4.1. Spectra in region II

This region plays a crucial role in wall turbulence, as it corresponds to the near-wall footprint of wall-attached eddies. According to Bradshaw (Reference Bradshaw1967) and Townsend (Reference Townsend1976), the effect of ‘distant’ wall-attached eddies on the near-wall region is mainly a low-frequency modulation of the wall shear stress, to which the local turbulence can readjust quasi-instantaneously, so that the law of the wall holds locally in time. Under this assumption, the resulting velocity fluctuations have a variance that can be expressed as

(4.1) \begin{equation} \langle u^2 \rangle ^{+} \sim \left (U^{+} + y^{+} \frac {\mathrm{d} U^{+}}{\mathrm{d} y^{+}}\right )^2 \equiv F(y^{+}) . \end{equation}

The key implication of this argument is that the spectral density associated with the footprint of attached eddies varies with wall distance only through the modulation function $F(y^+)$ . Hence, once the spectral distribution is known at a reference location, its variation with $y^+$ can be predicted by simple rescaling. This ansatz has been partially validated in previous studies (Jiménez Reference Jiménez2024; Pirozzoli Reference Pirozzoli2024). Assuming that the same relation applies to the spectral energy density of eddies with sizes $\lambda _{s} \leqslant \lambda \leqslant \lambda _{\ell }$ , the spectral density at any $y^{+}$ can be predicted once it is known at a reference wall distance $y_{0}^{+}$ , namely

(4.2) \begin{equation} (k^{+} E_{u}^{+})(\lambda ^{+},y^{+}) = (k^{+} E_{u}^{+})(\lambda ^{+},y_{0}^{+}) \frac {F(y^{+})}{F(y_{0}^{+})} . \end{equation}

Figure 3.

Pre-multiplied $u$ spectrum for flow case G (a) and reconstructed spectrum based on (4.2) (b). Iso-contours of $k^+_{\theta } E^+_{uu}$ are shown from $0.36$ to $3.6$ , at intervals of $0.36$ . The dashed horizontal line in panel (b) marks the reference location used for the reconstruction ( $y_0^{+} = 15$ ).

Figure 3. Long description

Two contour plots depict the pre-multiplied spectrum and reconstructed spectrum for flow case G. Panel A shows the pre-multiplied spectrum with iso-contours of a variable ranging from 10^-4 to 10^0 at intervals of 10^0.2. The x-axis is labeled lambda theta plus, and the y-axis is labeled y plus. Panel B shows the reconstructed spectrum based on equation 4.2, with the same iso-contours and axis labels. A dashed horizontal line in Panel B marks the reference location used for the reconstruction at y plus equals 100.

Equation (4.2) is tested in figure 3, where the original spectral map (a) is compared with the reconstructed map, obtained by extrapolating the spectral density from the centre of the buffer layer ( $y_0^{+} = 15$ , b). The figure shows remarkable agreement not only within region II, but throughout the entire region beneath the primary energetic ridge. This confirms the physical validity of assumption (4.2) and demonstrates that the near-wall region of wall-bounded flows can be largely interpreted in terms of ‘turbulent Stokes layers’, namely local turbulent boundary layers whose dynamics is controlled by the overlying eddies. In our view, this represents a central result in wall-turbulence theory, with significant implications for the subsequent analysis.

Figure 4.

Pre-multiplied spectra of $u$ at $y_0^{+} = 15$ . See table 1 for colour codes.

Whereas (4.2) predicts accurately the variation of the spectral density with wall-normal distance, a complete description also requires determining its dependence on wavelength at the reference position $y_0^{+}$ . The analysis reported by Pirozzoli (Reference Pirozzoli2024) addresses this very point. As shown in figure 4, the pre-multiplied spectral density at fixed $y^{+}$ exhibits an intermediate (overlap) range with a clear power-law behaviour

(4.3) \begin{equation} (k^{+} E_{u}^{+})(\lambda ^{+},y_{0}^{+}) = A \, \left ( \lambda ^{+} \right )^{-\alpha } , \end{equation}

with $\alpha = 0.18$ and $A = 3.37$ at $y_{0}^{+} = 15$ . It is worth noting that (4.3) slightly (but decisively) departs from the classical theory of Perry & Chong (Reference Perry and Chong1982), which would suggest $\alpha =0$ . Combining equations (4.3) and (4.2) yields a complete theoretical prediction for the spectrum in region II.

Evaluation of the velocity variance in region II involves integrating the spectrum (4.2) over the appropriate wavelength range. Referring to figure 2 for guidance, for $y^+ \leqslant y_s^+$ one obtains

(4.4) \begin{equation} \lt u^2\gt ^+_{\textit{II}} (y^+) = \int _{\log \lambda _s^+}^{\log \lambda _{\ell }^+} k_{\theta }^+ E_u^+ \, \mathrm{d} \log \lambda _{\theta }^+ \approx \frac {A F(y^+)}{\alpha F(15)} \left [ \left (\lambda _s^+\right )^{-\alpha }-\left (\frac {\lambda _{\ell }}R\right )^{-\alpha } \textit{Re}_{\tau }^{-\alpha } \right ] \!; \end{equation}

whereas for $y_s^+ \leqslant y^+ \leqslant y_{\ell }^+$ one has

(4.5) \begin{align} \lt u^2\gt ^+_{\textit{II}} (y^+) = \int _{\log (y^+/\beta )}^{\log \lambda _{\ell }^+} k_{\theta }^+ E_u^+ \, \mathrm{d} \log \lambda _{\theta }^+ \approx \frac {A F(y^+)}{\alpha F(15)} \left [ \left (\frac {y^+}{\beta }\right )^{-\alpha }-\left (\frac {\lambda _{\ell }}R\right )^{-\alpha } \textit{Re}_{\tau }^{-\alpha } \right ] \!. \end{align}

Figure 5.

Streamwise velocity variance for region II. The DNS data are shown as solid lines; dashed lines correspond to the prediction of (4.4), and dash-dotted lines to (4.5). Vertical lines indicate the characteristic off-wall locations $y_s^{+}$ and $y_{\ell }^{+}$ (the latter for flow case G). See table 1 for colour codes.

These two formulas are tested in figure 5. The solid lines represent the velocity variance from region II extracted from the DNS data, the dashed lines correspond to the predictive (4.4) and the dot–dashed line to (4.5). The figure demonstrates good predictive capability of the present analysis, although the accuracy of (4.5) is slightly reduced, particularly just beyond the kink at the upper end of the distributions. Despite minor discrepancies in the shape, the results confirm the strong predictive performance of the present theory with respect to variations in Reynolds number – an even more significant outcome, as it indicates that the energetic contribution of the intermediate eddies can be confidently extrapolated beyond the currently accessible Reynolds-number range.

4.2. Spectra in region III

Figure 6.

Pre-multiplied spectral density of $u$ in region III for flow case G. Data are shown at five wall distances (listed in the figure inset) and plotted as a function of either $\lambda ^+$ (a) and of the similarity variable $\zeta = \beta \lambda / y$ (b). The grey line in (b) refers to the fitting function defined in (4.6).

The spectra in region III are seldom analysed in detail. In the AEM framework, this region, located at the top left of the main energetic ridge, can be associated with two mechanisms: (i) the influence of wall-attached eddies on points located farther from the wall (local in $\lambda$ , non-local in $y$ ); and (ii) the energy cascade of wall-attached eddies toward smaller scales (local in $y$ , non-local in $\lambda$ ). This region is likely unaffected by direct viscous effects, as well as by indirect effects associated with wall proximity. Possible viscous effects are nevertheless discussed in Appendix A. Hence, dimensional analysis suggests that the spectral density of $u$ depends only on the ratio $\lambda / y$ , rather than on $\lambda$ and $y$ separately, as originally argued by Perry & Chong (Reference Perry and Chong1982). Strong support for this assumption can also be drawn from experimental data (e.g. Baars & Marusic Reference Baars and Marusic2020a ). Geometrical evidence for this universal behaviour is provided in figure 2, which shows that in this region the iso-lines of the pre-multiplied spectral density are aligned along diagonal bands. This behaviour is confirmed more quantitatively in figure 6, where we report the pre-multiplied spectral density at five off-wall distances in the logarithmic layer for the flow case G. When shown as a function of $\lambda ^{+}$ (panel a), no universality is observed. However, the data clearly indicate the formation of two subranges with distinct power-law behaviour: a $(\lambda ^{+})^{2/3}$ scaling at smaller wavelengths and a $(\lambda ^{+})^{1}$ scaling at larger wavelengths. The former corresponds to an underlying $E_u^{+} \sim (k^{+})^{-5/3}$ Kolmogorov inertial behaviour, and therefore becomes apparent only at sufficiently large $y^{+}$ , as expected under the assumption of a local energy cascade at fixed $y^+$ . The latter corresponds to a pre-multiplied spectrum behaving as $E_u^{+} \sim (k^{+})^{-2}$ . To the best of our knowledge, this scaling has not been clearly reported in previous experimental studies, although it can also be identified in spectra from DNS of turbulent channel flow (Lee & Moser Reference Lee and Moser2015) (not shown here). Possible explanations for the emergence of this ‘odd’ scaling may be related to the prevalence of sharp interfaces, ramps and quasi-two-dimensional shear layers in the logarithmic region (Meinhart & Adrian Reference Meinhart and Adrian1995; de Silva, Hutchins & Marusic Reference de Silva, Hutchins and Marusic2016), whose likely spectral footprint is a $k^{-2}$ velocity spectrum (Pullin & Saffman Reference Pullin and Saffman1998). This interpretation likely deserves a dedicated follow-up study; however, the observed trend appears robust for the present purposes.

The same data are reported in figure 6(b) as a function of the similarity variable $\zeta = \beta \lambda / y$ , such that $\zeta = 1$ corresponds to the main energetic ridge in figure 2. Points with $\zeta \gt 1$ belong to region II and are therefore omitted from the present analysis. In this representation, a clear near universality of the spectra emerges at intermediate values of $\zeta$ . Deviations are observed at small $\zeta$ , where the dissipative range begins to influence the spectrum. Some scatter is also observed as $\zeta \to 1$ , where the energy is found to increase slightly with $y^{+}$ , consistent with the trends reported by Jiménez (Reference Jiménez2024) and with those identified in our analysis for region II. For the sole purpose of deriving an analytical prediction for the contribution of this region to the velocity variance, we fit the pre-multiplied spectra using a simple polynomial function

(4.6) \begin{equation} p(\zeta ) = A \left [ \zeta + (3 q - 2) \, \zeta ^{2} + (1 - 2 q) \, \zeta ^{3} \right ]\!, \end{equation}

which reproduces the observed $E_u^{+} \sim (k^{+})^{-2}$ scaling at small $\zeta$ and has zero derivative at $\zeta = 1$ , where the spectral density attains a peak. The fitting parameters determined from the DNS data are $A = 4.8$ and $q = 1/3$ . We note that this function does not reproduce the $E_u^{+} \sim (k^{+})^{-5/3}$ dissipative spectral range observed in figure 6(b) at small wavelengths. However, the contribution of the latter to the velocity variance is small, as shown in Appendix A, so that the resulting error in the modelled velocity variance is negligible.

Figure 7.

Streamwise velocity variance for region III. The DNS data are shown as solid lines; dashed lines correspond to the prediction of (4.7), and dash-dotted lines to (4.9). Vertical lines indicate the characteristic off-wall locations $y_s^{+}$ and $y_{\ell }^{+}$ (the latter for flow case G). See table 1 for colour codes.

Integration of the model spectrum (4.6) yields the contribution to the streamwise velocity variance from region III. As in region II, referring to figure 2, we must distinguish two sub-cases. For $y_s^+ \leqslant y^+ \leqslant y_{\ell }^+$ , integration gives

(4.7) \begin{equation} \overline {u^2}^+_{\textit{III}} (y^+) = \int _{\log \lambda _{s}^+}^{\log (y^+/\beta )} k_{\theta }^+ E_u^+ \, \mathrm{d} \log \lambda _{\theta }^+ \approx \int _{\beta \lambda _s^+ / y^+}^1 \frac {p(\zeta )}{\zeta } \, \mathrm{d} \zeta = P(1) - P(\beta \lambda _s^+ / y^+), \end{equation}

with

(4.8) \begin{equation} P(\zeta ) = A \left [ \zeta + (3 q - 2) \frac {\zeta ^2}2 + (1-2 q) \frac {\zeta ^3}3 \right ]\!; \end{equation}

whereas for $y^+ \geqslant y_{\ell }^+$ it yields

(4.9) \begin{align} \overline {u^2}^+_{\textit{III}} (y^+) = \int _{\log \lambda _s^+}^{\log \lambda _{\ell }^+} k_{\theta }^+ E_u^+ \, \mathrm{d} \log \lambda _{\theta }^+ \approx \int _{\beta \lambda _s^+ / y}^{\beta \lambda _{\ell }^+ / y^+} \frac {p(\zeta )}{\zeta } \, \mathrm{d} \zeta = P(\beta \lambda _{\ell }^+ / y^+) - P(\beta \lambda _s^+ / y^+). \end{align}

These two formulas are tested in figure 7, which shows the contribution to the streamwise velocity variance from region III in flow case G (solid line), along with the predictions of (4.7) and (4.9). Overall, the agreement is good, although somewhat less accurate than in region II. Predictions for the second sub-region are noticeably poorer, owing to the lack of reliable theoretical guidance for the velocity spectra beyond the logarithmic layer. Despite shortcomings far from the wall, the figure conveys that the increase of the velocity variance with Reynolds number is well captured.

4.3. Spectra in region I

Figure 8.

(a) Wall-normal traverse of the pre-multiplied $u$ spectrum at $\lambda ^+ = 136$ for flow case G; (b) streamwise velocity variance in region I for all flow cases. In (a) the solid grey line indicates the prediction given by (A3). In (b) the dashed line represents the fit given by (4.13), and the colour codes are listed in table 1.

Having quantified the contribution from the attached eddies to the intermediate band of wavenumbers, we now turn to the small-scale region, where viscous effects play a central role. The spectra in region I for $y \lesssim y_s$ are dominated by the strong buffer-layer peak, which is known to be very nearly universal when scaled in wall units. Farther from the wall, this region mainly includes eddies which results from the dissipative end of the cascade initiated from the wall-attached eddies, and which occurs for points at the left of the dashed red line in figure 2. Hence viscous effects are certainly important, which we discuss in Appendix A. Figure 8(a) shows the premultiplied spectra at a fixed wavelength ( $\lambda ^+ = 136$ , corresponding to the centre of the buffer-layer peak) for various Reynolds numbers, plotted as a function of wall distance. A clear universality of the inner-scaled spectra across Reynolds numbers is observed, characterised by two distinct behaviours. Specifically, near the wall the spectrum scales as ${y^+}^2$ , consistent with the asymptotic condition enforcing $u \sim y$ for small $y$ . Farther from the wall, beyond the buffer-layer energetic peak at $y^+ \approx 12$ , the data show good correspondence with the viscous corrected asymptotic formula (A3) reported in the Appendix A. At wall distances larger than about $y_{\ell }$ , the spectra deviate abruptly from the theoretical estimates.

The contribution to the streamwise velocity variance from region I can be obtained by integrating the spectral density over the associated wavenumber range

(4.10) \begin{equation} {\langle u^2 \rangle }^+_I (y^+) = \int _{-\infty }^{\log \lambda _s^+} k_{\theta }^+ E_u^+ \, \mathrm{d} \log \lambda _{\theta }^+ . \end{equation}

Given the observed near wall-normal behaviour of the spectral density and universality in wall units, the velocity variance is expected to behave asymptotically as $y^2$ , and DNS data fitting suggests

(4.11) \begin{equation} \langle u^2 \rangle ^+_I (y^+) \approx f_I(y^+) = A_s (y^+)^{2}, \end{equation}

with $A_s = 0.129$ . Farther from the wall, evaluation of the velocity variance would require integration of (A3), which cannot be obtained in explicit form. However, as shown in the Appendix B, the following asymptotic behaviour emerges (see (B5)):

(4.12) \begin{equation} \langle u^2 \rangle ^+_I (y^+) \approx f_O(y^+) = C_1 (y^+)^{-3/4} + C_2 (y^+)^{-1}, \end{equation}

with $C_1 = 14.8$ , $C_2 = 94.4$ , constants determined from DNS data fitting. Guided by the asymptotic behaviours near and away from the wall, we identified a plausible combination of these two asymptotic regimes by taking their generalised harmonic mean

(4.13) \begin{equation} \langle u^2 \rangle ^+_I (y^+) \approx \left ( f_I(y^+)^{-\gamma } + f_O(y^+)^{-\gamma } \right )^{-1/\gamma }, \end{equation}

where fitting yields $\gamma = 0.86$ . Figure 8(b) shows that this expression indeed provides an excellent approximation of the contribution of the small scales to the streamwise velocity variance over a wide range of $y$ , up to approximately $y_{\ell }$ .

4.4. Spectra in region IV

Figure 9.

Wall-normal profiles of the streamwise velocity variance for region IV, compensated by $ \textit{Re}_{\tau }^{\alpha }$ ( $\alpha = 0.18$ ). The grey line represents the prediction of (4.15) with $D = 0.025$ . See table 1 for colour codes.

As previously argued for the spectral maps beneath the spectral ridge (see figure 3), the near-wall spectra in region IV follow the Townsend–Bradshaw scaling with respect to wall distance. Regarding the dependence on wavelength, Pirozzoli (Reference Pirozzoli2024) observed that, to remain consistent with the spectral overlap scaling (4.3), the spectra at large scales should decay as $ \textit{Re}_{\tau }^{-\alpha }$ for any fixed outer-scaled wavelength $\lambda /R$ .

Combining these theoretical inferences leads to the following expected functional form of the spectrum in region IV:

(4.14) \begin{equation} (k^{+} E_{u}^{+})(\lambda ^{+}, y^{+}) = \textit{Re}_{\tau }^{-\alpha } \, G(\lambda /R) \, F(y^{+}) , \end{equation}

with $F(y^{+})$ defined in (4.1). The contribution of the large scales to the velocity variance then follows from integration

(4.15) \begin{equation} \langle u^{2} \rangle ^{+}_{IV}(y^{+}) = \int _{\log \lambda _{\ell }^{+}}^{\infty } k_{\theta }^{+} E_{u}^{+} \, \mathrm{d} \log \lambda _{\theta }^{+} = D \, \textit{Re}_{\tau }^{-\alpha } \, F(y^{+}) , \end{equation}

with $D$ an undetermined constant.

This prediction is tested in figure 9, where the contribution to the velocity variance from region IV, extracted from the DNS (solid lines), is plotted in compensated form (i.e. multiplied by $ \textit{Re}_{\tau }^{\alpha }$ ) at various $ \textit{Re}_{\tau }$ . The figure confirms excellent universality across the Reynolds-number range, and shows that the trend with wall distance closely follows the theoretical trend. The DNS data further suggest that an appropriate value of the multiplicative constant is $D \approx 0.025$ . Deviations occur at distances larger than approximately $y_{\ell }^+$ , as also observed for the other regions.

4.5. The overall velocity variance

Figure 10.

Contributions to the streamwise velocity variance for flow case G from small scales (region I, dotted lines), large scales (region IV, dashed lines), intermediate scales (regions II + III, dash-dotted lines) and the overall variance (solid lines). Black lines represent DNS data, while blue lines show predictions from the present analysis. Vertical lines indicate the characteristic off-wall locations $y_s^{+}$ and $y_{\ell }^{+}$ .

Finally, the models developed for the various regions of the spectral map are combined to obtain a prediction for the overall streamwise velocity variance. This is illustrated in figure 10, where the DNS data for flow case G (black lines) are compared with the model predictions (red lines). The velocity variance is decomposed into contributions from the small scales (region I), the intermediate scales (regions II + III) and the large scales (region IV). Predictions are shown only up to $y_{\ell }^{+}$ , where, based on the results presented in the previous section, they are expected to remain reliable – hence roughly up to the middle of the logarithmic layer for the mean velocity profile.

The figure shows excellent agreement for the energy contribution from the small and large scales, and very good prediction of the intermediate-scale energy in the near-wall region (up to $y_{s}^{+}$ ). Farther from the wall, the agreement is only partial: both DNS and the model exhibit a peak of comparable magnitude, but its position is slightly shifted toward smaller $y^{+}$ in the model. As a result, the overall variance distribution matches the DNS data very well up to $y_{s}^{+}$ , although the outer ‘shoulder’ appears somewhat smoother. This discrepancy arises from an imperfect representation of the spectral maps in the immediate vicinity of the energetic ridge. We have verified that this can be corrected through empirical curve fitting; however, we deliberately avoid introducing additional assumptions in order to keep the model as simple and transparent as possible.

4.6. The asymptotic behaviour

Figure 11.

(a) Wall-normal profiles of the contributions to the streamwise velocity variance (thick solid lines) from small scales (region I, dotted lines), large scales (region IV, dashed lines) and intermediate scales (regions II + III, thin solid lines), as obtained from DNS data. (b) Wall-normal profiles of the streamwise velocity variance obtained from DNS (symbols) and from the present model (solid lines). See table 1 for colour codes.

Next, we examine the behaviour of the velocity variance across the Reynolds-number range. To gain insight, figure 11(a) shows the contributions to the velocity variance from the small, intermediate and large scales, as determined from DNS at various Reynolds numbers. The figure clearly indicates that, while the contribution from the small scales remains nearly universal, the large-scale contribution decreases approximately as $ \textit{Re}_{\tau }^{-\alpha }$ at fixed $y^{+}$ , whereas the intermediate-scale contribution increases following a defect power law, as predicted by our analysis. The latter observation is particularly significant, as it contrasts with common expectations for high-Reynolds-number wall turbulence. The figure further reveals unambiguously that the emergence of the outer peak in the velocity variance distribution is directly linked to the growing importance of the intermediate-scale contribution. Thus, the appearance of the outer peak does not reflect a new physical mechanism, but rather reflects the different growth rates of the energy associated with motions of different scales. Even without further analysis, it is evident that an outer peak must eventually develop at sufficiently high Reynolds number, located at a wall distance between $y_{s}^{+}$ and $y_{\ell }^{+}$ .

Finally, we assess the performance of the velocity variance model developed in this study across the Reynolds-number range. This is illustrated in figure 11(b), where the predicted distributions are shown up to the infinite-Reynolds-number limit. The first important message conveyed by the figure is that the agreement with DNS data in the accessible Reynolds-number range is consistently good, aside from the previously noted limitations. More ambitiously, the figure also provides an extrapolation of the trends to Reynolds numbers far beyond those currently attainable in experiments or simulations. Naturally, caution is required when interpreting extrapolations well outside the range over which the model has been calibrated. With this caveat in mind, the extrapolation suggests that a distinct outer peak does not clearly emerge until $ \textit{Re}_{\tau } \approx 10^{6}$ . Because the growth rate of this outer peak exceeds that of the inner peak, their magnitudes become comparable at very high Reynolds numbers. Our prediction is that the inner-peak intensity asymptotes to approximately $12.3$ (a reasonably robust estimate), whereas the outer peak approaches approximately $12.8$ (a more uncertain value).

4.7. Comparison with other models

The spectral model developed in this paper can be compared with several previously proposed models. At a fundamental level, it shares certain structural similarities with the model of Perry et al. (Reference Perry, Henbest and Chong1986), which was formulated for the logarithmic region in the inviscid limit (i.e. $\delta _\nu \ll y \ll R$ as $\textit{Re}_\tau \to \infty$ ). In the model of Perry et al. (Reference Perry, Henbest and Chong1986), the one-dimensional streamwise velocity spectrum is decomposed into five components: (i) an outer-scaling ( $R$ -scaling) region; (ii) a first overlap region between the outer- and inner-scaling ( $y$ -scaling) regions, where a $k^{-1}$ spectrum is expected; (iii) an inner-scaling region; (iv) a second overlap region between the inner- and Kolmogorov-scaling regions, corresponding to the inertial subrange exhibiting a $k^{-5/3}$ spectrum; and (v) the Kolmogorov-scaling dissipation range.

In the spectral model presented in this paper, for $y \leqslant y_l$ , regions IV and II correspond to the outer-scaling and first overlap regions, respectively, whereas regions III and I collectively represent the remaining components of Perry et al. (Reference Perry, Henbest and Chong1986). Although the overall structure is similar, the present model incorporates additional physical ingredients to account for finite-Reynolds-number effects and near-wall viscous influences. In particular, the spectra in regions VI and II, where Perry et al. (Reference Perry, Henbest and Chong1986) combined a $y$ -independent outer-scaling spectrum with a $k^{-1}$ spectrum, are modelled here using (4.1).

It is also worth noting that the absence of an explicit $k^{-1}$ spectrum in the present formulation does not preclude the emergence of an approximately logarithmic variation of the streamwise velocity variance with wall-normal location. Such behaviour naturally arises provided that the spectrum spans the range from the $R$ -scaling region (region VI) to the $y$ -scaling region (region III); see Hwang et al. (Reference Hwang, Hutchins and Marusic2022).

In addition to the comparison with the classical spectral model, Gustenyov et al. (Reference Gustenyov, Bayley and Smits2025) recently proposed a model for the streamwise wavenumber spectra in pipe flow. In their study, the pre-multiplied spectrum at each wall-normal location is decomposed as

(4.16) \begin{equation} \frac {k\,E_{u}(k;y)}{u_\tau ^2} = f_1 + f_2 + f_3 + f_{\textit{HIT}}, \end{equation}

where $f_1$ , $f_2$ and $f_3$ represent the spectral contributions associated with large-scale motions (LSMs), very-large-scale motions (VLSMs) and near-wall streaks, respectively, while $f_{\textit{HIT}}$ accounts for incoherent motions related to the energy cascade and turbulence dissipation. Furthermore, in order to ensure consistency with the classical AEM, the amplitudes of $f_1$ and $f_2$ were assumed to scale approximately with $\log (y/R)$ (see (2.6) and (2.8) in Gustenyov et al. Reference Gustenyov, Bayley and Smits2025).

The model developed in the present study shares several features with that of Gustenyov et al. (Reference Gustenyov, Bayley and Smits2025). First, both models are consistent with the classical AEH, in the sense that the streamwise velocity variance in the logarithmic region scales approximately with $\log (y/R)$ . Second, neither model explicitly relies on the emergence of a $k^{-1}$ spectrum, which is expected only in the inviscid limit. Third, both approaches account for the spectral contributions of near-wall streaks as well as for motions associated with the energy cascade and turbulence dissipation, features that were neglected in the original AEM of Townsend (Reference Townsend1956).

There are, however, several notable differences between the two approaches. The model of Gustenyov et al. (Reference Gustenyov, Bayley and Smits2025) is formulated in terms of the streamwise wavenumber spectrum, which enables an explicit separation between the contributions of LSMs and VLSMs. By contrast, the present study is formulated using the azimuthal (or spanwise) wavenumber spectrum, which limits such separation because LSMs and VLSMs share similar azimuthal length scales; see the observations of Kim & Adrian (Reference Kim and Adrian1999) and the alternative explanation based on the self-sustaining process proposed by de Giovanetti, Sung & Hwang (Reference de Giovanetti, Sung and Hwang2017). On the other hand, the present model explicitly incorporates near-wall viscous effects on both LSMs and VLSMs. This feature allows the model to capture the deviation of the near-wall turbulence-intensity peak from the classical $\log \textit{Re}_\tau$ scaling, as recently discussed by Pirozzoli (Reference Pirozzoli2024). More importantly, although both models predict the emergence of an outer peak at high Reynolds number, the underlying mechanisms are not identical. In the present model, the outer peak arises from the growing contribution of the intermediate wall-attached-eddy range (regions II + III), whereas the contribution of the largest scales (region IV) slightly decreases with Reynolds number. In contrast, Gustenyov et al. attribute the outer peak to the increasing contribution of LSMs and VLSMs in their spectral decomposition.