1.1 Townsend–Perry constant $A_{1}$ in the context of the turbulence intensity and spectra

Townsend (Reference Townsend1976) hypothesized that the energy-containing motions in TBLs are comprised of a hierarchy of geometrically self-similar eddying motions, that are inertially dominated (inviscid), attached to the wall and scalable with their distance from the wall (Marusic & Monty Reference Marusic and Monty2019). According to the classical model of attached eddies (Perry & Chong Reference Perry and Chong1982), the wall-normal extents of the smallest attached eddies scale with inner variables, e.g. $100\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}}$ , while the largest scale with $\unicode[STIX]{x1D6FF}$ . Consequently, $Re_{\unicode[STIX]{x1D70F}}$ is a direct measure of the attached-eddy range of scales. Following the attached-eddy modelling framework, the streamwise TI within the logarithmic region adheres to

(1.1) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u^{2}}^{+}=B_{1}-A_{1}\ln \left(\frac{z}{\unicode[STIX]{x1D6FF}}\right), & \displaystyle\end{eqnarray}$$

where $A_{1}$ and $B_{1}$ are constants; $A_{1}$ was dubbed the Townsend–Perry constant. A scaling of $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ (or a plateau in the premultiplied spectrum $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}$ ) is consistent with the presence of a sufficient range of attached-eddy scales. Such a spectral scaling for the energy-containing, inertial range of anisotropic scales can be predicted with the aid of dimensional analysis, a spectral overlap argument and an assumed type of eddy similarity (e.g. Perry & Abell Reference Perry and Abell1975; Davidson & Krogstad Reference Davidson and Krogstad2009). Perry, Henbest & Chong (Reference Perry, Henbest and Chong1986) related the plateau magnitude of the premultiplied spectrum back to (1.1), resulting in

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

An underlying assumption of (1.1), in combination with (1.2), is that all energy is induced by self-similar attached-eddy motions. And so, from detailed studies on the streamwise turbulence kinetic energy, from which profiles of $\overline{u^{2}}(z)$ and streamwise spectra $\unicode[STIX]{x1D719}_{uu}$ are available, the Townsend–Perry constant $A_{1}$ inferred via either (1.1) or (1.2) must be equal, of course provided that attached-eddy turbulence dictates the scaling.

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 ). Triangle ‘N’ refers to the $k_{x}^{-1}$ region identified by Nickels et al. (Reference Nickels, Marusic, Hafez and Chong2005). (b) Turbulence intensity profile and (1.1) with $A_{1}=1.26$ and $B_{1}=2.30$ .

Thus far, evidence for (1.2) has been inconclusive, mainly due to the limited spectral range over which this region may exist. It is instructive to present an energy 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 $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}$ in figure 1(a). 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 ). Near-wall streaks (Kline et al. Reference Kline, Reynolds, Schraub and Rundstadler1967) dominate the inner-spectral peak in the TBL spectrogram (identified with the $\times$ marker at $\unicode[STIX]{x1D706}_{x}^{+}=10^{3}$ and $z^{+}=15$ ), while large-scale organized motions induce a broad spectral peak in the log region, indicated 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). Nickels et al. (Reference Nickels, Marusic, Hafez and Chong2005) determined a $k_{x}^{-1}$ region as $z^{+}>100$ , $\unicode[STIX]{x1D706}_{x}>15.7z$ (wall-scaling) and $\unicode[STIX]{x1D706}_{x}<0.3\unicode[STIX]{x1D6FF}$ (outer-scaling) at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,000$ (triangular region ‘N’ in figure 1 a). This region satisfied (1.2) with $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}=A_{1}\approx 0.92$ . In Part 1 it was determined from a coherence analysis (Baars, Hutchins & Marusic Reference Baars, Hutchins and Marusic2017b ), relative to a wall-based reference, that wall-attached self-similar motions only become spectrally energetic at $\unicode[STIX]{x1D706}_{x}\gtrsim 14z$ . It is important to note here that this result was derived from a stochastic analysis of the streamwise velocity component only. It was furthermore suggested that $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ is unlikely for $Re_{\unicode[STIX]{x1D70F}}\lesssim 80\,000$ . This is consistent with the study of Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017), where experimentally acquired streamwise–spanwise two-dimensional spectra of $u$ at $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{4})$ were examined for a $k_{x}^{-1}$ . They concluded that an appreciable $k^{-1}$ scaling region can only appear for $Re_{\unicode[STIX]{x1D70F}}\gtrsim 60\,000$ . Moreover, even for the highest $Re_{\unicode[STIX]{x1D70F}}$ laboratory data, the presence of a $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ 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 Smits2015), while this region must grow with $Re_{\unicode[STIX]{x1D70F}}$ .

We now switch our attention to evidence for (1.1). A caveat in determining $A_{1}$ from $\overline{u^{2}}(z)$ profiles is that, generally, all turbulent scales are lumped together as one (integral of the entire spectrum). This approach inherently assumes that the attached-eddy structures dominate $\overline{u^{2}}$ . Now, if one accepts that the attached-eddy contribution to the overall turbulence intensity grows with $Re_{\unicode[STIX]{x1D70F}}$ , this assumption should become more valid. For this reason, Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) considered high $Re_{\unicode[STIX]{x1D70F}}$ data in the range $2\times 10^{4}<Re_{\unicode[STIX]{x1D70F}}<6\times 10^{5}$ (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Hutchins et al. Reference Hutchins, Chauhan, Marusic and Klewicki2012; Winkel et al. Reference Winkel, Cutbirth, Ceccio, Perlin and Dowling2012; Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015) and inferred that $A_{1}=1.26$ (see figure 1 b). It is worth noting that the value for $A_{1}$ has changed significantly over time. For instance, values for $A_{1}$ have been quoted as 1.03 (Perry & Li Reference Perry and Li1990), 1.26 (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Örlü et al. Reference Örlü, Fiorini, Segalini, Bellani, Talamelli and Alfredsson2017) and 1.65 (Yamamoto & Tsuji Reference Yamamoto and Tsuji2018). These variations in $A_{1}$ are largely due to the varying TI slope with $Re_{\unicode[STIX]{x1D70F}}$ and the different fitting regions for (1.1).

The previous discussion illustrates that $A_{1}$ values found from $\overline{u^{2}}$ profiles vary, while the AEH envisions a constant $A_{1}$ in (1.1): one that is invariant with $Re_{\unicode[STIX]{x1D70F}}$ . Moreover, $A_{1}$ values found from $\overline{u^{2}}$ profiles do not agree with values for $A_{1}$ inferred from spectra via (1.2), despite the fact that this is expected per the attached-eddy model (Perry et al. Reference Perry, Henbest and Chong1986). A central facet of this mismatch is the simple fact that (1.1) and (1.2) are restricted to attached-eddy turbulence only, while in measures of the total streamwise turbulence kinetic energy more contributions are present. For a quantitative insight into what portion of the turbulence kinetic energy is representative of attached-eddy turbulence, a spectral decomposition method was introduced in Part 1 (and is summarized next).

Figure 2.

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). Following figure 14 of Part 1.

1.2 Streamwise energy spectra and the triple decomposition

Data-driven spectral filters were empirically found with the aid of two-point measurements and a spectral coherence analysis. A first filter, denoted as $f_{{\mathcal{W}}}$ , was based on a reference position deep within the near-wall region (or at the wall). Such a reference position allows us to determine the degree of coherence between the $u$ fluctuations within the TBL and the fluctuations that are present at the reference position. The other filter, $f_{{\mathcal{L}}}$ , was based on a reference position in the logarithmic region. It was verified that both spectral filters were universal for $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3})-O(10^{6})$ . Filter $f_{{\mathcal{W}}}$ was formulated as

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

Subscript ${\mathcal{W}}$ signifies the wall-based reference, on which this filter is based, and the three constants are: $C_{1}=0.3017$ , $R=14.01$ and $T_{n}=10$ (table 2, Part 1). A smooth filter $f_{{\mathcal{W}}}(z;\unicode[STIX]{x1D706}_{x})$ was generated by convoluting (1.3) with a log-normal distribution, $g(\unicode[STIX]{x1D706}_{x})$ , spanning six standard deviations, corresponding to 1.2 decades in $\unicode[STIX]{x1D706}_{x}$ (details are provided in Part 1). Filter $f_{{\mathcal{W}}}(z;\unicode[STIX]{x1D706}_{x})\in [0,1]$ and equals a wavelength-dependent fraction of energy that is stochastically coherent with the near-wall region. Consequently, $(1-f_{{\mathcal{W}}})$ is the incoherent energy fraction. Filter $f_{{\mathcal{L}}}$ employs a reference position $z_{{\mathcal{L}}}$ in the logarithmic region,

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

Filter constants are $C_{1}^{\prime }=0.3831$ , $R^{\prime }=13.18$ and $T_{n}=10$ . A smooth filter $f_{{\mathcal{L}}}(z_{{\mathcal{L}}};\unicode[STIX]{x1D706}_{x})$ was formed in a similar way as $f_{{\mathcal{W}}}(z;\unicode[STIX]{x1D706}_{x})$ . Of the fraction of energy that is stochastically coherent with the near-wall region (via $f_{{\mathcal{W}}}$ ), a sub-fraction of that energy is also coherent with $z_{{\mathcal{L}}}$ in the logarithmic region (and this fraction is prescribed by $f_{{\mathcal{L}}}$ ).

A triple decomposition for $\unicode[STIX]{x1D719}_{uu}$ was formed from $f_{{\mathcal{W}}}$ and $f_{{\mathcal{L}}}$ (§ 5.1, Part 1), following

(1.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}(z;\unicode[STIX]{x1D706}_{x})\equiv \unicode[STIX]{x1D719}_{uu}(z;\unicode[STIX]{x1D706}_{x})f_{{\mathcal{L}}}(z;\unicode[STIX]{x1D706}_{x}), & \displaystyle\end{eqnarray}$$

(1.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}(z;\unicode[STIX]{x1D706}_{x})\equiv \unicode[STIX]{x1D719}_{uu}(z;\unicode[STIX]{x1D706}_{x})[1-f_{{\mathcal{W}}}(z;\unicode[STIX]{x1D706}_{x})], & \displaystyle\end{eqnarray}$$

(1.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}(z;\unicode[STIX]{x1D706}_{x})\equiv \unicode[STIX]{x1D719}_{uu}(z;\unicode[STIX]{x1D706}_{x})[f_{{\mathcal{W}}}(z;\unicode[STIX]{x1D706}_{x})-f_{{\mathcal{L}}}(z;\unicode[STIX]{x1D706}_{x})]. & \displaystyle\end{eqnarray}$$

Consequently, $\unicode[STIX]{x1D719}_{uu}=\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}+\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}+\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ and figure 2 illustrates this decomposition for $Re_{\unicode[STIX]{x1D70F}}\approx 14\,100$ (duplicate of figure 14, Part 1). The three energy spectrograms of (1.5)–(1.7) are overlaid on the premultiplied energy spectrogram $k_{x}^{+}\unicode[STIX]{x1D719}_{uu}^{+}$ . Here, $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ and the triple decomposition is performed for $z<z_{{\mathcal{L}}}$ . In the near-wall region, here taken as $z^{+}\lesssim z_{T}^{+}$ (nominally $z_{T}^{+}=80$ is used, approximately the wall-normal position at which the near-wall spectral peak becomes indistinguishable from the spectrogram), $f_{{\mathcal{W}}}$ is $z$ -invariant and taken as $f_{{\mathcal{W}}}(z_{T}^{+};\unicode[STIX]{x1D706}_{x})$ . Throughout this work, the exact value of $z_{T}^{+}$ is of secondary importance, since small variations in this location do not affect the conclusions, given a lower bound of the logarithmic region in viscous-scaling, $z_{T}=O(100\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}})$ . In § 2.2 of Part 1 we discussed that a classical scaling of the lower limit of the logarithmic region (as opposed to a mesolayer-type scaling via $z^{+}\propto Re_{\unicode[STIX]{x1D70F}}^{0.5}$ ) should not be discarded. In fact, in this paper we show that when we accept such a classical scaling, the growth of the near-wall peak in $\overline{u^{2}}$ can be explained via the increasingly intense energetic imprint of the Reynolds-number-dependent outer motions onto the near-wall region.

Component $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ (figure 2 a) comprises the energy that is coherent via $f_{{\mathcal{L}}}$ : large-scale wall-attached energy that is coherent with $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ . This component includes spectral imprints of self-similar, wall-attached structures reaching beyond $z_{{\mathcal{L}}}$ and non-self-similar wall-attached structures that are coherent with $z_{{\mathcal{L}}}$ (e.g. some very large-scale motions (VLSMs)). Component $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ (figure 2 c) is formed from the $f_{{\mathcal{W}}}$ -based incoherent energy. This small-scale energy is wall detached and includes detached (non)-self-similar motions, such as phase-inconsistent attached eddies, incoherent VLSMs, etc. The remaining component, $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ , equals the wall-coherent energy below $z_{{\mathcal{L}}}$ and consists of self-similar and non-self-similar contributions. However, the non-self-similar contributions are likely to reside at large $\unicode[STIX]{x1D706}_{x}$ (reflecting global modes, Bullock, Cooper & Abernathy Reference Bullock, Cooper and Abernathy1978; del Álamo & Jiménez Reference del Álamo and Jiménez2003).

1.3 Present contribution and outline

Coming back to § 1.1, we can now argue that $A_{1}$ can be inferred from $\overline{u^{2}}(z)$ profiles via (1.1), as long as the streamwise TI contributions, other than that from the self-similar wall-attached motions, are removed. This step is crucial, because Part 1 already addressed that the other contributions (e.g. $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ in figure 2 a and $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ in figure 2 c) result in additions to the streamwise TI that constitute a clear $z$ -dependence. And, the Reynolds-number-dependent outer-spectral peak seems to mask a possible $\unicode[STIX]{x1D719}_{uu}\propto k_{x}^{-1}$ (see spectra in Morrison et al. Reference Morrison, Jiang, McKeon and Smits2002; Nickels et al. Reference Nickels, Marusic, Hafez and Chong2005; Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010b ; Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2017; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). When re-assessing $A_{1}$ in this paper, both the spectral view and $\overline{u^{2}}(z)$ are considered simultaneously, while recognizing that $A_{1}$ must solely be associated with the turbulence that obeys the AEH.

Next, in §§ 2.1–2.2, decompositions of the streamwise TI are presented for a range of $Re_{\unicode[STIX]{x1D70F}}$ . Data used are the same as in Part 1 (Baars & Marusic Reference Baars and Marusic2020, § 3.2). Findings on the Townsend–Perry constant $A_{1}$ are reconciled in § 2.3, after which its relation to the near-wall TI growth, with $Re_{\unicode[STIX]{x1D70F}}$ , is presented in § 3. Empirical trends within the wall-normal profiles for all three additive sub-components of the streamwise TI are presented in § 4, together with a discussion of their scalings.

2.1 Methodology and logarithmic scalings

Figure 3.

(a) Triple-decomposed energy spectrum at $z^{+}\approx 101$ and $Re_{\unicode[STIX]{x1D70F}}\approx 14\,100$ , reproduced from figure 13(e) in Part 1. (b) Streamwise TI profile with the three TI sub-components following (2.2). (c) Similar to (b) but for all wall-normal locations.

Figure 3(a) shows the three sub-components $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ , $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ and $\unicode[STIX]{x1D719}_{{\mathcal{W}}}^{i}$ for the spectrum at $z^{+}\approx 101$ (slice through figure 2). When integrated, these sub-components form three contributions to the streamwise TI, being $\overline{u^{2}}_{{\mathcal{L}}}$ , $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ and $\overline{u^{2}}_{{\mathcal{W}}}$ , respectively. In summary,

(2.1)

(2.2)

Figure 3(b) presents these three sub-components of the TI at $z^{+}\approx 101$ , together with $\overline{u^{2}}(z)$ (open diamonds). Wall-normal profiles of the three sub-components are obtained when this integration is performed for each $z$ (figure 3 c). Note that the contributions are shown in a cumulative format; the bottom profile (squares) represents $\overline{u^{2}}_{{\mathcal{W}}}$ , the intermediate profile (circles) encompasses $\overline{u^{2}}_{{\mathcal{W}}}+\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ , whereas the final profile (diamonds), $\overline{u^{2}}_{{\mathcal{W}}}+\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}+\overline{u^{2}}_{{\mathcal{L}}}$ , equals $\overline{u^{2}}$ by construction. Regarding the full $\overline{u^{2}}(z)$ profile, it is well known that the near-wall streamwise TI is attenuated due to spatial resolution effects of hot-wires (Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009). Here the spanwise width of the hot-wire sensing length was $l^{+}\approx 22$ . A corrected profile for the streamwise TI is superposed in figure 3(b) with filled diamonds, following the method of Smits, McKeon & Marusic (Reference Smits, McKeon and Marusic2011). Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) confirmed that this correction scheme is valid for Reynolds numbers up to $Re_{\unicode[STIX]{x1D70F}}\approx 20\,000$ . Because the TI above the near-wall region (say $z>z_{T}$ ) is unaffected by spatial resolution issues, we proceed our analysis without hot-wire corrections.

The wall-incoherent component, $\overline{u^{2}}_{{\mathcal{W}}}$ , exhibits an increase in its energy magnitude with increasing $z$ throughout the logarithmic region. Section 4 addresses the wall-normal trend of this TI component in more detail. Components $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ and $\overline{u^{2}}_{{\mathcal{L}}}$ have to be considered in relation to one another. Figure 4(a) illustrates the dependence of the two TI sub-components on $z_{{\mathcal{L}}}$ , by presenting $\overline{u^{2}}_{{\mathcal{W}}}$ (squares) and $\overline{u^{2}}_{{\mathcal{W}}}+\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ (lines) for a range of $z_{{\mathcal{L}}}$ (indicated with the vertical lines). Part 1 addressed how their spectral equivalents, $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ and $\unicode[STIX]{x1D719}_{{\mathcal{L}}}^{c}$ , varied with $z_{{\mathcal{L}}}$ and here we describe what implications that has on the streamwise TI. At low $z_{{\mathcal{L}}}$ , the wall-attached motions smaller than $z=z_{{\mathcal{L}}}$ contribute to $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ , but the wall-normal range of $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ is limited (per definition, $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ is non-existent above $z_{{\mathcal{L}}}$ ). With increasing $z_{{\mathcal{L}}}$ , the range of wall-attached motions increases, but global modes (or imprints of non-self-similar VLSMs/superstructures) that are restricted to $z<z_{{\mathcal{L}}}$ also start to contribute significantly to $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ (due to the inherent difficulty in spectrally decomposing the two, see § 5.2 in Part 1). Hence, $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ does not just contain energy from wall-attached self-similar motions. When $z_{{\mathcal{L}}}$ resides in the intermittent region, all global modes are being assigned to $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ (and thus to $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ ). This is reflected by the highest $z_{{\mathcal{L}}}$ profile in figure 4(a); in the process of increasing $z_{{\mathcal{L}}}$ , a hump has appeared in the streamwise TI (approaching $\overline{u^{2}}$ for $z_{{\mathcal{L}}}\rightarrow \unicode[STIX]{x1D6FF}$ ).

Figure 4.

(a) Streamwise TI at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,100$ . Component $\overline{u^{2}}_{{\mathcal{W}}}$ is shown with the square symbols, while the sequence of lines with increasing colour intensity represents $\overline{u^{2}}_{{\mathcal{W}}}+\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ for increasing $z_{{\mathcal{L}}}$ ; locations of $z_{{\mathcal{L}}}$ are indicated with the vertical lines. (b) Profiles of $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}^{+}$ .

We now focus exclusively on $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ as this sub-component is closely aligned with the scaling following (1.1). Figure 4(b) shows $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ for $z^{+}>z_{T}^{+}$ (the near-wall TI is irrelevant in this discussion). Although it was pointed out above that wall-normal profiles of $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ do comprise a signature of wall-attached non-self-similar motions, two trends of its statistics are reflective of wall-attached self-similar motions:

Figure 5.

(a) Value of $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}^{+}$ at $z_{T}^{+}=80$ from figure 4(b); the line, following (2.3), is the fit to the data at $z_{{\mathcal{L}}}<0.15\unicode[STIX]{x1D6FF}$ (its slope $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ is listed). (b) Value of $A_{{\mathcal{W}}{\mathcal{L}}}$ , as indicated in figure 4(b), superposed on the line corresponding to its mean for data at $z_{{\mathcal{L}}}<0.15\unicode[STIX]{x1D6FF}$ (value listed).

Figure 6.

(a) Streamwise TI profiles for $Re_{\unicode[STIX]{x1D70F}}\approx 2800$ , 3900 and 7300 (Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009) and $Re_{\unicode[STIX]{x1D70F}}\approx 13\,000$ and 19 000 (Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). (b–f) Profiles of (a) decomposed into various TI sub-components: $\overline{u^{2}}_{{\mathcal{W}}}$ is shown with the square symbols, while the sequence of lines with increasing colour intensity represents $\overline{u^{2}}_{{\mathcal{W}}}+\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ for increasing $z_{{\mathcal{L}}}$ (similar to figure 4 a).

1. (i) First, the magnitude of $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ at $z_{T}$ is displayed in figure 5(a), with $z_{{\mathcal{L}}}$ forming the abscissa (with a finer $z_{{\mathcal{L}}}$ -discretization than used in figure 4). When assuming that the non-self-similar, large-scale motions have a negligible influence on the TI-trend at $z_{T}$ , and that $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ obeys Townsend’s AEH, we arrive at

   (2.3) $$\begin{eqnarray}\displaystyle \overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}\propto A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }\ln \left(\frac{z_{{\mathcal{L}}}}{\unicode[STIX]{x1D6FF}}\right). & & \displaystyle\end{eqnarray}$$
   That is, an increase of $z_{{\mathcal{L}}}$ mimics an increase in $Re_{\unicode[STIX]{x1D70F}}$ through the inclusion of more wall-attached scales in $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ . Data in figure 5(a) adhere to (2.3) for approximately one decade in $z_{{\mathcal{L}}}$ and fitting of the data at $z_{{\mathcal{L}}}<0.15\unicode[STIX]{x1D6FF}$ results in $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }\approx 0.956$ (note that $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ is often taken as the upper edge of the logarithmic region).

2. (ii) A second measure that is reflective of self-similar wall-attached motions is the decay of $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}(z)$ following (1.1), which is now used to quantify the trend in $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ . It is impossible to perform a direct fit of a logarithmic decay to the data of $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ in figure 4(b), because of the aforementioned issues (for large $z_{{\mathcal{L}}}$ , the profiles are influenced by non-self-similar, global-mode turbulence). Instead, a logarithmic slope $A_{{\mathcal{W}}{\mathcal{L}}}$ is defined from the two endpoints of the profile, $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}(z_{T})$ and $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}=0$ , via

   (2.4) $$\begin{eqnarray}\displaystyle A_{{\mathcal{W}}{\mathcal{L}}}\equiv \frac{\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}(z_{T}^{+})}{\ln \left(\left.z^{+}\right|_{\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}=0}\right)-\ln (z_{T}^{+})}. & & \displaystyle\end{eqnarray}$$
   Figure 4(b) displays the logarithmic slope for one profile (discrete point measurements were interpolated to exactly $z_{T}^{+}=80$ and the $z^{+}$ position at which $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ becomes zero). Data in figure 5(b), and their mean value $A_{{\mathcal{W}}{\mathcal{L}}}\approx 0.973$ , are in close agreement to $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }\approx 0.956$ from figure 5(a). This is expected when $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ obeys an attached-eddy scaling.

2.2 Reynolds-number variation

We now assess how the identified logarithmic scalings via (2.3) and (2.4) depend on the Reynolds number. Single-point hot-wire measurements at a range of Reynolds numbers were employed in § 6 of Part 1 to address the Reynolds-number variation of the triple-decomposed energy spectrograms. These same single-point hot-wire data are here processed via the procedure described previously (§ 2.1). First, the $\overline{u^{2}}(z)$ profiles for these data are shown in figure 6(a). For the three lowest Reynolds numbers ( $Re_{\unicode[STIX]{x1D70F}}\approx 2800$ , 3900 and 7300: Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009)), data were corrected for spatial attenuation effects (Smits et al. Reference Smits, McKeon and Marusic2011), whereas the two other profiles ( $Re_{\unicode[STIX]{x1D70F}}\approx 13\,000$ and 19 300: Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018)) comprise fully resolved measurements. An energy growth in the outer region presents itself through the emergence of a local maximum in $\overline{u^{2}}$ (Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), whereas at the same time, the near-wall TI grows with $Re_{\unicode[STIX]{x1D70F}}$ (Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017).

Data of each Reynolds-number case are spectrally decomposed to generate a similar output as presented in figure 4(a). For each of the five $Re_{\unicode[STIX]{x1D70F}}$ profiles in figure 6(a), the result is shown in figure 6(b–f). Additionally, with the aid of (2.3) and (2.4), figure 5(a,b) is constructed for each of the five Reynolds numbers, as shown in figure 7(a,b).

Figure 7.

(a) Value of $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}^{+}$ at $z_{T}^{+}=80$ from figure 6(b–f); lines, following (2.3), are fit to the data at $z_{{\mathcal{L}}}<0.15\unicode[STIX]{x1D6FF}$ (their slopes $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ are listed). (b) Value of $A_{{\mathcal{W}}{\mathcal{L}}}$ , superposed on the line corresponding to its mean value for data at $z_{{\mathcal{L}}}<0.15\unicode[STIX]{x1D6FF}$ (values listed). Each subsequent $Re_{\unicode[STIX]{x1D70F}}$ case is vertically offset by 2 and 0.6, starting with the second from the bottom, in (a) and (b), respectively.

Especially at the two largest Reynolds numbers ( $Re_{\unicode[STIX]{x1D70F}}\approx 13\,000$ and 19 300), there is a consistent agreement between $A_{{\mathcal{W}}{\mathcal{L}}}$ and $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ , which is indicative of the slopes being a reflection of attached-eddy type turbulence. At the two lowest Reynolds numbers ( $Re_{\unicode[STIX]{x1D70F}}\approx 2800$ and 3900), the $A_{{\mathcal{W}}{\mathcal{L}}}$ slope extracted from the two profile endpoints of $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ exhibits a decreasing trend (top two profiles in figure 7 b). This is ascribed to the fact that the upward trend of $\overline{u^{2}}_{{\mathcal{W}}}$ (square symbols in figures 6 b,c) changes rapidly near the upper edge of the logarithmic region: its magnitude starts to decrease around $z/\unicode[STIX]{x1D6FF}\approx 0.15$ in order to merge with the TI profiles in the wake region. Because of this decrease, there is a less rapid decay of the $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ profiles near $z/\unicode[STIX]{x1D6FF}\approx 0.15$ . When slope $A_{{\mathcal{W}}{\mathcal{L}}}$ is determined from the two profile endpoints, it causes a decreased slope. Generally, the limited scale separation in the triple-decomposed spectrograms at low Reynolds numbers exacerbates this issue (see also the spectrograms in figure 18 of Part 1).

2.3 Reconciling $A_{1}$ from trends in the turbulence intensity and spectra

Having re-assessed the wall-normal decay of the TI sub-component associated with Townsend’s attached eddies (§§ 2.1–2.2), we can now proceed with reconciling the status quo. Recall that (1.1) is restricted to the streamwise TI that is generated by inviscid, geometrically self-similar and wall-attached eddies only. Because both $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ and $A_{{\mathcal{W}}{\mathcal{L}}}$ were inferred by considering the sub-component of the TI that complies with Townsend’s assumptions only, those slopes are interpreted as $A_{1}$ . Figure 8 displays $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ , for all Reynolds numbers, with the open square symbols. Uncertainty estimates are shown with the error bars and are based on 95 % confidence bounds from the fitting procedure of (2.3). Alongside, with the solid square symbols, values of $A_{{\mathcal{W}}{\mathcal{L}}}$ are shown with the uncertainty estimates based on the 95 % confidence interval of the data points residing at $z_{{\mathcal{L}}}<0.15\unicode[STIX]{x1D6FF}$ in figures 5(b) and 7(b). Numerical values are summarized in table 1. To complete quantification of (1.1) by considering $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ energy only, offset $B_{1}$ can be determined. For this we have to introduce a new quantity $\overline{u^{2}}_{AE}$ , which is the TI decay with a pure logarithmic decay,

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u^{2}}_{AE}^{+}(z/\unicode[STIX]{x1D6FF})=B_{1}(z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF})-A_{1}\ln \left(\frac{z}{\unicode[STIX]{x1D6FF}}\right),\quad \text{for }z^{+}\geqslant z_{T}^{+}. & \displaystyle\end{eqnarray}$$

Although offset $B_{1}$ depends on $z_{{\mathcal{L}}}$ (see figure 4 b), we only have to consider the scenario for one specific $z_{{\mathcal{L}}}$ to infer its Reynolds-number trend (here we take $z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF}$ ). Values for $B_{1}(z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF})$ are shown on the bottom of figure 8. Mean values for both $A_{1}$ and $B_{1}$ are found from the mean values of $A_{1}=A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ and $B_{1}$ in table 1, resulting in

(2.6a,b ) $$\begin{eqnarray}\displaystyle A_{1}=0.975,\quad B_{1}(z_{{\mathcal{L}}}=0.15\unicode[STIX]{x1D6FF})=-2.267. & & \displaystyle\end{eqnarray}$$

To indicate the effect of the variation in $A_{1}$ and $B_{1}$ with $Re_{\unicode[STIX]{x1D70F}}$ , six lines according to formulation (2.5), with the six values of $A_{1}=A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ and $B_{1}$ (down to $z_{T}^{+}=80$ ) are shown in the inset in figure 8, together with the TI profiles of figure 6(a). The scatter in $A_{1}$ and $B_{1}$ (as well as the uncertainty estimates from the fitting procedure, listed in table 1), result in indistinguishable logarithmic trends in relation to typical experimental uncertainty in the TI profiles (e.g. Winkel et al. Reference Winkel, Cutbirth, Ceccio, Perlin and Dowling2012; Vincenti et al. Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013; Marusic et al. Reference Marusic, Baars and Hutchins2017; Örlü et al. Reference Örlü, Fiorini, Segalini, Bellani, Talamelli and Alfredsson2017; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). Both $A_{1}$ and $B_{1}$ are thus considered to be Reynolds-number invariant for $2800\lesssim Re_{\unicode[STIX]{x1D70F}}\lesssim 19\,300$ .

Figure 8.

Values of $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ and $A_{{\mathcal{W}}{\mathcal{L}}}$ , alongside the peak values of $k_{x}^{+}\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}^{+}$ (duplicated from figure 20, Part 1). Values of $B_{1}$ are plotted at the bottom. TI profiles of figure 6(a) are shown in the inset, together with (2.5) for all six cases of $A_{1}=A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ and  $B_{1}$ .

Table 1.

Values of $A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ , $A_{{\mathcal{W}}{\mathcal{L}}}$ and $B_{1}$ , which were inferred from the $\overline{u^{2}}_{{\mathcal{W}}{\mathcal{L}}}$ profiles. Uncertainty estimates for $A_{1}=A_{{\mathcal{W}}{\mathcal{L}}}^{\prime }$ and $B_{1}$ are based on 95 % confidence bounds from the fitting of (2.5), while the uncertainty estimates for $A_{{\mathcal{W}}{\mathcal{L}}}$ are based on the 95 % confidence interval of the data points residing at $z_{{\mathcal{L}}}<0.15\unicode[STIX]{x1D6FF}$ in figures 5(b) and 7(b).

A last set of data points in figure 8 comprises the peak values of $k_{x}^{+}\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}^{+}$ at $z^{+}=100$ , duplicated from figure 20 in Part 1. These data consider $A_{1}$ in the context of the energy spectra. Because the scale separation in spectral space is still relatively limited at these Reynolds numbers, the peak value in the associated spectra ( $k_{x}^{+}\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}^{+}$ ) keeps maturing with $Re_{\unicode[STIX]{x1D70F}}$ (detailed in § 6.2, Part 1). At $Re_{\unicode[STIX]{x1D70F}}\approx 19\,300$ there is a consistency between the value for $A_{1}$ found from the TI trend and the peak/plateau in the associated spectrum. However, it is acknowledged that no complete similarity has been observed in the associated $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ spectra (e.g. figure 19, Part 1). Furthermore, the three Princeton HRTF data points of the peak in the $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ spectrum are not conclusive on whether the spectral peak plateaus. The difficult conditions in the Princeton HRTF and the challenging aspects 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) require more research to settle this issue. Only fully resolved, higher-Reynolds-number data at $z^{+}=O(100)$ can provide definitive answers on whether a distinguished $k_{x}^{-1}$ region develops (in which case the peak has to plateau to a Reynolds-number-invariant $A_{1}$ , consistent with the TI trends). In the interim, figure 8 does not exclude that possibility: a rough extrapolation of the $\unicode[STIX]{x1D719}_{{\mathcal{W}}{\mathcal{L}}}$ peak values approaches $A_{1}\approx 1$ , consistent with (2.6). Finally, to the authors’ knowledge, our current work indicates for the first time that a Reynolds-number-invariant $A_{1}$ could be consistent with a potential $k_{x}^{-1}$ at ultra-high $Re_{\unicode[STIX]{x1D70F}}$ . Previously, $A_{1}=1.03$ (Perry & Li Reference Perry and Li1990; Marusic & Kunkel Reference Marusic and Kunkel2003) found from $\overline{u^{2}}(z)$ profiles was in close agreement with the spectral-based value of $A_{1}\approx 0.95$ by Nickels et al. (Reference Nickels, Marusic, Hafez and Chong2005), but this was strictly coincidental. The former was obtained at significantly lower Reynolds numbers (highest $Re_{\unicode[STIX]{x1D70F}}\approx 4400$ ) than the latter ( $Re_{\unicode[STIX]{x1D70F}}\approx 14\,000$ ).