2.1. Data collapse near the wall

Figure 1 shows the profiles of $\langle uu \rangle ^+$, for the channel (figure 1a,b), the pipe (figure 1c,d) and the TBL (figure 1e,f). While figure 1a,c,e displays marked $Re_\tau$ variations, figure 1b,d,f illustrates excellent data collapse after normalization by peak values; that is

(2.2)\begin{equation} \langle u u\rangle^+(y^+,Re_\tau)=\langle u u\rangle^+_p(Re_\tau) f(y^+). \end{equation}

Note that according to CS, the peak location is an invariant at $y^+_p\approx 15$. This is generally accepted as correct (at least since Sreenivasan (Reference Sreenivasan1989)), see Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021). On the other hand, Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) commented that the invariant peak location is violated by their pipe data, which show that $y^+_p$ slightly increases from 14.28 at $Re_\tau \approx 500$ to 15.14 at $Re_\tau =6000$. Nevertheless, using finer near-wall resolutions than in Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), Yao et al. (Reference Yao, Rezaeiravesh, Schlatter and Hussain2023) found no such variation of $y_p^+$ with $Re_\tau$ ($y^+_p=15.07, 15.03, 15.50$ for $Re_\tau =180,2000,5000$, respectively). This small variation is typically found by others as well (Moser, Kim & Mansour Reference Moser, Kim and Mansour1999; Jimenez et al. Reference Jimenez, Hoyas, Simens and Mizuno2010; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014) but it is non-systematic, possibly owing to secondary reasons such as the grid and probe resolutions.

Figure 1.

Wall-normal dependence of streamwise velocity fluctuation scaled in viscous units (abscissa in logarithmic scale) for a series of $Re_\tau$s in (a,b) channels, (c,d) pipes and (e,f) TBL flows. (a,c,e) Here $\langle u u\rangle ^+$ versus $y^+$. (b,d,f) Here $\langle u u\rangle ^+$ normalized by its (inner) peak value $\langle u u\rangle ^+_p$, showing very good collapse. Coloured lines indicate DNS data at different $Re_\tau$s marked in the figure legends, for channels by Lee & Moser (Reference Lee and Moser2015), for pipes by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) and for TBLs by Schlatter et al. (Reference Schlatter, Örlü, Li, Brethouwer, Fransson, Johansson, Alfredsson and Henningson2009, Reference Schlatter, Li, Brethouwer, Johansson and Henningson2010).

As for (2.2), data collapse for the spanwise velocity fluctuation is achieved via

(2.3)\begin{equation} \langle w w\rangle^+(y^+,Re_\tau)=\langle w w\rangle^+_p(Re_\tau) h(y^+), \end{equation}

where $\langle w w\rangle ^+_p$ is the peak value, and $h$ is a $y^+$-dependent function. As shown in figure 2, different $Re_\tau$ curves are in close agreement with each other, with the self-preserving range from the wall to the peak (at $y^+\approx 45$) or beyond. We note a marginal $Re_\tau$ dependence on this peak location; it is hard to tell whether they arise from numerical uncertainty or physical modulation by outer flow structures. Since this variation is marginal, we shall not consider this any further.

Figure 2.

The same plots as in figure 1 but for the spanwise velocity fluctuations: (a,b) channels, (c,d) pipes and (e,f) TBLs; (a,c,e) $\langle ww\rangle ^+$, (b,d,f) $\langle w w\rangle ^+/\langle ww\rangle ^+_p$ versus $y^+$. Lines represent the same DNS data as in figure 1.

Coming now to pressure fluctuations, figure 3(a,c,e) shows $Re_\tau$ dependence of $p'^+$. The best collapse is obtained by plotting $p'^+_{p}-p'^+$, as shown in figure 3(b,d,f). On this basis, we may write

(2.4)\begin{equation} p'^+(y^+, Re_\tau)=p'^+_{p}(Re_\tau)-j(y^+), \end{equation}

where $j$ is (in general) a $y^+$-dependent function. The collapse extends from the wall to the peak (at $y^+\approx 30$), with $p'^+_p-p'^+_w$ a constant around 0.4. This constancy inspires us to postulate (2.4). Instead, if one plots $p'^+/p'^+_{p}$, data would not collapse (not shown here). The reason is clear from (2.4), i.e. $p'^+/p'^+_{p}=1-j(y^+)/p'^+_{p}(Re_\tau )$: an increasing $p'^+_{p}$ with $Re_\tau$ would eventually spoil the data collapse of $p'^+/p'^+_{p}$. That is the reason why (2.2) or (2.3) is not applied to $p'^+$. Note that Panton et al. (Reference Panton, Lee and Moser2017) attempted another data collapse by using $\langle pp\rangle ^+_w-\langle pp\rangle ^+$, but it is not as satisfactory as (2.4) in figure 3, as also discussed later in § 3.

Figure 3.

Wall-normal dependence for the r.m.s. of pressure fluctuation $p'^+=\langle pp\rangle ^{+1/2}$ in (a,b) channels, (c,d) pipes and (e,f) TBL flows: (a,c,e) $p'^+$; (b,d,f) $p'^+_p-p'^+$ versus $y^+$. Lines are the same DNS data as in figure 1.

2.2. Summary for the near-wall scaling

The above comparisons demonstrate that the $Re_\tau$ and $y^+$ dependencies could be decoupled after a proper normalization by peak values. Recalling (1.2) for the $Re_\tau$-scaling of the peak values and substituting it for $\langle uu\rangle ^+_p$, $\langle ww\rangle ^+_p$ and $p'^+_p$ into (2.2), (2.3) and (2.4), respectively, one has a uniform expansion, as reinforced for $\langle uu\rangle ^+$ also by Monkewitz (Reference Monkewitz2022), i.e.

(2.5)\begin{equation} \phi(y^+)=\phi_0(y^+)+\phi_1(y^+)/Re^{1/4}_\tau, \end{equation}

where $\phi _0(y^+)=\phi _\infty f(y^+)$ and $\phi _1(y^+)=-c_{\phi, \infty } f(y^+)$ for $u$; $\phi _0(y^+)=\phi _\infty h(y^+)$ and $\phi _1(y^+)=-c_{\phi, \infty } h(y^+)$ for $w$; $\phi _0(y^+)=\phi _\infty -j(y^+)$ and $\phi _1(y^+)=-c_{\phi, \infty }$ for pressure fluctuations.

Note that (2.5) is a specific case of

(2.6)\begin{equation} \phi(y^+,Re_\tau)=f_0(y^+)+f_1(y^+)g(Re_\tau), \end{equation}

which is a second-order truncation of (2.1), considered also by Spalart & Abe (Reference Spalart and Abe2021) and Monkewitz (Reference Monkewitz2022). If $f_0=0$, (2.6) reduces to

(2.7)\begin{equation} \phi(y^+,Re_\tau)=f_1(y^+)g(Re_\tau), \end{equation}

which is the scaling proposed by Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021) for $\langle uu \rangle ^+$ with $g(Re_\tau )=\epsilon ^+_{x-w}$ (streamwise wall dissipation). However, Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021) found that their proposal did not work as well for $\langle w w\rangle ^+$, which they speculated was due to different superposition and modulation enforced by outer flow structures. Here, we show that replacing wall dissipation by peak value, i.e. $g=\phi _p$, (2.7) applies for both $\langle uu \rangle ^+$ and $\langle ww \rangle ^+$. Even so, (2.7) is not proper for pressure due to a constancy $p'^+_p-p'^+_w$ as explained earlier. The non-zero $f_0$ needed in (2.6) has been missed in the past.

Finally, we recall that from (2.6), Monkewitz (Reference Monkewitz2022) developed a composite model for $\langle uu\rangle ^+$, which shows that $g=Re^{-1/4}_\tau$ yields a better data description than the alternative $g=\ln Re_\tau$ considered by Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021). The gauge function $g=Re^{-1/4}_\tau$ in (2.6) restores wall scaling for asymptotically high $Re_\tau$; we will use it below to derive an outer decay profile, which has not been achieved before.

3.1. Matching for the defect law for fluctuations

Based on the above inner expansion and outer similarity, we now develop a matching procedure to derive the analytical form in the intermediate zone. One may invoke the derivation of Millikan (Reference Millikan1938) for the log law in the mean velocity distribution, which has been extended for $\langle uu\rangle ^+$ by Hultmark (Reference Hultmark2012) and for $\langle p p\rangle ^+$ by Panton et al. (Reference Panton, Lee and Moser2017). Nevertheless, different orders of matching would lead to different scaling proposals, and here we present a short account of matching to obtain the defect law for fluctuations.

As a starting point, we note an exact matching between the inner and outer flows for the total stress in channels or pipes. That is, $\tau ^+(y^+,Re_\tau )=1-y^+/Re_\tau$, which can be rewritten as $\tau ^+(y^\ast,Re_\tau )=1-y^\ast$. The first version of $\tau ^+(y^+,Re_\tau )$ corresponds to the two-term inner expansion of (2.6) with $\phi =\tau ^+$, $f_0(y^+)=1$, $f_1(y^+)=y^+$ and $g=Re^{-1}_\tau$; the second version of $\tau ^+(y^\ast,Re_\tau )$ corresponds to the one-term expansion in (3.1), or to the outer similarity in (3.2) with $\phi =\tau ^+$ and $F_0(y^\ast )=1-y^\ast$. Thus, the total stress satisfies a straightforward matching between (2.6) and (3.2), i.e. $f_0(y^+)+f_1(y^+)g(Re_\tau )=F_0(y^\ast )$, valid for the entire flow, quite unlike Millikan's matching analysis that requires $1/Re_\tau \ll y^\ast \ll 1$ (or $1\ll y^+\ll Re_\tau$) and dictates a term-by-term balance.

Therefore, we match (2.6) directly with (3.2), i.e. $\phi (y^+,Re_\tau )=F_0(y^\ast )$, and obtain

(3.3)\begin{align} \phi(y^+,Re_\tau)&=f_{0}(y^+)+f_{1}(y^+)/Re^{1/4}_\tau\nonumber\\ &=f_{0}(y^+)+{h}_{1}(y^+)y^{{\ast1/4}}\nonumber\\ &=F_0(y^\ast) \end{align}

(where $g=Re_\tau ^{-1/4}$ is used and ${h}_{1}(y^+)=f_1(y^+)/y^{+1/4}$). Note that $f_{0}=\phi _0$ and $h_{1}=\phi _{1}/y^{+1/4}$ as the wall is approached, so that (3.3) approaches (2.5), but $f_0$ and $h_1$ towards the outer region are to be determined. At this stage, as $F_0(y^\ast )$ in (3.3) depends only on $y^\ast$, we have $\partial F_0(y^\ast )/\partial y^+=0$ in (3.3), and hence

(3.4a)\begin{gather} f_{0}(y^+)=c_0, \end{gather}

(3.4b)\begin{gather} {h}_{1}(y^+)=c_1, \end{gather}

where $c_0$ and $c_1$ are constants independent of $y^\ast$, $y^+$ and $Re_\tau$, but may depend on the variable $\phi$. Denoting $c_0=\alpha _\phi$ and $c_1=-\beta _\phi$, we obtain (1.3) from (3.3) and (3.4). It is readily verified that (1.3) matches (3.2) in the outer region and (2.6) in the inner, hence offers a common description connecting the two.

In the above procedure, we have assumed that the two-term expansion $f_{0}(y^+)+f_{1}(y^+)g(Re_\tau )$ extends to the outer flow, so that it matches with $F_0(y^\ast )$ and satisfies the outer similarity as well. As explained above, the procedure works exactly for the total stress in channels and pipes with $g=Re^{-1}_\tau$.

3.2. Comparison with the data

Figures 5–8 show comparisons of data with (1.1) and (1.3) for $\langle uu\rangle ^+$, $\langle ww\rangle ^+$ and $p'^+$, respectively. Table 1 collects all the parameters for the three profiles, arising from these fits to the data, which we now discuss.

Table 1.

Parameters in (1.3), for different fluctuations. Superscripts ‘CH’, ‘Pipe’ and ‘TBL’ represent channel, pipe and boundary layer flows, respectively. Note that both $\alpha _\phi$ and $\beta _\phi$ vary only modestly among different flows, implying that essentially the same mechanisms applies for all flows. Moreover, $\beta _\phi$ is quite close to $\alpha _\phi$, as $\phi$ at the boundary layer edge $y^\ast =1$ is fairly small.

Similar to figures 1–3, figures 5(a,b), 6(a,b), 7(a,b) and 8(a,b) are for channel, figures 5(c,d), 6(c,d), 7(c,d) and 8(c,d) for pipe and figures 5(e,f), 6(e,f), 7(e,f) and 8(e,f) for boundary layers; for figures 5(a,c,e), 6(a,c,e), 7(a,c,e) and 8(a,c,e) the abscissa are in linear units while for figures 5(b,d,f), 6(b,d,f), 7(b,d,f) and 8(b,d,f), logarithmic units are used. A difference from figures 1–3 is that the data in figures 5–8 are denoted by black symbols, so that (1.1) and (1.3) are better marked to guide the eye.

Figure 7.

Variance of spanwise velocity fluctuation in channel (a,b), pipe (c,d) and TBL (e,f) flows scaled in outer units $y^\ast =y/\delta$. Dashed (red) lines indicate the bounded decay (1.3) with parameters in table 1. Dotted (green) lines indicate the logarithmic decay (1.1), i.e. $1.08-0.387\ln y^\ast$ for channel and pipe, and $1.23-0.387\ln y^\ast$ for TBL. Symbols with lines are the same DNS data as in figure 1, i.e. channel from Lee & Moser (Reference Lee and Moser2015), pipe from Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), TBL from Schlatter et al. (Reference Schlatter, Örlü, Li, Brethouwer, Fransson, Johansson, Alfredsson and Henningson2009, Reference Schlatter, Li, Brethouwer, Johansson and Henningson2010).

Figure 8.

The r.m.s. of pressure fluctuation in channel (a,b), pipe (c,d) and TBL (e,f) flows scaled in the outer unit $y^\ast =y/\delta$. Dashed (red) lines indicate the bounded decay (1.3) with parameters in table 1. Dotted (green) lines indicate the logarithmic decay in (3.5), i.e. $\sqrt {0.27-2.56\ln y^\ast }$ for channel, $\sqrt {0.6-2.45\ln y^\ast }$ for pipe and $\sqrt {2.5-2.45\ln y^\ast }$ for TBL. Symbols with lines are the same DNS data as in figure 7.

Particularly for $\langle uu\rangle ^+$, to avoid distractions by data scatter at small $Re_\tau$, we collect in figure 5 only high $Re_\tau$ profiles from DNS, namely, $Re_\tau$ from $2000$ to $10^4$ for channel; 2000 to 6000 for pipe; and 1270 to 1990 for TBL. Compared with figure 4(a,d,g), it is clear in figure 5(a,c,e) that all high $Re_\tau$ profiles collapse on each other in the flow range $0.1\lesssim y^\ast \lesssim 1$, thus bearing witness to outer similarity. This is confirmed again by experimental data in figure 6 corresponding to higher $Re_\tau$.

Note that the logarithmic behaviour advanced in the literature (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) is indicated by a green dotted line. Although it characterizes data in the region from $0.1\lesssim y^\ast \lesssim 0.3$, the value of the intercept $B_\phi$ needs to be adjusted for the three flows from 1.61 to 2.2, while $A_\phi$ holds constant around 1.26. In contrast, the red dashed line represents (1.3) with the same $\alpha _\phi =10$ and $\beta _\phi =9.3$, which reproduces data well for channel, pipe and TBL flows, covering not only the logarithmic range (vis-à-vis the mean flow) but also the so-called wake region, almost all the way to the centreline of channel and pipe flows.

One may imagine that the data in figure 5 have not reached the asymptotic state and that (1.1) might agree with data better for higher $Re_\tau$. But experimental data from Princeton pipes (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012) with $Re_\tau$ covering one more decade, e.g. $Re_\tau$ from approximately $5000$ to $10^5$, do not show any improvement of the fit to (1.1) (see figure 6a,b). A similar observation is also true for the TBL, as shown in figure 6(c–f).

Likewise, for $\langle ww\rangle ^+$ and $p'^+$, (1.3) extends almost all the way to the centreline of channel and pipe flows. Here, data in figures 7 and 8 are the same DNS groups as in figure 1 and contain those low $Re_\tau$ profiles. The agreement with (1.3) is excellent at the smallest $Re_\tau \approx 500$ for channel and pipe, in contrast to the log variation that agrees with data only for $Re_\tau \gtrsim 2000$. Therefore, in both $y$ and $Re_\tau$ ranges, (1.3) covers a wider range than (1.1).

Three further points will now be discussed. First, the difference between (1.3) and (1.1) is more vital for asymptotically high $Re_\tau$. For (1.1), an infinitely large of $\phi \propto \ln Re_\tau$ would arise as $y^\ast \rightarrow 0$ and $Re_\tau \rightarrow \infty$. In contrast, (1.3) assigns a plateau of $\phi \approx \alpha _\phi$ in the same limit. Such an asymptotic plateau implies that turbulent eddies in the bulk would be in a quasiequilibrium state in the sense that their contribution to $\phi$ is invariant when $y$ changes. Note also that according to (1.3), the outer peak of $\langle uu\rangle ^+$, if it exists, should be bounded by $\langle uu\rangle ^+\approx 10$. Clarification of such differences of perspectives in the asymptotic state require data at higher Reynolds numbers.

Second, while (1.3) adheres closely with TBL data of Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015) up to $y^\ast =1$ (figure 6c), it is slightly and uniformly higher than the TBL data of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) for $y^\ast >0.6$ (figure 6e). This is due to the fact that the former set of data are obtained for flow over a flat plate mounted in the same pipe in which data of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012) are measured. Therefore, the data of Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015) in its outer wake region resemble the centre behaviour of pipe data by Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012), both in agreement with (1.3) up to $y^\ast =1$. In contrast, TBL data of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) are measured in the Melbourne wind tunnel with the standard free stream boundary condition, so that a vanishing $\langle uu\rangle ^+\approx 0$ towards the boundary layer edge is observed (figure 6e,f), which is lower than (1.3). This difference reflects the wake influence on $\langle uu\rangle ^+$ in TBL. In fact, the wake influence is much sharper for $\langle ww\rangle ^+$ and $p'^+$ in TBL (figures 7e and 8e). This issue will be addressed in § 4.2.

The third point is that, for $p'^+$, the green dotted line in figure 8 represents

(3.5)\begin{equation} p'^+(y^\ast)=\sqrt{ B_{\phi}-A_{\phi} \ln y^\ast }, \end{equation}

which is the square root of (1.1) for pressure variance

(3.6)\begin{equation} \langle pp\rangle^+(y^\ast)={ B_{\phi} - A_{\phi} \ln y^\ast}. \end{equation}

This equation is obtained by Panton et al. (Reference Panton, Lee and Moser2017) via an inner–outer matching (i.e. a viscous inner layer overlapping with an inviscid outer layer). It is almost indistinguishable from (1.3) in figure 8. Nevertheless, as shown in figure 9(a–c), $\langle pp\rangle ^+_w-\langle pp\rangle ^+$ versus $y^+$ produces no data collapse for $y^+>5$. Particularly for the trough located at approximately $y^+=30$, the data are markedly lower for increasing $Re_\tau$, thus creating a challenge for the inner–outer matching analysis. To reconcile this challenge, Panton et al. (Reference Panton, Lee and Moser2017) introduced two logarithmic slopes, i.e. $A^{CP}=2.56$ for the common part of presumed log profile in the overlap layer, and another $A^{w}=2.24$ for the $Re_\tau$ variation of the wall pressure. Following this fix, one can estimate

(3.7)\begin{equation} \langle pp\rangle^+_p-\langle pp\rangle^+_w\propto (A^{CP}-A^{w})\ln Re_\tau\approx 0.32\ln Re_\tau, \end{equation}

which would break the wall scaling completely.

Figure 9.

Wall-normal dependence of the variance of pressure fluctuations is shown in the plots of $\langle pp\rangle ^+_w-\langle pp\rangle ^+$ versus $y^+$, where $\langle pp\rangle ^+_w$ is the wall-value of $\langle pp\rangle ^+$. Lines are the same DNS data as in figure 3: (a) channels; (b)  pipes; (c) TBLs. Note that lines depart markedly from each other with increasing $Re_\tau$ in the region $y^+>5$. (d) Difference between the peak and wall values of pressure variance, i.e. $\langle pp\rangle ^+_p-\langle pp\rangle ^+_w$, for channel, pipe and TBL flows for a series of $Re_\tau$ values. Dotted line (green) indicates the logarithmic growth by (3.7), i.e. $0.32\ln Re_\tau +0.33$. Dashed line (red) indicates the bounded variation of (3.8) according to CS. Symbols are DNS data, squares for channel (Lee & Moser Reference Lee and Moser2015), circles for pipe (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) and diamonds for TBL (Schlatter et al. Reference Schlatter, Örlü, Li, Brethouwer, Fransson, Johansson, Alfredsson and Henningson2009). Note that DNS channel data at $Re_\tau =150, 300, 400$ of Iwamoto, Suzuki & Kasagi (Reference Iwamoto, Suzuki and Kasagi2002), and $Re_\tau =180, 550, 944, 2000$ of Hoyas & Jimenez (Reference Hoyas and Jimenez2006) are also included here.

As a comparison, figure 3(b,d,f) show that data of $p'^+_w-p'^+$ collapsed well up to the trough, better than $\langle pp\rangle ^+_w-\langle pp\rangle ^+$ in figure 9(a–c). Moreover, via the bounding relation $p'^+_w\approx 4.4-10.5/Re^{1/4}_\tau$ and $p'^+_p\approx 4.84-10.5/Re^{1/4}_\tau$ given in CS, we have

(3.8)\begin{equation} \langle pp\rangle^+_p-\langle pp\rangle^+_w=p'^{{+}2}_p-p'^{{+}2}_w\approx4.07-9.24/Re_\tau^{1/4}, \end{equation}

which depicts data satisfactorily over a wider $Re_\tau$ range than (3.7) in figure 9(d).

Finally, to evaluate the goodness of the theoretical fits with the data, taking the channel at $Re_\tau =5200$ (Lee & Moser Reference Lee and Moser2015) for example, we have calculated the variance of difference between DNS data ($\phi _{Data}$) and theoretical predictions ($\phi _{Eq}$), i.e. $\sigma ^2 = \sum _{0.1 \le {y^\ast _i} \le 1} {{{[\phi _{Data}({y^\ast _i}) - \phi _{Eq} ({y^\ast _i})]}^2}} /N$ where $N$ is the total number of data points in the range from $y^\ast =0.1$ to $y^\ast =1$. For (1.3), $\sigma ^2$ is $0.007$ for $\langle uu\rangle ^+$, $0.002$ for $\langle ww\rangle ^+$ and $0.002$ for $p'^+$; while for (1.1), $\sigma ^2$ is $0.26$, $0.14$ and $0.004$ for $\langle uu\rangle ^+$, $\langle ww\rangle ^+$ and $p'^+$, respectively. The larger $\sigma ^2$ of (1.1) for $\langle uu\rangle ^+$ and $\langle ww\rangle ^+$ is not surprising because of their deviation from data towards the centreline. But we should remember that (1.1) is derived for asymptotically high $Re_\tau$ and is not supposed to depict the data behaviour towards $y^\ast =1$. In this sense, $1/Re_\tau \ll y^\ast \ll 1$ might be better for the competition between the two proposals, but that would lose sight of the advantage that (1.3) covers a wider outer domain.

3.3. Brief critique of the matching arguments

Two further points will be considered here. Despite the agreement with the data, one should note that, according to (1.3), $\partial \phi /\partial y^\ast =-\beta _\phi /4$ at $y^\ast =1$, which is against the mirror symmetry of $\partial \phi /\partial y^\ast =0$ at the centreline of channel and pipe. This indicates that there is a discrepancy between (1.3) and data towards the centreline, for which a centre core layer or the wake modification is needed, as shown in Monkewitz (Reference Monkewitz2023).

Secondly, a perusal of the fits with data suggests the existence of a gap between the inner peak and the outer data collapse, particularly for $\langle uu\rangle ^+$ (figure 1b,d,f). This indicates a modest $Re_\tau$-dependence even after normalization by the peak. This may also be the influence of higher-order terms in (2.1), as explained here by the two-term expansion. From (2.6), we expand $\phi (y^+)/\phi _p(y^+)$ by the parameter $g(Re_\tau )$; that is,

(3.9)\begin{equation} \frac{{\phi ({y^ + })}}{{\phi (y_p^ + )}} = \frac{{{f_0}({y^ + })}}{{{f_0}(y_p^ + )}}\left\{ {1 + \left[ {\frac{{{f_1}({y^ + })}}{{{f_0}({y^ + })}} - \frac{{{f_1}(y_p^ + )}}{{{f_0}(y_p^ + )}}} \right]g({{Re}_\tau }) + \cdots} \right\}. \end{equation}

When $f_0(y^+)$ and $f_1(y^+)$ are proportional to each other, one has ${f_1}({y^ + })/{f_0}({y^ + }) - {f_1}({y^ + _p})/{f_0}({y^ + _p}) = 0$, and hence $\phi ({y^ + })/\phi ({y^ +_p })$ from (3.9) is independent of $g(Re_\tau )$. This case corresponds to the data collapse of $\langle uu\rangle ^+$ from the wall to the peak. However, when $f_0(y^+)$ and $f_1(y^+)$ are no longer proportional to each other, the gauge function $g(Re_\tau )$ in (3.9) indicates that (3.9) still has $Re_\tau$-dependence, which corresponds to the gap between the inner peak and the outer approximation (1.3). Therefore, the proportionality or otherwise of $f_0(y^+)$ and $f_1(y^+)$ in (2.6) would determine the data collapse or slight mismatch when plotting via $\phi ({y^ + })/\phi ({y^ +_p })$. There is no stipulation here that the ratio of $f_0(y^+)$ to $f_1(y^+)$ is constant in matching (3.3). In fact, according to (3.4), $f_0(y^+)/f_1(y^+)=c_0/(c_1y^{+1/4})$. Therefore, even if $f_0$ and $f_1$ are not proportional to each other, we can still match (2.6) with (3.2) to obtain (3.3).