2. The inner asymptotic expansion of $\langle uu\rangle ^+$ and its inner peak

The inner asymptotic expansion of the streamwise normal stress $\langle uu\rangle ^+$ for large $Re_{\tau }$ is extracted from the channel DNS of table 1 in a similar fashion as the mean velocity expansion in Monkewitz (Reference Monkewitz2021), i.e. without recourse to a model.

Table 1.

Channel DNS profiles used to determine the first two terms of the asymptotic expansion (2.1).

Generalizing the Reynolds number dependence of the gauge function in (3.1) of Monkewitz (Reference Monkewitz2021) from $Re_{\tau }^{-1}$ to $\varPhi (Re_{\tau })$, the first two terms of the asymptotic expansion

(2.1)\begin{equation} \langle uu\rangle^+_{{DNS}} = f(y^+) + \varPhi(Re_{\tau}) g(y^+), \end{equation}

are obtained for various gauge functions $\varPhi$ from pairs of DNS profiles at different $Re_{\tau }$. A good collapse of the $f$ and $g$ values from different profile pairs signifies that the gauge function $\varPhi$ has been properly chosen and that higher-order terms in the expansion (2.1) are small or absent.

It turns out that $\varPhi = Re_{\tau }^{-1/4}$ and $\varPhi = 1/\ln Re_{\tau }$ both produce a good collapse of the functions $f$ and $g$ obtained from all possible profile pairs in table 1, with $f(y^+)$ the finite limit of $\langle uu\rangle ^+$ for $Re_{\tau }\to \infty$. The choice of $\varPhi = Re_{\tau }^{-1/4}$ for the present paper, proposed by CS2021, is based on the following considerations:

1. (i) Arguments in favour of $\varPhi = Re_{\tau }^{-1/4}$: the decomposition (2.1) with $\varPhi = Re_{\tau }^{-1/4}$ is shown in figure 1 and the collapse from different profile pairs on the fit (2.2) is seen to be rather good up to $y^+$ of approximately 200. Furthermore, the leading term of the Taylor expansion (2.3) of (2.2) about the wall, shown in figure 1 as dotted lines, corresponds to an upper bound of $1/4$ for the dissipation rate at the wall, as argued by CS2021. They have furthermore argued, that the coefficient of $(y^+)^2$ in the Taylor expansion about the wall and the inner peak height $\langle uu\rangle ^+_{{IP}}$ are strictly proportional. Without having imposed this constraint on the fit (2.2), it yields a near perfect proportionality between (2.4) and the coefficient of $(y^+)^2$ in (2.3), the proportionality factor being $45(1 + 0.11/Re_{\tau })$. This proportionality has recently received strong support from the extensive data analyses of Hultmark & Smits (Reference Hultmark and Smits2021) and Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021), who found a ratio of 46, independent of Reynolds number.

2. (ii) Arguments against $\varPhi = 1/\ln Re_{\tau }$: the decomposition (2.1) with $\varPhi = 1/\ln Re_{\tau }$, on the other hand, is shown in figure 1 of the supplementary material and is seen to produce an equally good collapse of the $f$ and $g$ functions from different DNS pairs. The Taylor expansion of $\langle uu\rangle ^+$ about the wall, $(0.30 - 0.86/\ln Re _{\tau })(y^+)^2 + \cdots$ is also in good agreement with Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021). However, as the coefficient of $(y^+)^2$ exceeds the limit of $1/4$ inferred by CS2021, this scaling is not pursued further.

3. (iii) Arguments against $\varPhi = \ln Re_{\tau }$: the choice of $\varPhi = \ln Re_{\tau }$ in (2.1), finally, which corresponds to the inner peak scaling of the attached eddy model (Marusic & Monty Reference Marusic and Monty2019), produces no comparable collapse of the $f$ and $g$ values from different DNS pairs of table 1, as seen in figure 2 of the supplementary material, and, of course, $\langle uu\rangle ^+_{{IP}} \to \infty$ for $Re_{\tau }\to \infty$. For the simplest case of channel flow, the transport equation for $\langle uu\rangle ^+$ (see for instance Hinze (Reference Hinze1975), (4) and (5)) is in all likelihood unbalanced with this scaling: based on the available profiles, notably those of Lee & Moser (Reference Lee and Moser2015) at $Re_{\tau } = 5186$, the viscous transport term $\mathcal {D}^+ = (1/2)[\textrm {d}^2 \langle uu\rangle ^+/(\textrm {d} y^+)^2]$ becomes negative beyond $y^+ \approxeq 5$. In the neighbourhood of $y^+_{{IP}} \approxeq 15$, it may be approximated by $-\langle uu\rangle ^+_{{IP}} (y^+_{{IP}})^{-2}$. Hence, with $\langle uu\rangle ^+_{{IP}}$ scaling as $\ln Re_{\tau }$, the viscous transport term in the neighbourhood of the inner peak goes to negative infinity for $Re_{\tau }\to \infty$. For the DNS of Lee & Moser (Reference Lee and Moser2015) at $Re_{\tau } = 5186$, all the other terms of the transport equation are negative around $y^+_{{IP}}$, except the production term, which is limited to $1/4$ (see e.g. Pope (Reference Pope2000), exercise 7.6). With the reasonable assumption that, between $Re_{\tau } = 5186$ and infinity, no term of the transport equation for $\langle uu\rangle ^+$ becomes positive unbounded in a neighbourhood of $y^+_{{IP}}$, the equation is clearly asymptotically unbalanced. The choice of $\varPhi = \ln Re_{\tau }$ is therefore abandoned at this point.

Figure 1.

The ${O}(1)$ (a) and ${O}(1/Re_{\tau }^{1/4})$ (b) components of $\langle uu\rangle ^+$ extracted from DNS pairs of table 1: (red) profiles 1 and 2 (—), 1 and 3 (- - -), 1 and 4 ($\cdots$), 1 and 5 ($-\cdot -$); (green) 2 and 3 (—), 2 and 4 (- - -), 2 and 5 ($\cdots$); (blue) 3 and 5 (—), 4 and 5 (- - -). Panel (a) (black) — and $-\cdot -, {O}(1)$ part of (2.2) with and without hump. Panel (b) (black) — and $-\cdot -, {O}(1/Re_{\tau }^{1/4})$ part with and without hump; (black) - - -, logarithmic slopes modified by $\pm 5\,\%$.

Proceeding with the asymptotic sequence $\{1, Re_{\tau }^{-1/4},\ldots \}$, the functions $f$ and $g$ of figure 1 need to be fitted in order to generate $\langle uu\rangle ^+$ profiles for any Reynolds number. Two features of $f$ and $g$ will prove to be important for the construction of the complete inner asymptotic expansion: the expected ‘hump’ at $y^+\approxeq 15$, and the rather clear logarithmic region between $y^+\approx 60$ and 200, with a logarithmic slope that decreases as $Re_{\tau }^{-1/4}$ from its maximum of 0.85 at infinite $Re_{\tau }$ (figure 1a). For the ${O}(1)$ part $f$, the fit $\mathcal {M}_2$ (A1), constructed from a Padé approximant for the derivative, analogous to the construction of the Musker mean velocity profile (Musker Reference Musker1979), has been developed and is supplemented by the ‘hump’ function $\mathcal {H}$ (A4). The ${O}(Re_{\tau }^{-1/4})$ part $g$ is well fitted by the ‘corner function’ $\mathcal {C}$ (A5) plus a ‘hump.’ Hence, the near-wall stress, up to higher-order terms (HOT), is described by

(2.2)\begin{align} \langle uu\rangle^+_{{wall}} &= \mathcal{M}_2(y^+; 0.25, 373, 1.7, 8.8616) + \mathcal{H}(y^+; 4, 1.3, 15) + \nonumber\\ &\quad + \frac{1}{Re_{\tau}^{1/4}}\{\mathcal{C}(y^+; 3.1623, -8.4, 2) + \mathcal{H}(y^+; -3.7, 1, 13)\} + \mathrm{HOT.} \end{align}

To go beyond ${O}(Re_{\tau }^{-1/4})$, the Reynolds number range of available DNS and their mutual consistency are insufficient.

From (2.2) and (A3), one readily obtains the Taylor expansion of $\langle uu\rangle ^+$ about the wall as

(2.3)\begin{align} \langle uu\rangle^+_{{wall}}(y^+\to 0) = (0.25 - 0.42 Re_{\tau}^{{-}1/4})(y^+)^2- 0.02(1 - Re_{\tau}^{{-}1/4})(y^+)^4 + \cdots .\end{align}

As noted above in point (i), the coefficient of $(y^+)^2$ in (2.3) fits all the $\overline {b_1^2}$ in table 1 of Hultmark & Smits (Reference Hultmark and Smits2021) to within less than $1\,\%$, which is not surprising as they used the same DNS data. The upper limit of 1/4 for the coefficient of $(y^+)^2$ in (2.3) is also consistent, within uncertainty, with the value of 0.26, obtained by Monkewitz & Nagib (Reference Monkewitz and Nagib2015, figure 6 and (2.19)) for the ZPG TBL.

Next, the inner peak height $\langle uu\rangle ^+_{{IP}}$ at $y^+ \approxeq 15$ is obtained from (2.2) or figure 1 as

(2.4)\begin{equation} \langle uu\rangle^+_{{IP}} = 11.3 - 17.7 Re_{\tau}^{{-}1/4} , \end{equation}

to be compared in figure 2 with some laboratory and computational data for channel, pipe, ZPG TBL and Couette flows. Also included in the figure are the correlation $11.5 - 19.3Re_{\tau }^{-1/4}$ of CS2021, which stays within 2 % of (2.4) for all $Re_{\tau }$, and $3.54 + 0.646\ln Re _{\tau }$ of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), which starts to deviate more than +2 % beyond a $Re_{\tau }$ of 30 000.

Figure 2.

Inner peak height $\langle uu\rangle ^+_{{IP}}$ vs. $Re_{\tau }$: (red) —, $\cdots$, (2.4) $\pm 2\,\%$; (black) - - -, $11.5 - 19.3Re_{\tau }^{-1/4}$ of CS2021; (black) $-\cdot \cdot -, 3.54 + 0.646\ln Re _{\tau }$ of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). Channel DNS ($\bullet$): (red) DNS of table 1, (dark red) DNS of Bernardini, Pirozzoli & Orlandi (Reference Bernardini, Pirozzoli and Orlandi2014), (yellow) DNS of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014). Pipe ($\blacklozenge$): (pink) DNS of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), (blue) Superpipe NSTAP data of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012), (purple) corrected and uncorrected ($\lozenge$) CICLoPE hot-wire data of Fiorini (Reference Fiorini2017). ZPG TBL ($\blacksquare$): (green) Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013), (orange) Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). Couette ($\blacktriangle$): (green) Kraheberger et al. (Reference Kraheberger, Hoyas and Oberlack2018).

While most of the experimental and DNS data are within $\pm$2 % of (2.4), there are two notable exceptions: the Couette data point of Kraheberger, Hoyas & Oberlack (Reference Kraheberger, Hoyas and Oberlack2018), for which no explanation can be offered at this time, and the Superpipe NSTAP data for $Re_{\tau } > 5400$. A tentative explanation for the low $\langle uu\rangle ^+_{{IP}}$ values in the Superpipe is given in Appendix B.

With (A2) and (A5) of Appendix A, the large $y^+$ asymptote of the profile (2.2) is the logarithmic law

(2.5)$$\begin{gather} \langle uu\rangle^+_{{wall}}(y^+{\gg} 1) = S_{{wall}}\ln y^+{+}C_{{wall}} \quad \mathrm{with} \end{gather}$$

(2.6)$$\begin{gather}S_{{wall}} = 0.85 - 8.4 Re_{\tau}^{{-}1/4},\quad C_{{wall}} = 5.251 + 9.671 Re_{\tau}^{{-}1/4}. \end{gather}$$

The logarithmic asymptotes (2.5) of $\langle uu\rangle ^+_{{wall}}$ for different $Re_{\tau }$ are visualized in figure 3 by the fan of straight dashed lines intersecting at $y^+\approx 3$, and allow the completion of the inner expansion even though they are followed by the data only over a short interval. The logarithmic slope $S_{{wall}}$ of these asymptotes, negative at low $Re_{\tau }$, is seen to become positive at $Re_{\tau } \approx 10^4$. A short log law of $\langle uu\rangle ^+$, albeit with a fixed slope, has already been seen by Hultmark (Reference Hultmark2011, (4.3)) in the same $y^+$-range of the Superpipe, but the role of probe corrections remains an issue (see Appendix B). More recently, Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018, figure 4) have developed a logarithmic fit of $\langle uu\rangle ^+$ for the region leading up to the outer peak, which, at $Re_{\tau }=20\,000$, is within 4 % of (2.5) over the interval $10^2\leq y^+\leq 10^3$. This establishes the relation between the change of sign of $S_{{wall}}$ in (2.6) and the emergence of an outer peak in the $\langle uu\rangle ^+$ profile, known for over 20 years to appear at high $Re_{\tau }$ (see, for instance, Fernholz & Finley Reference Fernholz and Finley1996).

Figure 3.

Solid lines: composite profiles of $\langle uu\rangle ^+$ for $Re_{\tau } = 1000$ (green), 1995 (blue), 5186 (red), 20 250 (green), 98 190 (blue), $10^6$ (red), $10^7$ (green), $10^9$ (blue), $\infty$ (red). - - -, DNS profiles 5 (green), 4 (blue) and 1 (red) of table 1. Symbols: $\vartriangle \vartriangle \vartriangle$, NSTAP Superpipe profiles of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012) at $Re_{\tau } = 1985$, 5411, 20 250 and 98 187; $\square$, ZPG TBL profile of Sillero et al. (Reference Sillero, Jiménez and Moser2013) for $Re_{\tau }= 1989$ (blue) and of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) for $Re_{\tau } = 14\,500$ (orange). Note that TBL profiles have not been used for the construction of the composite expansion and are shown for comparison only. Long grey dashes (red for $Re_{\tau } = \infty$), wall log laws (2.5) for the $11\ Re_{\tau }$; short grey dashes (red for $Re_{\tau } = \infty$), corresponding overlap log laws (2.9) with $\circ$ marking their ‘end points’ at $y^+ = Re_{\tau }$; $\blacklozenge$, intersections of wall and overlap log laws at $y^+_\times$ (2.7); — (grey), fit (3.3) of $\langle uu\rangle ^+_{{CL}}$, with $\bullet$ marking the fit at the $Re_{\tau }$ values of the profiles shown; $\cdots$, leading term $0.25(y^+)^2$ of the Taylor expansion about the wall; $- \cdot -$, logarithmic slope of $-1.26$.

To actually form such an outer peak, the wall asymptote (2.5) has to cross over to a decay law at some $y^+_\times$. This cross-over location $y^+_\times$ and the decay law beyond $y^+_\times$ could in principle be extracted from DNS data in a manner similar to the determination of $\langle uu\rangle ^+_{{wall}}$. Due to the complexity of the expansion and the limitations of the DNS data, this has not been possible. A first indication on the value of $y^+_\times$ comes from figure 1, which shows that the channel DNS closely follows the wall log law (2.5) up to $y^+\approx 200$, implying that $y^+_\times$ must be larger than 200.

Turning to a straight fit of $y^+_\times$, all the data between $Re_{\tau } = 10^3$ and $10^5$ are seen in figure 3 to be compatible with $y^+_\times$ equal to a simple constant

(2.7)\begin{equation} y^+_\times{=} 470\quad \text{and}\quad \langle uu\rangle^+_\times{\equiv} \langle uu\rangle^+_{{wall}}(y^+_\times) = 10.48 - 42.0 Re_{\tau}^{{-}1/4}, \end{equation}

with the corresponding $\langle uu\rangle ^+_\times$ following from ((2.5), (2.6)). To guide the eye, the points $(y^+_\times, \langle uu\rangle ^+_\times )$ are marked by $\blacklozenge$ for the profiles of figure 3. It is important to note here, that $\langle uu\rangle ^+_\times$ is not the outer maximum nor $y^+_\times$ its location, but the intersection of the logarithmic asymptote (2.5) with the asymptotic logarithmic decay law of the overlap region. As the actual outer peak height and its location depend on the slopes of both asymptotes, and on the manner the corner between them is smoothed, its detailed discussion is postponed to the short § 4, after the complete composite expansion is established in § 3.

The adoption of a constant $y^+_\times$ for all $Re_{\tau }$ means that the inner expansion, which cannot end at a finite value of the inner coordinate, extends beyond the cross-over point $y^+_\times$ into the region of logarithmic decay, where it overlaps with the outer expansion. Hence, the complete inner expansion is obtained by adding a corner function (A5) to $\langle uu\rangle ^+_{{wall}}$ of (2.2)

(2.8)\begin{equation} \langle uu\rangle^+_{{in}} = \langle uu\rangle^+_{{wall}} + \mathcal{C}(y^+; y^+_\times, \Delta S, 2), \end{equation}

and its limit for $y^+ \gg y^+_\times \gg 1$ yields the asymptotic logarithmic overlap layer, i.e. the common part of inner and outer expansions

(2.9)\begin{align} \langle uu\rangle^+_{{cp}} \equiv \langle uu\rangle^+_{{in}}(y^+{\gg} y^+_\times{\gg} 1) &= S_{{cp}}[\ln y^+{-} \ln y^+_\times ] + \langle uu\rangle^+_\times{+} \mathrm{HOT} \nonumber\\ \mathrm{with}\quad S_{{cp}} &= S_{{wall}} + \Delta S. \end{align}

The logarithmic slope $S_{{cp}}$ of the overlap log law (2.9) must be determined by matching to the outer expansion in § 3. At this point it can only be said that $S_{{cp}}$ must be negative to form an outer peak at large $Re_{\tau }$. Furthermore, it must go to zero at infinite $Re_{\tau }$, as shown in figure 3, because $\langle uu\rangle ^+$ near the wall has been shown to remain finite for all $Re_{\tau }$. This implies that $\Delta S$ is of the form $\Delta S = -0.85 + \Delta S'$, with $\Delta S'$ vanishing for $Re_{\tau }\to \infty$, to compensate the ${O}(1)$ contribution to $S_{{wall}}$ in (2.6).

3. The outer and composite expansions of $\langle uu\rangle ^+$

Moving on to the outer expansion, it is written as a logarithmic part matching the common part (2.9) for small $Y$ and satisfying the symmetry condition on the centreline $Y=1$, plus a wake part $\mathcal {W}(Y)$, which goes to zero for $Y\to 0$

(3.1)\begin{equation} \langle uu\rangle^+_{{out}} = S_{{out}}\ln\left[Re_{\tau} Y \left(1 - \frac{Y}{2}\right)\right] - S_{{out}}\ln y^+_\times{+} \langle uu\rangle^+_\times{+} \mathcal{W}(Y) + \mathrm{HOT.} \end{equation}

The matching of outer and inner expansions furthermore requires $S_{{out}}$ to be identical to $S_{{cp}}$ in (2.9).

To obtain the logarithmic slope $S_{{out}} = S_{{cp}}$, the outer expansion (3.1) is evaluated at $Y=1$ and identified with the fit (3.3) of centreline stress

(3.2)\begin{align} \langle uu\rangle^+_{{CL}} &= S_{{out}}\ln\left(\frac{Re_{\tau}}2\right) - S_{{out}}\ln y^+_\times{+} \langle uu\rangle^+_\times{+} \mathcal{W}(1) \end{align}

(3.3)\begin{align} &= 0.55 + [0.1007 + 33Re_{\tau}^{{-}1/4}]^{{-}1} . \end{align}

As seen in figure 3, this fit reproduces the channel and Superpipe centreline data up to $Re_{\tau } = 10^5$. At higher $Re_{\tau }, \langle uu\rangle ^+_{{CL}}$ increases to the infinite Reynolds number limit of $\langle uu\rangle ^+_{\times } = 10.48$, such that $\langle uu\rangle ^+_{{out}}$ becomes a simple constant throughout the channel or pipe. Note that the fit (3.3), which relies strongly on the outer Superpipe data of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012), implies that differences are relatively minor between the outer expansions for pipe and channel (see comments in § 5).

What is still missing for the determination of the outer logarithmic slope $S_{{out}}$ is the wake function $\mathcal {W}(Y)$. The fit with the requisite symmetry properties

(3.4)\begin{equation} \mathcal{W}(Y) = S_{{out}}\ln\left[\frac{1}4 + \frac{3}4(1-Y)^2\right], \end{equation}

is again developed from both channel DNS and Superpipe NSTAP data, while ZPG TBLs obviously require a different $\mathcal {W}(Y)$. However, it appears from figure 3, that the TBL overlap log law remains close to (2.9) developed for channel and pipe.

Equation (3.4), together with (3.2) and (3.3), finally allows one to determine $S_{{out}} = S_{{cp}}$ in (2.9). The resulting logarithmic slope is found to scale as

(3.5)\begin{equation} S_{{out}} = \sigma \ln^2Re_{\tau} Re_{\tau}^{{-}1/4}, \end{equation}

with $\sigma$ a weak function of $Re_{\tau }$, varying between −0.19 and −0.15 in the interval $Re_{\tau } \in [10^3, 10^{10}]$. While the $Re_{\tau }^{-1/4}$ dependence is directly related to the scaling of $\langle uu\rangle ^+_{{in}}$, obtained without model assumptions, the factor $\ln ^2Re_{\tau }$ in (3.5) may depend on the details of how $S_{{out}}$ has been determined. However, as long as $\langle uu\rangle ^+_\times$ remains finite, $S_{{out}}$ must go to zero for $Re_{\tau } \to \infty$.

With the determination of $S_{{out}}$, the composite expansion

(3.6)\begin{equation} \langle uu\rangle^+_{{comp}} = \langle uu\rangle^+_{{in}} + \langle uu\rangle^+_{{out}} - \langle uu\rangle^+_{{cp}}, \end{equation}

is complete up to ${O}(Re_{\tau }^{-1/4})$. This final result is compared in figure 4 with the DNS data of table 1, and the differences between composite expansion and DNS are seen to be at most $1.5\,\%$ of the inner peak height (2.4). It is also noted that, in the region $0 \leq y^+ \lessapprox 10^2$, the difference between DNS and composite expansion is principally due to an imperfect ‘hump’ function (A4). However, no improvement is pursued here, as the deviations from DNS are barely larger than the line thickness in figure 1 and no additional insight would be gained from a more complex $\mathcal {H}$.

Figure 4.

Difference between the DNS profiles $\langle uu\rangle ^+$ of table 1 ( 1–5: red, pink, violet, blue, green) and their complete composite fit (3.6).

The composite expansion (3.6) allows a unified comparison with complete $\langle uu\rangle ^+$ profiles of different origins, as well as an extrapolation of $\langle uu\rangle ^+$ to truly large $Re_{\tau }$. Figure 3 shows the close correspondence, over the entire $Re_{\tau }$ range of $10^3$ to $10^5$, between composite profiles and both DNS and several more recent high Reynolds number laboratory data. Only the Superpipe data of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012) are seen to progressively fall below the composite expansion close to the wall, which is also reflected in the low inner peak heights for the Superpipe in figure 2. An explanation for these discrepancies is proposed in Appendix B.

Figure 3 also shows the asymptotic ‘skeleton’ of the composite expansion: the fan of logarithmic asymptotes (2.5) of $\langle uu\rangle ^+_{{wall}}$ (2.2) and the corresponding asymptotic overlap log laws (2.9), together with their intersections (2.7), marked by $\blacklozenge$. The considerable difference, at the lower Reynolds numbers, between this asymptotic logarithmic ‘skeleton’ and the composite expansion, is already noted here. Similarly, the data are seen to closely approach the overlap log law (2.9) – the short-dashed grey lines between $\blacklozenge$ and $\circ$ in figure 3 – only beyond a $Re_{\tau }$ of approximately $10^5$. Below this $Re_{\tau }$, the wake region is reached before the condition $y^+\gg y^+_\times$ is satisfied. See also the comments in the concluding § 5 and the comparison with the patched asymptotic expansion of Marusic & Kunkel (Reference Marusic and Kunkel2003) in figure 3 of the supplementary material.

4. The outer peak

The scaling of the outer peak $\langle uu\rangle ^+_{{OP}}$ and its location $y^+_{{OP}}$ have given and still give rise to extended debates, because they are not yet accessible to DNS and in experiments are typically seen at a wall distance where probe corrections are often an important issue. Probably the first, clean characterization of this peak has been provided by Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) for the ZPG TBL at $Re_{\tau }$ values up to 20’000. The Reynolds number dependence of the outer peak, observed by Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), appears to clash with the choice of a constant $y^+_\times$ in (2.7). A close look at figure 3 shows, however, that the location $y^+_{{OP}}$ of the actual outer peak depends strongly on the slopes of the wall and overlap log laws (2.5) and (2.9), as well as on how the transition between the two is fitted. This sensitivity to fitting details is comparable to the sensitivity to measurement errors in experiments (see also Appendix B).

The evolutions of the outer peak height $\langle uu\rangle ^+_{{OP}}$ and location $y^+_{{OP}}$ with $Re_{\tau }$, obtained from the present composite expansion (3.6), are shown in figure 5; $\langle uu\rangle ^+_{{OP}}$, which is always below $\langle uu\rangle ^+_\times$ (2.7), is seen in panel (a) to be fully consistent with other experimental data, as well as with the outer peak correlations $2.82+0.42\ln Re _{\tau }$ and $0.33+0.63\ln Re _{\tau }$ of Pullin et al. (Reference Pullin, Inoue and Saito2013), up to $Re_{\tau }$ values well above $10^6$.

Figure 5.

(a) Outer peak height $\langle uu\rangle ^+_{{OP}}$ vs $Re_{\tau }$. Symbols as in figure 2, except for: green $\blacklozenge$, pipe data point of Morrison et al. (Reference Morrison, McKeon, Jiang and Smits2004); red $\bullet$, outer peak height of full composite expansion (3.6) at selected $Re_{\tau }$ values; red —, $\langle uu\rangle ^+_\times$ of (2.7); black - - - and $- \cdot - \cdot -$, outer peak correlations $2.82+0.42\ln Re _{\tau }$ and $0.33+0.63\ln Re _{\tau }$ of Pullin, Inoue & Saito (Reference Pullin, Inoue and Saito2013). (b) Corresponding outer peak locations $y^+_{{OP}}$. Symbols as in panel (a), except for: red —, $y^+_\times = 470$; blue - - -, correlation of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012); red $\cdots$, fit $1200 - 1900Re_{\tau }^{-0.06}$; yellow $- \cdot - \cdot -$, correlation $32.66\ Re_{\tau }^{0.27}$ of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018, (3.3b)) for the intersection of tangents outside of $y^+_{{OP}}$.

Panel (b) shows the location $y^+_{{OP}}$ vs $Re_{\tau }$. In this panel, the uncertainty of the experimental points is large and could be as high as 100 % at the lower $Re_{\tau }$ values. Up to $Re_{\tau }=10^5$ the outer peak locations for both the data and the present composite expansion are seen to be compatible with the correlation $\propto Re_{\tau }^{0.67}$ of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012). Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018, (3.3b)), on the other hand, give a correlation of $32.66Re_{\tau }^{0.27}$ for the intersection of two logarithmic tangents to the data, which is located outside of the outer peak, and provides an upper bound on $y^+_{{OP}}$. Together with the outer peak location of the present composite expansion, which is well fitted by the ad hoc correlation $y^+_{{OP}} = 1200 - 1900Re_{\tau }^{-0.06}$, the outer peak is firmly placed in the inner asymptotic region, meaning that, in terms of the usual intermediate or overlap variable $\eta =y^+/Re_{\tau }^{1/2}$, the outer peak approaches $\eta = 0$ in the limit of $Re_{\tau } \to \infty$.