2 Results

As illustrated in figure 1, $y_{p}$ in TBL is notably larger than that in CP at a common $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ , and our prediction implies that this critical layer of $y_{p}$ in TBL becomes increasingly thicker than that of CP with increasing $Re_{\unicode[STIX]{x1D70F}}$ . This has a practical significance, as $y_{p}$ can be an engineering-relevant quantity: in the drag control of wall flows, a recent study by Yao, Chen & Hussain (Reference Yao, Chen and Hussain2018) shows that the optimal controlling parameter, i.e. the height of large-scale swirls excited by actuators, follows the $Re_{\unicode[STIX]{x1D70F}}$ scaling of  $y_{p}$ .

Figure 1.

Ratio between the viscous shear force VF ( $\unicode[STIX]{x2202}S^{+}/\unicode[STIX]{x2202}y^{+}$ ) and turbulent inertial TI ( $\unicode[STIX]{x2202}W^{+}/\unicode[STIX]{x2202}y^{+}$ ) showing the inner and outer flow regions demarcated by $y^{+}=y_{p}^{+}$ where $\unicode[STIX]{x2202}W^{+}/\unicode[STIX]{x2202}y^{+}=0$ . Here, $W^{+}=-\overline{u^{\prime }v^{\prime }}^{+}$ and $S^{+}=\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+}$ . Symbols are DNS data: stars from Schlatter et al. (Reference Schlatter, Li, Brethouwer, Johansson and Henningson2010); squares from Wu & Moin (Reference Wu and Moin2008); circles from Lee & Moser (Reference Lee and Moser2015).

Recall the near-wall motion described by viscous units $y^{+}=yu_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ , and the outer inertial-dominated flow region by $y/\unicode[STIX]{x1D6FF}$ . Sreenivasan (Reference Sreenivasan1988) proposed the $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ scaling of $y_{p}^{+}$ for CP at high $Re_{\unicode[STIX]{x1D70F}}$ , validated by many others later (Klewicki Reference Klewicki2013; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014; Lee & Moser Reference Lee and Moser2015; Ahn et al. Reference Ahn, Lee and Sung2017). This can be derived as follows. Integrating (1.2) with respect to $y^{+}$ yields

(where the total stress $\unicode[STIX]{x1D70F}^{+}=1-y^{+}/Re_{\unicode[STIX]{x1D70F}}$ ); further provided with the log law:

it readily yields

by setting $\text{d}W^{+}/\text{d}y^{+}=0$ ( $\unicode[STIX]{x1D705}$ is the von Kármán constant with its value taken around 0.45; see She et al. (Reference She, Chen and Hussain2017) and Chen et al. (Reference Chen, Hussain and She2018)). However, equation (2.3) is not accurate for CP at moderate $Re_{\unicode[STIX]{x1D70F}}$ (Ahn et al. Reference Ahn, Lee and Sung2017), as seen in figure 2, showing a clear scaling transition.

Figure 2.

Scaling transition for the critical layer ( $y_{p}^{+}$ ) of momentum cascade. Solid line denotes (2.5) for low $Re$ and dashed line denotes (2.3) for high $Re$ . Open symbols, DNS channel and pipe; solid symbols, experimental pipe.

This discrepancy of the classical scaling analysis is now well resolved here, through a specification of the buffer layer beneath the log layer, yielding a scaling transition towards lower $Re$ . Specifically, the viscous buffer layer extends from $y_{sub}^{+}\approx 9.7$ to $y_{buf}^{+}\approx 41$ , and we introduce a stress length ( $\ell ^{+}=\sqrt{W^{+}}/S^{+}$ ) which takes the following analytic form (She et al. Reference She, Chen and Hussain2017; Chen et al. Reference Chen, Hussain and She2018):

This allows us to examine the scaling of $y_{p}$ for small $Re_{\unicode[STIX]{x1D70F}}$ , such that $y_{p}^{+}\lesssim y_{buf}^{+}$ .

Substituting the approximation $S^{+}\approx 1/\ell ^{+}$ with (2.4c ) into (2.1), we obtain (2.3) for high $Re_{\unicode[STIX]{x1D70F}}$ ; but for small $Re_{\unicode[STIX]{x1D70F}}$ , with (2.4b ), we obtain:

The existence of the buffer layer $y^{+2}$ scaling is the key to the present discovered scaling transition; this scaling arises from a local quasi-balance between the residue viscous effect $\unicode[STIX]{x1D700}$ and turbulent production $SW$ , as follows. Introduce a shear eddy length: $\ell _{\unicode[STIX]{x1D708}}=(W/S)^{3/4}\unicode[STIX]{x1D700}^{-1/4}$ ; expansion analysis shows that $\ell _{\unicode[STIX]{x1D708}}\propto y^{2}$ as $y\rightarrow 0$ (She et al. Reference She, Chen and Hussain2017). Note that $\ell ^{+}=\ell _{\unicode[STIX]{x1D708}}^{+}\unicode[STIX]{x1D6E9}^{1/4}$ , where $\unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D700}/(SW)$ . As shown in She et al. (Reference She, Chen and Hussain2017), in the sublayer, $\unicode[STIX]{x1D6E9}\ll 1$ and $\ell _{\unicode[STIX]{x1D708}}^{+}\ll \ell ^{+}$ , and near-wall motion is dominated by random fluctuations; Taylor expansion results in (2.4a ). However, in the buffer layer, $\unicode[STIX]{x1D6E9}$ reaches a maximum of order unity (due to dominant effect of coherent motion); then, $\ell ^{+}$ follows the scaling of $\ell _{\unicode[STIX]{x1D708}}^{+}$ , which is (2.4b ). In the log layer where $\unicode[STIX]{x1D6E9}\approx 1$ , $\ell \approx \ell _{\unicode[STIX]{x1D708}}\propto y$ from homogeneous dilation symmetry (Chen et al. Reference Chen, Hussain and She2018), which is (2.4c ).

The realization of a buffer layer leads to a physical interpretation for the above scaling transition. Note that cascade is normally considered to result from the multiscale dynamics of turbulence (Frisch Reference Frisch1995). For the energy cascade, many studies have shown that it is caused by vortex stretching (for example, Goto Reference Goto and Eckhardt2009); for the momentum cascade, according to Jimenez (Reference Jimenez2012) and Yang & Lozano-Durán (Reference Yang and Lozano-Durán2017), it is carried out by a hierarchy of multiscale wall-attached eddies, which transfer momentum towards the wall until it is drained by the viscous effect. In Jimenez (Reference Jimenez2018), it is shown that the logarithmic eddies (or vortex clusters) scale as  $y$ , which yield a scaling as predicted by (2.4c ); moreover, the buffer layer eddies are viewed as in an autonomous cycle having smaller sizes but with a higher Corrsin shear parameter, in accordance with (2.4b ), which indicates a smaller $\ell$ and hence a larger  $S$ . Thus, approaching the wall from log to buffer layers, eddies with different spatial organizations, not surprisingly, would dictate different cascade processes.

We now validate the predicted scaling transition by empirical data. Note that unlike for DNS data, it is quite inaccurate to determine $y_{p}^{+}$ from the experimentally measured $\overline{u^{\prime }v^{\prime }}$ profile (for example, with large scatter); thus, an alternative way to determine $y_{p}^{+}$ is needed. Taking a $y^{+}$ -derivative of (2.1) at $y^{+}=y_{p}^{+}$ yields

which presents an equivalent way to find $y_{p}^{+}$ from $S^{+}(y^{+})=\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+}$ . It is more accurate since the twice $y^{+}$ -derivative of $U^{+}(y^{+})$ can be obtained more reliably. For the Princeton SuperPipe (McKeon et al. Reference McKeon, Li, Jiang, Morrison and Smits2004; Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2013), as $\overline{u^{\prime }v^{\prime }}$ data are absent, Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014) and Ahn et al. (Reference Ahn, Lee and Sung2017) determine $y_{p}^{+}$ in this way.

Figure 2 shows that (2.5) agrees well with data at smaller $Re_{\unicode[STIX]{x1D70F}}$ , which, together with (2.3), indicates that the $Re_{\unicode[STIX]{x1D70F}}^{0.372}$ scaling by data fitting over the $Re_{\unicode[STIX]{x1D70F}}$ range 500–5200 in Ahn et al. (Reference Ahn, Lee and Sung2017) is perhaps a misinterpretation. Furthermore, we predict that the scaling transition from $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ to $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ takes place around $Re_{\unicode[STIX]{x1D70F}}^{(crit)}=4\unicode[STIX]{x1D705}y_{buf}^{+2}\approx 3000$ for CP, by equating (2.5) with (2.3). Evidently, the $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ scaling in figure 2 in turn supports the intermediate scaling (2.4b ) in the buffer layer. Whilst the scaling transition in CP has been shown in Chen et al. (Reference Chen, Hussain and She2018), here we include more data sets (for $Re_{\unicode[STIX]{x1D70F}}$ covering three decades) to provide compelling evidence of scaling transition. More importantly, the theme of the current paper is to reveal and demonstrate, under the same analysis, the non-universal scaling transition among CP (internal) and TBL (external) flows, as we discuss below.

Applying the above analysis to TBL, a new element arises – that is, a small but finite mean wall-normal velocity  $V$ . The small $V$ changes the $U$ profile only slightly, but leads to its non-trivial streamwise development, and a different total stress, i.e. $\unicode[STIX]{x1D70F}^{+}=1-\int _{0}^{y^{+}}(U^{+}(\unicode[STIX]{x2202}V^{+}/\unicode[STIX]{x2202}y^{\prime })-V^{+}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{\prime }))\,\text{d}y^{\prime }$ . For determining  $V$ , we consider the mean kinetic energy equation of  $V$ , in which the dissipation term (that is, $\unicode[STIX]{x1D716}_{V}=\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}V/\unicode[STIX]{x2202}y)^{2}$ ) can be specified by a dimensional analysis using  $\unicode[STIX]{x1D708}$ , $\unicode[STIX]{x1D6FF}$ and free-stream wall-normal velocity  $V_{e}$ , yielding $\unicode[STIX]{x1D716}_{V}=(\unicode[STIX]{x1D708}V_{e}^{2}/\unicode[STIX]{x1D6FF}^{2})\unicode[STIX]{x1D6FA}(y/\unicode[STIX]{x1D6FF})$ . The self-similar function $\unicode[STIX]{x1D6FA}$ satisfies $\unicode[STIX]{x1D6FA}(0)=0$ , which suggests a simple linear approximation: $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D714}_{1}y/\unicode[STIX]{x1D6FF}$ , and then, $\unicode[STIX]{x1D716}_{V}=\unicode[STIX]{x1D714}_{1}(\unicode[STIX]{x1D708}V_{e}^{2}/\unicode[STIX]{x1D6FF}^{2})(y/\unicode[STIX]{x1D6FF})$ . Using the definition of $\unicode[STIX]{x1D716}_{V}$ and integrating it with respect to  $y$ , one obtains

where $\unicode[STIX]{x1D714}_{1}=(3/2)^{2}=9/4$ due to $V(0)=0$ and $V(\unicode[STIX]{x1D6FF})=V_{e}$ . Figure 3 shows that (2.7) agrees closely with DNS data, which thus well describes the data collapse found by Wei & Klewicki (Reference Wei and Klewicki2016).

Figure 3.

DNS profiles (Schlatter et al. Reference Schlatter, Li, Brethouwer, Johansson and Henningson2010) of total stress $\unicode[STIX]{x1D70F}^{+}$ (symbols), mean wall-normal velocity $V/V_{e}$ (symbols) and mean momentum flux (Reynolds shear stress) $W^{+}=-\overline{u^{\prime }v^{\prime }}^{+}$ (dashed lines) in TBL. Profiles of $\unicode[STIX]{x1D70F}^{+}$ and $V/V_{e}$ at four different cases $Re_{\unicode[STIX]{x1D70F}}=492$ , 671, 974 and 1271 collapse. Solid lines indicate our theories: equation (2.7) for $V$ and (2.8) for  $\unicode[STIX]{x1D70F}^{+}$ .

Based on (2.7), $\unicode[STIX]{x1D70F}^{+}$ in TBL is obtained:

To see this, we first denote $U^{+}\propto (y^{+})^{\unicode[STIX]{x1D709}}$ , where according to the log law (2.2), $\unicode[STIX]{x1D709}-1\approx -1$ or $\unicode[STIX]{x1D709}\approx 0$ (also see Barenblatt (Reference Barenblatt1993), where $\unicode[STIX]{x1D709}\propto 1/\ln Re$ ). Then, substituting $U^{+}\propto (y^{+})^{\unicode[STIX]{x1D709}}$ and $V^{+}\propto (y^{+})^{3/2}$ of (2.7) into $\unicode[STIX]{x1D70F}^{+}=1-\int _{0}^{y^{+}}(U^{+}(\unicode[STIX]{x2202}V^{+}/\unicode[STIX]{x2202}y^{\prime })-V^{+}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{\prime }))\,\text{d}y^{\prime }$ yields $1-\unicode[STIX]{x1D70F}^{+}\propto (y^{+})^{3/2+\unicode[STIX]{x1D709}}$ . As $\unicode[STIX]{x1D709}\approx 0$ , $1-\unicode[STIX]{x1D70F}^{+}\approx \unicode[STIX]{x1D70F}_{0}y^{+3/2}$ . Moreover, as $\unicode[STIX]{x1D70F}^{+}=0$ at $y^{+}=\unicode[STIX]{x1D6FF}^{+}$ (or $y^{+}=Re_{\unicode[STIX]{x1D70F}}$ ), $\unicode[STIX]{x1D70F}_{0}=Re_{\unicode[STIX]{x1D70F}}^{-3/2}$ , and thus (2.8). Compared to $1-\unicode[STIX]{x1D70F}^{+}=y^{+}/Re_{\unicode[STIX]{x1D70F}}$ in CP, $1-\unicode[STIX]{x1D70F}^{+}$ in TBL is higher at the same $y$ location due to  $V$ . This explains earlier observations in Degraaff & Eaton (Reference Degraaff and Eaton2000), well supported by DNS data in figure 3.

Figure 4.

The same plot as figure 2 but for TBL. Solid symbols, DNS; open symbols, experiments. Note that the open symbols of experiments are measured by using (2.10) – agreeing well with solid symbols of DNS data in the overlapped $Re_{\unicode[STIX]{x1D70F}}$ range (from 500 to 2000). For comparison, two dotted lines for CP are included, which show much smaller $y_{p}^{+}$ for high $Re$ .

Completing the previous analysis of $y_{p}^{+}$ using (2.8) yields

for large $Re_{\unicode[STIX]{x1D70F}}$ , and

for small $Re_{\unicode[STIX]{x1D70F}}$ . The relation defining $y_{p}^{+}$ from $\text{d}S^{+}/\text{d}y^{+}$ for TBL takes the following form:

in sharp contrast to (2.6) for CP. Again, equating (2.9b ) and (2.9a ), the scaling transition occurs around $Re_{\unicode[STIX]{x1D70F}}=2(9\unicode[STIX]{x1D705}^{2}y_{buf}^{+5})^{1/3}\approx 1200$ .

Figure 4 shows the good agreement between the prediction (2.9) and data. Note that $y_{p-TBL}^{+}$ between DNS data (Spalart Reference Spalart1988; Wu & Moin Reference Wu and Moin2009; Sillero, Jimnez & Moser Reference Sillero, Jimnez and Moser2014) – by pinpointing the maximum of the $W^{+}$ profile directly – and experimental data (Degraaff & Eaton Reference Degraaff and Eaton2000; Carlier & Stanislas Reference Carlier and Stanislas2005; Örlü Reference Örlü2009; Vallikivi, Ganapathisubramani & Smits Reference Vallikivi, Ganapathisubramani and Smits2015) – by using (2.10) – agree well with each other in the overlapped $Re_{\unicode[STIX]{x1D70F}}$ range, hence supporting the validity of (2.10). Moreover, we have checked that $W^{+}\approx 1-(y^{+}/Re_{\unicode[STIX]{x1D70F}})^{3/2}$ agrees closely with the available high- $Re_{\unicode[STIX]{x1D70F}}$ experimental $W^{+}$ profile in the outer flow region; thus (2.8) and hence (2.10) seem robust. Incidentally, a $Re_{\unicode[STIX]{x1D703}}^{0.63}$ scaling was reported for TBL based on fitting of experimental data (Fernholz & Finleyt Reference Fernholz and Finleyt1996), which is quite close to our (2.9a ) (as $Re_{\unicode[STIX]{x1D703}}/Re_{\unicode[STIX]{x1D70F}}\sim O(1)$ ; see Chen & She (Reference Chen and She2016), Monkewitz, Chauhan & Nagib (Reference Monkewitz, Chauhan and Nagib2007)). However, those TBL experimental data have too much scatter, and hence are not included here.

Figure 5.

Scaling transition for the maximum momentum flux ( $1-W_{p}^{+}=1+\overline{u^{\prime }v^{\prime }}_{p}^{+}$ ). Note that solid symbols are DNS TBL data, the same as in figure 4, while open symbols are DNS CP data, the same as in figure 2.

The non-universal scaling transition is also reflected in the maximum momentum flux $W_{p}^{+}(=-\overline{u^{\prime }v^{\prime }}_{p}^{+})$ , which is obtained by substituting $y_{p}^{+}$ in (2.1). For large $Re_{\unicode[STIX]{x1D70F}}$ , using (2.9a ) for TBL and (2.3) for CP, respectively, we have

for small $Re_{\unicode[STIX]{x1D70F}}$ , using (2.9b ) and (2.5), we have Figure 5 shows $1-W_{p}^{+}$ displaying a distinguished scaling transition, well described by (2.11) and (2.12). Note that $1-W_{p}^{+}$ and $y_{p}^{+}$ have different dependences on $Re_{\unicode[STIX]{x1D70F}}$ , and their transitions are continuous and not sharp. As a result, the transition values of $Re_{\unicode[STIX]{x1D70F}}$ estimated by the simply crossing of low- and high- $Re_{\unicode[STIX]{x1D70F}}$ scalings are different. For example, equating (2.11a ) with (2.12a ) we obtain a crossover $Re_{\unicode[STIX]{x1D70F}}$ for $1-W_{p}^{+}$ in TBL (i.e. $Re_{\unicode[STIX]{x1D70F}}\approx 300$ ) – much smaller than 1200 for the transition of  $y_{p}^{+}$ . That is why for the same DNS data sets, scaling transitions are observed for $1-W_{p}^{+}$ in figure 5, but not for $y_{p}^{+}$ in figure 4.

Figure 6.

Compensated plots of $y_{p}^{+}$ scaling for channel and pipe flows (a), and for TBL flows (b). Compensated plots of $1-W_{p}^{+}$ scaling for channel and pipe flows (c), and for TBL flows (d). Lines are scaling predictions; symbols are data – the same as in figure 2 (for channel and pipe) and figure 4 (for TBL).

To better display the scaling transition, we show in figure 6 the compensated plots of both $y_{p}^{+}$ and $1-W_{p}^{+}$ . The plateaus in figure 6(a,b) justify the small- $Re$ scaling $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ for CP and $Re_{\unicode[STIX]{x1D70F}}^{3/7}$ for TBL, respectively; data departing from the plateaus indicate the scaling transition with increasing $Re_{\unicode[STIX]{x1D70F}}$ . For example, for channel flows, most DNS data follow the $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ scaling; however, DNS data at $Re_{\unicode[STIX]{x1D70F}}=4000$ (Yamamoto & Tsuji Reference Yamamoto and Tsuji2018), 5200 (Lee & Moser Reference Lee and Moser2015) and 8000 (Yamamoto & Tsuji Reference Yamamoto and Tsuji2018) agree better with (2.3) than with (2.5), hence clearly displaying the predicted scaling transition. Moreover, figure 6(c,d) show the transition in $1-W_{p}^{+}$ , where most DNS data are in the large-scaling regimes. It is unknown why the scaling transition for $1-W_{p}^{+}$ is less sharp and occurs earlier than that for  $y_{p}^{+}$ , possibly due to $1-W_{p}^{+}$ being a subleading-order behaviour of the momentum cascade (the leading-order behaviour is $W^{+}\approx 1$ as $Re\rightarrow \infty$ ), which varies mildly but sensitively with increasing $Re$ . A further examination on the vorticity transport of $W^{+}$ (Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014) is perhaps helpful to answer this question. Note the deviation from (2.11a ) for the two largest $Re$ data points in figure 6(d), which may suggest a new scaling trend but may also be caused by insufficiently large simulation domains – an issue to clarify when high-quality data at large $Re$ are available.

According to the DNS data trend and our predictions in figure 5, $1-W_{p}^{+}<0.03$ for CP at $Re_{\unicode[STIX]{x1D70F}}=10^{4}$ and $1-W_{p}^{+}<0.01$ at $Re_{\unicode[STIX]{x1D70F}}=10^{5}$ ; the deviation from unity is even smaller for TBL (i.e. $1-W_{p}^{+}<0.01$ at $Re_{\unicode[STIX]{x1D70F}}=10^{4}$ and $1-W_{p}^{+}<0.003$ at $Re_{\unicode[STIX]{x1D70F}}=10^{5}$ ). These small deviations are far beyond the experimental resolutions, and hence we use DNS data to validate our predictions of $1-W_{p}^{+}$ . It is clear from figures 5 and 6(c,d) that the predicted large- $Re$ scalings are well supported by both CP and TBL DNS data; while for small $Re$ , there are only a few data points in these figures, showing mild deviations from our large- $Re$ predictions, but they all follow closely our scaling predictions for small $Re$ . One can of course fit the solid symbols in figure 5 (i.e. $1-W_{p}^{+}$ of TBL) by a single power law (for example, $6.2Re_{\unicode[STIX]{x1D70F}}^{-0.7}$ ), but then one cannot explain these fitting parameters. Moreover, the fitting may be another data misinterpretation, as there is a scaling transition according to our theory.

Finally, let us envisage the existence of an asymptotic flow state for CP and TBL by the probable full coupling between energy and momentum cascades. While the energy cascade is more developed in the bulk flow region ( $y^{+}>y_{Q}^{+}$ with $y_{Q}^{+}\sim 150{-}200$ ) where production quasi-balances dissipation, the momentum cascade is centred at $y^{+}=y_{p}^{+}$ , which is smaller than $y_{Q}^{+}$ at moderate $Re_{\unicode[STIX]{x1D70F}}$ but would overtake $y_{Q}^{+}$ as $Re_{\unicode[STIX]{x1D70F}}$ increases. Therefore, equating $y_{p}^{+}=y_{Q}^{+}$ , we obtain a critical $Re_{\unicode[STIX]{x1D70F}}$ indicating the onset of co-existence of two cascades in the bulk flow. It was speculated (Chen et al. Reference Chen, Wei, Hussain and She2015) that at this state the energy cascade will be modified, leading to anomalous energy dissipation. Taking $y_{Q}^{+}\approx 150$ for CP (Vincenti et al. Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013) yields $Re_{\unicode[STIX]{x1D70F}}^{(c)}=\unicode[STIX]{x1D705}y_{Q}^{+2}\approx 10\,100$ , which interestingly matches $Re_{\unicode[STIX]{x1D70F}}\approx 10\,481$ for the emergence of an outer $\overline{u^{\prime }u^{\prime }}$ peak in the Princeton SuperPipe (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2013). For TBL, $y_{Q}^{+}\approx 200$ (Schlatter et al. Reference Schlatter, Li, Brethouwer, Johansson and Henningson2010), using $y_{p}^{+}$ in (2.9) we obtain another critical $Re_{\unicode[STIX]{x1D70F}}^{(c)}\approx 5500$ , very close to $Re_{\unicode[STIX]{x1D70F}}\approx 6400$ where the outer $\overline{u^{\prime }u^{\prime }}$ peak in TBL is observed (Vincenti et al. Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013). This apparent link to the emergence of the $\overline{u^{\prime }u^{\prime }}$ outer peak is intriguing, which implies a possible asymptotic flow regime for Reynolds number larger than $Re_{\unicode[STIX]{x1D70F}}^{(c)}$ .