2.1 Review of empirically observed trends for $\unicode[STIX]{x1D719}_{uu}$ energy spectra

Figure 1. (a) Premultiplied energy spectrogram $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}$ (filled iso-contours 0.2:0.2:1.8) at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,100$ (Baars, Hutchins & Marusic Reference Baars, Hutchins and Marusic2017a ), alongside a spectrogram of the surrogate dissipation $k_{x}^{3}\unicode[STIX]{x1D719}_{uu}$ (at each $z$ , the dissipation spectrum is normalized by its maximum value; iso-contours 0.2:0.2:0.8). The dashed line tracks the ridge in the dissipation ( $\unicode[STIX]{x1D706}_{x}=35\unicode[STIX]{x1D702}$ ). Triangle ‘N’ refers to the $k_{x}^{-1}$ region identified by Nickels et al. (Reference Nickels, Marusic, Hafez and Chong2005), while ‘B’ labels a region where Baars, Hutchins & Marusic (Reference Baars, Hutchins and Marusic2017b ) contemplated a self-similar behaviour. Note: $\unicode[STIX]{x1D706}_{x}\equiv U(z)/f$ . (b) Saddle-type topology formed by the inner-peak imprint and outer-peak footprint.

Over the past two decades, wall-turbulence data at high $Re_{\unicode[STIX]{x1D70F}}$ have revealed new features of $\unicode[STIX]{x1D719}_{uu}$ . Figure 1(a) presents the streamwise energy spectra as a spectrogram: premultiplied spectra at 40 logarithmically spaced positions within the range $10.6\lesssim z^{+}\lesssim \unicode[STIX]{x1D6FF}^{+}$ are presented with iso-contours of $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}$ . These spectra were obtained from hot-wire measurements at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,100$ in Melbourne’s TBL facility (Baars et al. Reference Baars, Hutchins and Marusic2017a ).

Dominant small-scale features known as recurrent near-wall streaks (Kline et al. Reference Kline, Reynolds, Schraub and Rundstadler1967) form the inner-spectral peak in the TBL spectrogram and obey viscous-scaling (identified with the $\times$ marker at $\unicode[STIX]{x1D706}_{x}^{+}=10^{3}$ and $z^{+}=15$ in figure 1 a). Attached eddies play an important role in the log-layer, and likely take a form consistent with vortex packets (Head & Bandyopadhyay Reference Head and Bandyopadhyay1981; Kim & Adrian Reference Kim and Adrian1999; Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Wu & Christensen Reference Wu and Christensen2006; Adrian Reference Adrian2007; Balakumar & Adrian Reference Balakumar and Adrian2007). These packets, or large-scale motions (LSM), exhibit a forward inclined structure with $u<0$ within the packet and $u>0$ at either spanwise-flanked side of the packet (Favre, Gaviglio & Dumas Reference Favre, Gaviglio and Dumas1967; Blackwelder & Kovasznay Reference Blackwelder and Kovasznay1972; Brown & Thomas Reference Brown and Thomas1977; Wark & Nagib Reference Wark and Nagib1991; Ganapathisubramani, Longmire & Marusic Reference Ganapathisubramani, Longmire and Marusic2003; Tomkins & Adrian Reference Tomkins and Adrian2003). Larger motions, known as superstructures or very large-scale motions (VLSM), have also been described (Hutchins & Marusic Reference Hutchins and Marusic2007b ). In terms of their spectral signature, these large-scale organized motions appear most prominently as a broad spectral peak in the log-region, indicated in figure 1(a) with a $\times$ marker at $\unicode[STIX]{x1D706}_{x}=4\unicode[STIX]{x1D6FF}$ and $z^{+}=3.9Re_{\unicode[STIX]{x1D70F}}^{1/2}\approx 464$  (Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009). Hutchins & Marusic (Reference Hutchins and Marusic2007a ) observed the emergence of the outer-spectral peak for $Re_{\unicode[STIX]{x1D70F}}\gtrsim 2000$ and its energetic relevance increases with $Re_{\unicode[STIX]{x1D70F}}$ .

Figure 2. Premultiplied energy spectra ( $Re_{\unicode[STIX]{x1D70F}}\approx 14\,100$ ) at nine logarithmically spaced wall-normal positions in the range $100\lesssim z^{+}\lesssim 0.15\unicode[STIX]{x1D6FF}^{+}$ , with the abscissae in (a) outer-scaling and (b) wall-scaling, respectively. Note: $k_{x}\equiv 2\unicode[STIX]{x03C0}f/U(z)$ .

Figure 2 aids in assessing scalings of $\unicode[STIX]{x1D719}_{uu}$ in the logarithmic region. Individual spectra at logarithmically spaced positions in the range $100\lesssim z^{+}\lesssim 0.15\unicode[STIX]{x1D6FF}^{+}$ are shown in figure 2(a) with an outer-scaling on the logarithmic scale axis (and with a wall-scaling in figure 2 b). At the high wavenumber-end, spectra arguably resemble a $k_{x}^{-5/3}$ inertial sub-range scaling (figure 2 b and Saddoughi & Veeravalli (Reference Saddoughi and Veeravalli1994), Sreenivasan (Reference Sreenivasan1995), Mydlarski & Warhaft (Reference Mydlarski and Warhaft1996), Gamard & George (Reference Gamard and George2000), Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018)). A decrease in spectral separation between the isotropic scales and the range over which viscosity acts, narrows the $k_{x}^{-5/3}$ region with decreasing $z$ . In that regard, a second spectrogram in figure 1(a) displays $k_{x}^{3}\unicode[STIX]{x1D719}_{uu}$ , indicative of the surrogate dissipation rate of energy $\unicode[STIX]{x1D716}\equiv 15\unicode[STIX]{x1D708}\int k_{x}^{3}\unicode[STIX]{x1D719}_{uu}\,\text{d}\log (k_{x})$ (with an applied local isotropy hypothesis, valid only well into the logarithmic region). In the logarithmic region, $\unicode[STIX]{x1D716}\sim 1/z$ and the dissipation scales via $k_{x}z\propto z^{3/4}$ (production–dissipation balance); this scaling reflects the ridge of the dissipation spectrogram following $\unicode[STIX]{x1D706}_{x}\approx 35\unicode[STIX]{x1D702}$ , where $\unicode[STIX]{x1D702}\equiv (\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D716})^{1/4}$ . When spectra adhere to the $k_{x}^{-5/3}$ up to a constant wall-scaling, e.g.  $k_{x}z=O(1)$ (dashed line in figure 1 a), the dissipation and energy spectrograms reveal the relatively small range over which a $k_{x}^{-5/3}$ may be expected.

When concentrating on the larger scales, adjacent to the inertial sub-range, a combination of dimensional analysis, a spectral overlap argument and an assumed type of eddy similarity lead to a $k_{x}^{-1}$ dependence (e.g. Perry & Abell Reference Perry and Abell1975; Perry, Henbest & Chong Reference Perry, Henbest and Chong1986; Davidson & Krogstad Reference Davidson and Krogstad2009),

(2.1) $$\begin{eqnarray}\displaystyle k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}=A_{1}, & & \displaystyle\end{eqnarray}$$

where $A_{1}$ is a universal constant (Perry et al. Reference Perry, Henbest and Chong1986). In $(\unicode[STIX]{x1D706}_{x},z)$ -space, the $k_{x}^{-1}$ scaling region obeys wall-scaling and outer-scaling at the small- and large-wavelength ends of the spectrum, respectively. By way of visual inspection, Nickels et al. (Reference Nickels, Marusic, Hafez and Chong2005) identified a $k_{x}^{-1}$ region as $z^{+}>100$ , $\unicode[STIX]{x1D706}_{x}>15.7z$ and $\unicode[STIX]{x1D706}_{x}<0.3\unicode[STIX]{x1D6FF}$ at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,000$ (triangular region ‘N’ in figure 1 a). To date, convincing support for a $k_{x}^{-1}$ dependence has not been shown, mainly due to the limited range over which this region plausibly exists. Even for the highest $Re_{\unicode[STIX]{x1D70F}}$ laboratory data, the presence of a $k_{x}^{-1}$ has been inconclusive (Morrison et al. Reference Morrison, Jiang, McKeon and Smits2002; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013; Vallikivi, Ganapathisubramani & Smits Reference Vallikivi, Ganapathisubramani and Smits2015a ; Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2017), while theory suggests that this region should grow with $Re_{\unicode[STIX]{x1D70F}}$ . A $k_{x}^{-1}$ scaling has been reported for atmospheric surface layer (ASL) data (Högström, Hunt & Smedman Reference Högström, Hunt and Smedman2002; Calaf et al. Reference Calaf, Hultmark, Oldroyd, Simeonov and Parlange2013) and free shear flow (Beresh et al. Reference Beresh, Henfling, Spillers and Spitzer2018), however, an important issue here is that those spectra are typically presented in log–log format: $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ may appear to be more convincing (§ A.1). To emphasize the ambiguity of a $k_{x}^{-1}$ scaling, the portions of the spectra satisfying Nickels’ criterion are highlighted in figure 2. These portions plateau at $A_{1}\approx 0.92$  (Nickels et al. Reference Nickels, Marusic, Hafez and Chong2005), although this plateau may be caused by a transitioning from the imprint signature of the inner-spectral peak, to the broad outer-spectral peak. Furthermore, it is well known that the outer-spectral peak has a distinct energy footprint down to the wall, as a direct consequence of its strong linear coherence with the near-wall turbulence (Baars et al. Reference Baars, Hutchins and Marusic2017b ; Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017). These spectral features – which could enhance $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ while not being truly representative of (2.1) – are most clearly observed in a spectrogram and induce a saddle-type topology (figure 1 b). Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017) examined experimentally acquired streamwise–spanwise two-dimensional spectra of $u$ at $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{4})$ for a $k_{x}^{-1}$ , but have also faced challenges in identifying an unobstructed spectral view. It is worth noting that numerical studies (e.g. Jiménez & Hoyas Reference Jiménez and Hoyas2008; Lee & Moser Reference Lee and Moser2015; Hwang Reference Hwang2015) indicate a promising $k^{-1}$ structure in, for instance, spectra of the spanwise velocity component in channel flow, even at $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3})$ .

If a $k_{x}^{-1}$ scaling region does exist in $\unicode[STIX]{x1D719}_{uu}$ , it presumably extends beyond Nickel’s identified outer limit of $\unicode[STIX]{x1D706}_{x}=0.3\unicode[STIX]{x1D6FF}$ . Any large-scale turbulence features that would scale differently than $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ are thought to obscure a $k_{x}^{-1}$ scaling at the large-scale end of the spectrum. Those obscuring features may relate to the presence of detached eddies (Marusic & Perry Reference Marusic and Perry1995; Jiménez Reference Jiménez2012), the processes of spatial alignment (Adrian et al. Reference Adrian, Meinhart and Tomkins2000) and spectral aliasing (Davidson, Nickels & Krogstad Reference Davidson, Nickels and Krogstad2006; Davidson & Krogstad Reference Davidson and Krogstad2008), all of which accumulate energy in the broad outer-spectral peak. In fact, Baars et al. (Reference Baars, Hutchins and Marusic2017b ) determined that a geometrically self-similar wall-attached structure of turbulence may be ingrained within a much larger $(\unicode[STIX]{x1D706}_{x},z)$ -region than the one identified by Nickels et al. (Reference Nickels, Marusic, Hafez and Chong2005) – that new region is identified region ‘B’ in figure 1(a).

2.2 The spectral view of Perry et al. for streamwise velocity spectra

Townsend (Reference Townsend1976) hypothesized that: ‘The velocity fields of the main eddies, regarded as persistent, organized flow patterns, extend to the wall and, in a sense, they are attached to the wall’. For predictive purposes, Townsend further assumed a self-similar nature of the main eddies. Townsend’s AEH thus commonly implies a hierarchy of geometrically self-similar eddying motions that are inertially dominated (inviscid), attached to the wall and scalable with their distance to the wall (Marusic & Monty Reference Marusic and Monty2019). In the log-region, Perry & Chong (Reference Perry and Chong1982) showed that an attached-eddy model predicts a $k_{x}^{-1}$ in $\unicode[STIX]{x1D719}_{uu}$ , whereas various other theoretical arguments and interpretations lead to the same result (e.g. Tchen Reference Tchen1953; Nikora Reference Nikora1999; Hunt & Carlotti Reference Hunt and Carlotti2001; Davidson & Krogstad Reference Davidson and Krogstad2009; Katul, Porporato & Nikora Reference Katul, Porporato and Nikora2012). Following the classical scaling, viscosity can be neglected above a fixed, inner-scaled wall-normal location, say $z^{+}=O(100)$ (Perry & Chong (Reference Perry and Chong1982), Nagib, Chauhan & Monkewitz (Reference Nagib, Chauhan and Monkewitz2007), Zagarola & Smits (Reference Zagarola and Smits1998), among others). A $k_{x}^{-1}$ in $\unicode[STIX]{x1D719}_{uu}$ at such a lower limit of the logarithmic region was envisioned by Perry and co-workers, provided that contributions to the spectrum that do not follow a spectral overlap scaling are absent or do not overlap with a $k_{x}^{-1}$ range (e.g. no imprints from the inner-spectral peak, no contributions from the scales that result in Kolmogorov scaling, and no masking by large-scale contributions that obey a $\unicode[STIX]{x1D6FF}$ scaling in spectral space). It must be noted here that various studies have questioned the classical scaling and have arrived at mesolayer and mixed scaling concepts (see Long & Chen Reference Long and Chen1981; Wosnik, Castillo & George Reference Wosnik, Castillo and George2000; Klewicki, Fife & Wei Reference Klewicki, Fife and Wei2009). However, an unresolved issue in all scaling studies for $\overline{u^{2}}(z)$ is that unaltered profiles are considered, which thus cluster the energy contributions from all turbulent, spectral scales (while it is agreed upon that different parts of the turbulence spectrum scale differently). Hence, a mesolayer scaling of $z^{+}\propto Re_{\unicode[STIX]{x1D70F}}^{0.5}$ (the minimum wall-normal location at which viscosity acts) may not reflect the lower limit of the inviscid, logarithmic-region component of turbulence. And so, we here hypothesize following Perry and co-workers, that if one considers the scaling of sub-contributions to the streamwise velocity spectra separately, different wall-normal scalings can emerge depending upon which turbulent scales are included. We highlight that a classical scaling for the lower limit of the logarithmic region in the context of inviscid, self-similar and wall-attached motions of Perry and co-workers (reviewed here in § 2.2) should not be discarded. Generally, the relatively slow progress on the issue of spectral scalings in TBL flow is influenced by the still limited range of scales of the energy cascade, even at moderate $Re_{\unicode[STIX]{x1D70F}}$ , thus making it complicated to observe extended scaling ranges along both wavenumber and wall-normal dimensions. For instance, with a smallest eddy of the order of $100\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}}$ , and a discrete hierarchy of 10 geometrically self-similar scales, it was shown by Perry & Chong (Reference Perry and Chong1982) that at least $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{5})$ is required for a $k_{x}^{-1}$ extending over one order of magnitude in $k_{x}$ .

Figure 3. Hypothetical structure of $k_{x}\unicode[STIX]{x1D719}_{uu}$ within the logarithmic region of TBLs, following Perry & Abell (Reference Perry and Abell1977) and Perry et al. (Reference Perry, Henbest and Chong1986) and reproduced from Marusic & Perry (Reference Marusic and Perry1995). Sketches for finite $Re_{\unicode[STIX]{x1D70F}}$ are hypothesized in (a) and (b), while an asymptotically high $Re_{\unicode[STIX]{x1D70F}}$ is envisioned in (c).

Perry & Abell (Reference Perry and Abell1977) and Perry et al. (Reference Perry, Henbest and Chong1986) contemplated that $\unicode[STIX]{x1D719}_{uu}$ encompasses three distinct contributions; a vision further refined later on (Marusic & Perry Reference Marusic and Perry1995). A conceptual sketch of these contributions is presented in figure 3(c) and are referred to as type ${\mathcal{A}}$ , ${\mathcal{B}}$ and ${\mathcal{C}}$ energy contributions. Central in this view is the ${\mathcal{A}}$ energy (only this energy fraction is captured by the attached-eddy model and would dominate the spectrum at ultra-high $Re_{\unicode[STIX]{x1D70F}}$ ). At the high wavenumber-end, the ${\mathcal{A}}$ contribution follows wall-scaling with boundary $W_{A}$ , starting at $k_{x}z=G$ (note that one fixed and inner-scaled location $z$ is assumed throughout figure 3). At the low wavenumber-end, ${\mathcal{A}}$ is bounded by $O_{A}$ , adhering to outer-scaling. A $k_{x}^{-1}$ region occurs where there is an overlap of the outer-scaling and wall-scaling. Note that both boundaries $W_{A}$ and $O_{A}$ are unknown and undefined, e.g. no physical arguments exist for formulating these, nor is there empirical evidence yet for these boundaries. Logically, the spectrum also includes energy from scales that are unresolved by the attached-eddy model and that energy thus scales differently than that of ${\mathcal{A}}$ . At low wavenumbers, ${\mathcal{B}}$ energy was envisioned to come from detached motions. Only through the availability of recent experimental data at high $Re_{\unicode[STIX]{x1D70F}}$ (Mathis et al. Reference Mathis, Hutchins and Marusic2009), it appears that this contribution is Reynolds-number dependent and that it induces an imprint on the near-wall region. Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) suggested that merging of self-similar LSM may be one of the mechanisms generating VLSM and superstructures. But, in a recent line of work, several studies – using linearized Navier–Stokes equations – have shown that long streaky motions can be amplified at all length scales (del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2006), Hwang & Cossu (Reference Hwang and Cossu2010a ,Reference Hwang and Cossu b ), among others), and this is consistent with the recent findings that the large-scale outer structures obey a self-similar character and involve self-sustaining mechanisms (Hwang Reference Hwang2015; de Giovanetti, Sung & Hwang Reference de Giovanetti, Sung and Hwang2017). Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2015) performed a modelling attempt of this low wavenumber-end of the spectrum at high $Re_{\unicode[STIX]{x1D70F}}$ , by including a $k_{x}^{-m}$ range. In the current context this can be interpreted as an attempt to model the ${\mathcal{B}}$ contribution. In addition, Srinath et al. (Reference Srinath, Vassilicos, Cuvier, Laval, Stanislas and Foucaut2018) related this modelling effort to the wide variation of streamwise lengths of wall-attached streamwise velocity structures. Finally, at the high wavenumber-end in figure 3(c), region ${\mathcal{C}}$ comprises Kolmogorov scales and small-scale detached eddies. This portion of the spectrum obeys $k_{x}^{-5/3}$ and is subject to the viscous roll-off beyond an $\unicode[STIX]{x1D702}$ -based scale (boundary with constant $M$ ).

Perhaps the most important feature in figure 3(c) is the complete spectral separation of ${\mathcal{B}}$ and ${\mathcal{C}}$ components at ultra-high $Re_{\unicode[STIX]{x1D70F}}$ , which has never been observed. The significant spectral overlap envisioned at low-Reynolds-number conditions, with a small range of energetic length scales, is schematically shown in figure 3(a,b). In essence, at low $Re_{\unicode[STIX]{x1D70F}}$ , the range of scales has not yet matured to a range where complete ${\mathcal{B}}/{\mathcal{C}}$ spectral-separation may occur (if at all existent). In light of the above, this article attempts to re-appraise the spectral view of figure 3 by way of utilizing data over a range of $Re_{\unicode[STIX]{x1D70F}}$ , in combination with a data-driven spectral decomposition technique.

2.3 Present contribution and outline

To summarize the above, a presence of geometrically self-similar, wall-attached motions, as hypothesized by Townsend (Reference Townsend1976), would imply the existence of $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ at high enough $Re_{\unicode[STIX]{x1D70F}}$ . This attached-eddy model behaviour only governs ${\mathcal{A}}$ in the spectral view hypothesized by Perry and co-workers. Empirical validation of a three-component structure in the velocity spectra $\unicode[STIX]{x1D719}_{uu}$ is non-existent, and boundaries $O_{A}$ and $W_{A}$ , as well as constants like $G$ , $N$ and $M$ , are unknown (figure 3). Moreover, a convincing $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ scaling has remained elusive as it has not been observed in high-fidelity data. In order to gain further insight into the structure of $\unicode[STIX]{x1D719}_{uu}$ , there is a need to examine the possible existence of a three-component spectral structure. This article presents a first attempt towards this, using a data-driven spectral decomposition method. Next, in § 3, a description of the multi-point synchronized experimental data is provided. Via linear systems theory (§ 4.1), data-driven spectral filters are derived (§§ 4.2–4.3) and allow for the spectral decomposition of $\unicode[STIX]{x1D719}_{uu}$ (§ 5.1). It also provides further insight into the required scale separation for the observance of $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ (§ 5.3). Subsequently, spectra at a range of $Re_{\unicode[STIX]{x1D70F}}$ are decomposed and interpreted according to the spectral structure of Perry and co-workers (§ 6). In Part 2 (Baars & Marusic Reference Baars and Marusic2020), the integrated energy in the $\unicode[STIX]{x1D719}_{uu}$ spectra, being the streamwise turbulence intensity $\overline{u^{2}}$ , is reassessed in the context of the spectral decomposition presented in this paper.

3.1 Two-point measurements at $Re_{\unicode[STIX]{x1D70F}}\approx 13\,500$

Two-point measurement data are the foundation for the spectral decomposition-methodology used. Data were acquired in Melbourne’s boundary layer facility (Nickels et al. Reference Nickels, Marusic, Hafez and Chong2005; Baars et al. Reference Baars, Squire, Talluru, Abbassi, Hutchins and Marusic2016b ) under nominal ZPG conditions. Pressure coefficient, $C_{p}$ , was constant to within $\pm 0.87\,\%$  (Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015), and free-stream turbulence intensities were less than 0.05 % of the free-stream at $x=0~\text{m}$  (Kulandaivelu Reference Kulandaivelu2011).

Four datasets were acquired using a hot-wire arrangement shown in figure 4. For each dataset, one hot-wire probe is positioned at a fixed location (reference $z=z_{R}$ ), while the other probe is traversed to map out a range of wall-normal positions, $z_{M}$ (probes were located at the same $(x,y)$ -position). One dataset (denoted by ${\mathcal{W}}$ ) employs a near-wall reference at $z_{R}^{+}\approx 4.4$ , while the reference probe was situated within the logarithmic region for the other three datasets ( ${\mathcal{L}}_{i}$ , $i=1\ldots 3$ , with reference positions $z_{R}^{+}\approx 246$ , $469$ and $1120$ , respectively). These four reference locations are from now on denoted as $z_{{\mathcal{W}}}$ and $z_{{\mathcal{L}}}$ for datasets ${\mathcal{W}}$ and ${\mathcal{L}}_{1-3}$ , respectively. Nominally, all datasets correspond to the same Reynolds-number condition ( $Re_{\unicode[STIX]{x1D70F}}\approx 13\,500$ ). Boundary layer parameters are summarized in table 1. The friction velocity was inferred from a direct survey of the wall-shear stress using a floating element drag balance, situated at the streamwise measurement location (Baars et al. Reference Baars, Squire, Talluru, Abbassi, Hutchins and Marusic2016b ), and a modified Coles law of the wake fit was used to determine $\unicode[STIX]{x1D6FF}$  (Jones, Marusic & Perry Reference Jones, Marusic and Perry2001). For reference, mean velocity profiles (figure 5 a) were fitted to a composite profile with log-law constants of $\unicode[STIX]{x1D705}=0.384$ and $A=4.17$  (Chauhan, Monkewitz & Nagib Reference Chauhan, Monkewitz and Nagib2009). This procedure resulted in values for $U_{\unicode[STIX]{x1D70F}}$ that deviated less than 2 % from the friction data-based values.

Table 1. Experimental parameters of two-point hot-wire data acquired in Melbourne’s boundary layer facility.

a For datasets ${\mathcal{L}}_{1-3}$ the reference and traversing wires were interchanged for $z_{M}<z_{R}$ and $z_{M}>z_{R}$ , see text for details.

For dataset ${\mathcal{W}}$ , the near-wall probe was a Dantec 55P30 type, while the traversing hot-wire consisted of a Dantec 55P15 boundary layer probe. Wollaston wires were used with a $d=5~\unicode[STIX]{x03BC}\text{m}$ and $d=2.5~\unicode[STIX]{x03BC}\text{m}$ core wire, for the near-wall and traversing probes, respectively. Both wires were etched to expose the core over a length of $l/d=200$  (Ligrani & Bradshaw Reference Ligrani and Bradshaw1987). In viscous-scaling, the lengths of $l^{+}\approx 42$ and $l^{+}\approx 22$ are of an acceptable length when the spatial resolution is concerned, since this study concentrates on the absolute energy in the logarithmic region only (Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). The traversing hot-wire was traversed to 40 logarithmically spaced positions in the range $10.7\lesssim z_{M}^{+}\lesssim 1.55\unicode[STIX]{x1D6FF}^{+}$ . For datasets ${\mathcal{L}}_{1-3}$ , both probes consisted of Dantec 55P15 type holders with $d=2.5~\unicode[STIX]{x03BC}\text{m}$ wires (also $l/d=200$ , resulting in $l^{+}\approx 21$ ). Each probe could be moved independently in $z$ using two separate traverse systems: a tunnel-specific traverse with a wall-implemented traverse situated underneath. Irrespective of $z_{R}$ , a mapping of 50 logarithmically spaced positions from $z_{M}^{+}\approx 10.5$ up to $z_{M}^{+}\approx 1.6\unicode[STIX]{x1D6FF}^{+}$ was carried out. In practice, positions below $z_{R}$ were mapped out with the probe connected to the bottom traverse (with the other probe at $z_{R}$ ). Subsequently, the probes switched positions, and the wall-normal range above $z_{R}$ was mapped out with the top traverse (with the bottom traverse-probe at  $z_{R}$ ). Since the geometry of the probe holders dictated a minimum separation distance of $\unicode[STIX]{x0394}z_{min}^{+}=|z_{M}-z_{R}|\approx 2.2~\text{mm}$ , 5 points ( ${\mathcal{L}}_{1}$ ), 2 points ( ${\mathcal{L}}_{2}$ ) and 1 point ( ${\mathcal{L}}_{3}$ ) were skipped in the 50-point profiles.

All hot-wire probes were operated in constant temperature mode, with an overheat ratio of 1.8, using in-house built anemometers. For each dataset, wires were sampled simultaneously at a rate of $\unicode[STIX]{x0394}T^{+}\equiv U_{\unicode[STIX]{x1D70F}}^{2}/\unicode[STIX]{x1D708}/f_{s}$ , where $f_{s}$ is the sampling frequency (acquisition rates, $\unicode[STIX]{x0394}T^{+}$ , were around unity or less, see Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009)). To prevent aliasing, the signals were passed through fourth-order Butterworth filters – with a spectral cut-off set at $f_{s}/2$ – prior to analogue to digital conversion using a 16-bit Data Translation DT9836 module. Relatively long signals were acquired with lengths of $TU_{\infty }/\unicode[STIX]{x1D6FF}>20\times 10^{3}$ , allowing for converged spectral statistics at the largest energetic wavelengths. Both hot-wire probes were calibrated, with a correction method for hot-wire voltage drift (Talluru et al. Reference Talluru, Kulandaivelu, Hutchins and Marusic2014). Boundary layer profiles of $U$ and $\overline{u^{2}}$ are plotted in figures 5(a) and 5(b), respectively. All mean velocity profiles are compared with $U^{+}=1/\unicode[STIX]{x1D705}\ln (z^{+})+A$ in the logarithmic region (thin blue lines), with $\unicode[STIX]{x1D705}=0.384$ and $A=4.17$ and show minimum effect from the intrusive reference probe. Likewise, the agreement of the $\overline{u^{2}}$ profiles can be gleaned from their comparison to $\overline{u^{2}}^{+}=B_{1}-A_{1}\ln (z/\unicode[STIX]{x1D6FF})$ in the logarithmic region (thin blue lines), with $A_{1}=1.26$ and $B_{1}=2.30$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013); profiles ${\mathcal{L}}_{1-3}$ comprise a slightly lower energy in the outer-region than the ${\mathcal{W}}$ data, which is ascribed to the minor variation in $Re_{\unicode[STIX]{x1D70F}}$ . Since the near-wall streamwise turbulence intensity is attenuated due to the hot-wire’s spatial resolution, corrected profiles via the method of Smits et al. (Reference Smits, McKeon and Marusic2011) are also shown. Peak-values at $z^{+}\approx 15$ agree with the expected behaviour following $\overline{u^{2}}_{max}=0.63\ln (Re_{\unicode[STIX]{x1D70F}})+3.80$  (Lee & Moser Reference Lee and Moser2015; Marusic et al. Reference Marusic, Baars and Hutchins2017; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018).

Figure 5. Boundary layer profiles of the streamwise (a) mean velocity and (b) turbulence intensity of the four two-point hot-wire datasets listed in table 1.

3.2 Additional data for investigating Reynolds-number dependence

A number of other TBL datasets are employed for investigating Reynolds-number trends, ranging from direct numerical simulation (DNS) up to ASL data. For all datasets, $\unicode[STIX]{x1D6FF}$ , $U_{\unicode[STIX]{x1D70F}}$ and $Re_{\unicode[STIX]{x1D70F}}$ were recomputed with the modified Coles law of the wake fit. The DNS data of a ZPG TBL by Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) correspond to $Re_{\unicode[STIX]{x1D70F}}\approx 1992$ . Streamwise/wall-normal planes of data, spanning the entire TBL in $z$ and extending ${\sim}11.9\unicode[STIX]{x1D6FF}$ in $x$ , are used. This streamwise extension is chosen to include large-scale motions, while still maintaining an acceptable Reynolds number increase from $Re_{\unicode[STIX]{x1D703}}\equiv \unicode[STIX]{x1D703}U_{\infty }/\unicode[STIX]{x1D708}\approx 5110$ to $6010$ , where $\unicode[STIX]{x1D703}$ is the momentum thickness ( $Re_{\unicode[STIX]{x1D70F}}\approx 1992$ at the streamwise centre). Around this Reynolds-number condition, the spanwise resolution of the DNS is $\unicode[STIX]{x0394}y^{+}\approx 3.72$ .

Five singe-point hot-wire datasets of $u$ are also included throughout this work (all acquired in Melbourne’s TBL facility). Data at $Re_{\unicode[STIX]{x1D70F}}\approx 2800$ , $3900$ and $7300$ were taken using a regular hot-wire set-up (Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009). Data at $Re_{\unicode[STIX]{x1D70F}}\approx 13\,000$ and $19\,300$ are taken from Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), where Princeton’s NSTAP probes were used for making fully resolved measurements. Our highest Reynolds-number data at $Re_{\unicode[STIX]{x1D70F}}\approx 1.4\times 10^{6}$ encompass the $u$ fluctuation in the ASL under near-neutrally buoyant conditions at the surface layer turbulence and environmental science test facility (known as SLTEST) on the great salt lakes of Utah. These data were synchronously acquired using a wall-normal array of five sonic anemometers and one purpose-built wall-shear stress sensor situated under the array (Heuer & Marusic Reference Heuer and Marusic2005; Marusic & Heuer Reference Marusic and Heuer2007).

4.2 Coherence relative to a near-wall reference

4.2.1 Coherence spectrogram and physical interpretation

Figure 7. Dataset ${\mathcal{W}}$ . Coherence spectrogram $\unicode[STIX]{x1D6FE}_{l}^{2}(z,z_{{\mathcal{W}}};\unicode[STIX]{x1D706}_{x})$ , relative to the near-wall reference location $z_{{\mathcal{W}}}^{+}\approx 4.4$ (iso-contours 0.1:0.1:0.9), superposed on its associated premultiplied energy spectrogram $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}$ (filled iso-contours 0.2:0.2:1.8), reproduced from figure 1. Baars et al. (Reference Baars, Hutchins and Marusic2017b ) contemplated a self-similar behaviour in the triangular region (here drawn with $R\approx 14$ ).

Using (4.1), coherence spectra can be computed from the ${\mathcal{W}}$ dataset for all $z$ within the TBL. A coherence spectrogram, formed by presenting all individual coherence spectra as iso-contours of $\unicode[STIX]{x1D6FE}_{l}^{2}$ , is presented in $(\unicode[STIX]{x1D706}_{x},z)$ -space and superposed on the $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}$ energy spectrogram in figure 7. A horizontal cut through the $\unicode[STIX]{x1D6FE}_{l}^{2}$ contour at $z_{O}^{+}\approx 473$ results in figure 6(b). Iso-contours of $\unicode[STIX]{x1D6FE}_{l}^{2}$ increase in value, with increasing $\unicode[STIX]{x1D706}_{x}$ , and follow lines of constant $\unicode[STIX]{x1D706}_{x}/z$ within the logarithmic region. Only a portion of the energy spectrogram below non-zero contours of $\unicode[STIX]{x1D6FE}_{l}^{2}$ is stochastically coherent with $z_{{\mathcal{W}}}$ . A dashed line at the large-wavelength end indicates the lowest frequency resolved.

Figure 8. (a–d) A visual road map conceptualizing a physical underpinning for a coherence spectrogram with a near-wall reference location. Precise forms of the structures in (a) and (b), drawn as a hairpin vortex and packets of hairpins, are of secondary importance in the context of a hierarchical distribution of some wall-attached eddies, and solely serves for visual purposes.

A physical interpretation for the $\unicode[STIX]{x1D6FE}_{l}^{2}$ trend is considered here with the aid of a road map in figure 8(a–d). Here we assume the existence of a wall-attached structure with an embedded self-similar hierarchy of scales (Baars et al. Reference Baars, Hutchins and Marusic2017b ). Figure 8(a) visualizes a wall-attached coherent feature as a hairpin vortex with a forward-leaning inclination angle of ${\sim}45^{\circ }$  (Adrian et al. Reference Adrian, Meinhart and Tomkins2000). Note that the exact types of structures do not matter in this discussion. Streamwise and wall-normal extents of such a structure are approximately equal, implying a streamwise/wall-normal aspect ratio of $O(1)$ . When the hairpins form packets (figure 8 b), their streamwise extent grows more than their wall-normal region of influence (and a characteristic angle reduces to ${\sim}12^{\circ }$ , e.g. Christensen & Adrian (Reference Christensen and Adrian2001)). The wall-attached structures have a characteristic wavelength $\unicode[STIX]{x1D706}_{x}^{H}$ and wall-normal extent $\unicode[STIX]{x1D6FF}_{H}$ , so that a streamwise/wall-normal aspect ratio arises as . The forward-leaning nature of the structures is neglected since a consistent phase shift is irrelevant in the context of the LCS and spectra. Townsend’s self-similarity implies that each structure hierarchy is associated with the same   and two structures are shown in figure 8(c) of heights $\unicode[STIX]{x1D6FF}_{1}^{H}$ and $\unicode[STIX]{x1D6FF}_{i}^{H}$ . Due to the structures’ random repetitions in space (or time) (Woodcock & Marusic Reference Woodcock and Marusic2015), we here assume a non-zero coherence for $\unicode[STIX]{x1D706}_{x}>\unicode[STIX]{x1D706}_{x}^{H}$ : thus, $\unicode[STIX]{x1D6FE}_{l}^{2}>0$ for $z<\unicode[STIX]{x1D6FF}_{i}^{H}$ and   in the case of hierarchy $i$ (bottom of figure 8 c). The magnitude of $\unicode[STIX]{x1D6FE}_{l}^{2}$ may vary with $\unicode[STIX]{x1D706}_{x}$ and $z$ , but a constant magnitude is drawn for ease with a single greyscale. A non-uniform magnitude results in the same final structure of the reconstructed $\unicode[STIX]{x1D6FE}_{l}^{2}$ spectrogram when $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$ (an infinite number of hierarchies). Here the magnitude of the coherence intensity function is less than unity and is attributed to an inherent three-dimensionality of wall-attached structures (only coherence in the $(x,z)$ -plane is considered), stochastically inconsistent motions (e.g. random meandering or flapping) and its continuous evolution (growth, saturation, decay). These factors contribute to the trivial cause of less-than-perfect coherence: the coexistence of detached (wall-incoherent) energy in $(\unicode[STIX]{x1D706}_{x},z)$ -space. This incoherent energy does add to the total energy in the denominator of (4.1). Finally, when superposing each hierarchy of self-similar scales to reconstruct a coherence spectrogram (figure 8 d), $\unicode[STIX]{x1D6FE}_{l}^{2}$ increases in the region where individual coherence intensity functions overlap, allegedly caused by the growing ratio of attached energy, relative to detached energy (supported by the data and here visualized by the increasing greyscale of the superposed transparent hierarchy of rectangles from figure 8 c). In a region bounded by $z=l$ , a constant $\unicode[STIX]{x1D706}_{x}/z$ and a constant $\unicode[STIX]{x1D706}_{x}$ , the $\unicode[STIX]{x1D6FE}_{l}^{2}$ iso-contours thus obey a constant $\unicode[STIX]{x1D706}_{x}/z$ via

(4.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{l}^{2}=C_{1}\ln \left(\frac{\unicode[STIX]{x1D706}_{x}}{z}\right)+C_{2}=C_{1}\ln \left(\frac{\unicode[STIX]{x1D706}_{x}}{z}\frac{1}{R}\right), & & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D6FE}_{l}^{2}$ is bounded: $\unicode[STIX]{x1D6FE}_{l}^{2}\in [0,1]$ . Constant $R$ is the aspect ratio at which the coherence falls to zero, e.g. $R=\unicode[STIX]{x1D706}_{x}/z|_{\unicode[STIX]{x1D6FE}_{l}^{2}\rightarrow 0}=\exp (-C_{2}/C_{1})$ . This example with $l^{+}\approx 80$ and $z\approx 0.71\unicode[STIX]{x1D6FF}$ for the upper bound of the triangular region (Baars et al. Reference Baars, Hutchins and Marusic2017b ), and with the assumption that each of the 10 discrete hierarchies are subject to a doubling in size, yields $Re_{\unicode[STIX]{x1D70F}}=80\times 2^{(10-1)}/0.71\approx 58\,000$ .

Figure 9. (a) Coherence spectra of dataset ${\mathcal{W}}$ with the wavelength axis predivided by  $z$ . The thicker portions of the curves correspond to $3Re_{\unicode[STIX]{x1D70F}}^{1/2}<z^{+}<0.15Re_{\unicode[STIX]{x1D70F}}$ and $20z<\unicode[STIX]{x1D706}_{x}<6\unicode[STIX]{x1D6FF}$ . Note that the inclined plane with the solid boundary reflects (4.7) with the constants stated in the text. (b,c) Linear coherence spectra $\unicode[STIX]{x1D6FE}_{l}^{2}(z,z_{{\mathcal{W}}};\unicode[STIX]{x1D706}_{x})$ for wall-normal positions within the range $3Re_{\unicode[STIX]{x1D70F}}^{1/2}<z^{+}<0.15Re_{\unicode[STIX]{x1D70F}}$ , for (b) dataset ${\mathcal{W}}$ and (c) the DNS and ASL data described in § 3.2. (d) Filter following (4.8) for $z^{+}=841$ .

4.2.2 Universality of the data-driven filter

The $\unicode[STIX]{x1D6FE}_{l}^{2}$ spectra of figure 7 are re-plotted in figure 9(a), with a wall-scaling $\unicode[STIX]{x1D706}_{x}/z$ , and highlights the region where Baars et al. (Reference Baars, Hutchins and Marusic2017b ) fitted (4.6), resulting in constants $C_{1}=0.302$ and $R=14.01$ . Figure 9(b) shows the spectra used for fitting, alongside (4.6), while figure 9(c) displays the same trend line, superposed on the coherence spectra of the DNS and ASL data (§ 3.2). Since (4.6) describes the $\unicode[STIX]{x1D6FE}_{l}^{2}$ growth for $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3})-O(10^{6})$ , it can be concluded that a wall-attached, self-similar structure is ingrained in $u$ (and that the constants in (4.6) are universal). Note that, however, for the ASL case, these constants depend on the temperature stratification (Krug et al. Reference Krug, Baars, Hutchins and Marusic2019). From a fit to the ${\mathcal{W}}$ data, Baars et al. (Reference Baars, Hutchins and Marusic2017b ) found that the large-scale limit of (4.6) is $\unicode[STIX]{x1D706}_{x}=T_{n}\unicode[STIX]{x1D6FF}$ (with $T_{n}\approx 10$ ), after which $\unicode[STIX]{x1D6FE}_{l}^{2}$ transitions to a constant value for all larger wavelengths. Following (4.5), (4.6) and its region of validity, a data-driven spectral filter is formulated in (4.7) as follows:

(4.7) $$\begin{eqnarray}\displaystyle f_{{\mathcal{W}}}^{p}(z;\unicode[STIX]{x1D706}_{x})=\left\{\begin{array}{@{}ll@{}}\displaystyle 0,\quad & \unicode[STIX]{x1D706}_{x}<Rz,\\ \displaystyle \min \left\{C_{1}\ln \left(\frac{\unicode[STIX]{x1D706}_{x}}{z}\frac{1}{R}\right),1\right\},\quad & Rz\leqslant \unicode[STIX]{x1D706}_{x}\leqslant T_{n}\unicode[STIX]{x1D6FF},\\ \displaystyle \min \left\{C_{1}\ln \left(\frac{T_{n}\unicode[STIX]{x1D6FF}}{z}\frac{1}{R}\right),1\right\},\quad & \unicode[STIX]{x1D706}_{x}>T_{n}\unicode[STIX]{x1D6FF}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

This filter is in essence a model for the linear coherence spectrum and can be evaluated at any value of $Re_{\unicode[STIX]{x1D70F}}$ , due to its confirmed $Re_{\unicode[STIX]{x1D70F}}$ universality.

Subscript ${\mathcal{W}}$ denotes the wall-based reference on which this filter is based, whereas superscript $p$ refers to its piecewise nature. To obtain smooth transitions as in the data, a logarithmic convolution of (4.7) with a log-normal distribution $g(\unicode[STIX]{x1D706}_{x})$ is performed,

(4.8) $$\begin{eqnarray}\displaystyle f_{{\mathcal{W}}}(z;\unicode[STIX]{x1D706}_{x})=f_{{\mathcal{W}}}^{p}(z;\unicode[STIX]{x1D706}_{x})\ast _{l}g(\unicode[STIX]{x1D706}_{x}). & & \displaystyle\end{eqnarray}$$

Here, $g(\unicode[STIX]{x1D706}_{x})$ spans six standard deviations, corresponding to 1.2 decades in $\unicode[STIX]{x1D706}_{x}$ , shown at the bottom right of figure 9(d) with a random amplitude. The displayed filter resembles the result for $z^{+}=841$ . In summary, filter $f_{{\mathcal{W}}}(z;\unicode[STIX]{x1D706}_{x})\in [0,1]$ indicates the wavelength-dependent energy fraction, for a logarithmic region-position $z$ , that is stochastically coherent with the near-wall region. Similarly, $(1-f_{{\mathcal{W}}})$ indicates the stochastically incoherent energy fraction. Coherent scales were characterised to appear at scales larger than $\unicode[STIX]{x1D706}_{x}=Rz$ (constant $R$ refers to a characteristic aspect ratio), with a transition to a constant value at $\unicode[STIX]{x1D706}_{x}=T_{n}\unicode[STIX]{x1D6FF}$ (constant $T_{n}$ is the nominal transition scale). Only at sufficient scale separation, the filter saturates (i.e. $f_{W}=1$ ) at a scale smaller than $\unicode[STIX]{x1D706}_{x}=T_{n}\unicode[STIX]{x1D6FF}$ ; this happens at $\unicode[STIX]{x1D706}_{x}=T\unicode[STIX]{x1D6FF}$ (note that subscript $n$ is now dropped). Filter characteristics are summarized in table 2.

Table 2. Constants of the two data-driven spectral filters. Nominally $z_{T}^{+}=80$ .

4.3 Coherence relative to a logarithmic-region reference

4.3.1 Coherence spectrogram and physical interpretation

Figure 10. Dataset ${\mathcal{L}}_{3}$ . Coherence spectrogram $\unicode[STIX]{x1D6FE}_{l}^{2}(z,z_{{\mathcal{L}}_{3}};\unicode[STIX]{x1D706}_{x})$ , relative to reference $z_{{\mathcal{L}}_{3}}^{+}\approx 1120$ (iso-contours 0.1:0.1:0.9), superposed on its associated premultiplied energy spectrogram $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}$ (filled iso-contours 0.2:0.2:1.8), reproduced from figure 1.

Thus far the coherence trend has been discussed in the context of wall-attached motions. This section is concerned with $z_{R}$ residing in the logarithmic region (a location $z_{{\mathcal{L}}}$ ), from which a data-driven filter emerges that is dubbed the ${\mathcal{L}}$ filter. This filter separates the wall-attached motions into those that are coherent and incoherent with $z_{{\mathcal{L}}}$ .

A graph similar to figure 7 is displayed in figure 10 for the ${\mathcal{L}}_{3}$ data (see figure 23 in § A.2 for all three datasets ${\mathcal{L}}_{1-3}$ ). To discuss the coherence in figure 10 it is beneficial to recall the conceptually reconstructed $\unicode[STIX]{x1D6FE}_{l}^{2}$ contour in figure 8(d). When only the wall-attached turbulence is responsible for any coherence throughout the TBL, the coherence spectrogram can be sub-divided into two components (figure 11). Ideally, for any given $z_{{\mathcal{L}}}$ , only the wall-attached turbulence that extends beyond $z_{{\mathcal{L}}}$ is coherent with $z_{{\mathcal{L}}}$ (its associated sub-component is re-shown in figure 11 b). Smaller wall-attached motions are not observed by $z_{{\mathcal{L}}}$ and do not contribute to the coherence relative to $z_{{\mathcal{L}}}$ (figure 11 c). Figure 10 reflects the trend of figure 11(b). Note that the $\unicode[STIX]{x1D6FE}_{l}^{2}$ contour is plotted with $\unicode[STIX]{x1D706}_{x}\equiv U(z_{{\mathcal{L}}})/f$ for $z<z_{{\mathcal{L}}}$ , since large-scale wall-attached structures (that are thus coherent with $z\geqslant z_{{\mathcal{L}}}$ ) comprise a convection velocity much larger than the mean velocity at locations in close proximity to the wall. For $z\ll z_{{\mathcal{L}}}$ , say $z<z_{{\mathcal{L}}}/8$ , the coherence spectra become non-zero for $\unicode[STIX]{x1D706}_{x}\gtrsim Rz_{{\mathcal{L}}}$ (the vertical line extending down to the abscissa). For $z\gg z_{{\mathcal{L}}}$ , coherence spectra tend towards the same trend as in figure 7. When $z\sim z_{{\mathcal{L}}}$ , coherence spectra reveal that even scales smaller than $\unicode[STIX]{x1D706}_{x}<Rz$ become coherent with $z_{{\mathcal{L}}}$ . This smaller-scale coherence is not associated with stochastically wall-attached motions, as there was no coherence in the $\unicode[STIX]{x1D706}_{x}<Rz$ region with the near-wall reference $z_{{\mathcal{W}}}$ (figure 7). Turbulent scales for which $\unicode[STIX]{x1D6FE}_{l}^{2}(z,z_{{\mathcal{L}}};\unicode[STIX]{x1D706}_{x})>\unicode[STIX]{x1D6FE}_{l}^{2}(z,z_{{\mathcal{W}}};\unicode[STIX]{x1D706}_{x})$ are thus wall-detached scales that are $z_{{\mathcal{L}}}$ -coherent (§ A.2). In § 5.1 it will become clear that for the purpose our work the small-scale coherence at $z\sim z_{{\mathcal{L}}}$ is irrelevant, as we work with the coherence model following figures 8 and 11.

Figure 11. Sub-dividing the (a) coherence spectrogram of wall-attached turbulence (figure 8 d) into a component that is (b) coherent and (c) incoherent with a logarithmic-region reference $z_{{\mathcal{L}}}$ .

4.3.2 Universality of the data-driven filter

Figure 12. (a) Coherence spectra of dataset ${\mathcal{L}}_{3}$ with the wavelength axis predivided by $z_{{\mathcal{L}}_{3}}$ . The thicker portions of the curves correspond to $30\lesssim z^{+}\lesssim z_{R}^{+}/8$ and $20z_{{\mathcal{L}}_{3}}<\unicode[STIX]{x1D706}_{x}<5\unicode[STIX]{x1D6FF}$ . Note that the inclined plane with the solid boundary reflects (4.10) with the constants stated in the text. (b,c) Linear coherence spectra $\unicode[STIX]{x1D6FE}_{l}^{2}(z,z_{{\mathcal{L}}};\unicode[STIX]{x1D706}_{x})$ for wall-normal positions within the range $30\lesssim z^{+}\lesssim z_{{\mathcal{L}}}^{+}/8$ , for (b) datasets ${\mathcal{L}}_{1-3}$ and (c) the DNS and ASL data. (d) Filter following (4.11) for $z_{{\mathcal{L}}}^{+}=1120$ .

Since this work focuses on revealing spectral signatures associated with attached eddies, a filter relative to $z_{{\mathcal{L}}}$ is considered, with large enough $\unicode[STIX]{x0394}z$ so that only the wall-attached turbulence produces coherence. In other words, this spectral filter can decompose the wall-attached portion of the turbulence at $z<z_{{\mathcal{L}}}$ into a $z_{{\mathcal{L}}}$ -coherent and $z_{{\mathcal{L}}}$ -incoherent component. Ideally, following figure 11(b), this filter equals $f_{{\mathcal{W}}}^{p}$ evaluated at $z=z_{{\mathcal{L}}}$ and is invariant with $z$ for $z<z_{{\mathcal{L}}}$ . This can be confirmed using the data. Coherence spectra for $30\lesssim z^{+}\lesssim z_{{\mathcal{L}}}^{+}/8$ and datasets ${\mathcal{L}}_{1-3}$ show excellent collapse in figure 12(b). Precise criteria for the minimum $\unicode[STIX]{x0394}z$ are non-trivial and not the main focus of this paper. Following (4.6) and § 4.2.1, fitting of

(4.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{l}^{2}=C_{1}^{\prime }\ln \left(\frac{\unicode[STIX]{x1D706}_{x}}{z_{{\mathcal{L}}}}\frac{1}{R^{\prime }}\right) & & \displaystyle\end{eqnarray}$$

to the ${\mathcal{L}}_{3}$ data (highlighted portions of the profiles in figure 12 a, described in the caption), yields $C_{1}^{\prime }=0.3831$ and $R^{\prime }=13.18$ . These constants are also seen to match the DNS and ASL data in figure 12(b), suggesting that (4.9) is universal. Two DNS-data based coherence spectra with $z_{{\mathcal{L}}}^{+}\approx 192$ ( $z_{{\mathcal{L}}}/\unicode[STIX]{x1D6FF}\approx 0.10$ ) and two ASL-data based spectra with $z_{{\mathcal{L}}}^{+}\approx 42\,300$ ( $z_{{\mathcal{L}}}/\unicode[STIX]{x1D6FF}\approx 0.03$ ) are plotted, all of which satisfy $30\lesssim z^{+}\lesssim z_{{\mathcal{L}}}^{+}/6$ . Note that, if the data adheres to the hypothesized underpinning in figures 8 and 11, the constants in (4.9) and (4.6) should be equal. The percentage variation of the constants (table 2) is attributed to the inherent simplification in using one convection velocity for all scales (wavelengths) and the – apparently different – contributions to the $z_{{\mathcal{W}}}$ and $z_{{\mathcal{L}}}$ coherence from non-self-similar VLSM (further discussed in § 5.1). From § 5 it will become clear that the present variation of these constants does not have a major impact on the spectral energy decomposition, nor the conclusions of this work. A data-driven spectral filter (valid for $z<z_{{\mathcal{L}}}$ ) is now given as

(4.10) $$\begin{eqnarray}\displaystyle f_{{\mathcal{L}}}^{p}(z_{{\mathcal{L}}};\unicode[STIX]{x1D706}_{x})=\left\{\begin{array}{@{}ll@{}}\displaystyle 0,\quad & \unicode[STIX]{x1D706}_{x}<R^{\prime }z_{{\mathcal{L}}},\\ \displaystyle \min \left\{C_{1}^{\prime }\ln \left(\frac{\unicode[STIX]{x1D706}_{x}}{z_{{\mathcal{L}}}}\frac{1}{R^{\prime }}\right),1\right\},\quad & R^{\prime }z_{{\mathcal{L}}}\leqslant \unicode[STIX]{x1D706}_{x}\leqslant T_{n}\unicode[STIX]{x1D6FF},\\ \displaystyle \min \left\{C_{1}^{\prime }\ln \left(\frac{T_{n}\unicode[STIX]{x1D6FF}}{z_{{\mathcal{L}}}}\frac{1}{R^{\prime }}\right),1\right\},\quad & \unicode[STIX]{x1D706}_{x}>T_{n}\unicode[STIX]{x1D6FF}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Similar to (4.8), a smooth filter is obtained with a logarithmic convolution,

(4.11) $$\begin{eqnarray}\displaystyle f_{{\mathcal{L}}}(z_{{\mathcal{L}}};\unicode[STIX]{x1D706}_{x})=f_{{\mathcal{L}}}^{p}(z_{{\mathcal{L}}};\unicode[STIX]{x1D706}_{x})\ast _{l}g(\unicode[STIX]{x1D706}_{x}), & & \displaystyle\end{eqnarray}$$

here $f_{{\mathcal{L}}}\in [0,1]$ and $f_{{\mathcal{L}}}$ is presented in figure 12(d).

To conclude, two data-driven, universal spectral filters have been presented: filter $f_{{\mathcal{W}}}$ , for extracting the spectral energy in the TBL that is stochastically coherent with the wall (e.g. wall-attached), and $f_{{\mathcal{L}}}$ , which reveals the sub-fraction of the wall-attached energy that is stochastically coherent with a position in the logarithmic region, $z_{{\mathcal{L}}}$ .

5.1 Triple decomposition via data-driven filters

Figure 13. Diagram of the triple decomposition of the energy spectrum at $z^{+}\approx 101$ ( ${\mathcal{W}}$ data). (a,b) The ${\mathcal{L}}$ and ${\mathcal{W}}$ filters overlaid on the energy spectrum. (c,d) Coherent (solid line) and incoherent (dashed line) components of the energy spectrum, via the ${\mathcal{L}}$ and ${\mathcal{W}}$ filters, respectively. (e) Resulting three spectral sub-components.

How the streamwise energy spectra are decomposed with the use of data-driven filters $f_{{\mathcal{W}}}$ and $f_{{\mathcal{L}}}$ is outlined in figure 13. A triple decomposition is performed, so that a single energy spectrum is split into three sub-components, each of which are interpreted in the context of the ${\mathcal{A}}$ , ${\mathcal{B}}$ and ${\mathcal{C}}$ components of Perry et al. (figure 3).

In figure 13(b), $f_{{\mathcal{W}}}$ is overlaid on the energy spectrum. At the high wavenumber-end the filter is bounded by $k_{x}z=2\unicode[STIX]{x03C0}/R\approx 0.45$ . At low wavenumbers, the filter-amplitude reaches 1 at scale $T$ . Figure 13(a) is similar to figure 13(b) but presents $f_{{\mathcal{L}}}$ with $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ . The filter is bounded by $k_{x}z=2\unicode[STIX]{x03C0}/R^{\prime }(z/z_{L})\approx 0.023$ ) and transitions at scale $T^{\prime }$ corresponding to $\unicode[STIX]{x1D706}_{x}=T_{n}\unicode[STIX]{x1D6FF}$ (and plateaus to a value less than one). Figure 13(c,d) show the coherent (solid line) and incoherent (dashed) components of the energy spectrum, per (4.5). Formulations (5.1)–(5.3) define the three spectral sub-components,

(5.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}\equiv \unicode[STIX]{x1D719}_{uu}f_{{\mathcal{L}}}, & \displaystyle\end{eqnarray}$$

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}\equiv \unicode[STIX]{x1D719}_{uu}(1-f_{{\mathcal{W}}}), & \displaystyle\end{eqnarray}$$

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}\equiv \unicode[STIX]{x1D719}_{uu}(f_{{\mathcal{W}}}-f_{{\mathcal{L}}}), & \displaystyle\end{eqnarray}$$

which yield the total spectrum when added: $\unicode[STIX]{x1D719}_{uu}=\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}+\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}+\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ .

A diagram leading to figure 13(e) demonstrates these definitions:

1. (i) At high wavenumbers, $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ is taken as the $f_{{\mathcal{W}}}$ -based incoherent component. This stochastically incoherent energy cannot be estimated using an LSE procedure with a wall-based input. Following the classification of Perry et al., the underlying turbulence of such a component may be thought of as the small-scale type ${\mathcal{C}}$ eddies. However, it can also include any (stochastically) detached (non)-self-similar motions, such as wall-incoherent VLSM.

2. (ii) At low wavenumbers, $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ is taken as the energy that is coherent via $f_{{\mathcal{L}}}$ . This large-scale wall-attached energy is also coherent with $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ . Physically, this wall-attached component may include self-similar structures reaching beyond $z_{{\mathcal{L}}}$ and non-self-similar structures that are coherent with $z_{{\mathcal{L}}}$ (e.g. VLSM).

3. (iii) A remaining component is dubbed $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ . Its energy equals the wall-coherent energy via $f_{{\mathcal{W}}}$ , $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{c}$ , minus its fraction that is coherent with $z_{{\mathcal{L}}}$ , being $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ (or equivalently, all $z_{{\mathcal{L}}}$ -incoherent energy, $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{i}$ , minus the wall-incoherent energy, $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ ). On account of the above, $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ is the wall-coherent energy that resides below $z_{{\mathcal{L}}}$ and can include both self-similar and non-self-similar components.

After performing the aforementioned decomposition for all $z$ we can overlay the three resulting energy spectrograms on the total energy spectrogram in figure 14(a–c). In addition, all spectral components within the range $100\lesssim z^{+}\lesssim 0.15\unicode[STIX]{x1D6FF}^{+}$ are plotted with wall-scaling and outer-scaling in figure 15. For figure 14, sub-components from the triple decomposition are only considered for $z<z_{{\mathcal{L}}}$ . In the near-wall region, here $z^{+}\lesssim z_{T}^{+}$ (with nominally $z_{T}^{+}=80$ ), $f_{{\mathcal{W}}}$ is $z$ -invariant and taken as $f_{{\mathcal{W}}}(z_{T}^{+};\unicode[STIX]{x1D706}_{x})$ . As this work is concerned with the spectral structure in the logarithmic region, the spectral sub-components within the near-wall region are not considered further.

Figure 14. Dataset ${\mathcal{W}}$ with $Re_{\unicode[STIX]{x1D70F}}\approx 14\,100$ . (a–c) Premultiplied energy spectrograms of the three spectral sub-components (for $z<z_{{\mathcal{L}}}$ ), each of them overlaid on the total energy spectrogram (filled iso-contours 0.2:0.2:1.8). Iso-contours corresponding to the three sub-components are shown with a twice-as-fine level spacing, 0.1:0.1:1.8.

Figure 15. Premultiplied energy spectra of figure 14(a–c) within the range $100\lesssim z^{+}\lesssim 0.15\unicode[STIX]{x1D6FF}^{+}$ . Total energy spectra are shown in each sub-figure with light grey. In each of the three rows, spectra corresponding to one specific spectral sub-component are superposed. Outer-scaling and wall-scaling are used in (a,c,e) and (b,d,f), respectively. Note: $k_{x}\equiv 2\unicode[STIX]{x03C0}f/U(z)$ .

Component $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ follows the unfiltered spectra at the smallest energetic scales. It is only at a wall-scale of approximately $\unicode[STIX]{x1D706}_{x}=Rz$ that $f_{{\mathcal{W}}}$ becomes active. The $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ energy roll-off at the large-scale end (small $k_{x}$ in figure 15 f) is generated by the product of the total spectrum and the ramp of $(1-f_{{\mathcal{W}}})$ within $Rz\lesssim \unicode[STIX]{x1D706}_{x}\lesssim T_{n}\unicode[STIX]{x1D6FF}$ . Since the filter ramp is universal in wall-scaling, and the measured spectra do not plateau in that region, the $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ spectra do not obey an unambiguous wall-scaling (figure 15 f). Maxima reside at $\unicode[STIX]{x1D706}_{x}\approx Rz$ , simply because the roll-off induced by the filter is steep enough to suppress any spectral increase at the large-scale end. Component $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ bounds $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ and its ramp-up at the small-scale end shows a reasonable collapse in wall-scaling (figure 15 d). Amplitudes of its peak decay rapidly with increasing $z$ position, due to $f_{{\mathcal{L}}}$ taking effect at an outer-scaling of $\unicode[STIX]{x1D706}_{x}\approx R^{\prime }z_{{\mathcal{L}}}$ (and due to the limited Reynolds number). An outer-scaling roll-off at the large-scale end of the spectrum is governed by the universal $f_{{\mathcal{L}}}$ ramp, multiplied by $\unicode[STIX]{x1D719}_{uu}$ . Because the measured spectra, $\unicode[STIX]{x1D719}_{uu}$ , do not obey perfect outer-scaling at the large wavelengths (with $z$ variation), the $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ spectra do not either. Finally, component $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ does comprise a certain degree of outer-scaling collapse at its small-scale end where $f_{{\mathcal{L}}}$ dictates its decay (figure 15 a). Before proceeding with an in-depth discussion on the spectral components, it is instructive to first concentrate on the limitations of the triple decomposition and the restricted scale separation at finite  $Re_{\unicode[STIX]{x1D70F}}$ .

5.2 A persistent difficulty in decomposing coherent scales

In the context of the spectral view of Perry et al. (figure 3), a component that is consistent with Townsend’s attached-eddies is bound by boundaries $W_{A}$ (satisfying wall-scaling) and $O_{A}$ (outer-scaling). Boundary $W_{A}$ follows from the coherence-based triple decomposition, as $f_{{\mathcal{W}}}$ describes the roll-off of the wall-attached motions (§ 4.2.1) at the small-scale end of the spectrum. In terms of energy, the boundary (at least at this one particular $Re_{\unicode[STIX]{x1D70F}}$ considered up to this point) is ill-defined, since the spectra do not collapse in the wavelength range of boundary $W_{A}$ (e.g. the roll-off of $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ spectra at small $k_{x}z$ in figure 15 f). One reason for this is the still limited scale separation at this Reynolds number: the broad outer-spectral peak resides in the same wavelength range as the roll-off of the $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ spectra. In §§ 5.3 and 6 it is described how a sufficient scale separation at higher $Re_{\unicode[STIX]{x1D70F}}$ can result in a scaling trend that follows figure 3 exactly.

Determining an $O_{A}$ boundary via the current data-driven, coherence-based decomposition technique remains difficult. Location $z_{{\mathcal{L}}}$ is subject to choice, and irrespective of $z_{{\mathcal{L}}}$ , the wall-attached (coherent) energy does supposedly include both self-similar and non-self-similar scales at all of its energetic wavelengths, meaning that a distinction between (i) self-similar wall-attached scales (envisioned as the ${\mathcal{A}}$ energy), and (ii) non-self-similar wall-attached scales (categorized in ${\mathcal{B}}$ ) is impossible via the current technique. While the conceptual reconstruction of the wall-coherence in figure 8 – with the adhering contours from the data – does imply a self-similar range of scales , the absolute energy in the wall-coherent region ( $\unicode[STIX]{x1D706}_{x}>Rz$ ) does include non-self-similar contributions, simply because the outer-spectral peak in the boundary layer spectrogram resides in that region and does not strictly adhere to simultaneous wall-scaling and outer-scaling. Choosing a small $z_{{\mathcal{L}}}$ does likely result in $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ containing lesser non-self-similar energy. But, an inherent drawback of a small $z_{{\mathcal{L}}}$ , is the limited range of self-similar wall-attached scales in $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ (namely scales reaching up to $z<z_{{\mathcal{L}}}$ ). Figure 16(a,b) visualizes how a different $z_{{\mathcal{L}}}$ re-distributes the energy between the $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ and $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ contributions (component $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ is unaffected). When moving up in $z_{{\mathcal{L}}}$ , more wall-attached energy is assigned to $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ , per the expense of what ends up in $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ . Since $f_{{\mathcal{L}}}$ only fully matures when $z_{{\mathcal{L}}}<T_{n}\unicode[STIX]{x1D6FF}/\exp (1/C_{1}^{\prime })/R^{\prime }\approx 0.056\unicode[STIX]{x1D6FF}$ ( $f_{{\mathcal{L}}}$ plateaus at 1 in figure 12 d), $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ will certainly contain the non-self-similar contributions from VLSM-type structures for $z>0.056\unicode[STIX]{x1D6FF}$ .

5.3 When can we possibly observe a $k_{x}^{-1}$ scaling region?

We consider the Reynolds-number dependence of the data-driven filters, in order to appraise the required scale separation for possibly observing a $k_{x}^{-1}$ region. Filters $f_{{\mathcal{W}}}$ and $f_{{\mathcal{L}}}$ are shown in figure 17(a,b) for $z^{+}=100$ at three different $Re_{\unicode[STIX]{x1D70F}}$ values, closely resembling the DNS and ASL data (§ 3.2), as well as the two-point laboratory data (§ 3.1). All filter curves start at $\unicode[STIX]{x1D706}_{x}=100\unicode[STIX]{x1D6FF}$ . Following the triple decomposition of § 5.1, the shaded area at the small wavenumber-end, cornered by $f_{{\mathcal{L}}}$ (here $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ ), represents the fraction of $k_{x}\unicode[STIX]{x1D719}_{uu}$ that forms $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ , while the shaded area bounded by $f_{{\mathcal{W}}}$ at the high wavenumber-end reflects $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ . The unshaded area in-between the filter curves equals the fraction of spectral energy that forms $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ , drawn with the black curves. Figure 17(d) presents figure 17(a–c) in the $Re_{\unicode[STIX]{x1D70F}}$ -continuum as three planes in $(f,Re_{\unicode[STIX]{x1D70F}},k_{x}z)$ -space, and has indicated the footprint of the filters in the $(Re_{\unicode[STIX]{x1D70F}},k_{x}z)$ -plane.

Figure 16. Dataset ${\mathcal{W}}$ with $Re_{\unicode[STIX]{x1D70F}}\approx 14\,100$ . (a,b) Similar to figure 14(a,b) but only with the 0.2 iso-contour shown for the premultiplied energy spectrograms of the spectral sub-components $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ and $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ (for $z<z_{{\mathcal{L}}}$ ). Different iso-contours correspond to a range of $z_{{\mathcal{L}}}$ ; their locations $z=z_{{\mathcal{L}}}$ are shown with the horizontal lines on the right-hand side. Note: $\unicode[STIX]{x1D706}_{x}\equiv U(z)/f$ .

Figure 17. (a–c) Filters $f_{{\mathcal{W}}}$ and $f_{{\mathcal{L}}}$ for $z^{+}=100$ and $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ at three different $Re_{\unicode[STIX]{x1D70F}}$ . The low-wavenumber boundaries of the curves on each plot correspond to $\unicode[STIX]{x1D706}_{x}=100\unicode[STIX]{x1D6FF}$ . (d) Variation of the filters with $Re_{\unicode[STIX]{x1D70F}}$ , including the filters’ onsets ( $R$ and $R^{\prime }$ ) and transitions ( $T$  and $T^{\prime }$ ).

Both filters transition to a constant value (a plateau) at wavelengths larger than $\unicode[STIX]{x1D706}_{x}=T_{n}\unicode[STIX]{x1D6FF}$ , unless their amplitude reaches 1 at smaller scales. With increasing $Re_{\unicode[STIX]{x1D70F}}$ , the plateau amplitude of $f_{{\mathcal{L}}}$ grows as $f_{{\mathcal{L}}}$ starts at a fixed inner-scaling (boundary $R$ ). At $Re_{\unicode[STIX]{x1D70F}}=(z^{+}/T_{n})R\exp (1/C_{1})\approx 3850$ the filter plateau reaches unity, corresponding to the point in figure 17(d) where the $T^{\prime }$ and $T$ boundaries separate. Boundary $T$ now becomes fixed in inner-scaling, while $T^{\prime }$ remains fixed in outer-scaling, because the onset of $f_{{\mathcal{L}}}$ is also outer-scaled, e.g. $\unicode[STIX]{x1D706}_{x}=R^{\prime }z_{{\mathcal{L}}}=0.15R^{\prime }\unicode[STIX]{x1D6FF}$ . At the low $Re_{\unicode[STIX]{x1D70F}}$ regime, $R$ and $R^{\prime }$ intersect at $Re_{\unicode[STIX]{x1D70F}}^{a}\equiv z^{+}/(z_{{\mathcal{L}}}/\unicode[STIX]{x1D6FF})\cdot (R/R^{\prime })\approx 708$ , meaning that $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ becomes non-existent (e.g. there is no wall-coherent energy at $z^{+}=100$ that is incoherent with $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ , simply because these positions merge: $z_{{\mathcal{L}}}^{+}=0.15Re_{\unicode[STIX]{x1D70F}}^{a}\approx 100$ . At higher $Re_{\unicode[STIX]{x1D70F}}$ , specifically at $Re_{\unicode[STIX]{x1D70F}}^{b}\equiv z^{+}/(z_{{\mathcal{L}}}/\unicode[STIX]{x1D6FF})\exp (1/C_{1})R/R^{\prime }\approx 19\,500$ , $f_{{\mathcal{L}}}$ only becomes active in a scale range where $f_{{\mathcal{W}}}$ has fully saturated. Though, because of the smooth filter transitions this does not happen until $Re_{\unicode[STIX]{x1D70F}}^{c}\approx 80\,000$ . And so, for $Re_{\unicode[STIX]{x1D70F}}>Re_{\unicode[STIX]{x1D70F}}^{c}$ (at $z^{+}=100$ and with $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ ), a range of scales starts to appear where all energy of $k_{x}\unicode[STIX]{x1D719}_{uu}$ would be assigned to $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ (only at $Re_{\unicode[STIX]{x1D70F}}\approx 10^{6}$ this region spans one decade of scales). Moving from coherence to energy spectra, ideally, the scales defining this region do not comprise any wall-incoherent energy, nor do they comprise energy of structures that are coherent with $z>z_{{\mathcal{L}}}$ . Thus, if the energy in this region is predominantly induced by self-similar attached eddies, this is the region where a $k_{x}^{-1}$ could be present. The current $Re_{\unicode[STIX]{x1D70F}}$ estimate at which $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ may be observed agrees reasonably well with the study by Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017), where they predicted that an appreciable $k^{-1}$ scaling region can only appear for $Re_{\unicode[STIX]{x1D70F}}\gtrsim 60\,000$ (from examination of two-dimensional, streamwise-spanwise $u$ spectra). Note that the aforementioned analysis holds for $z^{+}=100$ and that a $k_{x}^{-1}$ region is expected to shrink linearly with $z$ , meaning that $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{6})$ is required for a $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ at $z^{+}=1000$ .

6.1 Spectrograms, spectra and scaling behaviours

Figure 18. Each column (per $Re_{\unicode[STIX]{x1D70F}}$ quoted on top) is similar to figure 14: premultiplied energy spectrograms of the three spectral sub-components (for $z<z_{{\mathcal{L}}}$ , iso-contours 0.2:0.2:1.8), each of them superposed on the total energy spectrogram (filled iso-contours 0.2:0.2:1.8). Filter $f_{{\mathcal{L}}}$ , with $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ , is applied for $z\lesssim z_{{\mathcal{L}}}$ ; $f_{{\mathcal{W}}}$ is applied for $z_{T}^{+}\lesssim z^{+}\lesssim z_{{\mathcal{L}}}^{+}$ , while $f_{{\mathcal{W}}}$ is $z$ -invariant for $z^{+}\lesssim z_{T}^{+}$ and equal to $f_{{\mathcal{W}}}(z_{T}^{+};\unicode[STIX]{x1D706}_{x})$ . Data are from H09, Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) and S18, Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), see § 3.2. Note: $\unicode[STIX]{x1D706}_{x}\equiv U(z)/f$ .

Reynolds-number trends of the triple-decomposed spectrograms are presented in figure 18 from singe-point data at four Reynolds numbers, spanning $Re_{\unicode[STIX]{x1D70F}}\approx 2800$ to $13\,000$ . Similarly as in figure 14, $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ and the triangles indicate wall-scaling $\unicode[STIX]{x1D706}_{x}=Rz$ and outer-scaling $\unicode[STIX]{x1D706}_{x}=T_{n}\unicode[STIX]{x1D6FF}$ . Spectra at $z^{+}=100$ (interpolated from the spectrograms) are detailed in figure 19 for all five single-point hot-wire datasets (§ 3.2). Additionally, a spatial spectrum from the DNS data is superposed. Components $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ and $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ are now discussed, after which we focus on $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ in § 6.2.

Figure 19. Premultiplied energy spectra at $z^{+}=100$ for six values of $Re_{\unicode[STIX]{x1D70F}}$ . Total energy spectra are shown in each sub-figure with light grey. In each of the three rows, spectra corresponding to one specific spectral sub-component are superposed. Outer-scaling and wall-scaling are used in (a,c,e) and (b,d,f), respectively. Data are from S13, Sillero et al. (Reference Sillero, Jiménez and Moser2013), H09, Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) and S18, Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), see § 3.2.

Large-scale contribution $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ is approximately Reynolds-number independent at $z^{+}=100$ (figure 19 a). A minor decrease of energy within the largest wavelengths, with increasing $Re_{\unicode[STIX]{x1D70F}}$ , is an artefact of using a single convection velocity to generate $k_{x}\equiv 2\unicode[STIX]{x03C0}f/U_{c}$ , here $U_{c}=U(z^{+}=100)$ . Since the $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ component is associated with strong coherence in $z$ , from the wall up to at least $z_{{\mathcal{L}}}$ , these structures make up global modes (Bullock, Cooper & Abernathy Reference Bullock, Cooper and Abernathy1978; del Álamo & Jiménez Reference del Álamo and Jiménez2003) that are associated with an outer-scaled convection velocity (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2004). Hence, when the local mean velocity at $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ is employed for each Reynolds-number dataset to construct the scale-axis in figure 19(a), the large-scale end of the spectra collapse (see § A.3 for further details, as well as Samie (Reference Samie2017)). A wall-normal trend of $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ , together with its $Re_{\unicode[STIX]{x1D70F}}$ dependence, is apparent from figure 18(a,d,g,j). The intensity of this spectral component grows with $Re_{\unicode[STIX]{x1D70F}}$ as seen from the 0.2 contour spanning a wider range of scales from left-to-right. With the current $z_{{\mathcal{L}}}$ location, the peak of $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ (filled blue circles) resides at $\unicode[STIX]{x1D706}_{x}\approx 10\unicode[STIX]{x1D6FF}$ and is situated above the geometric centre between a fixed inner- and outer-scaling position, e.g.  $z_{gc}^{+}\equiv \sqrt{z_{T}^{+}\,0.15Re_{\unicode[STIX]{x1D70F}}}=3.46\sqrt{Re_{\unicode[STIX]{x1D70F}}}$ (with $z_{T}^{+}=80$ ), indicated with the blue open circles. Although the emergence and scaling of the broad outer-spectral peak has been documented in the literature (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007a ; Mathis et al. Reference Mathis, Hutchins and Marusic2009; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013; Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015a ), the analysis has always been approached from a total-energy perspective (observations from the measured spectra/spectrograms, $\unicode[STIX]{x1D719}_{uu}$ ). Clearly, when decomposing the spectrograms into physically relevant components, the characteristic scale and wall-normal location of a type- ${\mathcal{B}}$ contribution, as in figure 3, may change. That is, the peak in $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ does not match the peak in the spectrogram, because $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ (and also $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ at these still limited $Re_{\unicode[STIX]{x1D70F}}$ ) constitute an energy roll-off below the outer-spectral peak in $\unicode[STIX]{x1D719}_{uu}$ . Future work may benefit from reappraising scaling trends of the peak-features in the spectrogram in terms of its sub-components. Also, whether $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ will saturate to a constant state at ultra-high $Re_{\unicode[STIX]{x1D70F}}$ remains an open question. Clearly, $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ depends on $Re_{\unicode[STIX]{x1D70F}}$ and $z$ .

Figure 19(f) evidences the Reynolds-number independence of $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ . At the small-scale end, all spectra are in close agreement (expected per the law of the wall; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018; Ganapathisubramani Reference Ganapathisubramani2018). In particular, the two highest $Re_{\unicode[STIX]{x1D70F}}$ spectra ( $Re_{\unicode[STIX]{x1D70F}}\approx 13\,000$ and $19\,300$ , S18: Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018)) are in excellent agreement with the DNS spectrum ( $Re_{\unicode[STIX]{x1D70F}}\approx 2000$ , S13: Sillero et al. (Reference Sillero, Jiménez and Moser2013)), since those data correspond to fully resolved measurements (§ 3.2 and detailed in Samie (Reference Samie2017)). Because all spectra collapse for $k_{x}z\gtrsim 10^{-1}$ , the roll-offs of the experimental spectra at the low wavenumber-end also collapse because of the Reynolds-number invariant $f_{W}$ . There, the DNS spectrum has a lower amplitude than the hot-wire-based spectra. This may partially be caused by Taylor’s approximation re-distributing the VLSM spectral content in temporal spectra (Perry & Li Reference Perry and Li1990; del Álamo & Jiménez Reference del Álamo and Jiménez2009), but, the DNS spectrum is likely to be inaccurate, since it is based on spatially developing data (§ 3.2). In any case, the exact spectral distribution of VLSM energy should be viewed with caution, as it may be more broadband in spatial spectra – see figure 17 in Perry & Li (Reference Perry and Li1990) and figure 10 in del Álamo & Jiménez (Reference del Álamo and Jiménez2009).

6.2 Reynolds-number trend of the $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ spectrum

Figure 20. Peak value of the premultiplied spectral energy component $k_{x}^{+}\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}^{+}$ at $z^{+}=100$ as function of $Re_{\unicode[STIX]{x1D70F}}$ , corresponding to the peak values of the spectra in figure 19(c,d). Higher Reynolds-number data from the Princeton HRTF are also included (Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015a ).

With increasing $Re_{\unicode[STIX]{x1D70F}}$ , the $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ component grows, while the relative energy contribution of the other two spectral components, to the total energy, decays (figure 19). At $z^{+}=100$ , and at the highest $Re_{\unicode[STIX]{x1D70F}}$ in figure 19(d), the spectral roll-offs of $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ at the large- and small-wavelength ends still overlap at $\unicode[STIX]{x1D706}_{x}=0.15R^{\prime }\unicode[STIX]{x1D6FF}$ (recall the discussion of figure 17), meaning that $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ has not yet matured to a $k_{x}^{-1}$ region (if at all present). Our results are consistent with those of Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017), who demonstrated that a $k_{x}^{-1}$ is not expected for $Re_{\unicode[STIX]{x1D70F}}\lesssim 80\,000$ . Nevertheless, in order to quantify the growth of the $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ spectra in figure 19(d), their peak values are plotted in figure 20 as a function of $Re_{\unicode[STIX]{x1D70F}}$ . Seven peak values are plotted from the data of the current study, up to $Re_{\unicode[STIX]{x1D70F}}\approx 19\,300$ . For inspecting how the growth continues beyond $Re_{\unicode[STIX]{x1D70F}}\approx 19\,300$ , three data points are included from the Princeton high-Reynolds-number test facility (HRTF). Streamwise velocity spectra in a nominally ZPG TBL were presented by Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015a ) and from low-to-high $Re_{\unicode[STIX]{x1D70F}}$ , the three spectra were taken from data at $z^{+}=90$ , 105 and 98 (close to $z^{+}=100$ for a direct comparison to the other data). The raw spectra of Vallikivi et al. (Reference Vallikivi, Ganapathisubramani and Smits2015a ) were filtered via the same filtering procedure as used in generating the $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ spectra in figure 19(d), meaning that the raw spectra were multiplied by $(f_{{\mathcal{W}}}-f_{{\mathcal{L}}})$ following formulation (5.3). Subsequently, its peak value was determined and plotted in figure 20. The same filtering procedures ensure that, given the assumptions made in this work, the large-scale contribution $\unicode[STIX]{x1D719}_{{\mathcal{L}}}$ and small-scale portion $\unicode[STIX]{x1D719}_{{\mathcal{W}}}$ are excluded so that we reveal the scaling trend of the spectral component that obeys inner- and outer-scalings. From all data points in figure 20 it remains inconclusive if the peak in $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ plateaus to a constant value at ultra-high $Re_{\unicode[STIX]{x1D70F}}$ . Our current standing with the limited data – e.g. subject to the challenges associated with the acquisition of repeatable statistics with miniature hot-wire probes (e.g. Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) – highlights the ongoing need for additional high-fidelity spectral data spanning several orders of magnitude in $Re_{\unicode[STIX]{x1D70F}}$ . Only those kinds of data can provide definitive answers on whether a distinguished $k_{x}^{-1}$ region develops (in which case the points in figure 20 would plateau to a constant). Finally, it is worth noting that figure 20 presents the best case scenario that we have at our disposal with the available data, namely, the case of $z^{+}=100$ . If a $k_{x}^{-1}$ region does exist for the turbulence residing at the inertial scales, it would first be visible at a location close to the wall, as that location comprises the largest range of energetic spectral scales between a $z$ -scaling and outer-scaling. Here we presume that viscous effects only affect the inertial scales at a wall-normal range below a location fixed in viscous-scaling (§ 5.3), which seems reasonable given that the inner-peak is confined to approximately $z^{+}<80$ (e.g. from observation in figure 18 c,f,i,l). Following the criterion from above that $Re_{\unicode[STIX]{x1D70F}}\gtrsim 80\,000$ for examining a $k_{x}^{-1}$ at $z^{+}=100$ , in combination with the inner- and outer-scalings, it simply follows that $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{6})$ is required for potentially observing a $k_{x}^{-1}$ at $z^{+}=1000$ .