3.1. Kolmogorov scaling

The spectra in Kolmogorov scaling are shown in figure 1 for $0.001\leqslant y/{\it\delta}\leqslant 1.0$ at $\mathit{Re}_{{\it\tau}}\approx 20\,000$ . The Kolmogorov scaling collapses the higher wavenumber data well for both flows at all locations, and as $y/{\it\delta}$ increases a power law range with a slope close to $-5/3$ emerges, extending a maximum of two decades with an upper limit at $k_{x}{\it\eta}_{K}\approx 0.1$ . The energy at larger length scales (smaller $k_{x}$ ) increases with $y/{\it\delta}$ , until at $y/{\it\delta}>0.15$ the energy at large scales starts to decrease in the wake region. The extent of the power law region continues to increase with increasing wall distance, but for boundary layers at larger $y/{\it\delta}$ the spectra depart from this collapse at higher wavenumbers than in pipe flow, which is likely due to the large-scale intermittency in the outer part of the boundary layer.

Figure 1.

Kolmogorov spectra in wall scaling for boundary layer (a) and pipe flow (b) at wall-normal positions $y/{\it\delta}\approx 0.001$ , 0.005, 0.01, 0.05, 0.15, 0.3 and 0.5, at $\mathit{Re}_{{\it\tau}}\approx 20\,000$ . Solid lines indicate wall locations $y/{\it\delta}\leqslant 0.15$ , changing to dashed lines for $y/{\it\delta}>0.15$ . Arrow indicates increase in $y/{\it\delta}$ until $y/{\it\delta}=0.15$ .

Figure 2.

Kolmogorov spectra for boundary layer (a) and pipe flow (b) with $\mathit{Re}_{{\it\tau}}\approx 3\times 10^{3}$ , $5\times 10^{3}$ , $10\times 10^{3}$ , $20\times 10^{3}$ , $40\times 10^{3}$ and $70\times 10^{3}$ , at $y/{\it\delta}\approx 0.5$ .

Figure 3.

Kolmogorov spectra for at $\mathit{Re}_{{\it\tau}}\approx 3\times 10^{3}$ , $5\times 10^{3}$ , $10\times 10^{3}$ , $20\times 10^{3}$ , $40\times 10^{3}$ , and $70\times 10^{3}$ , at $y/{\it\delta}\approx 0.05$ (a) and $y/{\it\delta}\approx 0.5$ (b); ——, boundary layer; – – – –, pipe.

Figure 2 shows Kolmogorov spectra for all Reynolds numbers at $y/{\it\delta}=0.5$ . The behaviour of boundary layer and pipe spectra at higher wavenumber is essentially identical for $\mathit{Re}_{{\it\tau}}\geqslant 5000$ , indicating that the smaller wavelengths are largely independent of the flow geometry at these Reynolds numbers. To demonstrate this point further, the boundary layer and pipe spectra are plotted together for $y/{\it\delta}=0.05$ and $y/{\it\delta}=0.5$ in Figure 3. We see that the curves collapse very well for the higher wavenumbers, but that they have an exponent that is closer to $-1.5$ or 1.52 than to $-5/3$ . This observation agrees with the work of Mydlarski & Warhaft (Reference Mydlarski and Warhaft1996), Gamard & George (Reference Gamard and George2000), McKeon & Morrison (Reference McKeon and Morrison2007) and George & Tutkun (Reference George and Tutkun2009), among others, who propose that viscous effects continue to be important in the inertial region, and that a slope of $-5/3$ is only reached at infinite Reynolds number. For example, Mydlarski & Warhaft (Reference Mydlarski and Warhaft1996) found an empirical relation for the slope given by $(5/3)\;(1{-}3.15\mathit{Re}_{{\it\lambda}}^{-2/3})$ , where $\mathit{Re}_{{\it\lambda}}$ is the Reynolds number based on the Taylor microscale. For the current data set, $\mathit{Re}_{{\it\lambda}}$ varies from about 200 to 1000, where Mydlarski et al.’s relationship gives slopes of 1.5–1.6, which agrees well with our observations.

By assuming Kolmogorov scaling to be valid at all Reynolds numbers, the error in finding the mean dissipation ${\it\varepsilon}$ can be estimated by examining the lack of collapse of the experimental spectra at high wavenumbers. This presumed error was found to increase with $\mathit{Re}_{{\it\tau}}$ from 0.5 % to 4 % for the pipe, and from 1 % to 5 % for the boundary layer, with the exception of the boundary layer case at $\mathit{Re}_{{\it\tau}}\approx 70\,000$ where the error in ${\it\varepsilon}$ was about 20 %. The corresponding maximum error in ${\it\eta}_{K}$ was therefore always less than 5 %.

3.2. The $k_{x}^{-1}$ dependence

To examine the $k_{x}^{-1}$ dependence, spectra in the usual log–log form are given in figure 4 for $0.001\leqslant y/{\it\delta}\leqslant 0.5$ . The spectra are also shown in premultiplied form in figures 5 and 6 at $\mathit{Re}_{{\it\tau}}=5000$ and 70 000, respectively, and in this form the $k_{x}^{-1}$ region would show up as a plateau. At both Reynolds numbers, the pipe flow spectra show two peaks: one at lower wavenumbers associated with the VLSM, and another at higher wavenumbers associated with the LSM (Balakumar & Adrian Reference Balakumar and Adrian2007). For boundary layers, the lower wavenumber peak associated with the SS (Hutchins & Marusic Reference Hutchins and Marusic2007) is present at all Reynolds numbers, while the higher-wavenumber LSM peak only appears at some Reynolds numbers at some locations. This is illustrated more precisely in figure 7.

Figure 4.

Spectra in wall scaling for boundary layer (a) and pipe flow (b) at wall-normal positions $y/{\it\delta}\approx 0.001$ , 0.005, 0.01, 0.05, 0.15, 0.3 and 0.5, for $\mathit{Re}_{{\it\tau}}\approx 5\times 10^{3}$ , $20\times 10^{3}$ and $70\times 10^{3}$ . Arrow indicates increasing $y/{\it\delta}$ .

Figure 5.

Premultiplied spectra for boundary layer (a) and pipe flow (b) at $\mathit{Re}_{{\it\tau}}\approx 5\times 10^{3}$ with varying $y/{\it\delta}$ ; ——, turbulent wall region ( $200/\mathit{Re}_{{\it\tau}}<y/{\it\delta}<0.15$ ); – ⋅ – ⋅ –, wake region ( $0.15<y/{\it\delta}<0.7$ ); arrow indicates increasing $y/{\it\delta}$ ; – – – – –, relation proposed by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004), shown as solid line for $0.63<k_{x}y<6.3$ .

Figure 6.

Premultiplied spectra for boundary layer (a) and pipe flow (b) at $\mathit{Re}_{{\it\tau}}\approx 70\,000$ with varying $y/{\it\delta}$ ; ——, turbulent wall region ( $800/\mathit{Re}_{{\it\tau}}<y/{\it\delta}<0.15$ ); – ⋅ – ⋅ –, wake region ( $0.15<y/{\it\delta}<0.7$ ); arrow indicates increasing $y/{\it\delta}$ ; – – – – –, relation proposed by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004), shown as solid line for $0.63<k_{x}y<6.3$ .

The two Reynolds number cases shown in figures 5 and 6 demonstrate a reasonable collapse at lower wavenumbers in outer scaling ( ${\it\delta}$ ), and at higher wavenumbers in wall-normal scaling ( $y$ ), especially in the turbulent wall layer. However, no region exists where both scalings collapse the data simultaneously, as noted previously by Morrison et al. (Reference Morrison, McKeon, Jiang and Smits2004) and Rosenberg et al. (Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013) for pipe flow. Although a small plateau region may be present around $y/{\it\delta}=0.15$ , this behaviour disappears at points either closer to the wall, or further away. This plateau, its extent and its variation with Reynolds numbers is further discussed in § 5. Similar trends can be observed in the spectra with varying Reynolds number at different wall-normal locations, as in figure 7.

Morrison et al. (Reference Morrison, McKeon, Jiang and Smits2004) suggested that the lack of overlap in the scaling of the spectra may happen if $u_{{\it\tau}}$ is not the appropriate velocity scale for outer scaling. Also, del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) suggested that the large wall-attached motions do not scale with $u_{{\it\tau}}$ because their contribution to the Reynolds stress is limited by the impermeability of the wall. They proposed a logarithmic correction to the $k_{x}^{-1}$ spectrum given by $k_{x}{\it\Phi}_{uu}={\it\beta}u_{{\it\tau}}^{2}\log (2{\rm\pi}{\it\alpha}^{2}/(k_{x}y))$ , and with ${\it\alpha}=2$ and ${\it\beta}=0.2$ the correction agreed well with experimental and numerical spectra in the range $y<{\it\lambda}_{x}<10y$ ( $0.63<k_{x}y<6.3$ ). We see from figure 5 that for $\mathit{Re}_{{\it\tau}}=5000$ in pipe flow a small interval in wavenumber agrees with this relation over the suggested bounds $0.63<k_{x}y<6.3$ , but the boundary layer spectra seem to have a different slope. At $\mathit{Re}_{{\it\tau}}=70\,000$ (figure 6), this relation agrees with the data over a wider range than that suggested by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004), but for the boundary layer the slope ${\it\beta}$ may depend on Reynolds number ( ${\it\beta}=0.23$ and 0.19 were better fits for $\mathit{Re}_{{\it\tau}}=5000$ and 70 000, respectively). These observations provide some support for the suggestion that $u_{{\it\tau}}$ may not be the correct velocity scale for large scales, which infers that complete similarity (and the $k_{x}^{-1}$ region) cannot be expected to occur even at very high Reynolds numbers.

Hence, it appears that in boundary layer and pipe flows in the turbulent wall region there is no obvious $k_{x}^{-1}$ region that persists with Reynolds number, or with a change in wall-normal location, and the spectra do not exhibit a region that collapses both in inner and outer scaling. This brings into question the relationship between the spectral overlap arguments of Perry & Abell (Reference Perry and Abell1977) and Perry et al. (Reference Perry, Henbest and Chong1986) and the logarithmic variation of the variances. This point is further discussed in § 5.

Figure 7.

Premultiplied spectra for boundary layer (a) and pipe flow (b) at wall-normal locations $y/{\it\delta}=0.05$ , 0.1, 0.15 and 0.5. Arrow indicates increasing $\mathit{Re}_{{\it\tau}}$ from 3000 to 70 000 (as line color changes from blue to red in online version).

3.3. Scaling of spectral peaks in wavenumber space

In pipe flow, Rosenberg et al. (Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013) identified distinct Reynolds number independent scaling for the wavenumber location of the LSM and VLSM peaks in each wall-normal region. Near the wall at $y^{+}<12$ , a single peak was present at a wavenumber that scaled with $y$ , where $k_{x}y\approx 0.07$ . At $y^{+}=12$ , the low- and high-wavenumber loci bifurcated, where the location of the LSM peak scaled with the viscous length ${\it\nu}/u_{{\it\tau}}$ ( $k_{x}^{+}\approx 0.006$ ) over the range $12<y^{+}<67$ , and then for $y^{+}>67$ its location scaled with $y$ according to $k_{x}y\approx 0.4$ up until $y/{\it\delta}=0.15$ . Beyond $y/{\it\delta}=0.15$ , the LSM peak location scaled with ${\it\delta}$ according to $k_{x}{\it\delta}=2.6$ .

As to the VLSM peak, Rosenberg et al. (Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013) found that, somewhat surprisingly, its location also scaled with wall-normal distance $y$ in the turbulent wall region ( $k_{x}y\approx 0.045$ ), and not with ${\it\delta}$ as seen for the SS peak in boundary layers by Hutchins & Marusic (Reference Hutchins and Marusic2007). Rosenberg et al.’s (Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013) analysis for pipe flow was repeated here using every wall-normal location (the original study used only about a third of the data), and it was found that for $y^{+}>50$ the VLSM peak location was better described by $k_{x}{\it\delta}\approx 0.17\;({\it\delta}/y)^{0.67}$ , that is, it is more precise to say that the VLSM peak in the turbulent wall region in pipes follows a weak mixed scaling rather than a simple wall-normal scaling. Finally, for $y/{\it\delta}>0.15$ the location of the VLSM peak scaled as $k_{x}{\it\delta}\approx 0.45$ .

Here, we apply the methodology used by Rosenberg et al. (Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013) to find the location of the spectral peaks in the boundary layer. To estimate the wavenumber peak location, a Gaussian curve in $\log k_{x}$ was locally fitted to the data. At locations where the LSM peak was more difficult to identify, because it appeared more as a narrow shoulder rather than as a distinct peak (in the region $100<y^{+}<0.15$ ), a cubic spline was used to fit the data and the point of inflection was taken as an estimate of the peak location.

The results are shown in figures 8 and 9 in inner and outer variables, respectively. In the near-wall region, for $y^{+}<10$ , a single peak is observed, scaling with the wall-normal distance and located at about $k_{x}y\approx 0.05$ ( ${\it\lambda}_{x}\approx 125y$ ). This scaling is the result of the local mean velocity used as convection velocity when applying Taylor’s hypothesis. If no Taylor hypothesis were applied, the inner peak scales simply with the viscous time scale, being constant at $f^{+}=f{\it\nu}/u_{{\it\tau}}^{2}\approx 0.008$ or $t^{+}=1/f^{+}\approx 125$ .

Figure 8.

Spectral peak locations in inner coordinates for the boundary layer. Filled (red online) symbols: near-wall and LSM peaks. Open symbols: peak locations for $y/{\it\delta}>0.15$ . Filled grey symbols: SS peaks. Filled black symbols: outer spectral peak location. Other symbols as in table 1. – – – –, $y^{+}=10$ and $y^{+}=50$ .

Figure 9.

Spectral peak locations in outer coordinates for the boundary layer. Filled (red online) symbols: all peaks in region $50<y^{+}<0.15\mathit{Re}_{{\it\tau}}$ . Open symbols: peaks for $y^{+}<50$ . Filled black symbols: outer spectral peak location. Other symbols as in table 1. – – – – –, $y/{\it\delta}=0.15$ .

Near $y^{+}\approx 10$ , close to where the inner peak in the variance is located, there is a bifurcation in the loci of the peaks, so that the peaks associated with the LSM begin to follow $k_{x}^{+}\approx 0.005$ (or ${\it\lambda}_{x}\approx 1250{\it\eta}$ ). For $50<y^{+}<0.15\mathit{Re}_{{\it\tau}}$ , the location of the LSM peak scales with $y$ as $k_{x}y\approx 0.4$ . The location of the SS peak in the same region seems to follow $k_{x}{\it\delta}\approx 0.33({\it\delta}/y)^{0.5}$ , showing simultaneous dependence on wall-normal distance and the boundary layer thickness, as it did in pipe flow. This trend, when expressed in terms of the wavelength, gives ${\it\lambda}_{x}^{+}\approx 20\;(y^{+}\mathit{Re}_{{\it\tau}})^{0.5}$ , which suggests that in the region $50\lesssim y^{+}\lesssim 0.15\mathit{Re}_{{\it\tau}}$ the SS are associated with wavelengths ${\it\lambda}_{x}\sim (y{\it\delta})^{0.5}$ . At first sight, this result is consistent with the observations of Morrill-Winter & Klewicki (Reference Morrill-Winter and Klewicki2013) who found that the wall-normal intensities begin scaling with $\sqrt{\mathit{Re }_{{\it\tau}}}$ near $y^{+}\approx C\sqrt{\mathit{Re }_{{\it\tau}}}$ . However, the implications and the consequences of this consistency are unclear, as it will be shown later that the location of the peak in the spectra in the streamwise velocity component does not appear to scale as $\sqrt{\mathit{Re }_{{\it\tau}}}$ across the range of Reynolds numbers considered here. In the outer region, for $y/{\it\delta}>0.15$ , the SS peak is no longer evident and only the LSM peak survives, with its location scaling as $k_{x}{\it\delta}\approx 2$ . This value corresponds to ${\it\lambda}_{x}\approx 3{\it\delta}$ , as found by Guala et al. (Reference Guala, Hommema and Adrian2006).

Overall, it appears that the LSMs have similar characteristics in boundary layer and pipe flows, although the SSs and VLSMs display some differences, even at very high Reynolds numbers; in outer variables the trajectory of the peak for boundary layers varies as $({\it\delta}/y)^{0.5}$ , whereas for pipes it varies as $({\it\delta}/y)^{0.67}$ . In addition, the outer flow in the pipe continues to be characterized by LSMs and VLSMs, whereas in the boundary layer only a single peak associated with the LSM is present, suggesting that the SS organization is lost in the wake region.

3.4. Scaling of outer spectral peak

Here, we consider the scaling of the outer spectral peak in the region of turbulent wall flow ( $50\lesssim y^{+}\lesssim 0.15\mathit{Re}_{{\it\tau}}$ ). Recall that the outer spectral peak is the point where the VLSM or SS spectral peak has its maximum magnitude $(k_{x}{\it\Phi}_{uu})_{OSP}$ , and it is described by a specific physical location $y_{OSP}$ and wavenumber $k_{OSP}$ . In boundary layers, Hutchins & Marusic (Reference Hutchins and Marusic2007) and Mathis et al. (Reference Mathis, Hutchins and Marusic2009) found that the location of the outer spectral peak in inner variables, as well as its magnitude, varied with Reynolds number.

Figure 10.

Wall-normal location of outer spectral peak for boundary layer (a) and pipe flow with all data from Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012) (b). ▪, boundary layer with error bounds ${\rm\Delta}y^{+}=\pm 100$ ; ●, pipe with error bounds ${\rm\Delta}y^{+}=\pm 60$ ; (– – – – –), $y^{+}=2.68\mathit{Re}_{{\it\tau}}^{0.5}$ ; (– – – – –), $y^{+}=3.40\mathit{Re}_{{\it\tau}}^{0.5}$ ; ▫, $R_{m}=0$ location for boundary layer; ○, $R_{m}=0$ location for pipe; ▴, $R_{m}=0$ location for boundary layer from Mathis et al. (Reference Mathis, Hutchins and Marusic2009); ▪, location of outer peak in $u^{2+}$ for boundary layer; ●, location of outer peak in $u^{2+}$ for pipe; ♦, skewness zero crossing location from Vincenti et al. (Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013).

For the current boundary layer study, the locations of the outer spectral peak in [ $y_{OSP}$ , $k_{OSP}$ ] coordinates are shown by the black symbols in figures 8 and 9. Each location was found by locally fitting a Gaussian curve to the data and then nominating the closest available data point as $y_{OSP}$ . The variation of $y_{OSP}^{+}$ with Reynolds number is shown in Figure 10(a). The data were acquired in ${\rm\Delta}y^{+}$ increments of about 40–100, and the location of the peak therefore has a comparable uncertainty range (the error bars show uncertainty limits of ${\rm\Delta}y^{+}\pm 100$ ). As can be seen, the locations of the outer spectral peak in the boundary layer at lower Reynolds numbers follow $\mathit{Re}_{{\it\tau}}^{0.5}$ trend (dashed line shows best fit for $\mathit{Re}_{{\it\tau}}<20\,000$ ), but at higher $\mathit{Re}_{{\it\tau}}$ the trend is much weaker. In fact, for $\mathit{Re}_{{\it\tau}}\gtrsim 20\,000$ , its location seems to remain approximately fixed at $y^{+}\approx 300$ .

To compare these observations to those of Mathis et al. (Reference Mathis, Hutchins and Marusic2009), consider now the amplitude modulation correlation coefficient $R_{m}$ , representing the degree of modulation of the small-scale fluctuations by the large energetic scale motions. Mathis et al. found that where $R_{m}=0$ , that is, the location of the point where LSMs and small-scale motions are not correlated, seemed to correspond closely with the location of the outer spectral peak. The loci of these zero crossings are shown in figure 10 together with the $R_{m}=0$ locations found for the current data set, using a cutoff frequency $f^{+}=0.005$ for the boundary layer and 0.002 for the pipe. Also shown are the locations of the outer peak in the variances for current data, which mark approximately the point where the variances begin to follow a logarithmic behaviour (Vallikivi et al. Reference Vallikivi, Hultmark and Smits2015), and the skewness zero crossing locations from Vincenti et al. (Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013), a point that has also been observed to behave in similar manner to the peak in the spectra and variances. We see that all indicators follow a similar trend to the outer spectral peak location up to $\mathit{Re}_{{\it\tau}}\approx 20\,000$ , but the higher-Reynolds-number data all show a much slower rate of increase.

For the pipe, the outer spectral peak behaves in a similar manner to that in the boundary layer (see figure 10 b), but its location shows less variation, probably because the pipe data are available in smaller increments ( ${\rm\Delta}y^{+}\approx 30{-}60$ ), allowing for higher precision. Again, the lower-Reynolds-number range seems to agree well with $\mathit{Re}_{{\it\tau}}^{0.5}$ trend (dashed line shows best fit for $\mathit{Re}_{{\it\tau}}\lesssim 20\,000$ ), but at higher $\mathit{Re}_{{\it\tau}}$ the data show a slower increase in $y^{+}$ . Just as in boundary layers, for $\mathit{Re}_{{\it\tau}}\gtrsim 20\,000$ its location seems to remain approximately fixed, this time at $y^{+}\approx 400$ . This behaviour is shared by the location of the peak in $u^{2+}$ , but in contrast the $R_{m}=0$ location for the pipe continues to increase with Reynolds number, something not seen in the boundary layer data.

The magnitude of the outer spectral peak for each case is shown in figure 11. For both flows, the amplitude appears to approach a constant value at high Reynolds numbers where $k_{x}{\it\Phi}_{uu}/u_{{\it\tau}}^{2}\approx 1.24\pm 0.05$ , although the rate of approach is faster for the boundary layer. These observations and the significance of the value 1.24 will be discussed in more detail below, but we should note that there are considerable uncertainties in finding the magnitude of the peak (principally due to calibration issues and the difficulties in finding the friction velocity in boundary layers).

Figure 11.

Magnitude of the outer spectral peak: ▪, boundary layer; ●, pipe; – – – – –, $k_{x}{\it\Phi}_{uu}/u_{{\it\tau}}^{2}=1.24$ .

The wavenumber associated with the outer spectral peak, $k_{OSP}$ , is shown in figure 12. For increasing Reynolds number, it decreases in viscous units as $k_{OSP}^{+}\approx M\mathit{Re}_{{\it\tau}}^{-0.5}$ , while increasing in outer scaling as $k_{OSP}{\it\delta}\approx M\mathit{Re}_{{\it\tau}}^{0.5}$ , where $M=0.2$ . In terms of wavelength, this gives ${\it\lambda}_{OSP}^{+}\approx (M/2{\rm\pi})\mathit{Re}_{{\it\tau}}^{0.5}$ and ${\it\lambda}_{OSP}/{\it\delta}\approx (M/2{\rm\pi})\mathit{Re}_{{\it\tau}}^{-0.5}$ , respectively, suggesting that the wavelength associated with the large energetic scales neither scales with the inner length scale, nor with the outer length scale; ${\it\lambda}_{OSP}$ appears to emerge as an independent length scale. In addition, ${\it\lambda}_{OSP}\approx (M/2{\rm\pi})({\it\delta}{\it\nu}/u_{{\it\tau}})^{0.5}$ , so that at very large Reynolds numbers ${\it\nu}/u_{{\it\tau}}\ll {\it\lambda}_{OSP}\ll {\it\delta}$ .

Figure 12.

Wavenumber of the outer spectral peak in inner (a) and outer coordinates (b): ▪, boundary layer (uncertainty in $k_{OSP}^{+}$ is about $\pm 50\,\%$ ; ▫, boundary layer data from Mathis et al. (Reference Mathis, Hutchins and Marusic2009); (solid line), $k_{x}^{+}=0.02\mathit{Re}_{{\it\tau}}^{-0.5}$ (a); $k_{x}{\it\delta}=0.02\mathit{Re}_{{\it\tau}}^{0.5}$ (b); ♢, predicted wavenumber for $\mathit{Re}_{{\it\tau}}=$ 650 000.

The wavenumber of the outer spectral peak appears, therefore, to be a new scale representative of the log region. Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2014) recently suggested that for the Townsend–Perry attached eddy model to show the correct behaviour for the integral length scale at high Reynolds number, an additional length scale needs to appear, one that lies between the inner and outer scales. Our current observations are closely aligned with that proposal. In addition, the appearance of a new region described by the length scale ${\it\lambda}_{OSP}$ is consistent with the observation of Vallikivi et al. (Reference Vallikivi, Hultmark and Smits2015) who find a log region (in mean and intensity) only after the outer spectral peak has emerged.

Given the consensus in the literature that the outer spectral peak is associated with the SS, there are two potential ways to interpret the above observations. First, we could say that the SS are actually the manifestation of this region in the flow, rather than the largest scales of motion. This would lead to a rather startling implication that at lower Reynolds numbers the SS appear as very large scales (that is, ${\it\lambda}_{x}/{\it\delta}\approx 3{-}6$ ) due to a lack of scale separation, but at higher Reynolds numbers the length scale of the SS decreases. The length scale associated with the outer spectral peak in an atmospheric boundary layer at $\mathit{Re}_{{\it\tau}}\approx 650\,000$ is expected to be ${\it\lambda}_{x}/{\it\delta}\approx 0.4$ (as inferred from the correlation shown in figure 12), which is much smaller than the scales of the VLSM/SS. This suggests that this first interpretation is too stringent and may not truly represent the data.

An alternate and perhaps more suitable interpretation is that there is a need to decouple the scale and location of the outer spectral peak from that of the SS/VLSMs. It can be seen from the spectra that, although the OSP appears to move towards the wall and become smaller (in outer scaling) with increasing Reynolds number, there is still significant energy content in larger scales. Therefore, the SS (in boundary layers) and VLSMs (in pipes) continue to be dynamically relevant. This means that the wavelength of the outer spectral peak is in fact a new length scale that emerges in parallel with the VLSMs. The VLSMs start to appear at ‘lower’ Reynolds numbers (say $\mathit{Re}_{{\it\tau}}\approx 2000$ ), but this new length scale only emerges when there is sufficient scale separation even beyond the one required for the VLSMs ( $\mathit{Re}_{{\it\tau}}>20\,000$ ). At these higher Reynolds numbers, the VLSMs represent the plateau region in the spectra (instead of the peak) while the peak itself captures this new inertial length scale.

To show that there is at least some support for this notion at even higher Reynolds numbers, figure 13 reproduces the spectrogram obtained in the neutral atmospheric boundary layer by Mathis et al. (Reference Mathis, Hutchins and Marusic2009), where we see that there may indeed be an outer spectral peak whose location is proportional to $\ln \mathit{Re}_{{\it\tau}}$ and whose wavelength (in outer units) follows $\mathit{Re}_{{\it\tau}}^{-0.5}$ . The location and the wavelength of this prediction appears to be consistent with the peak found by Guala, Metzger & McKeon (Reference Guala, Metzger and McKeon2011) in the wall-normal range $y^{+}\approx 1000$ at a wavelength of about $0.45{\it\delta}$ . Note that these values have been inferred from figure 12 in Guala et al. (Reference Guala, Metzger and McKeon2011) that shows multiple peaks at different wall-normal locations and wavenumbers and the uncertainty in identifying a peak in the spectrogram (in wall-normal location) is stated to be about $\pm 1000$ wall-units.

Figure 13.

Contour plot of the spectra in the atmospheric boundary layer at $\mathit{Re}_{{\it\tau}}=650\,000$ , adapted from Mathis et al. (Reference Mathis, Hutchins and Marusic2009):  $+$ , location of inner spectral peak; ♦, speculated location of outer spectral peak; (solid lines), trends in the loci of LSM and SS peaks as described in § 3.3.

As noted earlier, pipe flows behave somewhat differently. Here, we expect that ${\it\lambda}_{OSP}/{\it\delta}\propto \mathit{Re}_{{\it\tau}}^{-0.67}$ and ${\it\lambda}_{OSP}^{+}\propto \mathit{Re}_{{\it\tau}}^{0.33}$ . Hence, the scale separation between inner and inertial ranges appears slower in pipes than in boundary layers, whereas the separation between outer and inertial scales is faster. The slower approach to the inner limit agrees well with the observation that the appearance of a logarithmic region occurs at a higher value of $y^{+}$ in pipes than in boundary layers.