2.1. NSTAP measurements

In the current study, two nanoscale thermal anemometry probes (NSTAPs) were used, one with filament length $\ell =60~{\rm\mu}\text{m}$ and cross-section $0.1~{\rm\mu}\text{m}\times 2~{\rm\mu}\text{m}$ , and a second one with $\ell =30~{\rm\mu}\text{m}$ and cross-section $0.08~{\rm\mu}\text{m}\times 1.5~{\rm\mu}\text{m}$ . The fabrication, validation and behaviour of these sensors are described in detail by Bailey et al. (Reference Bailey, Kunkel, Hultmark, Vallikivi, Hill, Meyer, Tsay, Arnold and Smits2010), Vallikivi et al. (Reference Vallikivi, Hultmark, Bailey and Smits2011) and Vallikivi & Smits (Reference Vallikivi and Smits2014).

The sensors were operated using a Dantec Streamline Constant Temperature Anemometry system with a 1:1 bridge, keeping the heated filament at a temperature of about 450 K. The frequency response, determined from a square wave test, was always above 150 kHz in still air, increasing to more than 300 kHz in the flow. The data were digitized using a 16-bit A/D board (NI PCI-6123) at a rate of 300 kHz and low-pass filtered using an eighth-order Butterworth filter at 150 kHz. The sensor was traversed in the wall-normal direction using a stepper motor traverse with a Numeric Jena LIK22 encoder with a resolution of $0.05~{\rm\mu}\text{m}$ . The initial wall-normal distance $y_{0}$ of the probe was measured using a depth-measuring microscope (Titan Tool Supply Inc.) with an accuracy of $5~{\rm\mu}\text{m}$ .

The NSTAPs were calibrated using a 0.4 mm Pitot tube together with two 0.4 mm static pressure taps located in the pipe wall at the same streamwise location. The Pitot tube measurements were corrected for static tap Reynolds number effects and viscous effects using the correlations proposed by McKeon & Smits (Reference McKeon and Smits2002) and McKeon et al. (Reference McKeon, Li, Jiang, Morrison and Smits2003), respectively. Fourteen calibration points were used, and the calibration was performed before and after each profile measurement. A fourth-order polynomial fit was used to find the calibration coefficients. The ambient fluid temperature change during a given profile ranged from 0.7 to 10.0 °C over the full Reynolds number range, and the data were corrected using the temperature correction outlined by Hultmark & Smits (Reference Hultmark and Smits2010).

NSTAP data were acquired for $2600<\mathit{Re}_{{\it\tau}}<72\,500$ , corresponding to $8400<\mathit{Re}_{{\it\theta}}<235\,000$ (where $\mathit{Re}_{{\it\theta}}=U_{\infty }{\it\theta}/{\it\nu}$ ). The tunnel was pressurized for all cases and the experimental conditions are listed in table 1 (cases 1–7). Here, $p_{a}$ denotes the ambient pressure, $\ell ^{+}$ is the wire length in viscous units ( $=\!\ell u_{{\it\tau}}/{\it\nu}$ ), $y_{0}$ is the initial wall-normal distance and $y_{0}^{+}=y_{0}u_{{\it\tau}}/{\it\nu}$ . The data were sampled for 60 s in cases 1–6 and for 20 s in case 7, corresponding to convection lengths of more than 20 000 and $5000{\it\delta}$ , respectively. Convergence was tested by comparing the magnitudes of the moments evaluated over the full sampling time with those obtained over half the sampling time. Even for the 10th-order moment, the differences were within 3 %.

Table 1.

Experimental conditions for boundary layer measurements: $p_{a}$ is the ambient tunnel pressure, $y_{0}$ is the initial distance from the wall, and $u_{{\it\tau}}$ is derived using the correlation proposed by Fernholz & Finley (Reference Fernholz and Finley1996). Cases 1–7: NSTAP data with sensor length $\ell$ . Cases 8–14: Pitot probe data with diameter $d_{p}$ .

It can be seen from table 1 that at the higher Reynolds number even the NSTAP probe has insufficient spatial resolution. To minimize bias errors due to spatial filtering, the correction by Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011b ) was applied to the variance. Smits et al. showed that this correction works well for $\ell ^{+}$ up to 153, considerably greater than the maximum value found here ( $=\!75$ ). Any data on the variance where the spatial filtering correction exceeded 5 % are shown with grey symbols in figure 4. Note that spatial resolution has its major effect in the near-wall region, and even at the highest Reynolds numbers the correction is less than 5 % for $200{\it\nu}/u_{{\it\tau}}<y=0.003{\it\delta}$ . For the higher-order moments, no equivalent correction method exists, so all higher-order data points where the correction on the variance exceeded 5 % were removed from the data set. In addition to spatial filtering, there is the uncertainty in initial wall location $y_{0}$ (about $5~{\rm\mu}\text{m}$ ), which needs to be borne in mind when examining the results for $y^{+}<100$ at high Reynolds numbers.

2.2. Pitot tube measurements

In addition to the NSTAP data, Pitot tube measurements were taken for $2800<\mathit{Re}_{{\it\tau}}<65\,000$ , corresponding to $9400<\mathit{Re}_{{\it\theta}}<223\,000$ . The experimental conditions for these cases are given in table 1 (cases 8–14). A Pitot probe with an outer diameter of $d_{p}=0.20~\text{mm}$ was used, in conjunction with two 0.4 mm static pressure taps in the plate. The pressure difference was measured using a DP15 Validyne pressure transducer with a 1.40 kPa range which was calibrated against a manometer standard. As for the NSTAP measurements, the initial wall distance of the Pitot probe $y_{0}$ was measured using a depth-measuring optical microscope and the probe was traversed in the wall-normal direction using a stepper motor traverse with a resolution of $0.05~{\rm\mu}\text{m}$ . The Pitot tube measurements were corrected following Bailey et al. (Reference Bailey, Hultmark, Monty, Alfredsson, Chong, Duncan, Fransson, Hutchins, Marusic, McKeon, Nagib, Örlü, Segalini, Smits and Vinuesa2013), including the static tap correction by McKeon et al. (Reference McKeon, Li, Jiang, Morrison and Smits2003), viscous and shear corrections by Zagarola & Smits (Reference Zagarola and Smits1998), and the near-wall correction by MacMillan (Reference MacMillan1957). The data for wall distances smaller than $2d_{p}$ were neglected, as in the pipe flow experiments described by Bailey et al. (Reference Bailey, Vallikivi, Hultmark and Smits2014). Further details on the experimental techniques are given by Vallikivi (Reference Vallikivi2014).

2.3. Friction velocity

To determine $u_{{\it\tau}}$ and the skin friction coefficient $C_{f}=2u_{{\it\tau}}^{2}/U_{\infty }^{2}$ , a number of different methods were used. First, the 0.2 mm Pitot probe when in contact with the wall was used as a Preston tube (Patel Reference Patel1965; Zagarola, Williams & Smits Reference Zagarola, Williams and Smits2001). Second, the Clauser chart technique (Clauser Reference Clauser1956) was used, where the log-law was fitted to the velocity profiles using the constants ${\it\kappa}=0.40$ and $B=5.1$ recommended by Coles (Reference Coles1956). These results were compared to the skin friction correlation proposed by Fernholz & Finley (Reference Fernholz and Finley1996), as well as data from one of the few direct measurements of skin friction using a drag plate at high Reynolds number (Winter & Gaudet Reference Winter and Gaudet1973) (see figure 2). For comparison, values from DeGraaff & Eaton (Reference DeGraaff and Eaton2000) found using the Clauser chart are also shown.

Figure 2.

Boundary layer skin friction coefficient $C_{f}$ : ○, Preston tube; ▫, Clauser fit for Pitot data sets; ▪, Clauser fit for NSTAP data sets (Clauser Reference Clauser1956); *, DeGraaff & Eaton (Reference DeGraaff and Eaton2000); solid line, Fernholz & Finley (Reference Fernholz and Finley1996); dashed line, Winter & Gaudet (Reference Winter and Gaudet1973). Error bars indicate $\pm 5\,\%$ .

All $C_{f}$ estimates except for the Preston tube data had a standard deviation less than 3 % compared to the Fernholz correlation. In addition, the Fernholz correlation closely matches the average value obtained by the other methods, and it agrees well with the force plate measurements by Winter & Gaudet (Reference Winter and Gaudet1973) over the same Reynolds number range. Hence, we used the value of $u_{{\it\tau}}$ determined from the Fernholz correlation for all subsequent data analysis.

2.4. Pipe flow data for comparison

We compare the turbulent boundary layer data with that from fully developed turbulent pipe flow obtained by Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013). The cases used for comparison are listed in table 2, and cover $3300\leqslant \mathit{Re}_{{\it\tau}}\leqslant 98\,000$ .

Table 2.

Pipe flow data for comparison, from Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2013).

3.1. Mean flow

The mean velocity profiles for the boundary layer are shown in figure 3. The agreement between the NSTAP and Pitot profiles is within 1.3 %, well within the uncertainty on $U$ (estimated to be ${<}\!2.2\,\%$ ). The mean velocity behaviour and scaling in boundary layers may be compared to the behaviour in pipe flow by referring to the extensive discussions of pipe flows given by Zagarola & Smits (Reference Zagarola and Smits1998), McKeon et al. (Reference McKeon, Li, Jiang, Morrison and Smits2004), Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2013) and Bailey et al. (Reference Bailey, Vallikivi, Hultmark and Smits2014), and so this will not be repeated here. Suffice it to say that both flows show an extended region of logarithmic behaviour, although for the boundary layer this behaviour starts closer to the wall compared to pipe flows, where the log-law only appears for $y^{+}>600{-}800$ . The middle of the log-layer, located at $y^{+}=3\mathit{Re}_{{\it\tau}}^{0.5}$ according to Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013), served as a conservative lower bound for fitting the logarithmic portion of the profile, and for all cases the log-layer was found to extend to about $0.15{\it\delta}$ . Due to the many uncertainties in the evaluation of the slope (and friction velocity in the boundary layer), it is not possible to determine any differences in the von Kármán constant between pipes and boundary layers (Bailey et al. Reference Bailey, Vallikivi, Hultmark and Smits2014). The constants proposed by Coles (Reference Coles1956) ( ${\it\kappa}=0.40$ and $B=5.1$ ) give an equally good fit for both flows.

Figure 3.

Boundary layer mean velocity profiles. (a) Inner coordinates; (b) inner coordinates, each profile shifted by ${\rm\Delta}U^{+}=5$ ; (c) outer coordinates. Symbols as given in table 1. In (a,b) solid black lines show (1.1) with ${\it\kappa}=0.40$ and $B=5.1$ ; thin solid lines show $U^{+}=y^{+}$ .

Table 3 lists the boundary layer thickness ${\it\delta}={\it\delta}_{99}$ , displacement thickness ${\it\delta}^{\ast }$ , momentum thickness ${\it\theta}$ , and shape factor $H={\it\delta}^{\ast }/{\it\theta}$ for each case. All bulk properties were found to decrease with Reynolds number in the expected manner, as observed by DeGraaff & Eaton (Reference DeGraaff and Eaton2000) and others, and they are discussed in more detail by Vallikivi et al. (Reference Vallikivi, Hultmark and Smits2013) and Vallikivi (Reference Vallikivi2014).

3.2. Variances

In figure 4, profiles of the streamwise variances $u^{2+}$ are shown in inner coordinates, with the boundary layer data on the left and the pipe flow data on the right. Points in the boundary layer data where the spatial filtering correction is greater than 5 % are indicated by grey symbols. The two flows show a broadly similar behaviour with a distinct inner peak at approximately the same wall-normal position. The inner peak appears to be invariant with Reynolds number, with a non-dimensional magnitude $u_{I}^{2+}=8.4\pm 0.8$ for the boundary layer data, which agrees with the pipe data within experimental error. However, the inner peak values for the boundary layer are only resolved for the three lowest Reynolds numbers tested ( $2622\leqslant \mathit{Re}_{{\it\tau}}\leqslant 8261$ ). Thus, the data cannot resolve the question regarding the scaling of the inner peak at higher Reynolds numbers.

Figure 4.

Variance profiles in inner coordinates: (a) boundary layer; (b) pipe. Symbols as given in tables 1 and 2. Grey symbols indicate boundary layer data where the spatial filtering exceeds 5 % according to the correction by Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011b ).

Table 3.

Boundary layer bulk and integral properties, with $C_{f}$ obtained using Fernholz & Finley (Reference Fernholz and Finley1996) correlation; see also table 1.

The data indicate that an outer peak emerges at approximately the same Reynolds number for the two flows. The outer peak magnitude, $u_{II}^{2+}$ , was found for each Reynolds number (for the three lowest Reynolds numbers there is no peak, and so the inflection point was used instead), and the results shown in figure 5 demonstrate that the magnitudes of the outer peaks are very similar in boundary layer and pipe flows.

Figure 5.

Magnitudes of the outer peak in $u^{2+}$ : ▫, boundary layer; ○, pipe; filled symbols, $\mathit{Re}_{{\it\tau}}>20\,000$ ; ♢, atmospheric boundary layer (Metzger, McKeon & Holmes Reference Metzger, McKeon and Holmes2007); - - - -, $u_{II}^{2+}=0.63\ln (\mathit{Re}_{{\it\tau}})+0.33$ (Pullin, Inoue & Saito Reference Pullin, Inoue and Saito2013); – ⋅ – ⋅ –,  $u_{II}^{2+}=0.42\ln (\mathit{Re}_{{\it\tau}})+2.82$ (Pullin et al. Reference Pullin, Inoue and Saito2013); 

, best fit to pipe data, given by $u_{II}^{2+}=0.49\ln (\mathit{Re}_{{\it\tau}})+1.7$ ; 

, best fit to boundary layer data, given by $u_{II}^{2+}=0.47\ln (\mathit{Re}_{{\it\tau}})+2.0$ .

Figure 6.

Variance profiles in outer coordinates: (a,c) boundary layer; (b,d) pipe. (a,b) All $\mathit{Re}_{{\it\tau}}$ ; (c,d) data for $\mathit{Re}_{{\it\tau}}\geqslant 20\,000$ and $y^{+}\geqslant 100$ . Symbols as given in tables 1 and 2. Solid line, (1.2) with $A_{1}=1.24$ and $B_{1}=1.48$ .

Pullin et al. (Reference Pullin, Inoue and Saito2013) presented an analysis that supports a logarithmic increase in the outer peak value $u_{II}^{2+}$ with Reynolds number, with two possible relations depending on the relation governing the location of the peak: either $u_{II}^{2+}=0.42\ln (\mathit{Re}_{{\it\tau}})+2.82$ , or $u_{II}^{2+}=0.63\ln (\mathit{Re}_{{\it\tau}})+0.33$ . Figure 5 shows that these relations are in generally good agreement with the data, although our analysis gives slightly different curve fits ( $u_{II}^{2+}=0.49\ln (\mathit{Re}_{{\it\tau}})+1.7$ in pipes and $u_{II}^{2+}=0.47\ln (\mathit{Re}_{{\it\tau}})+2.0$ in boundary layers). The data broadly support the hypothesis of Pullin et al. that at the limit of infinite Reynolds number the wall-normal turbulent transport of turbulent energy declines and the turbulence is asymptotically attenuated across the whole outer layer. Hence, with increasing Reynolds number, the locations of the inner peak (at constant $y^{+}$ ) and the outer peak (varying as $\ln \mathit{Re}_{{\it\tau}}$ ) are both moving closer to the wall in physical coordinates, and so the asymptotic state of the wall layer is a slip-flow bounded by a vortex sheet at the wall (Pullin et al. Reference Pullin, Inoue and Saito2013).

Figure 6 shows that in outer coordinates the variances do not collapse as well in boundary layers as they do in pipes, and a clear trend with increasing Reynolds numbers can be observed, especially at the lower Reynolds numbers. The mixed scaling $(u_{{\it\tau}}U_{\infty })^{0.5}$ , introduced for boundary layers by DeGraaff & Eaton (Reference DeGraaff and Eaton2000), did not noticeably improve the collapse, and so it was not pursued further.

Figure 7.

Mean velocity profiles (empty symbols) and turbulence intensities (filled symbols) for $\mathit{Re}_{{\it\tau}}\approx 20\,000$ (a,b), $Re_{{\it\tau}}\approx 40\,000$ (c,d), $Re_{{\it\tau}}\approx 70\,000$ (e,f) in the boundary layer (a,c,e) and the pipe (b,d,f). Symbols as given in tables 1 and 2. Lines: - - - -, limits of logarithmic region ( $y^{+}=400$ and $y^{+}=0.15\mathit{Re}_{{\it\tau}}$ for the boundary layer, $y^{+}=800$ and $y^{+}=0.15\mathit{Re}_{{\it\tau}}$ for the pipe); ——, (1.1) with ${\it\kappa}=0.40,B=5.1$ , and (1.2) with $A_{1}=1.24,B_{1}=1.48$ .

For $\mathit{Re}_{{\it\tau}}>20\,000$ , the logarithmic behaviour described by (1.2) is observed in the boundary layer profiles for $y^{+}\gtrsim 400$ . This logarithmic behaviour can be seen more clearly when the data are restricted to $\mathit{Re}_{{\it\tau}}>20\,000$ and $y^{+}>100$ . The Townsend–Perry constant $A_{1}=1.24$ and additive constant $B_{1}=1.48$ were taken from the results in pipe flow (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2013). The slope appears to be very similar in boundary layer and pipe flows, as found by Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) (who used $A_{1}=1.26$ ), but for boundary layers there seems to be some variation with Reynolds number in the additive constant. The uncertainty limits on these constants are unknown at present.

3.3. Logarithmic regions in the mean and the variance

Profiles of the mean velocity and the variances are shown cross-plotted together in figure 7. The dashed lines correspond to $y^{+}=400$ and $y/{\it\delta}=0.15$ in the boundary layer, and $y^{+}=800$ and $y/{\it\delta}=0.15$ in the pipe. Both the mean velocity and the variances follow a logarithmic behaviour within these limits, but the mean velocity in the boundary layer case seems to follow a logarithmic variation down to $y^{+}\approx 100$ . This result indicates the presence of an intermediate range below $y^{+}\approx 400$ in boundary layers, and below $y^{+}\approx 800$ in pipes, where the boundary layer variances deviate from the logarithmic behaviour in a manner very similar to that described in Hultmark (Reference Hultmark2012) for pipe flows. This region corresponds approximately to the mesolayer described by Afzal (Reference Afzal1982, Reference Afzal1984), George & Castillo (Reference George and Castillo1997), Sreenivasan & Sahay (Reference Sreenivasan, Sahay and Panton1997), Wosnik, Castillo & George (Reference Wosnik, Castillo and George2000) and Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005) in boundary layers, and the power law region described by McKeon et al. (Reference McKeon, Li, Jiang, Morrison and Smits2004) in pipes. George & Castillo (Reference George and Castillo1997) suggested that in this region the mean velocity has reached a seemingly logarithmic behaviour but the effects of viscosity are still evident in the behaviour of the turbulent stress terms. Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005) identified the mesolayer with the region where the stress gradients are close to zero, and the viscous force balances the pressure force in pipe flow or the mean advection in the turbulent boundary layer. The present results are at a sufficiently high Reynolds number to distinguish the mesolayer clearly from the logarithmic region where all viscous effects are negligible, but a more precise statement needs a consideration of the spectral behaviour, which is given by Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015).

3.4. Higher-order moments

We now consider the behaviour of the higher-order moments of the streamwise velocity fluctuations (up to 10th order). Because no established correction for spatial filtering on higher-order moments exists, data with more than 5 % spatial filtering on the variances, as indicated by the spatial filtering correction, were excluded from the analysis.

To make direct comparisons between boundary layer and pipe flows, six cases with matching Reynolds numbers were chosen: cases 1–3 and 5–7 given in table 1 for the boundary layer, and cases 1–6 given in table 2 for the pipe. These cases correspond to $\mathit{Re}_{{\it\tau}}\approx$ 3000, 5000, 10 000, 20 000, 40 000, and 70 000, and they are summarized in table 4. The higher-order moments for the pipe are reported here for the first time, although they are based on the data collected by Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2013).

Table 4.

Cases chosen for comparing boundary layer (current data) and pipe flow (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2013).

Figure 8 shows, for a representative boundary layer case, the probability density function $P(u)$ (p.d.f.), and the premultiplied probability density function $u^{2p}P(u)$ , where $2p=[2,6,10]$ indicates the $p$ th even moment. Each moment is the area under the corresponding curve, and the data appear to be statistically converged. We see a clear deviation around the maximum value in the p.d.f. from a Gaussian behaviour, as well as deviations from symmetry in the premultiplied p.d.f.s.

Figure 8.

Probability density functions for the boundary layer at $\mathit{Re}_{{\it\tau}}=70\,000$ , $y^{+}=800$ . (a) ●, $P(u)$ ; ——, Gaussian distribution. (b) $u^{2p}P(u)$ ; ▪, $2p=2$ ; ◂, $2p=6$ ; ▴, $2p=10$ .

The skewness $S=\langle u^{3}\rangle /\langle u^{2}\rangle ^{3/2}$ is shown in figure 9. Both flows exhibit a very similar behaviour, with the skewness slightly positive near the wall for $y^{+}<200$ and becoming negative further away from the wall. For the boundary layer, the skewness is well collapsed in inner coordinates over the region $100<y^{+}<0.15\mathit{Re}_{{\it\tau}}$ , and we see all profiles change sign at $y^{+}\approx 200$ and reach a value of $S\approx -0.1$ before becoming more negative in the wake, where they collapse well in outer coordinates. In contrast, the pipe flow profiles show a small Reynolds number dependence over the log region, and the collapse in the wake region in terms of outer layer coordinates is not as clean as that seen in the boundary layer data.

Figure 9.

Skewness in inner coordinates (a,b) and outer coordinates (c,d) for cases given in table 4. (a,c) Boundary layer; (b,d) pipe.

The kurtosis $K=\langle u^{4}\rangle /\langle u^{2}\rangle ^{2}$ is shown in figure 10. Again, the behaviour for the two flows is very similar, with perhaps some small dependence on the Reynolds number in the outer region (although this could also be the result of some unresolved spatial filtering effects).

Figure 10.

Kurtosis in inner coordinates (a,b) and outer coordinates (c,d) for cases given in table 4. (a,c) Boundary layer; (b,d) pipe.

The $p$ th roots of the $p$ th even moments $\langle (u^{+})^{2p}\rangle ^{1/p}$ are shown in figure 11 for $2p=2$ , 6, and 10 (the data for $2p=4$ , 8, and 12 show the same trends). In inner coordinates, the behaviour is qualitatively similar to that seen in the variances, with an inner peak at about $y^{+}=15$ that is either increasing very slowly with Reynolds number or not at all, a blending region over the range $30<y^{+}<300$ , followed by a logarithmic region. The results from the boundary layer and the pipe agree well throughout most of the flow, with the only differences appearing in the outer layer due to the different outer boundary conditions, as expected. Some minor differences can also be seen in the near-wall region, around $y^{+}\approx 15$ , where the pipe flow displays a slightly higher peak value, possibly due to smaller spatial filtering effects since $\ell ^{+}$ is smaller for the pipe than the boundary layer (see table 4). Finally, it appears that the inner limit of the logarithmic range for pipes occurs at higher values of $y^{+}$ than in boundary layers, similar to what was observed for the variances.

Figure 11.

Higher-order even moments $2p=2$ , 6, and 10 for boundary layer (filled symbols) and pipe flow (empty symbols). Data in inner coordinates (a,c,e) for all cases given in table 4. Same data in outer coordinates (b,d,f) for $\mathit{Re}_{{\it\tau}}\geqslant 20\,000$ and $y^{+}>100$ . Solid line, (1.3) with $A_{p}$ and $B_{p}$ derived from the pipe flow profile at $\mathit{Re}_{{\it\tau}}=70\,000$ (where $A_{p}=[1.13;2.48;3.44]$ and $B_{p}=[1.17;3.31;6.73]$ accordingly).

Figure 11 also displays the data in outer coordinates for the three highest Reynolds numbers ( $\mathit{Re}_{{\it\tau}}\geqslant 20\,000$ ). As seen in the variances, the higher moments show good agreement between boundary layer and pipe flows. With increasing moment, the agreement improves, and the range of logarithmic behaviour increases.

The constants $A_{p}$ and $B_{p}$ in (1.3) were found by regression fit to each profile, that is, separately for each flow, Reynolds number, and moment. To test the consequences of choosing a particular range for the curve fit, different ranges were used for fitting, with the inner limit varying as $y_{min}^{+}=[3\mathit{Re}_{{\it\tau}}^{0.5};200;400;600;800]$ while keeping a constant outer limit at $(y/{\it\delta})_{max}=0.15$ (the results were not sensitive to reasonable variations in the outer limit). A minimum of four points in each profile were used for determining the constants, otherwise the profile was discarded as not having a sufficiently extensive logarithmic region.

The variation of the slope $A_{p}$ with $p$ is shown in figure 12. For Gaussian statistics, $A_{p}$ would vary as $A_{1}[(2p-1)!!]^{1/p}$ , where $A_{1}$ is the Townsend–Perry constant and $!!$ denotes double factorial. It is evident that for boundary layer and pipe flows all the constants have a sub-Gaussian behaviour, as observed by Meneveau & Marusic (Reference Meneveau and Marusic2013) for boundary layers at lower Reynolds numbers. For smaller $y_{min}^{+}$ values, there is a clear Reynolds number dependence in $A_{p}$ between cases, as well as a dependence on $y_{min}^{+}$ . For both flows, $A_{p}$ was found to become independent of $\mathit{Re}_{{\it\tau}}$ for $y_{min}^{+}\gtrsim 400$ . The limit $y_{min}^{+}=3\mathit{Re}_{{\it\tau}}^{0.5}$ used by Meneveau & Marusic (Reference Meneveau and Marusic2013) underestimates the inner limit at low $\mathit{Re}_{{\it\tau}}$ while overestimating it at high $\mathit{Re}_{{\it\tau}}$ , and so it appears that either a constant inner limit or one with a very weak $\mathit{Re}_{{\it\tau}}$ dependence is more appropriate for both flows. A good representation of the asymptotic value of the slope is given empirically by $A_{p}\sim A_{1}(2p-1)^{1/2}$ for both pipe and boundary layer ( $A_{1}=1.24$ , as before).

Figure 12.

Constant $A_{p}$ for even moments $2p$ , for the fitting range $y^{+}=[y_{min}^{+},0.15Re_{{\it\tau}}]$ . Left: boundary layer; right: pipe. Line colors given in table 4. ——, expected Gaussian variation $A_{p}=A_{1}[(2p-1)!!]^{1/p}$ ; - - - -, empirical fit $A_{p}=A_{1}(2p-1)^{1/2}$ .

It appears that if a large enough value of $y_{min}^{+}$ is chosen, $A_{p}$ is independent of Reynolds number in pipe and boundary layer flows. The only outlier, the highest Reynolds number case for the boundary layer, has a slightly lower value of $A_{p}$ , but this could be due to experimental error, which is expected to be greatest at the highest Reynolds number.

The behaviour of the additive constant $B_{p}$ is more difficult to establish, due to its high sensitivity to the magnitude of the moments. In pipes the constant appears to be independent of Reynolds number, whereas in boundary layers a weak dependence is observed in most cases. In contrast, Meneveau & Marusic (Reference Meneveau and Marusic2013) found a much stronger dependence, possibly caused by using $3\mathit{Re}_{{\it\tau}}^{0.5}$ as the inner limit on the curve fit. Interestingly, for the current data set $B_{p}$ reached a constant value for the three highest Reynolds number cases when $y_{min}^{+}\geqslant 600$ was used as inner limit, which could suggest that high-order moments are still affected by viscosity for smaller $y^{+}$ . However, these trends are probably within experimental error, and no strong conclusions can be made.