Scaling of small-scale wall turbulence

Abstract

In the vicinity of walls, turbulence is anisotropic. Since the classical hypotheses of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303), Obukhov (Izv. Akad. Nauk SSSR Geogr. Geofiz, vol. 13, 1949, pp. 58–69) and Corrsin (J. Appl. Phys., vol. 22, 1951, pp. 469–473) require small-scale turbulence to be isotropic, they have only limited relevance to wall-bounded turbulent flows. Here, we put forward a hypothesis whereby small-scale near-wall statistics, when suitably normalized, are independent of the type of flow as well as of the Reynolds and Péclet numbers. The relatively large amount of available wall turbulence direct numerical simulations data, related mainly to second-order statistics, in a channel flow and a boundary layer provides good support for the independence with respect to the Reynolds number. To fully validate the hypothesis, more data are required for higher-order statistics as well as for other wall flows and for different surface conditions.

1. Introduction

One striking success in describing and predicting small-scale turbulent motion (Nelkin 1992) is the phenomenological picture for the velocity field introduced by Kolmogorov (1941). Kolmogorov's hypotheses were extended further by Obukhov (1949) and Corrsin (1951) to describe the behaviour of a passive scalar field within a turbulent flow. The Kolmogorov–Obukhov–Corrsin hypotheses are commonly referred to as the KOC theory in the literature. This theory requires that the Reynolds and Péclet numbers are sufficiently large so that the small-scale velocity and passive scalar fields are isotropic and statistically independent of the large scales (Warhaft 2000; Chassaing et al. 2002). However, wall turbulent flows are characterized by anisotropy (Antonia, Kim & Browne 1991; Kasagi, Tomita & Kuroda 1992; Antonia, Abe & Kawamura 2009; Vreman & Kuerten 2014a,b; Tang et al. 2015; Pumir, Xu & Siggia 2016; Kaneda & Yamamoto 2021), especially near the wall, thus the influence of the large scales on small-scale quantities may persist. Wall turbulence is of great importance in meteorological and oceanic flows and hence in a climatological context (Marusic, Mathis & Hutchins 2010; Smits & Marusic 2013). The flow behaviour in the near-wall region also determines the drag force on planes and ships, and the energy required to pump fluids through pipelines (Marusic et al. 2010; Smits & Marusic 2013). Pipe and channel flows and the boundary layer, including the atmospheric surface layer, are the most frequently investigated wall-bounded turbulent flows. The conventional scaling of wall turbulence is based on the wall parameters ${u_\tau }$, ${\theta _\tau }$ and $\nu$, where $u_\tau =(\tau _w/\rho )^{1/2}$ is the friction velocity ($\tau _w$ is the wall shear stress and $\rho$ is the density of the fluid), $\nu$ is the kinematic viscosity, and $\theta _\tau = {Q_w}/\rho {c_p}{u_\tau }$ is the friction temperature ($Q_w$ is the averaged wall heat flux and $c_p$ the specific heat at constant pressure). Several decades of research into small-scale wall turbulence have shown that the scaling of small-scale quantities, such as the mean turbulent energy dissipation rate $\bar {\varepsilon }$ ($\equiv \nu \overline {(\partial {u_i}/\partial {x_k})(\partial {u_i}/\partial {x_k})}$), the mean scalar dissipation rate $\bar {\varepsilon }_\theta$ ($\equiv \kappa \overline {(\partial \theta /\partial {x_k})(\partial \theta /\partial {x_k})}$) and their transport equations, based on ${u_\tau }$, ${\theta _\tau }$ and $\nu$, is untenable in the region near the wall (Antonia et al. 1991; Kasagi et al. 1992; Antonia & Kim 1994; Abe, Antonia & Kawamura 2009; Vreman & Kuerten 2014b; Gerolymos & Vallet 2016; Alcántara-Ávila, Hoyas & Pérez-Quiles 2021; Pirozzoli et al. 2021). The scaling of the energy spectra based on ${u_\tau }$ and $\nu$ near the wall is also untenable (see figure 5(a), which will be discussed later). Here, the streamwise ($x_1$), wall-normal ($x_2$) and spanwise ($x_3$) velocity fluctuations are denoted respectively by $u_1$, $u_2$ and $u_3$. Also, $\kappa$ is the thermal diffusivity, $\theta$ denotes the scalar fluctuation, and the overbar denotes an averaged value at a given $x_2$ location. To our knowledge, there is no theory that can describe and predict the behaviour of small-scale wall turbulence. The major objective of this paper is to develop a hypothesis that addresses this behaviour in an adequate manner.

2. A hypothesis of small-scale wall turbulence

If a proper theory of small-scale wall turbulence were to emerge, which has the prospect of being universal, then it first requires the identification of the proper scaling parameters for small-scale quantities, which, as noted above, are unlikely to be the conventional scaling of wall turbulence, i.e. ${u_\tau }$, ${\theta _\tau }$ and $\nu$. Another set of scaling parameters for small-scale turbulence provided by the KOC theory is $\bar {\varepsilon }$, $\bar {\varepsilon }_\theta$ and $\nu$. This requires local isotropy to be satisfied and the small scales to be statistically independent of the large scales. These two requirements are not satisfied in wall turbulence (Antonia et al. 1991, 2009; Kasagi et al. 1992; Abe et al. 2009; Vreman & Kuerten 2014a,b; Tang et al. 2015; Gerolymos & Vallet 2016; Pumir et al. 2016; Alcántara-Ávila et al. 2021; Kaneda & Yamamoto 2021), so the theory is invalidated. In wall turbulence, one expects that the small scales will deviate from isotropy and may be affected by the large-scale forcing associated with diffusion and mean shear phenomena. However, if this deviation and the effect of the large-scale forcing can become independent of the Reynolds and Péclet numbers, then the small-scale quantities, when suitably normalized, may become universal. If $\bar {\varepsilon }$, $\bar {\varepsilon }_\theta$ and $\nu$ are indeed the proper scaling parameters, then we can formulate a hypothesis for small-scale wall turbulence, as follows.

In wall turbulence at sufficiently high Reynolds and Péclet numbers, although the effect of the large-scale forcing on the small scales depends on the distance from the wall and on the turbulent scale (or wavenumber), it does not depend on the Reynolds and Péclet numbers when the normalization uses $\bar {\varepsilon }$, $\bar {\varepsilon }_\theta$ and $\nu$. Consequently, the small-scale statistics are universal. 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 universality applies. By ‘universality’ we mean independence with respect to the type of flow as well as the Reynolds and Péclet numbers.

Mathematically, this hypothesis predicts that in wall turbulence, the behaviour of any small-scale quantity, say ${\varPsi }$, can be expressed as

(2.1)\begin{equation} {\varPsi}(r^* \text{or}\ k^*,x_2^*)= f_\varPsi(r^* \text{or}\ k^*,x_2^*), \end{equation}

where the asterisk denotes normalization by $\bar {\varepsilon }$, $\bar {\varepsilon }_\theta$ and $\nu$ (or equivalently the Batchelor temperature scale ${\theta _B} = {({\bar {\varepsilon } _\theta }{(\nu /\bar {\varepsilon } )^{1/2}})^{1/2}}$ and the Kolmogorov velocity ${u_K} = {(\nu \bar {\varepsilon } )^{1/4}}$ and length $\eta = {({\nu ^3}/\bar {\varepsilon } )^{1/4}}$ scales), and the function $f_\varPsi$ is universal; $f_\varPsi$ should depend only on $x_2^*$ and the scale $r^*$ in physical space (or wavenumber $k^*$ in spectral space) once appropriate values of the Reynolds and Péclet numbers are reached. In particular, at a given scale $r^*$ (or $k^*$) in the small-scale range, ${\varPsi }(x_2^*)$ will depend only on $x_2^*$. We immediately recover the predictions of the KOC theory, provided that local isotropy is satisfied and the small scales are statistically independent of the large scales in the region far from the wall, viz. ${\varPsi }(r^*\ \text {or}\ k^*)= f_\varPsi (r^*\ \text {or}\ k^*)$; this is supported by the energy spectra in a channel flow, in a boundary layer and in a pipe flow (Saddoughi & Veeravalli 1994; Rosenberg et al. 2013; Tang et al. 2015; Vallikivi, Ganapathisubramani & Smits 2015). Further, for scales larger than the viscous scales but smaller than the integral scale $L$ (representing the scale at which turbulent energy is introduced), the KOC theory predicts that ${\varPsi }(r\ \text {or}\ k)$ are uniquely determined by $\bar {\varepsilon }$ and $\bar {\varepsilon }_\theta$ (no longer depending on $\nu$). In this case, a universal scaling law, i.e. the Kolmogorov $k^{-5/3}$ scaling of the energy spectrum, emerges. This is supported by the spectra of Rosenberg et al. (2013) on the pipe centreline at $Re_\tau =98\,190$, where one can observe approximately one decade of $k^{-5/3}$ behaviour. Evidently, the KOC predictions are compatible with those of the present hypothesis provided that the distance from the wall is sufficiently large, local isotropy is satisfied, and the small scales are statistically independent of the large scales. However, when the latter conditions do not apply, the present hypothesis extends the KOC theory in a significant way.

3. Test of the hypothesis

Figure 1(a) shows the distributions of $\bar {\varepsilon } ^ +$, where $+$ denotes normalization by $u_\tau$ and $\nu$, in a channel flow over a large range of $Re_\tau$ ($550\unicode{x2013}10^4$). Since $\bar {\varepsilon } ^ +=u_K^{4}/u_\tau ^4=\delta _\nu ^4/\eta ^{4}$, the purpose of showing the $\bar {\varepsilon } ^ +$ distributions is to examine whether they can collapse over a certain $x_2^+$ range. This would imply that the wall parameters and the Kolmogorov scales can be used interchangeably in this range. In the region $x_2^+\lesssim 20$, the magnitude of $\bar {\varepsilon } ^ +$ increases systematically with $Re_\tau$. Beyond this region, it decreases quickly with $x_2^+$ to approach a value close to 0 at $x_2^+\approx 10^3$. In order to examine this region more closely, figure 1(b) shows the same distributions premultiplied by $x_2^+$, i.e. $x_2^+\bar {\varepsilon } ^ +$; note that this does not affect the collapse of $\bar {\varepsilon } ^ +$, if it indeed exists. Evidently, the magnitude of $x_2^+\bar {\varepsilon } ^ +$ evolves systematically with $Re_\tau$ except for the distributions at $Re_\tau =5200$ and $10^4$, which collapse over the range $50\lesssim x_2^+\lesssim 200$. Further, the distributions of $x_2^+\bar {\varepsilon } ^ +$ in figure 1(b) do not exclude the possibility that the $x_2^+$ range over which $x_2^+\bar {\varepsilon } ^ +$ collapses may extend to larger and smaller values of $x_2^+$ as $Re_\tau$ keeps increasing. In particular, we underline that if $x_2^+\bar {\varepsilon } ^ +$ does not collapse, then scaling based on Kolmogorov scales will exclude the scaling based on wall parameters, and vice versa.

Figure 1. Distributions of (a) $\bar {\varepsilon } ^ +$ and (b) $x_2^+\bar {\varepsilon } ^ +$ in a channel flow. They are plotted using the data of Lee & Moser (2015, 2019) and Hoyas et al. (2022).

Before testing the scaling based on the Kolmogorov scales, we report in figure 2 the variation of $x_2^*$ with $x_2^+$ across the whole channel, over the ranges $x_2^+=0\unicode{x2013}100$ and $x_2^+=1\unicode{x2013}3$, respectively. This figure helps to identify the $x_2^+$ range that corresponds to the $x_2^*$ range over which the present hypothesis applies. Note that $x_2^*=x_2^+/\eta ^+=x_2^+\bar {\varepsilon } ^{+1/4}$. Strictly speaking, at a given $x_2^+$, if $\bar {\varepsilon }^+$ (or equivalently $x_2^+\bar {\varepsilon }^+$) does not collapse (figure 1), then $x_2^*$ also should not collapse. Indeed, the dependence of $x_2^*$ on $Re_\tau$ at a given $x_2^+$ can be observed from figures 2(a,b) and the inset, at large $x_2^+$ ($\gtrsim 500$), moderate $x_2^+$ (${\sim }100$) and small $x_2^+$ (${\sim }1$). We first test the scaling based on the wall parameters and the Kolmogorov scales in the context of the components of the dissipation rate tensor, i.e.

(3.1a–c)\begin{equation} {\mathsf{R}} _{1} = 2\nu \overline{\left( {\frac{{\partial {u_1}}}{{\partial {x_k}}}\,\frac{{\partial {u_1}}}{{\partial {x_k}}}} \right)}, \quad {{\mathsf{R}} _{1}} = 2\nu \overline{\left( {\frac{{\partial {u_2}}}{{\partial {x_k}}}\,\frac{{\partial {u_2}}}{{\partial {x_k}}}} \right)}, \quad{{\mathsf{R}} _{3}} = 2\nu \overline{\left( {\frac{{\partial {u_3}}}{{\partial {x_k}}}\,\frac{{\partial {u_3}}}{{\partial {x_k}}}} \right)}. \end{equation}

Since ${{\mathsf{R}}_1}+{{\mathsf{R}}_2}+{{\mathsf{R}}_3}=2\bar {\varepsilon }$, we show, as in figure 1(b), the premultiplied ${{\mathsf{R}}_i^+}$ in figure 3(a), i.e. ${x_2^+{\mathsf{R}}_i^+}$ for $Re_\tau =550\unicode{x2013}10^4$. The distributions of ${{\mathsf{R}}_i^*}$ are plotted in figure 3(b). The following comments can be made with regard to figure 3. (i) Like the distributions of $x_2^+\bar {\varepsilon } ^ +$, those of ${x_2^+{\mathsf{R}}_i^+}$ also collapse approximately at $Re_\tau =5200$ and $10^4$ in the range $50\lesssim x_2^+\lesssim 200$. (ii) In the region $x_2^+\lesssim 10$, ${x_2^+{\mathsf{R}}_2^+}\approx 0$ and thus the collapse is not surprising. (iii) In the region around $x_2^+\approx 23$, there is a reasonably good collapse for ${x_2^+{\mathsf{R}}_1^+}$ at all $Re_\tau$. Since $x_2^+({{\mathsf{R}}_1^+}+{{\mathsf{R}}_2^+}+{{\mathsf{R}}_3^+})=2x_2^+\bar \varepsilon ^+$ and the $x_2^+\bar \varepsilon ^+$ distributions around $x_2^+\approx 23$ do not collapse (see figure 1b), the collapse of ${x_2^+{\mathsf{R}}_1^+}$ around this location implies a relatively more rapid evolution for other components, which is indeed observed from the ${x_2^+{\mathsf{R}}_3^+}$ distributions. (iv) Except for the quantities at certain locations, as mentioned above, the other distributions evolve systematically with $Re_\tau$. (v) In contrast, there is a nearly perfect collapse for the distributions of ${{\mathsf{R}}_2^*}$ across the channel when $Re_\tau >1000$. The same feature can be observed in the ${{\mathsf{R}}_1^*}$ and ${{\mathsf{R}}_3^*}$ distributions when $Re_\tau >1000$ and $x_2^*\gtrsim 20$. (vi) In the region $x_2^*\lesssim 20$, the magnitudes of ${{\mathsf{R}}_1^*}$ and ${{\mathsf{R}}_3^*}$ appear to evolve, albeit slightly, with $Re_\tau$ for $Re_\tau \gtrsim 550$. This is consistent with the observation, also in a channel flow, of Gerolymos & Vallet (2016) that the peaks of ${{\mathsf{R}} _{1}}/(2\bar \varepsilon )-1/3$ and ${R _{3}}/(2\bar \varepsilon )-1/3$ vary with $Re_\tau$. However, a close look at the distributions of ${{\mathsf{R}}_1^*}$ and ${{\mathsf{R}}_3^*}$ in the region $x_2^*\lesssim 20$ shows that there is an approximate collapse when $Re_\tau >1000$; this is confirmed by figure 4, which shows the distributions of ${\mathsf{R}}_1^*$, ${\mathsf{R}}_2^*$ and ${\mathsf{R}}_3^*$ for $Re_\tau >1000$ in the channel flow. Note that the same feature can be observed also in the distributions of the Kolmogorov-normalized ${\mathsf{R}}_{12}=2\nu \overline {({({{\partial {u_1}}}/{{\partial {x_k}}})({{\partial {u_2}}}/{{\partial {x_k}}})})}$, i.e. ${\mathsf{R}}^*_{12}$, which are also shown in figure 4. (vii) Overall, the comparison between the wall-parameter-normalized and Kolmogorov-normalized distributions of ${{\mathsf{R}}_i}$ indicates that the scaling based on Kolmogorov scales is superior to that based on wall parameters. More importantly, there is an approximate collapse for all distributions across the channel when $Re_\tau >1000$ (figure 4). Also shown in figure 4 are the ${{\mathsf{R}}_i^*}$ and ${\mathsf{R}}^*_{12}$ distributions in a boundary layer for $Re_\tau >1000$. It is remarkable that the boundary layer data can collapse reasonably well with the channel data in the region $x_2^*\lesssim 70$. Evidently, figure 4 provides reasonably good support for the present small-scale wall turbulence hypothesis.

Figure 2. Variation of $x_2^*$ with $x_2^+$ (a) across the whole channel and (b) on the range $x_2^+=0\unicode{x2013}100$. The inset in (b) zooms in on the range $x_2^+=1\unicode{x2013}3$. The plots use the data of Lee & Moser (2015, 2019) and Hoyas et al. (2022).

Figure 3. Distributions of ${\mathsf{R}}_1$, ${\mathsf{R}}_2$ and ${\mathsf{R}}_3$ in a channel flow. Normalizations based on (a) wall parameters and (b) Kolmogorov scales. They are plotted using the data of Lee & Moser (2015, 2019) and Hoyas et al. (2022). Note that the ${\mathsf{R}}_i^+$ distributions in (a) have been premultiplied by $x_2^+$. The horizontal line in (b) indicates the isotropic ratio ($2/3$).

Figure 4. Distributions of (a) ${\mathsf{R}}_1^*$, and (b) ${\mathsf{R}}_2^*$ and ${\mathsf{R}}_3^*$, in a channel flow for $Re_\tau >1000$ for the data in figure 3(b); the corresponding distributions for the Kolmogorov-normalized ${\mathsf{R}}_{12}=2\nu \overline {( {({{\partial {u_1}}}/{{\partial {x_k}}})({{\partial {u_2}}}/{{\partial {x_k}}})})}$, i.e. ${\mathsf{R}}^*_{12}$, are also shown. For comparison, the boundary data of Simens et al. (2009), Jiménez et al. (2010), Borrell, Sillero & Jiménez (2013) and Sillero, Jiménez & Moser (2013) for $Re_\tau >1000$ are also shown. Note that as in figures 9(a) and 10(a), 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.

Before testing the hypothesis systematically in the context of energy spectra, it is desirable to compare the wall-parameter- and Kolmogorov-normalized spectra at a location very near the wall to evaluate which scaling is adequate. The results are shown in figures 5(a,b), which compare the wall-parameter-normalized spectrum $\phi _{u_2}(k^+_{x_3})$ at $x_2^+=2$ with the Kolmogorov-normalized spectrum $\phi _{u_2}(k^*_{x_3})$ at $x_2^*=1.5$ in a channel flow (Lee & Moser 2015, 2019). The comparison allows a critical evaluation of the two normalizations, using exaggerated horizontal and vertical axes. The same scales for the horizontal axis and two decades for the vertical axis are used in figures 5(a,b). These two locations ($x_2^+=2$ and $x_2^*=1.5$) are close to each other (see the inset of figure 2b). It can be seen that there is no collapse for $\phi _{u_2}(k_{x_3}^+)$, regardless of the wavenumber. Instead, the variation of $\phi _{u_2}(k_{x_3}^+)$ with $Re_\tau$ is systematic, implying that the scaling based on wall parameters is not tenable. In contrast, there is a good collapse for $\phi _{u_2}(k_{x_3}^*)$ in the high-wavenumber range (small-scale range). This is consistent with the prediction of the present hypothesis.

Figure 5. (a) Wall parameter-normalized spectrum $\phi _{u_2}(k^*_{x_3})$ at $x_2^+=2$ in a channel flow (Lee & Moser 2015, 2019). The arrow is in the direction of increasing $Re_\tau$. (b) Kolmogorov-normalized spectra at $x_2^*=1.5$, which is close to $x_2^+=2$ (see the inset in figure 2b). Note that when data were not obtained at exactly $x_2^+=2$ and $x_2^*=1.5$, interpolation was used, based on available spectra closest to these two locations.

We now test the hypothesis using spectra in both a channel flow and a boundary layer. According to (2.1), the spectra should be universal in the small-scale range, depending only on $x_2^*$ and $k^*$. In particular, at a given distance $x_2^*$ from the wall, energy spectra should depend only on $k^*$. Figure 6 shows the Kolmogorov-normalized $\phi _{u_1}(k^*_{x_3})$, $\phi _{u_2}(k^*_{x_3})$ and $\phi _{u_3}(k^*_{x_3})$ at $x_2^*=1.5$ in a channel flow and a boundary layer. At this very-near-wall location, all spectra collapse reasonably well in the small-scale range, except for $\phi _{u_1}(k^*_{x_3})$, which requires $Re_\tau \gtrsim 1000$ for $\phi _{u_1}(k^*_{x_3})$ to collapse over a large range of $k^*_{x_3}$. This implies that the threshold value of $Re_\tau$, beyond which the hypothesis is valid, may depend on the specific quantity investigated. For this reason, the spectra of ${u_1}$ in both the $x_1$ and $x_3$ directions at $Re_\tau =550$ will not be shown in figures 7–9. When $Re_\tau \gtrsim 1000$, the collapse of $\phi _{u_1}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ extends to $k_{x_3}^*\approx 0.015$ and 0.05, respectively. In contrast, there is adequate collapse at virtually all wavenumbers for $\phi _{u_2}(k_{x_3}^*)$ over the same range of $Re_\tau$ ($\gtrsim 1000$).

Figure 6. Kolmogorov-normalized spectra at $x_2^*=1.5$ in a channel flow and a boundary layer. The inset zooms in on the range $k_{x_3}^*=0.2\unicode{x2013}0.4$ for $\phi _{u_1}(k^*_{x_3})$ and $\phi _{u_3}(k^*_{x_3})$. Note that when data were not obtained at exactly $x_2^*=1.5$, interpolation was used, based on available spectra closest to this location.

Figure 7. Energy spectra in a boundary layer and a channel flow. Distributions of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ at (a) $x_2^*=10$ and (b) $x_2^*=50$. The plots use the wall-parameter-normalized spectra in a boundary layer (Simens et al. 2009; Jiménez et al. 2010; Borrell et al. 2013; Sillero et al. 2013) and the non-normalized spectra in a channel flow (Lee & Moser 2015, 2019). For clarity, the distributions at $x_2^*=50$ for $\phi _{u_2}({k_{x_3}^*})$ in (b) are shifted to the left by 0.5 of a decade. Note that the spectrum of ${u_1}$ at $Re_\tau =550$ is not shown for the reason mentioned in the context of figure 6. Note also that when data were not obtained at exactly $x_2^*=10$ and 50, interpolation was used, based on available spectra closest to these two locations.

Figure 8. Energy spectra in a channel flow. Distributions of $\phi _{u_1}(k_{x_1}^*)$, $\phi _{u_2}(k_{x_1}^*)$ and $\phi _{u_3}(k_{x_1}^*)$ at (a) $x_2^*=10$ and (b) $x_2^*=50$. The plots use the non-normalized spectra in a channel flow (Lee & Moser 2015, 2019). For clarity, the distributions for $\phi _{u_3}(k_{x_1}^*)$ in (a,b) are shifted to the left by 0.5 of a decade. Note that the spectrum of ${u_1}$ at $Re_\tau =550$ is not shown in (a,b) for the reason mentioned in the context of figure 6. Note also that when data were not obtained at exactly $x_2^*=10$ and 50, interpolation was used, based on available spectra closest to these two locations.

Figure 9. Energy spectra in a boundary layer and a channel flow. (a) Distributions of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ versus $x_2^*$ at $k_{x_3}^*=0.08$. (b) Distributions of $\phi _{u_1}(k_{x_1}^*)$, $\phi _{u_2}(k_{x_1}^*)$ and $\phi _{u_3}(k_{x_1}^*)$ versus $x_2^*$ at $k_{x_1}^*=0.06$. The plots use the wall-parameter-normalized spectra in a boundary layer (Simens et al. 2009; Jiménez et al. 2010; Borrell et al. 2013; Sillero et al. 2013) and the non-normalized spectra in a channel flow (Lee & Moser 2015, 2019). Note that the spectrum of ${u_1}$ at $Re_\tau =550$ is not shown in (a,b) for the reason mentioned in the context of figure 6. Note also that when data were not obtained at exactly $k_{x_3}^*=0.08$ and $k_{x_1}^*=0.06$, interpolation was used, based on available spectra closest to these two locations.

Figures 7(a,b) show distributions of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ in a channel flow and a boundary layer at two locations ($x_2^*=10$ and 50) over a wide range of $Re_\tau$ ($=550\unicode{x2013}5200$). Figures 8(a,b) show distributions of $\phi _{u_1}(k_{x_1}^*)$, $\phi _{u_2}(k_{x_1}^*)$ and $\phi _{u_3}(k_{x_1}^*)$ in a channel flow at the same locations. As expected, the collapse is nearly perfect over a wide range of relatively large wavenumbers after normalizing by the Kolmogorov scales (or equivalently, $\bar {\varepsilon }$ and $\nu$). When $Re_\tau \gtrsim 1000$, the collapse of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ extends to $k_{x_3}^*\approx 0.02$, 0.1 and 0.05, respectively, at $x_2^*=10$ (figure 7a). At a location further away from the wall ($x_2^*=50$, figure 7b), the collapse of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ can extend to $k_{x_3}^*\approx 0.03$, 0.02 and 0.02, respectively; note that $k_{x_3}^*\approx 0.02$ for $\phi _{u_2}(k_{x_3}^*)$ actually corresponds to $k_{x_3}^*\approx 0.04$ since the distributions of $\phi _{u_2}(k_{x_3}^*)$ are shifted to the left by 0.5 of a decade for clarity (see the caption of figure 7). A similar observation can be made for $\phi _{u_1}(k_{x_1}^*)$, $\phi _{u_2}(k_{x_1}^*)$ and $\phi _{u_3}(k_{x_1}^*)$ at $x_2^*=50$ in figure 8(b), which shows that the collapse can be observed over the range $k_{x_1}^*\gtrsim 0.01\unicode{x2013}0.03$ when $Re_\tau \gtrsim 1000$. It is worth mentioning that the collapse of $\phi _{u_1}(k_{x_1}^*)$ at $x_2^*=10$ in figure 8(a) can extend to $k_{x_1}^*\approx 0.001$, which is three decades larger than the Kolmogorov scales. It is clear that the wavenumber range over which the energy spectra collapse may depend not only on the specific small-scale quantity, but also on the location $x_2^*$. Another feature of figures 7 and 8 is that the collapse extends to increasingly smaller wavenumbers (larger scales) with increasing $Re_\tau$, except for $\phi _{u_2}(k_{x_1}^*)$ at $x_2^*=10$ (figure 8a), which collapses at all wavenumbers in the present $Re_\tau$ range. We recall that there is a strong departure from local isotropy in the context of the dissipation tensor in the region $x_2^*\lesssim 30$ (figure 3b). It is pertinent to identify the scales that characterize the anisotropy of $\bar {\varepsilon }$. Although not shown here, at $x_2^*=10$, the peaks of the dissipation spectra $k_{x_3}^{*2}\phi _{u_1}(k_{x_3}^*)$, $k_{x_3}^{*2}\phi _{u_2}(k_{x_3}^*)$ and $k_{x_3}^{*2}\phi _{u_3}(k_{x_3}^*)$ are located at $k_{x_3}^*\approx 0.11$, 0.19 and 0.09, respectively. On the other hand, a general definition of the Taylor microscale is ${\lambda } = {(5\nu )^{1/2}}\,{{{{\overline {{u_iu_i}} }^{1/2}}}}/{{{{\bar {\varepsilon } }^{1/2}}}}$. The estimated magnitude of ${\lambda }/\eta$ is $11.7\unicode{x2013}12.8$ at $x_2^*=10$ in the channel flow for $Re_\tau =550\unicode{x2013}5200$. Therefore, the locations of the peaks of the dissipation spectra are comparable to ${\lambda }$. Namely, the anisotropy of $\bar {\varepsilon }$ represents mainly the anisotropy of the scales around ${\lambda }$, at least at $x_2^*=10$.

Further, according to (2.1), the energy spectra should depend only on the location $x_2^*$ at a given small scale $k^*$. In order to demonstrate this, we report in figure 9(a) the distributions of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ versus $x_2^*$ in a channel flow and a boundary layer at a typical $k_{x_3}^*$ ($=0.08$). Similarly, figure 9(b) shows the distributions of $\phi _{u_1}(k_{x_1}^*)$, $\phi _{u_2}(k_{x_1}^*)$ and $\phi _{u_3}(k_{x_1}^*)$ versus $x_2^*$ in a channel flow at a typical $k_{x_1}^*$ ($=0.06$). Evidently, there is reasonably good collapse in the near-wall regions and, generally, the larger $Re_\tau$, the wider the $x_2^*$ range over which the collapse occurs. In order to quantify the $x_2^*$ range over which the spectra collapse at large $Re_\tau$, we report in figures 10(a,b) the energy spectra corresponding to figures 9(a,b) for only $Re_\tau \gtrsim 2000$. It is clear that the collapse can extend to $x_2^*\approx 200$ for $Re_\tau \gtrsim 2000$ in a boundary layer and a channel flow. Overall, figures 5(b) and 6–10 provide good support for the present hypothesis in the context of energy spectra.

Figure 10. Energy spectra in a boundary layer and a channel flow corresponding to figures 9(a,b) for only $Re_\tau \gtrsim 2000$. The insets zoom in on the range $x_2^*=100\unicode{x2013}800$.

Finally, we test the hypothesis in the context of the transport equations for $\bar {\varepsilon }$ and $\bar {\varepsilon }_\theta$ in a channel flow:

(3.2)\begin{align} &\underbrace { - \frac{\partial }{{\partial {x_2}}}\overline{{u_2}\varepsilon } }_{{T_u}}\underbrace {{}- 2\nu\,\overline{\frac{{\partial {u_i}}}{{\partial {x_2}}}\,\frac{{\partial {u_i}}}{{\partial {x_1}}}}\,\frac{{\partial {{\bar U}_1}}}{{\partial {x_2}}}}_{{P_U}}\underbrace {{}- 2\nu\, \overline{\frac{{\partial {u_1}}}{{\partial {x_k}}}\,\frac{{\partial {u_2}}}{{\partial {x_k}}}}\, \frac{{\partial {{\bar U}_1}}}{{\partial {x_2}}}}_{{P_m}}\underbrace {{}- 2\nu\, \overline{\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_l}}}{{\partial {x_k}}}\,\frac{{\partial {u_i}}}{{\partial {x_l}}}} }_{{P_u}}\nonumber\\ &\qquad \underbrace {{}- 2\nu\,\overline{{u_2}\,\frac{{\partial {u_1}}}{{\partial {x_2}}}}\, \frac{{{\partial ^2}{{\bar U}_1}}}{{\partial {x_2}\,\partial {x_2}}}}_{{P_g}}\underbrace {{}- 2\nu\, \frac{\partial }{{\partial {x_2}}}\overline{\frac{{\partial {u_2}}}{{\partial {x_k}}}\,\frac{{\partial p}}{{\partial {x_k}}}} }_\varPi \underbrace {{}- {\rm{2}}{\nu ^2}\,\overline{\frac{{{\partial ^2}{u_i}}}{{\partial {x_k}\,\partial {x_l}}}\,\frac{{{\partial ^2}{u_i}}}{{\partial {x_k}\,\partial {x_l}}}} }_D + \underbrace {\nu\,\frac{{{\partial ^{\rm{2}}}\bar \varepsilon }}{{\partial x_2^2}}}_{{T_\nu }}=0, \end{align}

(3.3)\begin{align} &\underbrace { - \frac{1}{Pr}\,\frac{{\partial \overline{{u_2}{{(\partial \theta /\partial {x_i})}^2}} }}{{\partial {x_2}}} }_{{T_{\theta u}}} \underbrace {{}- \frac{2}{Pr}\,\overline{\frac{{\partial {\theta }}}{{\partial x_i}}\,\frac{{\partial {\theta }}}{{\partial x_j}}\,\frac{{\partial u_j}}{{\partial x_i}}} }_{{P_{\theta u}}}\underbrace {{}- \frac{2}{Pr}\,\overline{\frac{{\partial {\theta }}}{{\partial x_2}}\,\frac{{\partial {\theta }}}{{\partial x_1}}}\,\frac{{\partial \bar U_1}}{{\partial x_2}}}_{{P_{\theta U}}}\underbrace {{}- \frac{2}{Pr}\,\overline{\frac{{\partial {\theta }}}{{\partial x_i}}\,\frac{{\partial {u_2}}}{{\partial x_i}}}\,\frac{{\partial {\varTheta }}}{{\partial x_2}}}_{{P_{\theta \varTheta }}}\nonumber\\ &\qquad \underbrace {{}- \frac{2}{Pr}\,\overline{u_2\,\frac{{\partial {\theta }}}{{\partial x_2}}}\, \frac{{{\partial ^2}{\varTheta }}}{{\partial x_2\,\partial x_2}}}_{{P_{\theta g}}}\underbrace {{}- \frac{2}{Pr^2}\,\overline{\frac{{{\partial ^2}{\theta }}}{{\partial x_j\,\partial x_i}}\,\frac{{{\partial ^2}{\theta }}}{{\partial x_j\,\partial x_i}}} }_{{D_\theta }} + \underbrace {\frac{1}{Pr^2}\,\frac{{{\partial ^2}{{\bar \varepsilon }_\theta }}}{{\partial x_2^{2}}}}_{{T_\kappa }}=0, \end{align}

where ${\bar U}_1$ is the mean velocity in the streamwise ($x_1$) direction, and $\varTheta$ is the mean scalar. In (3.2), the terms $T_u$, $\varPi$ and $T_\nu$ are the (large-scale) turbulent, pressure and viscous diffusion terms, respectively. The terms $P_U$, $P_m$ and $P_g$ are the (large-scale) mixed production, mean gradient production and gradient production terms, respectively. The terms $P_u$ and $D$ are the small-scale terms that represent the production of $\bar {\varepsilon }$ through the stretching of vorticity, and its destruction through the action of viscosity, respectively. Similarly, in (3.3), terms $T_{\theta u}$ and $T_\kappa$ are the (large-scale) turbulent and viscous diffusion terms, respectively. Terms $P_{\theta U}$, $P_{\theta \varTheta }$ and $P_{\theta g}$ are the (large-scale) mixed production, mean gradient production and gradient production terms, respectively. Terms $P_{\theta u}$ and $D_\theta$ are the small-scale terms that represent the production of $\bar {\varepsilon }_\theta$ due to stretching of the temperature field and its destruction by the thermal diffusivity, respectively. All terms in (3.2), after normalizing by wall parameters ($u_\tau$ and $\nu$) and Kolmogorov scales ($\eta$ and $u_K$; or equivalently, $\bar {\varepsilon }$ and $\nu$), are shown in figures 11(a,b), respectively. It can be observed from figure 11(a) that the magnitudes of both large- and small-scale terms in the region near the wall increase with $Re_\tau$, except for the gradient production term ($P_{g}$ term) and the pressure term ($\varPi$ term), which make only small contributions to (3.2). In contrast, the corresponding Kolmogorov-normalized distributions at ${Re}_{\tau }=180$ and 590 (figure 11b) follow each other closely and, for ${Re}_{\tau }\gtrsim 590$, collapse reasonably well in the region $x_2^*\lesssim 60$. All terms in (3.3), after normalizing by the wall parameters ($u_\tau$, $\theta _\tau$ and $\nu$) and the Batchelor–Kolmogorov scales ($\eta$, $\theta _B$ and $u_K$; or equivalently, $\bar {\varepsilon }$, $\bar {\varepsilon }_\theta$ and $\nu$), at $Pr=\nu /\kappa =0.71$ and four values of $Re_\tau$ ($=500, 1000, 2000, 5000$; the corresponding Péclet number is equal to $Re_\tau \,Pr$) are shown in figures 12(a,b). We can observe that there is a significant difference between the distributions normalized by wall parameters and those normalized by the Batchelor–Kolmogorov scales, for both large and small-scale terms. In the region near the wall, the magnitude of each term, normalized by wall parameters, increases systematically with $Re_\tau$ except for the gradient production term ($P_{\theta g}$ term), which makes only a small contribution to (3.3). However, the Batchelor–Kolmogorov normalized distributions collapse reasonably well in the region near the wall. As $Re_\tau$ increases, the $x_2^*$ range for this collapse widens. Figures 11(b) and 12(b) provide further strong support for the present hypothesis.

Figure 11. Distributions of terms in (3.2) in a channel flow. Normalizations based on (a) wall parameters and (b) Kolmogorov scales. They are plotted using the data of Vreman & Kuerten (2014a,b) and Kaneda & Yamamoto (2021). Note that for the data of Kaneda & Yamamoto (2021) at ${Re}_{\tau }=1000$, there are two simulations with effective grid spacings ${(\Delta {x_1})^ + } \times {(\Delta {x_2})^ + } \times {(\Delta {x_3})^ + }= 18.5 \times 0.6\unicode{x2013}0.8 \times 8.9$ and $7.8 \times 0.6\unicode{x2013}0.8 \times 4.2$. Only the latter dataset, corresponding to a higher resolution, is shown. Note that in order to obtain the Kolmogorov-normalized distributions in (b), the wall-parameter-normalized distributions in (a) are divided by $(u_K^ +/\eta ^+)^3$ since their relation in the context of transport equation for $\bar {\varepsilon }$ is given by $[\text {wall scaling}] = [\text {Kolmogorov scaling}](({{u_K^4}}/{{{\eta ^2}}})({{\delta _\upsilon ^2}}/{{u_\tau ^4}})) =[\text {Kolmogorov scaling}] ({{u_K^{ + 3}}}/{{{\eta ^{ + 3}}}}).$

Figure 12. Distributions of terms in (3.3) for $Pr=0.71$ in a channel flow. Normalizations based on (a) wall parameters and (b) Batchelor–Kolmogorov scales, using the information in Alcántara-Ávila et al. (2021), as explained in Appendix A.

4. Discussion

4.1. $x_2^*$ range over which present hypothesis applies

It is of interest to identify the $x_2^*$ range of validity for the present hypothesis. The following further comments can be made with regard to figures 4, 10, 11(b) and 12(b).

(i) Figure 4 shows that the distributions of ${\mathsf{R}}_i^*$ ($i=1,2,3$) in a boundary layer collapse well with the channel data in the range $x_2^*\lesssim 70$ (or equivalently $x_2^+\lesssim 213$ ($\approx 0.11\delta$) in the boundary layer at $Re_\tau =1988$). In contrast, in the channel flow, the collapse of the ${\mathsf{R}}_i^*$ distributions at $Re_\tau =2000$ and 5200 with the distribution at $Re_\tau =10^4$ can extend to the channel centreline. A close look at the ${\mathsf{R}}_i^*$ distributions in the channel flow shows that the departure from the isotropic value of $2/3$ is smaller than 10 % and 5 % when $x_2^*\gtrsim 70$ and 150, respectively. Evidently, the channel flow approaches local isotropy gradually, which is a requirement of the KOC theory, with increasing $x_2^*$.

(ii) It can be observed from figures 10(a,b) that the collapse of the energy spectra at two typical scales ($k_{x_3}^*=0.08$ and $k_{x_1}^*=0.06$) can extend to $x_2^*\approx 200$ for $Re_\tau \gtrsim 2000$ in a boundary layer and a channel flow. At $Re_\tau \approx 2000$, $x_2^*\approx 200$ corresponds to $x_2^+\approx 986$ ($\approx 0.49\delta$) in a channel flow and $x_2^+\approx 795$ ($\approx 0.40\delta$) in a boundary layer. We should stress that the $x_2^*$ range over which the energy spectra collapse can widen if we focus on a smaller scale (or larger wavenumber). In order to demonstrate this, we report in figure 13(a) the distributions of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ versus $x_2^*$ in a channel flow at $k_{x_3}^*=0.45$. We can observe that the collapse of the energy spectrum at $Re_\tau =2000$ with the spectrum at $Re_\tau =5200$ can extend to the channel centreline. In particular, in the region $x_2^*\gtrsim 60$, all the spectra become approximately constant. This implies that beyond this location, the energy spectra depend only on the wavenumber (and not on $x_2^*$). Consequently, all the energy spectra in the region $x_2^*\gtrsim 60$ should collapse, which is the prediction of the KOC theory. This is indeed supported by figure 13(b), which shows the distributions of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ versus $k_{x_3}^*$ at two locations ($x_2^*=60$ and 200) marked in figure 13(a); at $Re_\tau =2000$, $x_2^*\approx 60$ and 200 correspond to $x_2^+\approx 181$ ($\approx 0.09\delta$) and $x_2^+\approx 985$ ($\approx 0.50\delta$), whereas at $Re_\tau =5200$, $x_2^*\approx 60$ and 200 correspond to $x_2^+\approx 179$ ($\approx 0.03\delta$) and $x_2^+\approx 912$ ($\approx 0.18\delta$), respectively. This is also consistent with the collapse of the experimental data for $\phi _{u_1}(k_{x_1}^*)$ at various locations (away from the wall) in the pipe flow and the boundary layer at higher $Re_\tau$ (Rosenberg et al. 2013; Vallikivi et al. 2015). From the above discussion with respect to ${\mathsf{R}}_i^*$ and the energy spectra, it is obvious that the KOC theory represents a special case of the present hypothesis.

Figure 13. (a) Distributions of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ versus $x_2^*$ in a channel flow at $k_{x_3}^*=0.45$. Pink and green curves correspond to $Re_\tau =2000$ and 5200, respectively. The vertical lines indicate the locations at $x_2^*=60$ and 200, respectively. (b) Corresponding distributions of $\phi _{u_1}(k_{x_3}^*)$, $\phi _{u_2}(k_{x_3}^*)$ and $\phi _{u_3}(k_{x_3}^*)$ versus $k_{x_3}^*$ at $x_2^*=60$ (blue curves) and 200 (red curves), respectively. Dashed and solid curves correspond to $Re_\tau =2000$ and 5200, respectively. They are plotted using the non-normalized spectra in a channel flow (Lee & Moser 2015, 2019).

(iii) In figure 11(b), the collapse can be observed for all terms in (3.2) in the region $x_2^*\lesssim 60$ for $Re_\tau \gtrsim 590$. At $Re_\tau =590$, $x_2^*\approx 50$, 60 and 80 correspond to $x_2^+\approx 147$ ($\approx 0.25\delta$), $x_2^+\approx 191$ ($\approx 0.32\delta$) and $x_2^+\approx 296$ ($\approx 0.50\delta$), respectively.

(iv) In figure 12(b), the collapse of all terms in (3.3) can extend to $x_2^*\approx 85$, 140 and 240 for $Re_\tau \gtrsim 500$, 1000 and 2000, respectively. The corresponding $x_2^+$ values are $342$ ($\approx 0.68\delta$), 662 ($\approx 0.66\delta$) and $1360$ ($\approx 0.68\delta$) at $Re_\tau =500$, 1000, and 2000, respectively. From the above results, it is remarkable that the $x_2^*$ range over which the hypothesis is valid extends generally from the wall to a location in the outer region for the small-scale quantities examined here. This $x_2^*$ range depends on the specific small-scale quantity. Generally, for a given small-scale quantity, the larger the Reynolds number, the wider the $x_2^*$ range over which this quantity is $Re_\tau$-independent.

4.2. Relationships to wall scaling and outer scaling for small-scale quantities

As pointed out in the context of figure 1, the collapse of $x_2^+\bar {\varepsilon } ^ +$ over a certain $x_2^+$ range implies that the wall parameters and the Kolmogorov scales can be used interchangeably in this $x_2^+$ range in the channel flow. This can be extended in a more general way. If $x_2^+\bar {\varepsilon } ^ +$ and $x_2^+\bar {\varepsilon }_\theta ^ +$ can collapse onto unique curves in all turbulent wall flows over a certain $x_2^+$ range – which has not yet been established – then it is not difficult to conclude that the wall parameters and the Kolmogorov–Batchelor scales can be used interchangeably. However, if different wall flows have different $Re_\tau$-independent $x_2^+\bar {\varepsilon } ^ +$ (and $x_2^+\bar {\varepsilon }_\theta ^ +$) distributions at large $Re_\tau$, unlike the Kolmogorov–Batchelor normalized distributions, which are universal as shown in our paper, then the wall-parameter-normalized distributions should collapse onto different curves in different wall flows.

The same remark can be made with regard to the relationship between normalizations based on Kolmogorov–Batchelor and outer scales. In particular, it should be mentioned that if there is an overlap between the wall parameters scaling and outer scaling in the context of $\bar {\varepsilon }$ (Abe & Antonia 2016) (and also $\bar {\varepsilon }_\theta$; Abe & Antonia 2017), then at least for a given flow, all three types of scaling, i.e. the Kolmogorov–Batchelor scales, wall parameters and outer scales, can be used interchangeably. However, when the Kolmogorov–Batchelor normalized small-scale quantities are independent of $x_2^*$, such as the energy spectra at large $x_2^*$ discussed in the context of figure 13, the outer scales and the Kolmogorov–Batchelor scales can be interchanged only when $\bar {\varepsilon } \delta /u_\tau ^3$ (and also $\bar {\varepsilon }_\theta \delta /(u_\tau \theta _\tau ^2)$) is constant (independent of $x_2/\delta$). Similarly, when the Kolmogorov–Batchelor normalized small-scale quantities are independent of $x_2^*$, the wall parameter and the Kolmogorov–Batchelor scales are interchangeable when $\bar {\varepsilon } ^+$ (and also $\bar {\varepsilon }_\theta ^+$) is constant (independent of $x_2^+$).

4.3. Behaviour of $\overline {{u_i}{u_i}}^*$ and $\overline {{\theta }{\theta }}^*$ as $x_2^*\to 0$

The present hypothesis has been proposed for small-scale quantities in wall turbulence and thus does not require $\overline {{u_i}{u_i}}^*$ and $\overline {{\theta }{\theta }}^*$ to be universal. However, in the region $x_2^*\to 0$, $\overline {{u_i}{u_i}}$ and $\overline {{\theta }{\theta }}$ are related directly to ${\bar {\varepsilon } }$ and ${\bar {\varepsilon }}_\theta$ (e.g. Pope 2000; Alcántara-Ávila et al. 2021), i.e.

(4.1a,b)\begin{equation} \bar{\varepsilon} = \frac{1}{2}\,\upsilon\,\frac{{{\partial ^2}\overline{{u_i}{u_i}} }}{{\partial x_2^2}},\quad {\bar{\varepsilon} _\theta } = \frac{1}{2}\,\kappa\,\frac{{{\partial ^2}\overline{{\theta \theta}} }}{{\partial x_2^2}}. \end{equation}

One thus expects $\overline {{u_i}{u_i}}^*$ and $\overline {{\theta }{\theta }}^*$ to be universal as $x_2^*\to 0$. After normalization by the Batchelor–Kolmogorov scales, (4.1a,b) can be rewritten as

(4.2a,b)\begin{equation} \frac{{{\partial ^2}\overline{{u_i}{u_i}}^* }}{{\partial x_2^{*2}}}=2, \quad \frac{1}{Pr}\,\frac{{{\partial ^2}\overline{{\theta}{\theta}}^* }}{{\partial x_2^{*2}}}=2. \end{equation}

Note that we have assumed $u_K$, $\eta$ and $\theta _B$ to be approximately constant as $x_2^+\to 0$. These assumptions are justifiable. For example, figure 3 of Antonia et al. (1991) indicates that $\eta ^+$ is approximately constant for $x_2^+=0.1\unicode{x2013}1$ at $Re_\tau =180$ and 395, respectively. Although not shown here, the distributions of $\eta ^+$, $u_K^+$ and $\theta _K^+$ versus $x_2^+$ for the present channel data at $Re_\tau =500\unicode{x2013}10^4$ indicate that they are approximately constant for $x_2^+\lesssim 1$ at each $Re_\tau$. Integrating (4.2a,b) twice with respect to $x_2^*$ leads to

(4.3a,b)\begin{equation} \overline{{u_i}{u_i}}^* ={{ x_2^{*2}}}, \quad \overline{{\theta}{\theta}}^* =Pr\,x_2^{*2}. \end{equation}

Figure 14 shows that both $\overline {{u_i}{u_i}}^*$ and $\overline {{\theta }{\theta }}^*$ are independent of $Re_\tau$ in the range $x_2^*\lesssim 1$. In particular, (4.3a,b) are satisfied adequately in the region $x_2^*\lesssim 0.2$ for $\overline {{u_i}{u_i}}^*$ and $x_2^*\lesssim 1$ for $\overline {{\theta }{\theta }}^*$.

Figure 14. Distributions of $\overline {u_iu_i}^*/x_2^{*2}$ in a channel flow and a boundary layer. The solid pink, red, blue, black and cyan curves correspond to the channel data at $Re_\tau =550$, 1000, 2000, 5200 and $10^4$, respectively. The dash-dotted green, blue and pink curves correspond to the boundary layer data at $Re_\tau =$445, 1307 and 1988, respectively. Also shown are the distributions of $\overline {\theta \theta }^*/x_2^{*2}$ in a channel flow; the dashed black, red, blue and pink curves correspond to $Re_\tau =$500, 1000, 2000 and 5000, respectively. They are plotted using the data of Simens et al. (2009), Jiménez et al. (2010), Borrell et al. (2013), Sillero et al. (2013), Lee & Moser (2015, 2019) and Alcántara-Ávila et al. (2021). The horizontal lines indicate the values of 1 and $Pr$ ($=0.71$).

5. Concluding remarks

We have proposed a hypothesis that describes the behaviour of small-scale wall turbulence reasonably well. The evidence presented indicates that the larger the Reynolds number, the larger the distance from the wall and the range of turbulent scales over which this hypothesis applies. More specifically, at a given $x_2^*$, collapse of a given small-scale quantity generally extends to increasingly larger values of $r^*$ (or smaller wavenumbers) as the Reynolds number increases. Similarly, the collapse of a given small-scale quantity extends to increasingly larger values of $x_2^*$ as the Reynolds number increases. We stress that the present hypothesis has been tested and validated mainly in a channel flow and a boundary layer using a relatively large amount of data related mainly to second-order statistics. Direct numerical simulations (DNS) data corresponding to third-order statistics (figure 11b) also support the hypothesis. It is now desirable to test the hypothesis against other third- and higher-order statistics, e.g. the third- and fourth-order moments of different velocity derivatives. The statistics examined here have validated adequately the claim of independence with respect to $Re_\tau$. Unfortunately, the data have been obtained mainly in two smooth wall flows (the channel and the boundary layer). It would certainly be desirable to test the hypothesis further in a fully developed pipe flow, preferably using DNS data where $Re_\tau$ exceeds 500, and more generally in flows over different types of surfaces.

The present hypothesis should greatly reduce the complexity of wall turbulence and also enhance our understanding of the physical mechanisms in wall flows. We should underline that an important application of the hypothesis is the ability to describe adequately the behaviour of small-scale quantities in wall turbulence. The collapse of the small-scale statistics shown in figures 4, 5(b), 6–10, 11(b), 12(b) and 13 suggests that the present hypothesis will underpin future developments of new wall turbulence models and thus should lead to a significant improvement in wall turbulence calculations. Since the accuracy of the hypothesis can only improve as $Re_\tau$ keeps increasing, calculations based on this hypothesis should be especially suitable in the context of very-large-Reynolds-number engineering and meteorological flows. Finally, a significant amount of evidence has shown that the present hypothesis is valid for small-scale velocity quantities in wall turbulence. In view of the similarity between the small-scale velocity and scalar ($Pr\approx 1$) fields (e.g. Abe et al. 2009; Antonia et al. 2009), one would expect that the present hypothesis will also be satisfied by small-scale scalar quantities in wall turbulence. The collapse in figure 12(b) is encouraging in this respect and should spawn future investigations.