12}}$, and the corresponding quantities on the right-hand side of (3.3), should be zero as $x_2\to 0$. On the other hand, Tang & Antonia (Reference Tang and Antonia2022) showed that ${{\bar \varepsilon }^*_{11}}$ and ${{\bar \varepsilon }^*_{33}}$ become $Re_\tau$-independent when $Re_\tau >1000$ in the channel and boundary layer. One thus expects that, based on (3.3), $\overline {{u_i}{u_j}}^*$ should be $Re_\tau$-independent as $x_2^*\to 0$, at least for $i=j=1$ and $i=j=3$. Since $Re_\tau >1000$ is required for ${{\bar \varepsilon }^*_{11}}$ and ${{\bar \varepsilon }^*_{33}}$ to become $Re_\tau$-independent in the channel and boundary layer (Tang & Antonia Reference Tang and Antonia2022), we show in figure 8 the Kolmogorov-normalized distributions ${\overline {{u_1}{u_1}} ^*}$, ${\overline {{u_2}{u_2}} ^*}$, ${\overline {{u_3}{u_3}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ in these two flows for only $Re_\tau >1000$ for the data of Simens et al. (Reference Simens, Jiménez, Hoyas and Mizuno2009), Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010), Borrell et al. (Reference Borrell, Sillero and Jiménez2013), Sillero et al. (Reference Sillero, Jiménez and Moser2013), Lee & Moser (Reference Lee and Moser2015, Reference Lee and Moser2019) and Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022). For clarity, we show in a separate figure (figure 9) the channel data of Kaneda & Yamamoto (Reference Kaneda and Yamamoto2021) at $Re_\tau =2000$, 4000 and 8000; the distribution for the same flow at $Re_\tau =2000$ in figure 8 has been added to figure 9 for reference. Although the distributions for ${{\bar \varepsilon }^*_{11}}$ and ${{\bar \varepsilon }^*_{33}}$ in the pipe for $Re_\tau \gtrsim 2000$ are not available, we show in figure 9 the distributions of $\overline {{u_i}{u_j}}^*$ in the pipe for a similar Reynolds number range as in the channel, namely $Re_\tau \gtrsim 2000$. There is a nearly perfect collapse for all distributions in figure 8 at small $x_2^*$. Specifically, the collapse of ${\overline {{u_1}{u_1}} ^*}$, ${\overline {{u_2}{u_2}} ^*}$, ${\overline {{u_3}{u_3}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ extends to $x_2^*\approx 4$, 40, 4 and 40, respectively. The same observations can also be made from figure 9 for the channel and pipe. Namely, the $x_2^*$ range over which $\overline {{u_i}{u_j}}^*$ distributions collapse is approximately one order of magnitude larger for ${\overline {{u_2}{u_2}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ than for ${\overline {{u_1}{u_1}} ^*}$ and ${\overline {{u_3}{u_3}} ^*}$. This can also be observed in figure 10, which shows the magnitudes of ${\overline {{u_1}{u_1}} ^*}$ and ${\overline {{u_3}{u_3}} ^*}$ at $x_2^*=4$ and 9 versus $Re_\tau$, and those of ${\overline {{u_2}{u_2}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ at $x_2^*=40$ and 90 versus $Re_\tau$ in the three wall flows. Evidently, there is a clear $Re_\tau$ dependence for ${\overline {{u_1}{u_1}} ^*}$ and ${\overline {{u_3}{u_3}} ^*}$ at $x_2^*=9$, and for ${\overline {{u_2}{u_2}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ at $x_2^*=90$, when $Re_\tau$ is small to moderate (<5200); this dependence becomes very small when $Re_\tau >5200$. However, the distributions at $x_2^*=4$ for ${\overline {{u_1}{u_1}} ^*}$ and ${\overline {{u_3}{u_3}} ^*}$, and those at $x_2^*=40$ for ${\overline {{u_2}{u_2}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$, are approximately constant for $Re_\tau \gtrsim 2000$. The above observations indicate that the $x_2^*$ range over which ${\overline {{u_i}{u_j}} ^*}$ collapses depends on the specific quantity. It is worth mentioning that a close look at the distributions of ${\overline {{u_2}{u_2}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ in figure 9 shows that there is a very small difference between the channel and the pipe in the range $x_2^*\lesssim 5$. Note that the magnitudes of ${\overline {{u_2}{u_2}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ are approximately 1–2 orders of magnitude smaller than those of ${\overline {{u_1}{u_1}} ^*}$ and ${\overline {{u_3}{u_3}} ^*}$. This small difference is likely due to the different resolutions of the simulations between these two flows.

Figure 8. (a) Kolmogorov-normalized ${\overline {{u_1}{u_1}} ^*}$, ${\overline {{u_2}{u_2}} ^*}$, ${\overline {{u_3}{u_3}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ in the channel and boundary layer for $Re_\tau >1000$. They are plotted using the wall-parameter-normalized ${\overline {{u_i}{u_j}} ^+}$ in these two flows (Simens et al. Reference Simens, Jiménez, Hoyas and Mizuno2009; Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Borrell et al. Reference Borrell, Sillero and Jiménez2013; Sillero et al. Reference Sillero, Jiménez and Moser2013; Lee & Moser Reference Lee and Moser2015, Reference Lee and Moser2019; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022). (b) Corresponding distributions on linear–log scales.

Figure 9. (a) Kolmogorov-normalized ${\overline {{u_1}{u_1}} ^*}$, ${\overline {{u_2}{u_2}} ^*}$, ${\overline {{u_3}{u_3}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ in the channel and pipe for $Re_\tau \gtrsim 2000$. They are plotted using the wall-parameter-normalized ${\overline {{u_i}{u_j}} ^+}$ in these two flows (Kaneda & Yamamoto Reference Kaneda and Yamamoto2021; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). For reference, also shown (cyan curves) are the distributions of Lee & Moser (Reference Lee and Moser2015, Reference Lee and Moser2019) at $Re_\tau = 2000$ in figure 8. (b) Corresponding distributions on linear–log scales.

Figure 10. (a,b) Dependence of ${\overline {{u_1}{u_1}} ^*}$ and ${\overline {{u_3}{u_3}} ^*}$ on $Re_\tau$ at $x_2^*=9$ and $4$. (c,d) Dependence of ${\overline {{u_2}{u_2}} ^*}$ and $-{\overline {{u_1}{u_2}} ^*}$ on $Re_\tau$ at $x_2^*=90$ and $40$. Pipe: $\square$, green (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). Channel: $\circ$, blue (Lee & Moser Reference Lee and Moser2015, Reference Lee and Moser2019; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022); $\triangledown$, magenta (Kaneda & Yamamoto Reference Kaneda and Yamamoto2021). Boundary layer: $\circ$, red (Simens et al. Reference Simens, Jiménez, Hoyas and Mizuno2009; Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Borrell et al. Reference Borrell, Sillero and Jiménez2013; Sillero et al. Reference Sillero, Jiménez and Moser2013).

Another important feature of the distributions in figures 8–10 is the increased extent of the collapse with increasing $Re_\tau$. This can also be observed in figure 11, which shows the corresponding local slope of ${\overline {{u_i}{u_j}} ^*}$, i.e. $LS_{u_iu_j}(x_2^*)= {\rm d}\log {{\overline {{u_i}{u_j}} ^*}}(x_2^*) /{\rm d}\log x_2^*$ in the channel flow at $Re_\tau =$5200 and $10^4$, respectively. The local slope for ${\overline {{u_1}{u_1}} ^*}$ at $Re_\tau =10^4$ approaches $2/3$ over the range $35 \lesssim x_2^*\lesssim 70$. This behaviour is similar to that for ${\overline {{u_i}{u_i}} ^*}$ (see figure 6) since ${\overline {{u_1}{u_1}} ^*}$ is the largest contribution to ${\overline {{u_i}{u_i}} ^*}$ (figure 8). In contrast, ${\overline {{u_2}{u_2}} ^*}$ and ${\overline {{u_3}{u_3}} ^*}$ make relatively small contributions to ${\overline {{u_i}{u_i}} ^*}$ at small to moderate $x_2^*$, thus their behaviours are unlikely to affect ${\overline {{u_i}{u_i}} ^*}$ significantly. Accordingly, the distributions of $LS_{u_3u_3}$ do not approach $2/3$, while those of $LS_{u_2u_2}$ do not deviate significantly from $2/3$ over the range $70 \lesssim x_2^*\lesssim 170$ at $Re_\tau =10^4$. Note that the local slopes of ${\overline {{u_1}{u_1}} ^*}$, ${\overline {{u_2}{u_2}} ^*}$, ${\overline {{u_3}{u_3}} ^*}$ and ${\overline {{u_1}{u_2}} ^*}$ approach 2, 4, 2 and 3, respectively, as $x_2^*\to 0$. This is not surprising since $u_1 \sim u_3\sim x_2$ and $u_2\sim x_2^2$ as $x_2\to 0$ (e.g. Pope Reference Pope2000). Overall, figures 3–11 provide good support for the present hypothesis in the context of the Reynolds stresses and the turbulent kinetic energy. Evidently, it would be desirable to investigate further the finite Reynolds number effect on ${\overline {{u_i}{u_i}} ^*}$ and ${\overline {{u_i}{u_j}} ^*}$ at moderate $x_2^*$ by further increasing $Re_\tau$.

Figure 11. Local slope of ${\overline {{u_i}{u_j}} ^*}$, i.e. $LS_{u_iu_j}(x_2^*)= {\rm d}\log {{\overline {{u_i}{u_j}} ^*}}(x_2^*) /{\rm d}\log x_2^*$; for clarity, only the channel data at $Re_\tau =5200$ (dashed curves) and $10^4$ (solid curves) are shown. The green dotted line corresponds to the value $2/3$.

We conclude this section by examining the dependence on $Re_\tau$ and $Pr$ of ${\overline {{\theta }{\theta }} ^*}$. Figure 12 shows the distributions of ${\overline {{\theta }{\theta }} ^*}/Pr$ in the channel at $Re_\tau =500$, 1000, 2000 and 5000, and $Pr=0.01$, 0.1, 0.71, 2 and 7, respectively. For clarity, the corresponding local slopes of ${\overline {{\theta }{\theta }} ^*}$, i.e. $LS_{\theta }(x_2^*)= {\rm d}\log {{\overline {{\theta }{\theta }} ^*}}(x_2^*) /{\rm d}\log x_2^*$, for only $Pr=0.71$ and 7, and for $Re_\tau \geq 1000$, are shown in figure 13. We can observe from figure 12 that the relation ${\overline {{\theta }{\theta }} ^*}/Pr=x_2^{*2}$ (see (2.10b)) is satisfied adequately at all $Re_\tau$ and $Pr$ (note that the Péclet number is equal to $Re_\tau \,Pr$) over a significant range of $x_2^*$. An overall feature of this figure is that at a given $Pr$, the collapse extends to increasingly larger values of $x_2^*$ as $Re_\tau$ increases. In particular, at $Pr=7$, the collapse appears to extend to $x_2^*\approx 10$. More importantly, there is an approximate plateau, of value approximately 37.5, over the range $3\lesssim x_2^*\lesssim 9$, indicating an emergence of a viscous-convective range, which is weakly affected by ${\gamma }$. This can also be seen in figure 13, which shows that the corresponding local slope of ${\overline {{\theta }{\theta }} ^*}$ is close to 0 in the same $x_2^*$ range at $Pr=7$ and $Re_\tau \geq 1000$. Over the range $30\lesssim x_2^*\lesssim 100$, there is a clear $Re_\tau$ dependence for almost all distributions of ${\overline {{\theta }{\theta }} ^*}/Pr$ at any $Pr$ (see figure 12). The magnitude of $LS_{\theta }$ at $Pr=0.71$ also increases systematically with $Re_\tau$ in this range (figure 13). Even at the highest $Re_\tau$ ($=$5000), the $2/3$ power-law scaling predicted by the present hypothesis is not approached. Similarly, at a larger $Pr$ ($=$7) but a smaller $Re_\tau$ ($=$2000), the distribution of $LS_{\theta }$ does not yet exhibit a plateau corresponding to $2/3$. Evidently, larger values of $Re_\tau$ and $Pr$ are required to establish the $2/3$ power-law scaling for ${\overline {{\theta }{\theta }} ^*}$ unequivocally. It is worth mentioning that in figure 12, we have used the $\bar \varepsilon ^ +$ distributions of Lee & Moser (Reference Lee and Moser2015, Reference Lee and Moser2019) at nearly the same $Re_\tau$ (550, 1000, 2000 and 5200) for normalization since the values of $\bar \varepsilon ^ +$ for Alcántara-Ávila, Hoyas & Pérez-Quiles (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2018), Lluesma-Rodríguez, Hoyas & Perez-Quiles (Reference Lluesma-Rodríguez, Hoyas and Perez-Quiles2018), Alcántara-Ávila et al. (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021) and Alcántara-Ávila & Hoyas (Reference Alcántara-Ávila and Hoyas2021) are not available. We should stress that the use of a different $\bar \varepsilon ^ +$ distribution will not affect the results in figure 12; this has been discussed previously (Tang & Antonia (Reference Tang and Antonia2022), see their appendix A) in the context of the transport equation for $\bar \varepsilon _\theta$. Following Tang & Antonia (Reference Tang and Antonia2022), when we normalize the data in figure 12 by using the $\bar \varepsilon ^ +$ data of Bernardini, Pirozzoli & Orlandi (Reference Bernardini, Pirozzoli and Orlandi2014) at $Re_\tau$= 550, 1000 and 2000, the resulting distributions (not shown here) cannot be distinguished from those in figure 12.

Figure 12. Batchelor–Kolmogorov normalized ${\overline {{\theta }{\theta }} ^*}/Pr$ in a channel. They are plotted using the wall-parameter-normalized ${\overline {{\theta }{\theta }} ^+}$ (Alcántara-Ávila et al. Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2018, Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021; Lluesma-Rodríguez et al. Reference Lluesma-Rodríguez, Hoyas and Perez-Quiles2018; Alcántara-Ávila & Hoyas Reference Alcántara-Ávila and Hoyas2021). Cyan curve shows $x_2^{*2}$. The green dashed line corresponds to the value 37.5. Note that the $\bar \varepsilon ^ +$ distributions of Lee & Moser (Reference Lee and Moser2015, Reference Lee and Moser2019) at nearly the same $Re_\tau$ (550, 1000, 2000 and 5200) are used for normalization since the values of $\bar \varepsilon ^ +$ for Alcántara-Ávila et al. (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2018, Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021), Lluesma-Rodríguez et al. (Reference Lluesma-Rodríguez, Hoyas and Perez-Quiles2018) and Alcántara-Ávila & Hoyas (Reference Alcántara-Ávila and Hoyas2021) are not available.

Figure 13. Local slope of ${\overline {{\theta }{\theta }} ^*}$, i.e. $LS_{\theta }(x_2^*)= {\rm d}\log {{\overline {{\theta }{\theta }} ^*}}(x_2^*) /{\rm d}\log x_2^*$, corresponding to the data in figure 12; for clarity, only the data at $Pr=0.71$, 7 and for $Re_\tau \geq 1000$ are shown. The green dotted lines correspond to the values $2/3$ and 0.

4. Relationship to the wall scaling

We recall that the scaling of the mean velocity and mean scalar based on wall parameters near the wall has received strong support in the literature. This is, however, not the case for $\overline {u_iu_j}^+$, $\overline {u_iu_i}^+$ and $\overline {\theta \theta }^+$, at least at finite $Re_\lambda$. The evidence presented above indicates that near the wall, the scaling of $\overline {u_iu_j}$, $\overline {u_iu_i}$ and $\overline {\theta \theta }$ based on $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ is superior to that based on wall parameters. The relationship between the wall parameters and $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ can be quantified by the wall-parameter-normalized $\bar \varepsilon$ and $\bar \varepsilon _\theta$, i.e. $\bar \varepsilon ^+$ and $\bar \varepsilon _\theta ^+$. Figure 14(a) shows the distributions of $\bar \varepsilon ^ +$ and $x_2^+\bar \varepsilon ^ +$ in a pipe for $Re_\tau =495\unicode{x2013}6019$; the latter allows the collapse of $\bar \varepsilon ^ +$ distributions (if it exists) at large $x_2^+$ to be examined more closely. An overall feature of figure 14(a) is that the magnitudes of $\bar \varepsilon ^ +$ and $x_2^+\bar \varepsilon ^ +$ become increasingly close to each other over a large range of $x_2^+$ with increasing $Re_\tau$. The same feature can also be observed in the channel for $Re_\tau =550\unicode{x2013}10^4$ (see figure 1 of Tang & Antonia Reference Tang and Antonia2022). In contrast, there is a nearly perfect collapse for the distributions of $\bar \varepsilon ^ +$ and $x_2^+\bar \varepsilon ^ +$ over the range $10\lesssim x_2^+\lesssim 30$ for $Re_\tau =445\unicode{x2013}1988$ in the boundary layer (figure 14b). We report in figure 15 the distributions of $\bar \varepsilon _\theta ^ +$ and $x_2^+ \bar \varepsilon _\theta ^ +$ in the channel for $Re_\tau =500\unicode{x2013}5000$. We can observe that the $\bar \varepsilon _\theta ^ +$ and $x_2^+\bar \varepsilon _\theta ^ +$ distributions collapse over the range $20\lesssim x^+_2\lesssim 60$ for $Re_\tau \geq 500$. The collapse appears to extend to larger and smaller values of $x_2^+$ at higher $Re_\tau$. It is expected that the collapse of $\bar \varepsilon ^ +$ and $x_2^+\bar \varepsilon ^ +$ in the boundary layer should also extend to larger and smaller values of $x_2^+$ as $Re_\tau$ continues to increase (figure 14b). On the other hand, the $Re_\tau$ independence of $\overline {u_iu_i}^*$, ${\overline {{u_1}{u_1}} ^*}$, ${\overline {{u_2}{u_2}} ^*}$, ${\overline {{u_3}{u_3}} ^*}$, ${\overline {{u_1}{u_2}} ^*}$ and $\overline {\theta \theta } ^*$ first emerges at the smallest $x_2^*$ and subsequently at increasingly larger $x_2^*$ as $Re_\tau$ increases. Consequently, as $Re_\tau$ continues to increase, there should be an $x_2$ range over which $\overline {u_iu_j}^*$ and $\overline {\theta \theta }^*$ versus $x_2^*$, and $x_2^+\bar \varepsilon ^ +$ and $x_2^+\bar \varepsilon _\theta ^ +$ versus $x_2^+$, collapse. Over this range and irrespective of the flow, one should then be able to use the two types of normalization interchangeably.

Figure 14. Distributions of $\bar \varepsilon ^ +$ and $x_2^+\bar \varepsilon ^ +$ in (a) the pipe, and (b) the boundary layer. They are plotted using the data of Simens et al. (Reference Simens, Jiménez, Hoyas and Mizuno2009), Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010), Borrell et al. (Reference Borrell, Sillero and Jiménez2013), Sillero et al. (Reference Sillero, Jiménez and Moser2013) and Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021).

Figure 15. Distributions of $\bar \varepsilon _\theta ^ +$ and $x_2^+\bar \varepsilon _\theta ^ +$ in a channel. They are plotted using the data of Alcántara-Ávila et al. (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021).

Further, although the distributions of $x_2^+\bar \varepsilon ^ +$ in the boundary layer have become $Re_\tau$-independent in the range $10\lesssim x_2^+\lesssim 30$ for $Re_\tau \gtrsim 445$ (figure 14b), it is unclear if those of $x_2^+\bar \varepsilon ^ +$ in the pipe flow at $Re_\tau =6019$ and in the channel flow at $Re_\tau =10^4$ have become $Re_\tau$-independent in the same $x_2^+$ range. Nonetheless, it is of interest to compare the $\bar \varepsilon ^ +$ and $x_2^+\bar \varepsilon ^ +$ distributions at the highest $Re_\tau$ in the three flows (figure 16). It is remarkable that there is an approximate collapse in the range $12\lesssim x_2^+\lesssim 40$. In particular, if the collapse of $\bar \varepsilon ^ +$ and $\bar \varepsilon _\theta ^ +$ can extend to the wall, i.e. at all locations in the range $x_2\lesssim x_{2\alpha }$, at least in a given wall flow, then $\bar \varepsilon ^ +$ and $\bar \varepsilon _\theta ^ +$ may become bounded at the wall (when $Re_\tau \to \infty$); here, $x_{2\alpha }$ is the upper limit of the range for which there is collapse with respect to $\bar \varepsilon ^ +$ and $\bar \varepsilon _\theta ^ +$. In this situation, the Batchelor–Kolmogorov normalized mean velocity and mean scalar should be $Re_\tau$-independent in the $x_2$ range $x_2<\min (x_{2\alpha },x_{2\alpha _1})$; here, $x_{2\alpha _1}$ is the upper limit of the range in which the wall-parameter-normalized mean velocity and mean scalar collapse. Also, $\overline {u_iu_i}^+$ and $\overline {\theta \theta }^+$ versus $x_2^+$ should be $Re_\tau$-independent in the $x_2$ range $x_2<\min (x_{2\alpha },x_{2\alpha _2})$; here, $x_{2\alpha _2}$ is the upper limit of the range for which $\overline {u_iu_i}^*$ and $\overline {\theta \theta }^*$ versus $x_2^*$ are $Re_\tau$-independent. This is fully consistent with Chen & Sreenivasan (Reference Chen and Sreenivasan2021), who suggest that the peak value of $\overline {u_1u_1}^+$ approaches a finite limit as the Reynolds number becomes infinitely large because the dissipation rate at the wall is expected to be bounded.

Figure 16. Distributions of $\bar \varepsilon ^ +$ and $x_2^+\bar \varepsilon ^ +$ in a channel ($Re_\tau =10^4$; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022), pipe ($Re_\tau =6019$; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) and boundary layer ($Re_\tau =1988$; Sillero et al. Reference Sillero, Jiménez and Moser2013).

Since $\bar \varepsilon ^ +$ in a channel, boundary layer and pipe (figures 14 and 1 of Tang & Antonia Reference Tang and Antonia2022) and $\bar \varepsilon _\theta ^ +$ in a channel (figure 15) do not collapse as $x_2^+\to 0$ in the present $Re_\tau$ range (${\leq }10^4$), it is expected that the Batchelor–Kolmogorov normalized mean velocity ($\bar U_1^*$) and mean scalar ($\varTheta ^*$) should depend on $Re_\tau$ as $x_2^+\to 0$. As an example, we show in figures 17(a,b) the distributions of $\bar U_1^+$ and $\varTheta ^+$ in a channel at $Pr=0.71$. It is not surprising that the $x_2^+$ range over which both $\bar U_1^+$ and $\varTheta ^+$ collapse extends to increasingly larger values of $x_2^+$ as $Re_\tau$ increases. On the other hand, the distributions of $\theta _B ^+$ collapse at all $Re_\tau$ over the range $20\lesssim x_2^+\lesssim 60$, while the $u_K^+$ distributions appear to collapse approximately at larger $Re_\tau$ ($\gtrsim$5200) over the range $20\lesssim x_2^+\lesssim 200$. More importantly, the distributions in figure 17(a,b) implies that there should be an $x_2^+$ ($\gtrsim$20, or equivalently, $x_2^*\gtrsim 11$) range over which $\bar U_1^*$ and $\varTheta ^*$ distributions should be $Re_\tau$-independent. In order to confirm this, we report in figures 17(c,d) the distributions of $\bar U_1^*=\bar U_1^+/u_K^+$ and $\varTheta ^*=\varTheta ^+/\theta _B ^+$, respectively. Also shown are the same distributions premultiplied by $x^*_2$, i.e. $\bar U_1^*/x_2^*$ and $\varTheta ^*/x_2 ^*$, which allow the collapse of $\bar U_1^*$ and $\varTheta ^*$ at large $x^*_2$ to be examined more closely. As expected, $\bar U_1^*$ for $Re_\tau \gtrsim 5200$, and $\varTheta ^*$ at all $Re_\tau$ appears to collapse for $x_2^*\gtrsim 11$; the larger $Re_\tau$ is, the wider is the $x_2^*$ range over which $\bar U_1^*$ and $\varTheta ^*$ are $Re_\tau$-independent. Only when $\bar \varepsilon ^ +$ and $\bar \varepsilon _\theta ^ +$ (or equivalently, $u_K^+$ and $\theta _B ^+$) become $Re_\tau$-independent at the wall (when $Re_\tau \to \infty$) can one expect $\bar U_1^*$ and $\varTheta ^*$ to collapse at the wall. It is worth mentioning that if $u_K^+$ and $\theta _B ^+$ become $Re_\tau$-independent over an $x_2^+$ range, then $\overline {u_iu_j}^*$ and $\overline {\theta \theta } ^ *$ should be proportional to $\overline {u_iu_j}^+$ and $\overline {\theta \theta } ^ +$ at a given $x_2^+$.

Figure 17. (a,b) Distributions of $\bar U_1^+$ and $\varTheta ^+$ in a channel at $Pr=0.71$. Also shown are the corresponding distributions of $u_K^+$ and $\theta _B ^+$. (c,d) Distributions of $\bar U_1^*=\bar U_1^+/u_K^+$, $\bar U_1^*/x_2^*$, $\varTheta ^*=\varTheta ^+/\theta _B ^+$ and $\varTheta ^*/x_2 ^*$. Data of Lee & Moser (Reference Lee and Moser2015, Reference Lee and Moser2019), Alcántara-Ávila et al. (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021) and Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022).

It is well-known that as $Re_\tau \to \infty$, there should be an $x_2^+$ range (far away from the wall) over which $-\overline {u_1u_2}^+ =1$, which should be located in the log-law region. If $\bar \varepsilon ^ +$ collapses in the same $x_2^+$ range and there is an overlap region over which both $-\overline {u_1u_2}^+ =1$ and $\overline {u_1u_2}^*= C_{12} x_2^{*2/3}$ (inertial-convective range) are established unequivocally (again, this may require $Re_\tau \to \infty$), then we can write

(4.1)\begin{equation} {\overline{{u_1}{u_2}} ^*} = {C_{12}}\,x_2^{*2/3} = \frac{{{{\overline{{u_1}{u_2}} }^ + }}}{{u_K^{ + 2}}} = {C_{12}}{(x_2^ + u_K^ + )^{2/3}} \quad \Rightarrow \quad u_K^ += \frac{-1}{{C_{12}^{3/8}\,x_2^{ + 1/4}}}. \end{equation}

Note that the relation $\bar \varepsilon ^ +=u_K^{+4}=1/\eta ^{+4}$ has been used to derive (4.1), which is consistent with the relation $u_K^ + = {(\kappa x_2^ + )^{ - 1/4}}$ of Perry, Henbest & Chong (Reference Perry, Henbest and Chong1986) obtained from balancing production and turbulent energy dissipation in the log-law region. It is worth mentioning that one can obtain $\bar \varepsilon _\theta ^ +\sim 1/x_2^+$ by assuming that the production and dissipation rate of the scalar variance are in balance in the log-law region. If in the overlap region mentioned above (which should be a subrange of both the log-law region and the inertial-convective range) there is $2/3$ power-law scaling for $\overline {u_1u_1}^*$, $\overline {u_2u_2}^*$, $\overline {u_3u_3}^*$ and $\overline {\theta \theta } ^ *$, then it is not difficult to conclude that $\overline {u_1u_1}^+$, $\overline {u_2u_2}^+$, $\overline {u_3u_3}^+$, $\overline {u_iu_i}^+$ and $\overline {\theta \theta } ^ +$ should be constant. As examples, $\overline {u_iu_i}^+$ and $\overline {\theta \theta } ^ +$, after normalizing the first equations in (A5) and (A6), can be written as

(4.2a,b)\begin{equation} {\overline{{u_i}{u_i}} ^ + } = {C_q}\,{{\bar \varepsilon }^{ + 2/3}}\,x_2^{ + 2/3},\quad {\overline{\theta \theta } ^ + } = {C_\theta }\,{{\bar \varepsilon }^{ +- 1/3}}\, \bar \varepsilon _\theta ^ +\,x_2^{ + 2/3}. \end{equation}

Since $\bar \varepsilon ^+ \sim \bar \varepsilon _\theta ^ +\sim 1/x_2^+$ in the overlap region, we can obtain

(4.3)\begin{equation} {\overline{{u_i}{u_i}} ^ + } \sim {\overline{\theta \theta } ^ + } \sim const. \end{equation}

Finally, in the viscous-convective range at large $Pr$, if both $\bar \varepsilon ^ +$ and $\bar \varepsilon _\theta ^ +$ collapse in the same $x_2^+$ range, then $\overline {\theta \theta } ^ *=const$ implies that $\overline {\theta \theta } ^ +=const\,\theta _B^{+2}$.

All of the analysis above is based on the direct numerical simulations (DNS) data. As pointed out by Smits (Reference Smits2022), over the last two decades or so, there have been major advances in instrumentation, such as the nanoscale thermal anemometry probes developed at Princeton (Kunkel, Arnold & Smits Reference Kunkel, Arnold and Smits2006; Bailey et al. Reference Bailey, Kunkel, Hultmark, Vallikivi, Hill, Meyer, Tsay, Arnold and Smits2010; Vallikivi et al. Reference Vallikivi, Hultmark, Bailey and Smits2011; Vallikivi & Smits Reference Vallikivi and Smits2014). This has allowed higher Reynolds number experimental data to be obtained, up to $Re_\tau \sim 10^5$, and thus has greatly advanced our understanding of the scaling and structure of wall turbulence. It is hence desirable to examine experimental data in wall flows at higher Reynolds numbers. Figures 18(a,b) show the distributions of $\overline {u_1u_1}^+$ in a pipe and a boundary layer (Vallikivi et al. Reference Vallikivi, Hultmark, Bailey and Smits2011, Reference Vallikivi, Hultmark and Smits2015b; Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013; Vallikivi, Ganapathisubramani & Smits Reference Vallikivi, Ganapathisubramani and Smits2015a) – downloaded from https://smits.princeton.edu/data-sets. The corresponding values of the viscous-scaled sensor length $l^+=lu_\tau /\nu$, where $l$ is the sensor length, are shown in table 1. We can observe that the maximum $l^+$ is 45.5 in the pipe, and 75 in the boundary layer; $\overline {u_1u_1}^+$ will therefore be underestimated in the near-wall region (Smits et al. Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011a). For this reason, a spatial filtering correction, proposed by Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011a), has been applied to those data. For reference, we add in figures 18(a,b) the data of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) in the boundary layer at $Re_\tau =6000\unicode{x2013}20\,000$ with $l^+ = 2.4\unicode{x2013}3.5$, which allows all turbulence scales, including the smallest scales, to be resolved.

Figure 18. (a,b) Distributions of $\overline {u_1u_1}^+$ in a pipe and a boundary layer. Data of Vallikivi et al. (Reference Vallikivi, Hultmark, Bailey and Smits2011, Reference Vallikivi, Ganapathisubramani and Smits2015a,Reference Vallikivi, Hultmark and Smitsb), Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013) and Rosenberg et al. (Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013). Note that the values of the viscous-scaled sensor length $l^+=lu_\tau /\nu$ at each $Re_\tau$ are shown in table 1. For reference, also shown are the data of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) in a boundary layer at $Re_\tau =6000\unicode{x2013}20\,000$ with $l^+ = 2.4\unicode{x2013}3.5$ (thick dashed curves). (c,d) Distributions of $\overline {u_1u_1}^+$ corresponding to (a,b) for only $Re_\tau \gtrsim 20\,000$. The solid line in (c) is $\overline {u_1 u_1}^+=1.95-1.26\log (x_2/\delta )$. The horizontal dotted lines in (c) indicate the values 6.66 and 6.95 respectively. The horizontal dotted lines in (d) indicate the values 6.66, 6.93 and 7.35.

Table 1. Values of the viscous-scaled sensor length $l^+$ corresponding to the data in figures 18(a,b).

The following comments can be made with regard to figure 18.

(i) Smits (Reference Smits2022) concluded that an outer peak in $\overline {u_1u_1}^+$ emerges at high $Re_\tau$ in the pipe and boundary layer, and the inner peak increases with $Re_\tau$, up to $Re_\tau =20\,000$. Those features are indeed observed in figures 18(a,b).

(ii) Based on the analysis in the context of (4.1)–(4.3), there should be an $x_2^+$ range $\overline {u_1u_1}^+\sim const$, which can indeed be observed by some of the distributions in figures 18(a,b) for $Re_\tau \gtrsim 20\,000$. In order to examine this feature more closely, we report in figures 18(c,d) the $\overline {u_1u_1}^+$ distributions corresponding to figures 18(a,b) for only $Re_\tau \gtrsim 20\,000$. In the boundary layer, there is an approximate plateau at all $Re_\tau$, with values of approximately 6.66, 6.93 and 7.35, over the range $80\lesssim x_2^+\lesssim 300$ (except for $Re_\tau =40\,053$ which has a slightly narrower plateau over the range $100\lesssim x_2^+\lesssim 300$), indicating an emergence of $\overline {u_1u_1}^+\sim const$ (figure 18d). An approximate plateau, with a value of approximately 6.95, can also be observed in the pipe at $Re_\tau = 37\,690$ over the range $100\lesssim x_2^+\lesssim 300$ (figure 18c).

(iii) The boundary layer data of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) show that $\overline {u_1u_1}^+$ increases systematically with $Re_\tau$ for $Re_\tau =6000\unicode{x2013}20\,000$ at any given $x_2^+$ (figures 18a,b). However, the magnitude of $\overline {u_1u_1}^+$ at $Re_\tau =20\,250$ in the pipe is smaller than that in the boundary layer at comparable $Re_\tau$ ($=$20 000) for $x_2^+\lesssim 100$ (figure 18c). Also, the magnitudes of $\overline {u_1u_1}^+$ at $Re_\tau =68\,371\unicode{x2013}98\,190$ in the pipe are smaller than or close to that in the boundary layer at $Re_\tau =20\,000$ for $x_2^+\lesssim 70$. This may be due to the correction scheme of Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011a). Indeed, Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) compared the corrected $\overline {u_1u_1}^+$ distributions ($l^+=20\unicode{x2013}29$), using the scheme proposed by Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011a), with the fully resolved $\overline {u_1u_1}^+$ distributions, measured by a nanoscale thermal anemometry probe (NSTAP), for $Re_\tau =6000\unicode{x2013}20\,000$ (see their figure 9). Their results show that the correction scheme of Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011a) performs adequately up to $Re_\tau =14\,500$; however, at $Re_\tau =20\,000$ ($l^+=29$), the magnitudes of the corrected $\overline {u_1u_1}^+$ distributions slightly underestimated the fully resolved $\overline {u_1u_1}^+$ measured by the NSTAP. It is expected that at given $x_2^+$ in the near-wall region, this underestimation may increase as $Re_\tau$ and $l^+$ increase.

(iv) As discussed above, figure 18 shows that $\overline {u_1u_1}^+\sim const$ can be observed for the boundary layer data of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) over the range $80\lesssim x_2^+\lesssim 300$ at $Re_\tau =20\,000$. If $\bar \varepsilon \sim x_2^{-1}$ (or equivalently, $u_K^+\sim x_2^{-1/4}$) can be observed in the same $x_2^+$ range, then this would lead to $\overline {u_1u_1}^*\sim x_2^{*2/3}$, the prediction of the present hypothesis. Although figure 17 of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) shows that $u_K^+$ and $\eta ^+$ exhibit reasonable collapse in the range $10\lesssim x^+_2 \lesssim 500$ for $Re_\tau =6000\unicode{x2013}20\,000$, there is no discernible range of $x_2^+$ over which we can observe $u_K^+\sim x_2^{-1/4}$ or $\eta ^+\sim x_2^{1/4}$. One possible reason is that they assumed local isotropy to estimate the mean turbulent kinetic energy dissipation rate, i.e. ${\bar \varepsilon _{iso}} = 15\nu \int _0^\infty {k_x^2\,{\phi _{{u_1}}}(k_x)}\, {\rm d}{k_x}$, which would result in $\bar \varepsilon$ being underestimated. As an example, we show in figure 19(a) the ${\bar \varepsilon _{iso}} /\bar \varepsilon$ distribution in a channel at comparable $Re_\tau$ ($=$5200–8000). The magnitude of ${\bar \varepsilon _{iso}} /\bar \varepsilon$ increases from 0 to 1 when $x_2^+$ increases from the wall to the region close to the centreline. Interestingly, ${\bar \varepsilon _{iso}} /\bar \varepsilon$ appears to be independent of $Re_\tau$ at given $x_2^+$ when $x_2^+\gtrsim 30$. In order to obtain a more accurate estimation of $\bar \varepsilon$ for the data of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) at $Re_\tau =20\,000$, we assume that the ratio ${\bar \varepsilon _{iso}} /\bar \varepsilon$ for $x_2^+\gtrsim 30$ in the channel at $Re_\tau =8000$ also applies to that in a boundary layer at large $Re_\tau$. The $u_K^+$ distribution, based on the ‘new’ estimates of $\bar \varepsilon$, is shown figure 19(a) (pink curve). Also shown in figure 19(a) is the $u_K^+$ distribution, based on $\bar \varepsilon _{iso}$ of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) (black curve). It can be observed that the magnitude of $u_K^+$, based on $\bar \varepsilon$, is systematically larger than that based on $\bar \varepsilon _{iso}$ when $x_2^+\lesssim 400$. More importantly, $u_K^+\sim x_2^{-1/4}$ is satisfied adequately in the range $x_2^+\lesssim 700$, which will result in $\overline {u_1u_1}^*\sim x_2^{*2/3}$ in the overlap region between $u_K^+\sim x_2^{-1/4}$ and $\overline {u_1u_1}^+=const$; this is supported by the distribution in figure 19(b), where we can observe that $\overline {u_1u_1}^*\sim x_2^{*2/3}$ is indeed satisfied approximately over the range $32\lesssim x_2^*\lesssim 90$. Note that at $Re_\tau =20\,000$, $x_2^*=32$ and 90 correspond to $x_2^+=79$ and 312, respectively. For reference, also shown in figure 19(b) is the $\overline {u_1u_1}^*$ distribution of Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) in a channel at ${Re}_{\tau }=10^4$. It is worth mentioning that $\overline {u_1u_1}^*\sim x_2^{*2/3}$ in the range $35\lesssim x_2^*\lesssim 70$ for the channel data. This is consistent with the local slope shown in figure 11.

Figure 19. (a) On the left, distributions of $\bar \varepsilon _{iso}/\bar \varepsilon$ in a channel at ${Re}_{\tau }=5200$ (red curve) and ${Re}_{\tau }=8000$ (blue curve); data of Lee & Moser (Reference Lee and Moser2015, Reference Lee and Moser2019) and Kaneda & Yamamoto (Reference Kaneda and Yamamoto2021). On the right, distributions of $u_K ^ +$ based on $\bar \varepsilon _{iso}$ (black curve, which is reproduced from figure 17(b) of Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) and $\bar \varepsilon$ (pink curve) in a boundary layer at ${Re}_{\tau }=20\,000$ (see text). Dotted curve is $0.46x_2^{+-1/4}$. (b) Distributions of $\overline {u_1u_1}^*$ ($=\overline {u_1u_1}^+/u_K^{+2}$) in a boundary layer at ${Re}_{\tau }=20\,000$ (see text). For reference, also shown as a cyan curve is the $\overline {u_1u_1}^*$ distribution of Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) in a channel at ${Re}_{\tau }=10^4$. Dotted curve is ${\sim }x_2^{*2/3}$.

(v) We recall that the attached eddy hypothesis proposed by Townsend (Reference Townsend1976) predicts

(4.4)\begin{equation} \overline{u_1 u_1}^+=B_1-A_1 \log(x_2/\delta), \end{equation}

where $B_1$ and $A_1$ are constants. Equation (4.4) has been supported strongly by both the experimental and numerical data at high $Re_\tau$ (see the review of Marusic & Monty Reference Marusic and Monty2019). We add the distribution of (4.4) in figure 18(c) with $B_1=1.95$ and $A_1=1.26$ (Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). It is interesting that (4.4) predicted by the attached eddy hypothesis, and the present prediction $\overline {u_1u_1}^+\sim const$, are located in different $x_2^+$ ranges at $Re_\tau =20\,000$. Namely, (4.4) and $\overline {u_1u_1}^+\sim const$ can co-exist.

5. Relationship to the scaling of small-scale wall turbulence

We recall that Tang & Antonia (Reference Tang and Antonia2022) have shown that an overall feature of the Kolmogorov-normalized energy spectra $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ is that their collapse extends to increasingly smaller wavenumbers (larger scales) with increasing $Re_\tau$. Rather than discussing the behaviour of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$, we focus here on the second-order velocity structure function $\overline {{{(\delta {u_i})}^2}} (r_z^*)$, which is closely related to $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ since (e.g. Dickey & Mellor Reference Dickey and Mellor1979; Monin & Yaglom Reference Monin and Yaglom2007)

(5.1)\begin{equation} \overline{{{(\delta {u_i})}^2}} (r_z^*) = 2\int_0^\infty {{\phi _{{u_i}}}(k_z^*)}\,(1 - \cos (k_z^*r_z^*))\,{\rm d} k_z^*, \end{equation}

where $r_z$ is the separation between two points in the $x_3$ direction; note that the repeated index $i$ does not indicate summation in (5.1a,b) since we are interested in the behaviour of different components. Equation (5.1a,b) is obtained based on the relation $\overline {{{(\delta {u_i})}^2}} (r_z) = 2(\overline {u_i^2} - {u_i}{u_i'}) = 2[{\int _0^\infty {{\phi _{{u_i}}}(k_z)}\,{\rm d} k_z - \int _0^\infty {{\phi _{{u_i}}}(k_z)} \cos (k_zr_z)\,{\rm d}{k_z}}]$ (e.g. Dickey & Mellor Reference Dickey and Mellor1979). The advantage of $\overline {{{(\delta {u_i})}^2}} (r_z^*)$ is that it leads to $2\overline {u_1^{*2}}$, $2\overline {u_2^{*2}}$ and $2\overline {u_3^{*2}}$ as $r_z^*\to L^*$ (where $L$ is the integral scale). We select four typical locations, i.e. $x_2^*=5$, 16, 50 and 150. The minimum $r_z^*$ at different locations is calculated based on the Kolmogorov-normalized resolution of the mesh in the $x_3$ direction. Note that when data were not obtained at exactly $x_2^*=5$, 16, 50 and 150, interpolation was used, based on available spectra closest to these locations. Then $\overline {{{(\delta {u_i})}^2}} (r_z^*)$ is calculated based on (5.1a,b). Figures 20(a–d) show the distributions of $\overline {{{(\delta {u_i})}^2}} (r_z^*)$ at $x_2^*=150$, 50, 16 and 5, respectively.

Figure 20. Kolmogorov-normalized second-order structure function $\overline {{{(\delta {u_i})}^2}} (r_z^*)$ at $x_2^*=5$, 16, 50 and 150. Dotted and solid/dotted curves correspond to the boundary layer data at $Re_\tau =1307$ and 1988, respectively. Dash-dotted, dashed and solid curves correspond to the channel data at $Re_\tau =1000$, 2000 and 5200, respectively. Note that as in figure 8, only the data for $Re_\tau \geq 1000$ are shown.

The following comments can be made with regard to figure 20.

(i) Local isotropy requires $\overline {{{(\delta {u_1})}^2}} (r_z^*) =\overline {{{(\delta {u_2})}^2}} (r_z^*)$. There is a strong departure from local isotropy at $x_2^*=5$ and 16 in the scale range $r_z^*\lesssim 10$. The departure gradually decreases as $x_2^*$ increases, and appears to be negligible at $x_2^*=150$, suggesting an approach towards local isotropy in the context of $\overline {{{(\delta {u_1})}^2}} (r_z^*) =\overline {{{(\delta {u_2})}^2}} (r_z^*)$.

(ii) At $x_2^*=5$, $\overline {{{(\delta {u_1})}^2}} (r_z^*)$ and $\overline {{{(\delta {u_2})}^2}} (r_z^*)$ collapse at all $r_z^*$, whereas $\overline {{{(\delta {u_3})}^2}} (r_z^*)$ appears to collapse only for $r_z^*\lesssim 30$. Consequently, the degree of collapse is better for $\overline {u_1^{*2}}$ and $\overline {u_2^{*2}}$ than for $\overline {u_3^{*2}}$.

(iii) At $x_2^*=16$, only $\overline {{{(\delta {u_2})}^2}} (r_z^*)$ collapses at all $r_z^*$. An important feature of the other distributions is that at a given $x_2^*$, the collapse extends to increasingly larger scales with increasing $Re_\tau$. One thus expects that as $Re_\tau$ is increased to a sufficiently large value, the collapse would extend to all $r_z^*$. In this situation, the sketch of the predictions for $\overline {u_i^{*2}}$ shown in figure 2 can be further extended to the $r_z^*$ direction, which is shown in figure 21. Note that $r_z^*$ has been divided by a Kolmogorov-normalized integral scale $L^*$ so that $\overline {{{(\delta {u_i})}^2}} (r_z^*)/2$ leads to $\overline {u_i^{*2}}$ as $r_z^*/L^*\to 1$ (or equivalently, $r_z/L\to 1$). For clarity, distributions of $\overline {{{(\delta {u_i})}^2}} (r_z^*)/2$ for only three values of $x_2^*$ are shown. We emphasize that as $r_z^*/L^*\to 1$ and $x_2^*\to \delta ^*$ (sufficiently far away from the wall), $\overline {{{(\delta {u_i})}^2}} (r_z^*)/2$ should go to $\infty$ as $Re_\tau \to \infty$. However, $\overline {{{(\delta {u_i})}^2}} (r_z^*)/2$ should be independent of the Reynolds number both in the region $x_2^*\ll \delta ^*$ (including all scales) and in the scale range $r_z^*/L^*\ll 1$ as $x_2^*\to \delta ^*$, as $Re_\tau \to \infty$; the latter corresponds to the prediction of the classical hypotheses of Kolmogorov (Reference Kolmogorov1941). Although not discussed here, the same remark can be made with regard to $\overline {{{(\delta {u_i})}^2}} (r_x^*)/2$ and the scalar structure function $\overline {{{(\delta {\theta })}^2}} /2$ in both the $r_x^*$ and $r_z^*$ directions.

Figure 21. Sketch of $\overline {{u_i}^{*2}}$ versus $x_2^*$ (red curve) and $\overline {{{(\delta {u_i})}^2}} (r_z^*) /2$ versus $r_z^*/L^*$ (blue curves) on log–log scales. The red arrow corresponds to the location $r_z^*/L^*=1$. Note that $\overline {{{(\delta {u_i})}^2}} (r_z^*)/2$ reaches $\overline {u_i^{*2}}$ as $r_z^*/L^*$ approaches 1.

Further, the collapse of $x_2^+\bar \varepsilon ^ +$ distributions in the boundary layer (figure 14b) in the range $10\lesssim x_2^+\lesssim 30$ implies that the wall parameters and the Kolmogorov scales can be used interchangeably for quantities such as the energy spectra and the second-order velocity structure function in the same $x_2^+$ range. Indeed, Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) have shown that the wall-parameter-normalized premultiplied energy spectra at $x_2^+=15$ and 24, 92, 178 and 277 collapse reasonably well for the small wavelength ranges in the boundary layer at $Re_\tau =6000$, 10 000, 14 500 and 20 000, respectively; the collapse of the premultiplied energy spectra extends to larger $x_2^+$, up to 277, suggesting that the $\bar \varepsilon ^ +$ distributions for $Re_\tau \gtrsim 6000$ in the boundary layer have extended to at least $x_2^+\approx 277$. This can be inferred from figure 17 of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), which shows that the wall-parameter-normalized Kolmogorov length and velocity scales (the energy dissipation rate was estimated by integrating the one-dimensional streamwise dissipation spectrum) exhibit reasonable collapse in the range $10\lesssim x_2^+\lesssim 500$.

6. Exploratory predictions for high-order moments at all scales

The predictions of the present hypothesis can be extended to high-order moments. For simplicity, we consider only the velocity and passive scalar statistics $\overline {u_i^n}^*$ and $\overline {\theta ^n}^*$ for $n\geq 3$, which can be expressed as

(6.1a,b)\begin{equation} \overline{u_i^{n}}^*= f_n(x_2^*), \quad \overline{\theta^{n}}^*= f_{\theta n}(x_2^*), \end{equation}

where the functions $f_n$ and $f_{\theta n}$ are independent of the Reynolds and Péclet numbers once appropriate values of the Reynolds and Péclet numbers are reached. In the inertial-convective range $0\ll x_2^*\ll \delta ^*$,

(6.2a,b)\begin{equation} \overline{u_i^{n}}^*= C_n x_2^{*n/3}, \quad \overline{\theta^{n}}^*= C_{\theta n} x_2^{*n/3}. \end{equation}

Finally, at large $Pr$, there exists a viscous-convective range for $\overline {\theta ^{n}}^*$ between the inertial-convective and the viscous-diffusion ranges over which

(6.3)\begin{equation} \overline{\theta^{n}}^*= const. \end{equation}

Combining all predictions of the present hypothesis with those of the small-scale wall turbulence hypothesis proposed by Tang & Antonia (Reference Tang and Antonia2022), we can formulate a more complete hypothesis for turbulent statistics in wall turbulence, as follows.

In wall turbulence at sufficiently high Reynolds and Péclet numbers, except for the large-scale quantities in the region far away from the wall, all turbulent statistics are independent of the Reynolds and Péclet numbers when the normalization uses $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$. They depend only on the turbulent scale (or wavenumber) and the distance from the wall. The larger the Reynolds and Péclet numbers, the larger the distance from the wall and the range of turbulent scales over which this hypothesis applies. In particular, there exist two inertial-convective ranges over which the scaling parameters are $\bar \varepsilon$ and $\bar \varepsilon _\theta$, and two viscous-convective ranges (at large $Pr$) over which the effect of $\nu$ is important. For large-scale quantities such as $\overline {u_i^n}^*$ and $\overline {\theta ^n}^*$, the inertial-convective range is located in the region $0\ll x_2^*\ll \delta ^*$. The other pertains to the small-scale quantities and is located in the region $0\ll r^*\ll L^*$ as $x_2^*\to \delta ^*$. Also, there are two viscous-convective ranges for the scalar variance and the scalar structure function, located between the viscous-diffusive and inertial-convective ranges.

The predictions of the above complete hypothesis are sketched in figure 22, which shows the approximate locations and scale ranges over which the scaling based on $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ is tenable. We recall that figure 21 is for the second-order velocity structure function $\overline {{{(\delta {u_i})}^2}} (r_z^*)$, whereas figure 22 is for any quantity (which can be a function of the turbulent scale $r^*$ and $x_2^*$). If we focus on the quantity $\overline {{{(\delta {u_i})}^2}} (r_z^*)$, then figure 22 can be considered, at least approximately, as the projection of the three-dimensional plot of figure 21 onto the $r_z^*$–$x_2^*$ plane, after $r_z^*/L^*$ in figure 21 is multiplied by $L^*$. We believe that the predictions in figure 22 could be one possible ultimate statistical state of wall turbulence. It is evident that this complete hypothesis for wall turbulence needs to be tested against data, especially for moments of order higher than 2, in future investigations.

Figure 22. Summary of predictions, based on the present hypothesis, for turbulent statistics in wall turbulence when the normalization uses $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ at sufficiently high Reynolds number. The red line is the upper limit for $x_2^*$ and $r^*$ over which the scaling based on $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ holds. Note that the two viscous-convective ranges apply only to the passive scalar at large $Pr$. Note also that the present scaling does not apply in the region $x_2^*\to \delta ^*$ and $r^*\to L^*$. The dashed curve is the upper limit of the viscous-dominated range. Here, $r$ can be either $r_x$ or $r_z$.

Finally, we stress that as $Re_\tau \to \infty$, if both $\bar \varepsilon ^ +$ and $\bar \varepsilon _\theta ^ +$ collapse onto unique curves at all locations in the range $x_2^+\lesssim x^+_{2\alpha }$ in all wall flows, then the wall parameters and the scaling parameters $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ can be used interchangeably in the same $x_2^+$ range. In this case, the scaling of wall turbulence based on wall parameters should also be tenable in the range $x_2^+\lesssim x^+_{2\alpha }$ (figure 23), which shows the approximate locations over which the scaling based on wall parameters is tenable. This is consistent with the perspective of Chen & Sreenivasan (Reference Chen and Sreenivasan2022) whereby wall-parameter-normalized statistics displaying non-zero wall values or near-wall peaks may become bounded at infinitely large Reynolds numbers. In particular, in the overlap region between $\bar \varepsilon \sim \bar \varepsilon _\theta \sim x_2^{-1}$ and the inertial-convective range for large-scale quantities such as $\overline {u_i^n}^*$ and $\overline {\theta ^n}^*$, wall-parameter-normalized large-scale quantities should approach constant values, as indicated in figure 23. We can demonstrate this by using $\overline {u_i^n}^*$ as an example, i.e.

(6.4)\begin{align} {\overline{u_i^n} ^*} &= {C_n}x_2^{*n/3} = \frac{{{{\overline{u_i^n} }^ + }}}{{u_K^{ + n}}} = {C_n}{(x_2^ + u_K^ + )^{n/3}} \nonumber\\ &\Rightarrow {\overline{u_i^n} ^ + } = {C_n}x_2^{ + n/3}u_K^{ + 4n/3} = {C_n}x_2^{ + n/3}{{\bar \varepsilon }^{ + n/3}} \sim {C_n}. \end{align}

We expect that a wall-parameter-based scaling could be another possible ultimate statistical state of wall turbulence. Again, this needs to be tested in the future.

Figure 23. Summary of predictions for turbulent statistics in wall turbulence in the range $x_2^+\lesssim x^+_{2\alpha }$ when the normalization uses wall parameters at sufficiently high Reynolds number. The red line is the upper limit for $x_2^+$ and $r^+$ over which the scaling based on wall parameters holds. Note that this scaling requires both $\bar \varepsilon ^ +$ and $\bar \varepsilon _\theta ^ +$ to collapse onto unique curves at all locations in the range $x_2^+\lesssim x^+_{2\alpha }$, and does not apply in the region $x_2^+>x^+_{2\alpha }$. The dashed curve is the upper limit of the viscous-dominated range. Here, $r$ can be either $r_x$ or $r_z$.

We recall that a region where $\overline {u_1u_1}^+\sim const$ can be observed in some of the distributions in figures 18(c,d) for $Re_\tau \gtrsim 20\,000$. However, the magnitude of the constant continues to depend on $Re_\tau$. This implies that a value of $Re_\tau$, even when equal to 72 526 (in figure 18d), may not be ‘sufficiently high’ for $\overline {u_iu_i}^+=const$ (independently of $Re_\tau$) to be established unequivocally. In order to demonstrate this further, we now focus on the behaviour of $-\overline {u_1u_2}^+$ at large $Re_\tau$. A well-known relation for the total shear stress in the channel is

(6.5)\begin{equation} {\tau _w}\left(1 - \frac{{{x_2}}}{\delta }\right) = \rho \nu\, \frac{{{\rm d}{\bar U_1}}}{{{\rm d}{x_2}}} - \rho\,\overline{{u_1}{u_2}}, \end{equation}

which, after trivial manipulations, can be rewritten as

(6.6)\begin{equation} - {\overline{{u_1}{u_2}} ^ + } = 1 - \frac{{x_2^ + }}{{{Re_\tau }}} - \frac{{{\rm d} \bar U_1^ + }}{{{\rm d} x_2^ + }}. \end{equation}

If we focus on the region away from the wall and channel centreline, then ${{\rm d} \bar U_1^ + /{\rm d} x_2^ + }$ may be replaced with the log-law region, i.e.

(6.7)\begin{equation} - {\overline{{u_1}{u_2}} ^ + } = 1 - \frac{{x_2^ + }}{{{Re_\tau }}} - \frac{1}{{\kappa x_2^ + }}. \end{equation}

Smits (Reference Smits2022) concluded that $\kappa =0.40 \pm 0.02$ based on a large amount of data measured in the Princeton SuperPipe using both Pitot tubes and hot wires. Figure 24(a) shows the distributions of $- {\overline {{u_1}{u_2}} ^ + }$ based on (6.6) at $Re_\tau =10^4$, 25 062, 40 053 and 72 526. To plot (6.6), we used the DNS data of ${{\rm d} \bar U_1^ + /{\rm d} x_2^ + }$ at $Re_\tau =10^4$ in a channel (Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) and the experimental data at $Re_\tau =25\,062\unicode{x2013}72\,526$ in a boundary layer (Vallikivi et al. Reference Vallikivi, Hultmark, Bailey and Smits2011, Reference Vallikivi, Ganapathisubramani and Smits2015a,Reference Vallikivi, Hultmark and Smitsb; Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013), respectively; they are shown in the inset. The use of ${{\rm d} \bar U_1^ + /{\rm d} x_2^ + }$ in a boundary layer should be reasonable since it is well-accepted that $\bar U_1^ +$ is universal in the inner layer of wall flows. Values $Re_\tau =25\,062$, 40 053 and 72 526 correspond to the three largest values of $Re_\tau$ in figures 18(b,d). We can observe that the variation of $- {\overline {{u_1}{u_2}} ^ + }$ with $Re_\tau$ is systematic in the region $x_2^ +\gtrsim 150$. In the range $x_2^ +<150$, the distributions at $Re_\tau =25\,062\unicode{x2013}72\,526$ exhibit some scatter. This reflects the scatter of ${{\rm d} \bar U_1^ + /{\rm d} x_2^ + }$ in the same $x_2^ +$ range (see the inset). Figure 24(b) shows the distributions of $- {\overline {{u_1}{u_2}} ^ + }$ based on (6.7) with $\kappa =0.42$ (Smits Reference Smits2022) at the same $Re_\tau$ as in figure 24(a). Also shown are the distributions based on (6.6) at $Re_\tau =25\,062\unicode{x2013}72\,526$, after replacing ${{\rm d} \bar U_1^ + /{\rm d} x_2^ + }$ in the range $x_2^ +<150$ with the DNS data of Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) at $Re_\tau =10^4$. There is adequate agreement between (6.6) and (6.7). Finally, the $- {\overline {{u_1}{u_2}} ^ + }$ distribution of Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) at $Re_\tau =10^4$ is shown in figures 24(a,b). It agrees reasonably well with (6.6) and (6.7). It is evident that even at $Re_\tau =72\,526$, $- {\overline {{u_1}{u_2}} ^ + }=1$ is not established (figure 24). We next quantify, albeit approximately, the values of $Re_\tau$ required for $- {\overline {{u_1}{u_2}} ^ + }=1$ to be established. Figure 25 shows the distributions of $- {\overline {{u_1}{u_2}} ^ + }$, based on (6.7) with $\kappa =0.42$ (Smits Reference Smits2022), at $Re_\tau =10^4$, $10^5$ and $10^6$. It appears that $Re_\tau \sim 10^6$ is needed for $- {\overline {{u_1}{u_2}} ^ + }\approx 1$ to be established in an $x_2^ +$ range of approximately two decades ($2\times 10^2\lesssim x_2^ +\lesssim 10^4$).

Figure 24. (a) Distributions of $- {\overline {{u_1}{u_2}} ^ + }$ (curves) based on (6.6) at $Re_\tau =10^4$, 25 062, 40 053 and 72 526. Note that the calculation of (6.6) is based on the DNS data of ${{\rm d} \bar U_1^ + /{\rm d} x_2^ + }$ in a channel at $Re_\tau =10^4$ (Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) and the experimental data at $Re_\tau =25\,062\unicode{x2013}72\,526$ (Vallikivi et al. Reference Vallikivi, Hultmark, Bailey and Smits2011, Reference Vallikivi, Ganapathisubramani and Smits2015a,Reference Vallikivi, Hultmark and Smitsb; Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013), respectively (see the inset). (b) Distributions of $- {\overline {{u_1}{u_2}} ^ + }$ (curves) based on (6.7) with $\kappa =0.42$ (Smits Reference Smits2022) at $Re_\tau =10^4$, 25 062, 40 053 and 72 526. Also shown are the calculations based on (6.6) at $Re_\tau =25\,062$ ($\circ$, blue), 40 053 ($\circ$) and 72 526 ($\circ$, green), after replacing ${{\rm d} \bar U_1^ + /{\rm d} x_2^ + }$ in the range $x_2^ +<150$ with the DNS data of Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) (see text). For comparison, also shown in (a,b) is the $- {\overline {{u_1}{u_2}} ^ + }$ distribution ($\circ$, red) of Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) at $Re_\tau =10^4$.

Figure 25. Distributions of $- {\overline {{u_1}{u_2}} ^ + }$ based on (6.7) with $\kappa =0.42$ (Smits Reference Smits2022) at $Re_\tau =10^4$, $10^5$ and $10^6$.

7. Concluding remarks

In summary, a hypothesis has been proposed to describe the behaviour of the Reynolds stresses, turbulent kinetic energy and scalar variance in wall turbulence. The major conclusions can be summarized as follows.

(i) When the normalization is based on $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$, the independence on the Reynolds and Péclet numbers predicted by the present hypothesis is first satisfied at very small $x_2^*$ before gradually extending to larger $x_2^*$ as $Re_\tau$ increases.

(ii) At large but not infinitely large $x_2^*$, the magnitudes of the Kolmogorov-normalized Reynolds stresses and turbulent kinetic energy depend on $Re_\tau$. Further, the scalar variance, when normalized by $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$, depends not only on $Re_\tau$ at a given $Pr$, but also on the Péclet number $Re_\tau Pr$ at a given $Re_\tau$. The normalized quantities vary with $Re_\tau$ and $Re_\tau Pr$ systematically, suggesting the importance of the finite $Re_\tau$ and $Re_\tau Pr$ effects in the context of establishing the inertial-convective range, i.e. the emergence of the $2/3$ power-law scaling. Although independence from the Reynolds and Péclet numbers of those quantities has not yet been observed in the present $Re_\tau$ (${\leq }10^4$) and $Pr$ (${\leq }7$) ranges at moderate $x_2^*$, a trend towards a $2/3$ power-law scaling is discernible for some quantities, such as the turbulent kinetic energy, and the streamwise and wall-normal components of the Reynolds stresses.

(iii) At the highest $Pr$ ($=7$), there is a trend towards a plateau for the scalar variance when normalized with $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$. The plateau falls between the inertial-convective and viscous-diffusive ranges for $Re_\tau =500\unicode{x2013}2000$, suggesting an emergence of a viscous-convective range.

The relationship between the wall-parameter normalization and the normalization based on $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ has been discussed in the context of $\bar \varepsilon ^+$ and $\bar \varepsilon _\theta ^+$. At finite $Re_\lambda$ and $Pr\,Re_\lambda$, the scaling based on $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ is superior to that based on wall parameters in the context of $\overline {u_iu_j}$, $\overline {u_iu_i}$ and $\overline {\theta \theta }$. We should stress that if the collapse of the distributions of $\bar \varepsilon ^+$ and $\bar \varepsilon _\theta ^+$ in wall flows extends to the wall, then both the wall parameters and scaling parameters $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$ can be used interchangeably in the near-wall region in the context of the Reynolds stresses, turbulent kinetic energy, scalar variance, mean velocity and mean scalar; this may require $Re_\lambda \to \infty$. In particular, as $Re_\lambda \to \infty$, there should be an overlap range (far away from the wall) over which both $-\overline {u_1u_2}^+ =1$ and $\overline {u_1u_2}^*= C_{12} x_2^{*2/3}$ (inertial-convective range) are established unequivocally, and $\bar \varepsilon ^ +$ collapses perfectly. Further, in this overlap range, $\overline {u_1u_1}^+$, $\overline {u_2u_2}^+$, $\overline {u_3u_3}^+$, $\overline {u_iu_i}^+$ and $\overline {\theta \theta } ^ +$ should be constant. In the viscous-convective range at large $Pr$, it is expected that $\overline {\theta \theta } ^ +=const\,\theta _B^{+2}$, when both $\bar \varepsilon ^ +$ and $\bar \varepsilon _\theta ^ +$ collapse in the same $x_2^+$ range.

Finally, the main message of figure 20 is that in the wall region, the $Re_\tau$ independence of $\overline {{{(\delta {u_i})}^2}} (r_z^*)$ is first established at all locations for small scales, and extends to increasingly larger scales with increasing $Re_\tau$. In the region very near the wall, this $Re_\tau$ independence can extend to all scales and to increasingly larger values of $x_2^*$ with increasing $Re_\tau$; in the $Re_\tau$ range for the currently available data, the $x_2^*$ range over which $\overline {{{(\delta {u_i})}^2}} (r_z^*)$ collapses depends on the specific quantity investigated. As $Re_\tau \to \infty$, $\overline {{{(\delta {u_i})}^2}} (r_z^*)$ should be $Re_\tau$-independent both in the near-wall region $x_2^*\ll \delta ^*$ (at all scales) and in the small-scale range $r_z^*/L^*\ll 1$ as $x_2^*\to \delta ^*$. This picture leads to one possible ultimate statistical state of wall turbulence in the context of the second-order velocity structure function (figure 21). Conceptual summary sketches of the predictions of the present hypothesis, combined with those of the small-scale wall turbulence hypothesis (Tang & Antonia Reference Tang and Antonia2022), are given in figure 22 when the normalization is based on $\bar \varepsilon$, $\bar \varepsilon _\theta$ and $\nu$, and in figure 23 when the normalization uses wall parameters. The latter is consistent with the restoration of wall scaling at asymptotically high $Re_\tau$ (Chen & Sreenivasan Reference Chen and Sreenivasan2021, Reference Chen and Sreenivasan2022). Finally, we should stress that large-scale turbulent statistics, normalized by wall parameters, are approximately constant over the range $0\ll x_2^+\ll \delta ^+$ (figure 23). This requires $\bar \varepsilon \sim \bar \varepsilon _\theta \sim x_2^{-1}$.