2.1. Basic model formulation

We construct the model for the longitudinal one-dimensional velocity spectrum $E_{11}$ in the streamwise, $x$ -aligned, wavenumber domain $k_1$ , using functions that depend on both $k_1$ and $y$ , the distance from the wall. The model has contributions from four eddy populations, broadly representing the coherent LSM ( $f_1$ ), VLSM ( $f_2$ ) and near-wall streaks ( $f_3$ ), as well as the incoherent homogeneous isotropic turbulence (HIT) arising from the breakdown of these organised motions as they transport away from the wall ( $f_{HIT}$ ). These distributions are modelled as premultiplied forms of $E_{11}$ (i.e. $k_1E_{11}$ ) that overlap in wavenumber space (reflecting the broad range of spectral content in each class of coherent motions), and the energy contained at each wavenumber is taken to be a linear summation over all eddy types. That is, we propose

(2.1) \begin{equation} k_1E_{11}/u_\tau ^2=f_1+f_2+f_3+f_{\it{HIT}}, \end{equation}

where $\overline {u_1^2}$ can then be obtained by integrating $E_{11}$ over all $k_1$ . The additive structure of (2.1) was selected because the energy content attributed to different modes of motion appeared to be simply superimposed in the premultiplied energy spectrum. A similar decomposition of the premultiplied energy content into different eddy structures was proposed schematically for turbulent boundary layers by Hutchins & Marusic (Reference Hutchins and Marusic2007).

For the coherent motions ( $f_1$ , $f_2$ and $f_3$ ), we will assume that their premultiplied energy distribution can be approximated using a log-normal distribution in wavenumber. This formulation is motivated by observations of multiple peaks in the premultiplied spectrum (which, in turn, motivated the decomposition of long-wavelength streamwise energy content into separate contributions that were identified as LSM and VLSM or superstructures, and observations that this bimodal structure can be reproduced by a superposition of two log-normal distributions (Kim & Adrian Reference Kim and Adrian1999; Smits Reference Smits2010). Note that existing model spectra representing HIT at high Reynolds number, e.g. the model presented by Pope (Reference Pope2000), also take on a nearly log-normal form when presented in premultiplied one-dimensional form. We thus treat each of $f_1$ , $f_2$ and $f_3$ as an individual eddy population as with a broad distribution of associated energy produced at a characteristic wavenumber, and use the Pope (Reference Pope2000) model for $f_{HIT}$ .

Hence we propose that

(2.2) \begin{equation} f_{n}=A_{n}\exp \left [ -\left (\frac {\log _{10}(k_1y)-\log _{10}(\mu _n)}{\log _{10}(\sigma _n)} \right )^2\right ], \end{equation}

where $n=1$ , 2, 3 indicate the distributions corresponding to high-wavenumber (LSM), lower-wavenumber (VLSM) and near-wall (streaks) energy distributions, respectively. The energy content is thus distributed around the (non-dimensional) wavenumbers at which each eddy type is produced, $\mu _n$ , with the wavenumber range of the distribution described by $\sigma _n$ , and the amplitude by $A_{n}$ . The three terms $A_n$ , $\mu _n$ and $\sigma _n$ depend on $y$ and introduce the wall-dependent scaling of $f_1$ , $f_2$ and $f_3$ , with their precise formulation described in the following subsections.

Two points need to be made. First, (2.2) is implicitly unable to reproduce the predicted $k^{-1}$ scaling region predicted by the attached-eddy hypothesis. However, as noted by Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015), the existing data have not revealed an extended region of $k^{-1}$ scaling even at very high Reynolds numbers, and as will be presented later, the log-normal distributions used here are able to reproduce many key features of the streamwise energy content without imposing this constraint. This is particularly relevant for the present model, because it covers even the lowest Reynolds numbers, where $k^{-1}$ scaling is not expected to hold under any circumstances. Second, we cannot expect an accurate reproduction of the high-wavenumber universal equilibrium range, therefore this model is currently limited to reproducing the energy-containing range of the streamwise velocity fluctuations. That said, for the ‘incoherent motions’, we will use

(2.3) \begin{equation} f_{\it{HIT}}=\big(k_1E^{\it{HIT}} _{11}/u_\tau ^2\big)\left(y/\delta \right)^{n}, \end{equation}

where $E^{\it{HIT}}_{11}$ is found from transformation of the three-dimensional spectrum model of Pope (Reference Pope2000) to a one-dimensional form. The function $f_{\it{HIT}}$ is intended to account for incoherent motions that form as a result of the natural breakdown of coherent motions. This energy content will have an increased prominence farther away from the wall, and the $(y/\delta )^n$ dependence with $n=1.1$ was found to capture this dependence for pipe flows.

2.2. Development and validation datasets

To develop and validate the model, we use experimental data covering $180 \le {Re}_\tau \le 1\,00\,000$ (see table 1). We mainly use the Superpipe data acquired at Princeton University (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013; Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015), encompassing a Reynolds number range $2000 \le {Re}_\tau \le 100\,000$ . NSTAP probes (Vallikivi & Smits Reference Vallikivi and Smits2014) were used for these experiments, with non-dimensional sensor lengths $\ell ^+=\ell u_\tau /\nu$ as given in the table. Where appropriate, the spatial filtering correction of Smits, McKeon & Marusic (Reference Smits, McKeon and Marusic2011a ) was applied, and Taylor’s hypothesis (Taylor Reference Taylor1938) was used to transform the time-dependent measurement to the spatial domain, so that $k_1\approx 2 {\unicode[Arial]{x03C0}} f/\overline {U_1}$ , where $f$ is the frequency, and $\overline {U_1}$ is the mean velocity. Because Taylor’s hypothesis can potentially introduce abnormalities in the spectrum (e.g. Moin Reference Moin2009), for additional validation we use DNS data in the range $180 \le {Re}_\tau \le 6000$ (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), and the Reynolds stress profiles obtained from particle image velocimetry (PIV) measurements in the CICLoPE facility in the range $5400 \le {Re}_\tau \le 40\,000$ (Willert et al. Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017). Additional statistical information from CICLoPE measurements provided by Fiorini (Reference Fiorini2017) and from DNS described by Ahn et al. (Reference Ahn, Lee, Jang and Sung2013, Reference Ahn, Lee, Lee, Kang and Sung2015), Wu & Moin (Reference Wu and Moin2008), El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013), Chin, Monty & Ooi (Reference Chin, Monty and Ooi2014) and Yao et al. (Reference Yao, Rezaeiravesh, Schlatter and Hussain2023) is also used.

Table 1. Superpipe flow experimental conditions; from Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013).

2.3. Overview

Before describing the model in detail, we give an example to illustrate how each of the functions $f_n$ contributes to the premultiplied wavenumber spectrum, and how the model spectrum compares to that measured in a high-Reynolds-number pipe flow experiment. Figure 1 shows the measured premultiplied spectrum for six $y$ -locations at ${Re}_\tau =10\,500$ . As already noted, the appearance of double peaks in the spectrum has been linked previously to the contributions made by the LSM and VLSM by Kim & Adrian (Reference Kim and Adrian1999) and Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015), among others. This characteristic structure was observed at all Reynolds numbers considered, over a relatively wide range of wall-normal distances. The figure also shows the corresponding model spectra, and the contributions made by each of the four eddy functions $f_1$ to $f_{\it{HIT}}$ . Although the model does not reproduce this bimodal feature of the premultiplied spectra, it captures well the broad energy content over the entire range of wall-normal distances shown. It may be possible to reproduce the spectra more accurately by using other representations for the eddy populations instead of the relatively simple log-normal populations used here, but this would require additional complexity that we sought to avoid. In addition, we note that for the spectrum nearest the wall, at $y^+=16$ , the model appears to over-predict the energy content. However, at this location $y/\ell = O(1)$ , and some degree of spatial filtering of the experimental energy content is to be anticipated (Smits Reference Smits2022). In terms of the separate contributions to the streamwise energy, $f_1$ and $f_2$ are important at almost all wall-normal locations, $f_3$ is significant only near the wall, and $f_{\it{HIT}}$ has its greatest contribution in the outer region of the flow.

Figure 1. Premultiplied energy spectra for pipe flow at ${Re}_\tau =10\,500$ : (a) $y/\delta =0.001536$ ( $y^+=16$ ), (b) $y/\delta =0.011$ ( $y^+=116$ ), (c) $y/\delta =0.1$ ( $y^+=1053$ ), (d) $y/\delta =0.314$ ( $y^+=3303$ ), (e) $y/\delta =0.6$ ( $y^+=6328$ ), (f) $y/\delta =0.979$ ( $y^+=10\,299$ ).

A comparison of the measured and model spectra at the same wall-normal locations in log-log form is presented in figure 2(a). This presentation highlights the differences between the measured and model spectra at low wavenumber, where there is little energy content, and at high wavenumber, where the log-normal basis functions used to construct the model are unable to reproduce the highest wavenumber content within the universal equilibrium range. That said, the superposition of eddy populations appears to reproduce the $k_1^{-5/3}$ slope of the inertial subrange in the outer layer. This is further illustrated in figure 2(b), which presents the measured and modelled spectrum at $y^+=390$ , or $y/\delta =0.037$ – a location within the expected range of the overlap layer (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). Here, the attached-eddy hypothesis predicts that an overlap of $y$ -scaled and $\delta$ -scaled eddies will produce a $k_1^{-1}$ spectral roll-off (e.g. Perry & Abell Reference Perry and Abell1977). Figure 2(b) shows that such a region appears to be present, at least approximately, in both the measured and modelled spectra at this location. However, a $k_1^{-1}$ region would appear as a plateau in the premultiplied form of the spectra, and as noted by Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015) among others, there is little evidence of such a plateau in the premultiplied form of the measured spectrum (as also shown in figure 1).

Figure 2. Energy spectra for pipe flow at ${Re}_\tau =10\,500$ . (a) Same $y$ locations as shown in figure 1, with the spectrum for $y^+=16$ on the bottom, and increasing $y$ locations shown successively shifted by two orders of magnitude. Solid black lines indicate measurement, and red lines indicate model prediction. (b) Energy spectrum at $y/\delta =0.037$ ( $y^+=390$ ) shown in both log-log and premultiplied form. The solid line indicates the measurement, and the dashed line indicates the model prediction. In both (a) and (b), the dash-dotted line indicates the $k_1^{-5/3}$ slope, and the dotted line indicates the $k_1^{-1}$ slope.

2.4. The LSM and VLSM ( $f_1$ and $f_2$ )

We now consider the detailed formulation of $f_1$ (LSM) and $f_2$ (VLSM), i.e. how the specific functions $\mu _{1,2}$ , $A_{1,2}$ and $\sigma _{1,2}$ were established as a function of wavenumber, wall distance and Reynolds number. The general approach was to use anticipated scaling behaviour whenever known, but allowing for alternative forms when they better approximated the available empirical evidence. The final forms of these coefficients are compromises that best describe the energy content contributed by each of the modelled motions while maintaining as much as possible prior theoretical and empirical considerations.

As a starting point, we seek expressions to describe the $y$ dependence of $\mu _1$ and $\mu _2$ . To do so, we find the wavenumbers corresponding to the LSM and VLSM peaks in the experimental spectra by smoothing the premultiplied spectra using a moving average, and then fitting two log-normal distributions to the data, following the approach used by Rosenberg et al. (Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013) and Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015). The results are shown in figures 3 and 4, respectively, and they are similar in all respects to the results obtained earlier by Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015).

Figure 3. Wall-normal dependence of the peak associated with LSM in (a) outer scaling and (b) inner scaling. The green lines in (a) indicate $\mu _1\delta /y$ for ${Re}_\tau =2000$ , $20\,400$ and $1\,00\,000$ , and the black line shows the peak of $f_{\it{HIT}}$ (see (2.3)). In (b), the green lines show $\mu _1/y^+$ for ${Re}_\tau =2000$ , $20\,400$ and $1\,00\,000$ , and the pink line marks $\mu _3/y^+$ (see (2.12)). The symbols and colours are given in table 1, with the filled symbols used to indicate when $y^+\gt 30$ and $y/\delta \leq 0.3$ in (a), and when $y^+\leq 30$ in (b).

Figure 4. Wall-normal dependence of the peak associated with VLSM in outer scaling. The red line marks $\mu _2\delta /y$ , where $\mu _2$ is given by (2.4). The symbols and colours are given in table 1.

For the VLSM peak (figure 4), the wall-normal dependence of the peak location was then modelled using outer scaling as

(2.4) \begin{equation} \mu _2=0.002+0.9({y}/{\delta })^{0.9}. \end{equation}

This formulation follows the power-law dependence observed by Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015), although with slightly different coefficients – a consequence of the scatter in the observed spectral peak near the wall. However, the contribution of the $f_2$ eddy population to the energy spectrum in this region is relatively small, and (2.4) was found to provide a reasonable representation of the VLSM peak variation with wall distance.

The behaviour of the LSM peak is more complex, as shown in figure 3. For $y^+\gt 30$ and $y/\delta \leq 0.3$ , it follows outer scaling and is well represented by $y/\delta ^{0.25}$ (figure 3 a), but near the wall, the peak is better represented by inner scaling, as also observed by Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015). Hence we propose

(2.5) \begin{equation} \mu _1=\big[1-(\exp[-y^+])^{0.02}\big](y/\delta)^{0.25}, \end{equation}

where the inner-scaled term is introduced to shift $\mu _1$ to lower wavenumbers when $y^+\lessapprox 200$ , to approximate the increased dependence on inner scaling near the wall. Even with this modification, as illustrated in figure 3(b), there are still significant differences between (2.5) and the experimentally observed peak wavenumber for $y^+\,{\lt}\,100$ . Nevertheless, when $f_1$ was combined with $f_3$ , the LSM peak location in this range was found to be well approximated by (2.5), as will be shown later. In addition, we see that for $y/\delta \gt 0.3$ , the location of the $f_1$ peak deviates from (2.5). However, at these wall-normal distances, the LSM and VLSM peaks are more difficult to distinguish (Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015), and we can anticipate a large contribution to energy from incoherent eddies, which are represented here by $f_{\it{HIT}}$ .

With $\mu _{1,2}$ established, $A_{1,2}$ and $\sigma _{1,2}$ can be determined. For the amplitude of $A_1$ (LSM), since $f_1$ represents the structures most closely related to the attached structures of Perry (Reference Perry1982), we can anticipate a logarithmic contribution to the Reynolds stress in outer scaling. Hence we propose that

(2.6) \begin{align} A_{1} =-0.18m_v\ln ({{y}/{\delta }}),\end{align}

where

(2.7) \begin{align} \quad m_v =(1-\exp [-0.25y^+])^6 \end{align}

and $m_v$ is introduced to reflect the viscous damping introduced by the wall. The expression for $m_v$ was chosen to match the near-wall decay in energy content and is not based on any established theoretical model.

In addition, since the VLSM are connected to the LSM eddies, as suggested by Adrian (Reference Adrian2007), Monty et al. (Reference Monty, Stewart, Williams and Chong2007) and others, we can expect that the wall-normal dependence of the VLSM would also take a logarithmic form. However, $f_2$ provides the largest contribution to the streamwise energy content from wavelengths larger than  $\delta$ , and therefore to the outer-layer signal $u^+_{\it{OL}}$ in (1.1). Following (1.2), and neglecting the nonlinear interactions, we can therefore anticipate the near-wall contribution by the VLSM to provide a good approximation for the level of outer-layer influence described by (1.2). Using this guidance, we propose that

(2.8) \begin{align} A_2 = -0.24 \, m_v \, \alpha ^2 \ln {[(y/\delta)+0.005]},\end{align}

where

(2.9) \begin{align} \alpha = 1-0.36 \exp [-0.01y^+].\end{align}

The formulation of $\alpha$ was determined through a fit applied to the values measured by Mathis et al. (Reference Mathis, Hutchins and Marusic2011), and the leading-order outer-scale dependence is damped by $m_v$ in the identical way used for $A_1$ . Although Mathis et al. (Reference Mathis, Hutchins and Marusic2011) determined $\alpha (y^+)$ from measurements taken in turbulent boundary layer measurements (see also Mathis et al. Reference Mathis, Monty, Hutchins and Marusic2009b ), we expect the near-wall behaviour to be similar for all canonical geometries, therefore we also apply it unchanged to pipe flows.

Finally, we consider the widths of these eddy populations, $\sigma _n$ . We have little guidance here, aside from the expectation that they follow outer scaling (Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015), therefore these values were determined empirically in a manner that best reconstructed the measured spectra. This procedure gave

(2.10) \begin{align} \sigma _{1} = 16+\ln {(y/\delta )}\end{align}

and

(2.11) \begin{align} \sigma _{2} = 1+10(y/\delta )^{0.1}.\end{align}

Taken together, we expect $f_1$ and $f_2$ to largely dictate the streamwise Reynolds stress distributions for $y^+\gt 30$ and $y/\delta \lt 0.7$ , and result in the formation of a logarithmic region in the turbulence within this range (as will be presented in § 3.2).

2.5. Near-wall streaks ( $f_3$ )

For $y^+\lt 30$ , the near-wall production mechanism becomes the main contributor to the overall velocity variance. Hence we expect $f_3$ to become increasingly important in this range, and that it will scale on inner variables. Furthermore, this function can also be considered to represent the universal signal $u^*$ in (1.1). Following the procedure used for $f_1$ and $f_2$ , we identify $\mu _3$ using the location of the near-wall peak in the measured spectra. As illustrated in figure 3(b), for $y^+\gt 8$ , the near-wall peak occurs at $k_1^+=0.007$ , closely corresponding to the ${\sim}1000\nu /u_\tau$ wavelength attributed to the near-wall streaks (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). Therefore, we let

(2.12) \begin{equation} \mu _3=0.007 y^{+}. \end{equation}

To describe the wall-normal and wavenumber dependence of the inner spectral peak, we use

(2.13) \begin{align} A_{3} = 1.6 \exp \big [-\big (0.945\ln \big(0.11y^{+}\big)\big )^2\big ]\end{align}

and

(2.14) \begin{align} \sigma _3 = 5,\end{align}

as determined by examining measured spectra that approximate the form of a joint log-normal distribution of energy in both inner-scaled wavenumber and wall-distance.

2.6. Incoherent motions $(f_{\it{HIT}})$

We anticipate that the coherent eddies represented by $f_1$ , $f_2$ and $f_3$ co-exist with disorganised (i.e. incoherent) turbulence that develops due to the breakdown of the coherent eddies and is transported away from the wall via turbulent advection. We model this contribution to the one-dimensional spectrum as being approximately homogeneous and isotropic, even though these eddies are expected to be anisotropic at low wavenumber. We begin with the Pope (Reference Pope2000) three-dimensional spectrum, where

(2.15) \begin{equation} E^{\it{HIT}}(k)=C\overline {\varepsilon }^{2/3}k^{-5/3}f_Lf_{\eta }. \end{equation}

Here, $\overline {\varepsilon }$ is the mean dissipation rate, $k$ is the magnitude of the three-dimensional wavenumber vector, and $C$ is the Kolmogorov constant. The primary feature of this model is the inertial subrange distinguished by the $k^{-5/3}$ scaling. The function $f_L$ describes the energy-containing range using $L$ as a length scale characterising large eddies such that

(2.16) \begin{equation} f_L=\left (\frac {kL}{[(kL)^2+C_L]^{1/2}}\right )^{5/3+p_0}, \end{equation}

where $C_L$ is a positive constant and $p_0=4$ . At small wavenumbers, (2.16) approximates the von Kármán (Reference von Kármán1948) interpolation equation with $E(k)\sim k^4$ , whereas at large $k$ , (2.16) approaches unity such that (2.15) transitions to the inertial subrange. The function $f_\eta$ describes the small-scale dissipation range, and scales with the Kolmogorov length scale $\eta = ( {\nu ^3}/{\overline {\varepsilon }} )^{1/4}$ . Specifically, $f_\eta$ is defined as

(2.17) \begin{equation} f_\eta =\exp \big(-\gamma \big(\big[\!\left(k\eta \right)^4+C_{\eta }^4\big]^{1/4}-C_\eta \big)\big), \end{equation}

and tends to unity at small $k$ , and zero at large $k$ . In (2.17), both $\gamma$ and $C_{\eta }$ are positive constants. For a specified $L$ and $\eta$ , the three-dimensional model spectrum can be determined using (2.15), (2.16) and (2.17) with $C=1.5$ and $\gamma =5.2$ (Saddoughi & Veeravalli Reference Saddoughi and Veeravalli1994), while constants $C_L$ and $C_\eta$ are dictated by the requirement that $E^{\it{HIT}}(k)$ and $2\nu k^2\,E^{\it{HIT}}(k)$ integrate to the turbulent kinetic energy and $\overline {\varepsilon }$ , respectively, at the centreline. Here, the best agreement with the measured values was found when $C_L=3.2$ and $C_{\eta }=0.41$ . The eddy population representing these motions is thus found from (2.15), (2.16) and (2.17) using $y$ -dependent values for $L$ , and $\eta$ . The result is then transformed to a one-dimensional spectrum using

(2.18) \begin{equation} E^{\it{HIT}}_{11}(k_1)=\int _{k_1}^{\infty } \frac {E^{\it{HIT}}(k)}{k}\left (1-\frac {k_1^2}{k^2}\right ){\rm d}k, \end{equation}

and then added to (2.1) using (2.3).

We therefore need wall-normal-dependent expressions for $L$ and $\overline {\varepsilon }$ . We focus on the outer layer because we expect $f_{\it{HIT}}$ to have the most influence in that region due to the breakdown of near-wall coherent eddies and subsequent transport of the resulting incoherent turbulence away from the wall.

For the $y$ dependence of $L$ , we use

(2.19) \begin{equation} L\approx \Big({\textstyle {\frac {3}{2}}}\overline {u_1^2}^+\Big)^{3/2}u_\tau ^3/{\overline {\varepsilon }}. \end{equation}

Bailey & Witte (Reference Bailey and Witte2016) demonstrated that this length scale is a better choice than the integral scale in describing the low-wavenumber bound of the inertial subrange as the Reynolds number and wall-normal distance vary. The wall dependence of $\overline {\varepsilon }$ was estimated from the measured time series using one-dimensional approximations (Gustenyov et al. Reference Gustenyov, Egerer, Hultmark, Smits and Bailey2023). As illustrated in figures 5(a) and 5(b), $\overline {u_1^2}^+$ and $\overline {\varepsilon }$ scale with outer variables over the full extent of the outer layer (where $f_{\it{HIT}}$ becomes increasingly important) over the entire Reynolds number range available. We see that

(2.20) \begin{equation} \overline {u_1^2}^+ \approx 0.15\big(2.5-\left(y/\delta \right)^{0.6}\big)^{4.2} \end{equation}

and

(2.21) \begin{equation} \frac {\overline {\varepsilon } \delta }{u_\tau ^3} \approx 1.6\left(y/\delta +0.035\right)^{-1.2} \end{equation}

provide reasonable approximations for the wall dependence of $\overline {u_1^2}^+$ and $\overline {\varepsilon }$ , although other expressions are likely to be as effective. Equations (2.19)–(2.21) can then be used to determine $L$ and $\eta$ , and hence $f_{\it{HIT}}$ from (2.3).

Figure 5. (a) Inner-scaled streamwise velocity variance with (2.20) shown using a solid red line. (b) Normalised mean dissipation rate $\overline {\varepsilon }$ with (2.21) shown using a red line. Symbols and colours are consistent with table 1.

3.1. Comparisons at ${Re}_\tau =10\,500$

The contour maps of the premultiplied spectra are given in figure 6. Figure 6(a) shows how the energy content provided by $f_1$ , $f_2$ and $f_3$ in the inner layer transitions to being mostly provided by $f_1$ and $f_2$ in the overlap layer, and how $f_{\it{HIT}}$ plays an increasingly important role as $y/\delta$ approaches unity. Figure 6(b) shows the total distribution found by summing all four terms, and figure 6(c) displays the corresponding contour map obtained by experiment. Overall, the model appears to capture the energy distribution of the measured spectra well at this Reynolds number. Most notably, the inner peak centred on $k_1\delta =75,\ y^+=15$ is represented accurately. Also reproduced well is the outer peak in the spectral map, appearing near $k_1\delta =2.8,\ y/\delta \approx 0.05$ , although the details are not as sharp as those observed in the experiment. As previously mentioned, this is a consequence of the log-normal form of the eddy population, which, although simple to implement from a modelling perspective, is less likely to capture some of the details of the spectral distributions. In particular, as noted earlier in the discussion of figure 1, the model is not able to reproduce the bimodality of the spectral content evident in figure 6(c) for $y/\delta \gt 0.1$ .

Figure 6. (a,b,c) Contours of $k_1E_{11}(k_1,y)/u_\tau ^2$ at ${Re}_\tau =10\,500$ : (a) contributions by the four eddy functions $f_1, f_2$ , $f_3$ and $f_{\it{HIT}}$ ; (b) model representation (summation of $f_1, f_2$ , $f_3$ and $f_{\it{HIT}}$ ); (c) superpipe results (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2013). Thick blue, black and red lines represent $\mu _1$ , $\mu _2$ and $\mu _3$ , respectively. Contour spacing is 0.1 for all graphs except for $f_{\it{HIT}}$ , where it is 0.02. (d) Comparison of the modelled $\overline {u_1^2}^+$ profile to the data of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2013), and the individual contributions from $f_1, f_2$ , $f_3$ and $f_{\it{HIT}}$ .

By integrating the energy spectra, we obtain the wall-normal distributions of $\overline {u_1^2}^+$ , and the individual contributions made by the four eddy functions. The model output is compared to the corresponding Superpipe profile in figure 6(d), and we see very good agreement for the overall energy content. The near-wall peak is seen to be formed by the superposition of $f_1$ , $f_2$ and $f_3$ , and, consistent with the concept of detached eddies, the energy contribution from $f_2$ reduces near the wall. For $y^+\gt 100$ , the energy content is largely provided by $f_1$ and $f_2$ , which contribute an equal amount of streamwise energy content, while rolling off logarithmically with increasing $y$ . Within the outer layer, $f_{\it{HIT}}$ becomes increasingly important and provides all the streamwise energy content at $y=\delta$ (the pipe centreline).

3.2. Reynolds number dependence of the stress distributions

The Reynolds number dependence of the modelled profiles is tested in figure 7 by comparisons with pipe flow data over Reynolds numbers ranging from 180 to 1 00 000. As indicated earlier, the data were drawn from DNS for $180 \le {Re}_\tau \le 6000$ (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), the Superpipe experiments for $2000 \le {Re}_\tau \le 1\,00\,000$ (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2013), and the near-wall PIV CICLoPE profiles for $6000 \le {Re}_\tau \le 40\,000$ (Willert et al. Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017).

Figure 7. Wall-normal dependence of the $\overline {u^2_1}^+$ values predicted by the model compared in inner scaling to (a) pipe flow DNS of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), (b) NSTAP measurements in the Superpipe of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2013), and (c) PIV measurements in CICLoPE of Willert et al. (Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017). The Superpipe comparison is also shown in (d) using outer scaling and compared to (3.1) for wall-normal locations $y^+\gt 100$ . In all plots, the lines indicate the model prediction, and the symbols indicate the reference data.

Despite being formulated in wavenumber space, figure 7(a–c) illustrate how well the integrated model spectra match the energy content relative to the available data over three orders of magnitude in Reynolds number. Some discrepancies are evident, such as a shift towards the wall in the modelled near-wall peak at the lowest Reynolds number that is not seen in the data, but the comparisons illustrate that certain key features of distributions are produced well by the model. Specifically, we refer to the logarithmic scaling in the overlap layer, the formation and growth of a near-wall peak around $y^+=15$ , and the development of an outer peak in the range $100\lt y^+\lt 1000$ . We now consider these features in more detail.

First, consider the logarithmic variation. We expect that at a sufficiently high Reynolds number, in the region where the overlap layer forms in the mean velocity profile, $\overline {u_1^2}^+$ follows a logarithmic variation given by

(3.1) \begin{equation} \overline {u_1^2}^+ \approx B-C\ln (y/\delta ) \end{equation}

(Townsend Reference Townsend1976; Perry et al. Reference Perry, Henbest and Chong1986), where $B=1.56$ and $C=1.26$ (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). Figure 7(d) shows how well the model reproduces this logarithmic dependence. Although this result is not unexpected given the formulation of $A_1$ and $A_2$ , figures 7(b) and 7(d) also illustrate that the range of wall-normal locations over which this behaviour is apparent is also well reproduced, despite not being explicitly defined within the model. However, figure 7(a) does indicate that the logarithmic dependence is over-enforced at low Reynolds number.

Second, consider the Reynolds number dependence of the inner peak at $y^+ \approx 15$ . Figure 7 shows that through the superposition of $f_1$ and $f_2$ with $f_3$ , the model is able to reproduce the expected behaviour of the inner peak even though it is not explicitly defined in the model. To provide a more quantitative comparison, the ${Re}_\tau$ dependence of the maximum value of the inner peak, $(\overline {u_1^2}^+)_{max}$ , is compared to the available DNS and experimental data in figure 8. The model closely follows the fit presented by Lee & Moser (Reference Lee and Moser2015) up to ${Re}_\tau \approx 6000$ , but then it conforms more closely with the trend observed by Willert et al. (Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017). Although experiments at higher Reynolds numbers are often affected by spatial filtering (e.g. Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009; Smits et al. Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011b ; Miller, Estejab & Bailey Reference Miller, Estejab and Bailey2014; Smits Reference Smits2022), the data from Willert et al. (Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017) were measured using PIV in a geometrically large facility and have a combined uncertainty of spatial filtering and noise of 1.4 % at ${Re}_\tau =11\,600$ , 2.7 % at at ${Re}_\tau =20\,000$ , 4.9 % at ${Re}_\tau =28\,000$ , and 7.8 % at ${Re}_\tau =40\,000$ . These uncertainties cannot explain the peak value falling below the extrapolation of Lee & Moser (Reference Lee and Moser2015) except at the highest ${Re}_\tau$ values, and even then only if the true value was at the upper limit of the uncertainty bound.

Figure 8. Velocity variance inner peak, $({\overline {u_1^2}}^+)_{max}$ , for pipe flow as a function of ${Re}_\tau$ . Blue symbols indicate values from DNS; black symbols indicate values from experiments; the red line indicates model prediction; the blue dashed line indicates Lee & Moser (Reference Lee and Moser2015) channel flow correlation $({\overline {u_1^2}}^+)_{max} \approx 3.66 + 0.642 \ln {Re}_\tau$ .

From the perspective of the model, the observation that $(\overline {u_1^2}^+)_{max}$ follows a logarithmic dependence only up to a threshold Reynolds number was found to be caused by a transition away from low ${Re}_\tau$ behaviour in which the outer-scaled attached eddies provide a significant contribution near the wall that increases with ${Re}_\tau$ . As the scale separation increases with ${Re}_\tau$ , the wall damping of the detached $f_2$ eddies, encapsulated by the inner-scaled $\alpha$ contribution, increasingly shifts the contribution of $f_2$ to being inner-scaled and thereby minimises the ${Re}_\tau$ dependence of the $f_2$ contribution to the streamwise energy content at $y^+=15$ . In other words, the near-wall universal inner-scaled behaviour becomes increasingly important near the wall, as the outer-scaled detached eddies move further from the wall in inner units.

Third, we examine the appearance of an outer peak in $\overline {u_1^2}^+$ at high ${Re}_\tau$ near $y^+\!\approx \!300$ (e.g. see Fernholz et al. Reference Fernholz, Krause, Nockemann and Schober1995; Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015; Willert et al. Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017; Smits Reference Smits2022). Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2015) showed that this peak can occur if there is a range of wavenumbers where the streamwise energy content scales on an intermediate length scale, specifically at wavenumbers corresponding to those of the LSM and VLSM (see also Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015). Figure 7(b,c) indicate that the model predicts the formation of an outer peak, but its peak value $(\overline {u_1^2}^+)_{\it{OP}}$ and location $(y^+)_{\it{OP}}$ display some discrepancies with experiment. Figure 9(a) indicates that although the model predicts a higher peak magnitude than that measured in the Superpipe, it is close to that measured in CICLoPE. Furthermore, the rate at which $(\overline {u_1^2}^+)_{\it{OP}}$ increases with ${Re}_\tau$ observed in experiment appears to be closely reproduced by the model. The Reynolds number dependence in $(y^+)_{\it{OP}}$ is also reproduced well by the model, but it predicts the formation of the peak at a lower $y^+$ location than seen in experiment.

Figure 9. Streamwise Reynolds stress (a) outer peak magnitude, and (b) location.

Within the model, the outer peak forms as a consequence of the inner-scaled influence on $f_2$ , represented by $\alpha$ , which produces a maximum in the contribution of $f_2$ to $\overline {u_1^2}^+$ (as illustrated in figure 6 d). As ${Re}_\tau$ increases, and the ratio of streamwise energy contained in the outer-scaled eddies increases relative to inner-scaled eddies, the importance of this wall influence on detached eddies results in the emergence of the outer peak. This is consistent with the hypothesis presented by Alfredsson, Segalini & Örlü (Reference Alfredsson, Segalini and Örlü2011), who suggested that the outer peak forms as a consequence of scale separation creating a difference in available energy between eddies away from the wall and those near the wall. This is also consistent with the mechanism proposed by Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2015), although this implies that their intermediate integral scale is not a new scale but rather an indication of the simultaneous presence of inner-scaled and outer-scaled streamwise energy content at the same wall-normal location. Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2015) also predict that $(\overline {u_1^2}^+)_{\it{OP}}$ will increase logarithmically with ${Re}_\tau$ , and that $(y^+)_{\it{OP}}$ will increase with ${Re}_\tau$ following a power law. The model prediction does not follow either of these trends, which most likely can be attributed to the bimodal nature of its $k_1$ dependence.

3.3. Spectral maps

Figure 10 provides an overview of the Reynolds number dependence of the energy content in the form of spectral maps of $k_1\,E_{11}(k_1,y)$ , where the model predictions are compared to Superpipe data at ${Re}_\tau =2000$ , $20\,400$ and $70\,000$ . The experimental results show how at ${Re}_\tau =2000$ , the spectral map is dominated by an inner spectral peak, corresponding to the peak in streamwise Reynolds stress, and although the spatial filtering increases, as ${Re}_\tau$ increases, the peak moves to increasingly higher $k_1\delta$ values. At higher ${Re}_\tau$ , the increased scale separation both broadens the energy content near the wall, and results in a systematic shift in the spectral energy distribution, appearing as a tilt in the spectral map with respect to the distance from the wall. As previously noted, however, the model is unable to reproduce the details of these spectral maps, but it does capture these effects of scale separation and ${Re}_\tau$ dependence.

Figure 10. Premultiplied spectral maps of streamwise energy content $k_1E_{11}/u_\tau ^2$ for (a,c,e) Superpipe data, (b,d, f) corresponding model prediction, with (a,b) ${Re}_\tau =2000$ , (c,d) ${Re}_\tau =20\,400$ , (e, f) ${Re}_\tau =70\,000$ . White dashed lines in (c) and (e) indicate wall positions below which spatial filtering corrections according to Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2013) exceed $3\,\%$ of the uncorrected value.

Furthermore, the outer peak in the premultiplied spectra of canonical wall-bounded flows, observed when ${Re}_\tau \gt 2000$ (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007; Mathis et al. Reference Mathis, Hutchins and Marusic2009a ; Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009), is also apparent in figure 10. Figure 10(b,d, f) demonstrate that the model predicts the emergence of an outer-spectral peak at wall-normal positions and wavenumbers similar to those observed in experiment, although a distinct peak becomes apparent only for ${Re}_\tau \gt 4000$ . As shown in figure 11(a), for ${Re}_\tau \lt 30\,000$ , the model predicts the location of the outer-spectral peak, $(y^+)_{\it{OSP}}$ , a little closer to the wall than observed in experiments, but with a similar logarithmic trend. Previous studies suggested that $(y^+)_{\it{OSP}}$ scales with ${Re}_\tau ^{0.5}$ (Mathis et al. Reference Mathis, Hutchins and Marusic2009a ; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013), but the model gives a trend closer to ${Re}_\tau ^{0.4}$ . For ${Re}_\tau \gt 20\,000$ , the data suggest that $(y^+)_{\it{OSP}}$ becomes independent of Reynolds number, which is not captured in the model, although spatial filtering may have been important for the experiments at the highest Reynolds numbers (see figure 10 e). Figure 11(b) presents the wavenumber of the outer-spectral peak in inner scaling, $(k_1^+)_{\it{OSP}}$ , while figure 11(c) presents it in outer scaling, $(k_1 \delta )_{\it{OSP}}$ . Although inner scaling captures the general location in wavenumber space over a broad range of ${Re}_\tau$ , the model predicts that $(k_1 \delta )_{\it{OSP}}$ is independent of Reynolds number, in contrast to the experiments. Nevertheless, the magnitude of the outer-spectral peak $(k_1E_{11}/u_\tau ^2)_{\it{OSP}}$ is well reproduced by the model, as shown in figure 11(d).

Figure 11. Outer-spectral peak (OSP): (a) wall-normal location; (b) corresponding wavenumber in inner coordinates; (c) wavenumber in outer coordinates; (d) peak magnitude. Red line: model prediction. Open circles: Superpipe data (Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015). Crosses: CICLoPE data (Fiorini Reference Fiorini2017).

In the context of the model, the outer peak arises due to the overlap in wavenumber space of the LSM and VLSM contributions (as illustrated in figure 6(a), for example). Whereas the VLSM energy content modelled using $f_2$ is constrained by inner-scaled wall influence, the LSM energy content is outer-scaled down to $y^+\approx 20$ . As the scale separation increases with Reynolds number, the LSM energy content overlap with the VLSM energy content increases, resulting in the increase in $(y^+)_{\it{OSP}}$ with ${Re}_\tau$ . This interpretation suggests that the outer-spectral peak and the outer peak in $\overline {u_1^2}^+$ are related through their source (i.e. contributions from both LSM and VLSM), despite the outer peak in $\overline {u_1^2}^+$ not becoming evident until a much higher ${Re}_\tau$ . With respect to the Reynolds number dependence of $(k_1\delta )_{\it{OSP}}$ , the inability of the model to capture the data is due to the wavenumber ranges of $f_1$ and $f_2$ being defined in outer scaling. This suggests that there are viscous inhibitions on the wavenumber content, at least up to $(y^+)_{\it{OSP}}$ , that are not captured by the model. Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015) also noted that the wavenumber of the outer-spectral peak follows neither inner nor outer scaling, suggesting that, following Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2015), an alternate integral scale emerges in this region at high ${Re}_\tau$ that is required to scale the attached eddies. However, as observed in our discussion of the outer peak in $\overline {u_1^2}^+$ , this intermediate scale is perhaps better described through a co-existence of inner-scaled and outer-scaled eddies in this wavenumber range.