3.1 Measurement of $S_{4}$

Difficulties in measuring small-scale quantities such as $S_{4}$ (often denoted by $F$ in the literature; we will hereafter use $F$ and $S_{4}$ interchangeably) have been discussed in detail by Antonia et al. (Reference Antonia, Satyaprakash and Hussain1982). Therefore, before discussing the results, it is important to comment on the accuracy of the measurements of $F$ (Zhou & Antonia Reference Zhou and Antonia2000; Pearson & Antonia Reference Pearson and Antonia2001; Antonia et al. Reference Antonia, Zhou and Romano2002; Zhou et al. Reference Zhou, Antonia and Chua2005; Tang et al. Reference Tang, Antonia, Djenidi, Abe, Zhou, Danaila and Zhou2015a ) that are used here. There are three major ‘systematic’ error sources when measuring $F$ : the uncertainty associated with the spatial/time resolutions of the hot wire, and the duration of the recorded signal. As an example, for the plane jet data, the spatial resolution of the wire is adequate even at the largest $Re_{\unicode[STIX]{x1D706}}$ ( $=1109$ ; Pearson & Antonia Reference Pearson and Antonia2001) due to the fact that the wire diameter is $1.27~\unicode[STIX]{x03BC}\text{m}$ (the wire length is ${\sim}3.1\unicode[STIX]{x1D702}$ ). In fact, for a single hot-wire measurement, the magnitude of $F$ is mainly affected by the temporal resolution and is practically insensitive to the spatial resolution (Burattini, Lavoie & Antonia (Reference Burattini, Lavoie and Antonia2008) demonstrated this in the context of $S_{3}$ ). Figure 2(a) shows that the magnitude of $F$ increases with $\mathit{f}_{c}/\mathit{f}_{\unicode[STIX]{x1D702}}$ (  $\mathit{f}_{c}$ and $\mathit{f}_{\unicode[STIX]{x1D702}}=U/2\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}$ are the filter cutoff frequency and the Kolmogorov frequency, respectively), approximately reaches a maximum at $\mathit{f}_{c}/\mathit{f}_{\unicode[STIX]{x1D702}}\approx 1.0$ and remains constant at higher frequencies. For most of our data, $\mathit{f}_{c}$ was chosen close to or slightly larger than the Kolmogorov frequency $\mathit{f}_{\unicode[STIX]{x1D702}}$ (i.e. $\mathit{f}_{c}/\mathit{f}_{\unicode[STIX]{x1D702}}\geqslant 1$ ), which is adequate for estimating $F$ (see figure 2 a). The final choice of $f_{c}$ was dictated by the onset of the electronic noise in the anemometer circuit. The record duration is also adequate since both $z^{2}\,\text{p.d.f.}(z)$ and $z^{4}\,\text{p.d.f.}(z)$ go to zero at large $|z|$ (figure 2 b). There remains the possibility of a random error associated mainly with the propagation of errors due mostly to the measurement of $u$ . An estimate was obtained from three single wires in the vorticity probe and shown as error bars in figures 3 and 4 for the data of Antonia et al. (Reference Antonia, Zhou and Romano2002) and Zhou et al. (Reference Zhou, Antonia and Chua2005).

Figure 3.

Distributions of $F$ for various flows at relatively low $Re_{\unicode[STIX]{x1D706}}$ . Centreline of a fully developed channel flow: ▫, estimated from the data of Tang et al. (Reference Tang, Antonia, Djenidi, Abe, Zhou, Danaila and Zhou2015a ). SFPBT: $\times$ , Kerr (Reference Kerr1985). Grid turbulence: ▵, estimated from the data of Zhou & Antonia (Reference Zhou and Antonia2000). Wakes: ○ and ▿ correspond to data in wakes generated by two different initial conditions, namely a solid circular cylinder and a screen strip, estimated from the data of Antonia et al. (Reference Antonia, Zhou and Romano2002). The error bars for wake data are also shown; note that they are very small. To facilitate visual comparison with data in other flows, the ordinate scale used in this figure is the same as that in figures 4, 5, 6, 7 and 8(b).

3.2 Velocity derivative flatness $F$ at relatively small $Re_{\unicode[STIX]{x1D706}}$

Antonia et al. (Reference Antonia, Tang, Djenidi and Danaila2015) showed that, although $S_{3}$ approaches a constant when $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large, the magnitude of $S_{3}$ is flow-dependent when $Re_{\unicode[STIX]{x1D706}}$ is small or moderate. There is no a priori reason why $S_{4}$ should behave differently than $S_{3}$ when $Re_{\unicode[STIX]{x1D706}}$ varies. The similarity between (1.4) and (2.16) suggests that $S_{4}$ should also depend on the type of flow at small to moderate $Re_{\unicode[STIX]{x1D706}}$ . To verify this, we report (figure 3) values of $F$ in various turbulent flows at relatively small $Re_{\unicode[STIX]{x1D706}}$ , e.g. grid turbulence, along the centreline of a fully developed channel flow, and ‘stationary’ forced periodic box turbulence (SFPBT) at relatively low $Re_{\unicode[STIX]{x1D706}}$ . In these flows, local isotropy (LI), which is a key ingredient of K41 and K62, is satisfied closely (see Antonia et al. Reference Antonia, Tang, Djenidi and Danaila2015; Tang et al. Reference Tang, Antonia, Djenidi, Abe, Zhou, Danaila and Zhou2015a ,Reference Tang, Antonia, Djenidi and Zhou b ); along the axis in the far field of a plane jet, LI is also satisfied closely (Antonia et al. Reference Antonia, Djenidi, Danaila and Tang2017). The data show a clear trend: $F$ increases as $Re_{\unicode[STIX]{x1D706}}$ increases for all the flows considered. The magnitude of $F$ is larger along the channel centreline than for either grid turbulence or SFPBT. Note that the distributions of $F$ obtained in two different grid turbulence experiments agree well over the same $Re_{\unicode[STIX]{x1D706}}$ range. Also included in figure 3 are the data in wakes generated by two different initial conditions: a solid circular cylinder and screen strip; they are estimated from the data of Antonia et al. (Reference Antonia, Zhou and Romano2002). Thiesset, Danaila & Antonia (Reference Thiesset, Danaila and Antonia2013) used the same wake data to assess LI through the isotropic relation between second-order structure functions of the lateral and longitudinal velocity increments, viz.

(3.1) $$\begin{eqnarray}\overline{(\unicode[STIX]{x1D6FF}v)^{2}}_{iso}=\left(1+\frac{r}{2}\frac{\text{d}}{\text{d}r}\right)\overline{(\unicode[STIX]{x1D6FF}u)^{2}}.\end{eqnarray}$$

They showed that (3.1) is satisfied in the dissipative range ( $r^{\ast }<20$ ), although the range of separations over which (3.1) is satisfied depends on the initial conditions. Figure 3, which shows that $F$ is multivalued, suggests that $F$ can, for a given flow, depend on the initial conditions. It can also vary from flow to flow.

Figure 4.

Dependence of $F$ on $Re_{\unicode[STIX]{x1D706}}$ in two types of flows and qualitative comparison with (2.16). Plane jet: $\star$ , Antonia, Anselmet & Chambers (Reference Antonia, Anselmet and Chambers1986); ▾ (black), Sreenivasan & Antonia (Reference Sreenivasan and Antonia1997); ▪, Antonia et al. (Reference Antonia, Djenidi, Danaila and Tang2017), without identifying the data sources. Circular jet: ▵, Mi, Xu & Zhou (Reference Mi, Xu and Zhou2013); ♦, Xu et al. (Reference Xu, Antonia and Rajagopalan2001); ●, Kahalerras, Malecot & Gagne (Reference Kahalerras, Malecot and Gagne1998); ▾ (blue), Pearson & Antonia (Reference Pearson and Antonia2001); ▴, Friehe, Atta & Gibson (Reference Friehe, Van Atta and Gibson1971); ▿, Kuo & Corrsin (Reference Kuo and Corrsin1971); ▫, Antonia et al. (Reference Antonia, Chambers and Satyaprakash1981) (see also Antonia et al. Reference Antonia, Satyaprakash and Hussain1982); ▹, Burattini (private communication). The blue and black curves are least-squares fits with (2.16), corresponding to circular jet ( $\unicode[STIX]{x1D6FE}=-11$ and $\unicode[STIX]{x1D6FD}=1000$ ) and plane jet ( $\unicode[STIX]{x1D6FE}=-11$ and $\unicode[STIX]{x1D6FD}=1600$ ), respectively. The inset shows all the data using log–log coordinates. Also shown in the inset is the K62 prediction (red curve): $F=0.91Re_{\unicode[STIX]{x1D706}}^{0.39}$ (Gylfason, Ayyalasomayajula & Warhaft Reference Gylfason, Ayyalasomayajula and Warhaft2004).

3.3 Derivative flatness factor $S_{4}$ at large $Re_{\unicode[STIX]{x1D706}}$

We now consider data for $S_{4}$ obtained by various authors in a wide range of flows. Separate plots are used (figures 4, 6, 7 and 8 b), with the same ordinate scale. We note that almost all the previous assessments of the $Re_{\unicode[STIX]{x1D706}}$ dependence of $F$ have included the high- $Re_{\unicode[STIX]{x1D706}}$ ASL data and led to the observation that $F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$ (K62), e.g. figure 6 of Sreenivasan & Antonia (Reference Sreenivasan and Antonia1997). For reasons that will be discussed in detail later (in the context of figure 8 a), we have excluded the ASL data from figures 4, 6, 7 and 8(b).

Recently, Antonia et al. (Reference Antonia, Djenidi, Danaila and Tang2017) showed that, along the axis of the far field of a plane jet, $F$ is practically constant over a range of $Re_{\unicode[STIX]{x1D706}}$ ( $500<Re_{\unicode[STIX]{x1D706}}<1100$ ) for several datasets, which are reproduced in figure 4 without identifying the data sources. Also included in figure 4 are the data of Antonia et al. (Reference Antonia, Anselmet and Chambers1986) and Sreenivasan & Antonia (Reference Sreenivasan and Antonia1997) also for the plane jet at lower $Re_{\unicode[STIX]{x1D706}}$ . In order to examine the boundedness of $F$ , we have added in figure 4 data obtained along the axis of the far field of a circular jet (Friehe et al. Reference Friehe, Van Atta and Gibson1971; Kuo & Corrsin Reference Kuo and Corrsin1971; Antonia et al. Reference Antonia, Chambers and Satyaprakash1981, Reference Antonia, Satyaprakash and Hussain1982; Kahalerras et al. Reference Kahalerras, Malecot and Gagne1998; Pearson & Antonia Reference Pearson and Antonia2001; Xu et al. Reference Xu, Antonia and Rajagopalan2001; Mi et al. Reference Mi, Xu and Zhou2013). Antonia et al. (Reference Antonia, Satyaprakash and Hussain1982) measured $F$ on the centreline of a circular jet at $x/d=70$ , 80, 90 and 120 at a fixed $Re_{d}$  ( $=U_{j}d/\unicode[STIX]{x1D708}=55\,600$ , where $U_{j}$ is the jet exit velocity and $d$ the nozzle diameter) on the jet axis (see their figure 7) and showed that $F$ is approximately constant ( ${\approx}10$ ) at $x/d=80$ , 90 and 120, respectively, which would be consistent with self-preservation at least for the small scales, and is also in good agreement with other data at comparable $Re_{\unicode[STIX]{x1D706}}$ in this flow (see figure 4). Their magnitude of $F$ at $x/d=70$ ( ${\approx}8.4$ ) appears to be underestimated, compared to $F$ for $x/d>70$ . For this reason, this data point is not shown in figure 4.

Equation (1.4) allows the FRN effect on $S_{3}$ to be estimated quantitatively in various flows, i.e. it allows an analytical prediction for $S_{3}$ . Evidently, it would be desirable to investigate the FRN effect on $F$ with (2.16) in each flow. However, the estimation of $\unicode[STIX]{x1D6FE}$ involves the simultaneous measurement of pressure and velocity fluctuations. Unfortunately, these quantities are not yet available and we only know the constant for $\unicode[STIX]{x1D6FD}$ in decaying HIT. Thus, we cannot predict $F$ analytically. However, in order to highlight the $Re_{\unicode[STIX]{x1D706}}$ dependence of $F$ , we apply a least-squares fit based on (2.16) to the data reported in figure 4; we assume that $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FD}$ are constants with the expectation that $\unicode[STIX]{x1D6FD}$ should vary between different flows to reflect the difference of FRN effect in each flow. As mentioned in the introduction, $S_{3}$ appears to reach the same constant value ( ${\approx}-0.53$ ) when $Re_{\unicode[STIX]{x1D706}}\geqslant 300$ ; further, for $Re_{\unicode[STIX]{x1D706}}$ in the range 100–300, $S_{3}$ changes by less than 20 % in different flows (Antonia et al. Reference Antonia, Djenidi, Danaila and Tang2017). We have therefore chosen $S_{3}\approx -0.53$ for curve fitting to these two flows. The fitted curves are shown in figure 4. The following comments can be made.

1. (i) The data in these two flows can be reasonably fitted with (2.16), with the same value of $\unicode[STIX]{x1D6FE}~(=-11)$ but different values of $\unicode[STIX]{x1D6FD}$ (1000 for the circular jet and 1600 for the plane jet, respectively).

2. (ii) The fitted curves show the same trend as the data: the flow dependence when $Re_{\unicode[STIX]{x1D706}}$ is small to moderate ( ${<}600$ ) becomes smaller when $Re_{\unicode[STIX]{x1D706}}\geqslant 600$ .

3. (iii) The trend shown by the data and the curves is clear: $F$ appears to approach a limiting value, possibly slightly smaller than 12, when $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large. In general, $Re_{\unicode[STIX]{x1D706}}$ needs to exceed approximately 500 for $F$ to be considered constant in these two flows. The way this constant is approached is flow-dependent.

4. (iv) The larger $\unicode[STIX]{x1D6FD}$ , the slower the rate at which the final value is reached. Evidently, the approach is slower on the axis of the plane jet ( $\unicode[STIX]{x1D6FD}=1600$ ) than on the axis of the circular jet ( $\unicode[STIX]{x1D6FD}=1000$ ).

The constancy of $F$ at large $Re_{\unicode[STIX]{x1D706}}$ is consistent with the analytical prediction of Qian (Reference Qian1986), who used a closure theory, the analytical results of Djenidi et al. (Reference Djenidi, Antonia, Danaila and Tang2017b ) and the prediction of K41. But it cannot be reconciled with K62, which predicts $F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$ . This can be observed clearly from the inset of figure 4, which shows that K62 (as an example, we show a K62 prediction (red curve): $F=0.91Re_{\unicode[STIX]{x1D706}}^{0.39}$ of Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004)) is not supported by the experimental data in both circular and plane jets. In fact, the K62 prediction is at best tangential to the plane jet data and does not represent at all the circular jet data. Since FRN-affected data (collected in different flows) as well as the ASL data (possibly also affected by the FRN effect) were included in almost all the previous examinations of $F$ versus $Re_{\unicode[STIX]{x1D706}}$ , it is not surprising that a consensus has not been reached for the value of $\unicode[STIX]{x1D6FC}_{4}$ . Indeed, the exponent $\unicode[STIX]{x1D6FC}_{4}$ can vary between 0.31 and 0.41 (e.g. Van Atta & Antonia Reference Van Atta and Antonia1980; Antonia et al. Reference Antonia, Satyaprakash and Hussain1982; Ishihara et al. Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007), depending on the intermittency models. The previous discussion leads to the conclusion that the difference in the values of $\unicode[STIX]{x1D6FC}_{4}$ reflects the difference in the way $F$ is affected by the different large-scale motions.

The DNS data for $F$ in SFPBT are shown figure 5. It can be seen from this figure that there is good collapse for all the DNS data when $Re_{\unicode[STIX]{x1D706}}\leqslant 200$ ; the same can be observed for $S_{3}$ in the same flow; see figure 5 of Antonia et al. (Reference Antonia, Tang, Djenidi and Danaila2015). For $Re_{\unicode[STIX]{x1D706}}>200$ , however, the two sets of data obtained by Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007) ( $k_{max}^{\ast }=2$ and $k_{max}^{\ast }=1$ , respectively) indicate that $F$ increases as $Re_{\unicode[STIX]{x1D706}}$ increases (the Gotoh et al. (Reference Gotoh, Fukayama and Nakano2002) data, for $k_{max}^{\ast }=1$ , give the impression that $F$ has almost become constant at $Re_{\unicode[STIX]{x1D706}}=460$ ). The authors claim agreement with $F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$ ( $\unicode[STIX]{x1D6FC}_{4}=0.31$  (Hill Reference Hill2002) and 0.39 (Gylfason et al. Reference Gylfason, Ayyalasomayajula and Warhaft2004), respectively, which have also been shown in the inset of figure 5), but in fact it is clear that the rate of increase of $F$ with $Re_{\unicode[STIX]{x1D706}}$ decreases as $Re_{\unicode[STIX]{x1D706}}$ increases (see the linear plot) and the relation $F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$ does not really represent the trend of the Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007) data (see the log–log plot). We can only surmise that Ishihara et al.’s (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007) claim that $F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$ was influenced by the inclusion of the ASL data. We contend that their data do not exclude the possibility that it will approach a constant at large $Re_{\unicode[STIX]{x1D706}}$ . We have already commented (Antonia et al. Reference Antonia, Tang, Djenidi and Danaila2015) on the behaviour of their Kolmogorov-normalized spectra, in particular the systematic increase of $k^{\ast 4}E^{\ast }(k^{\ast })$ (for $k^{\ast }>0.5$ ) with $Re_{\unicode[STIX]{x1D706}}$ (Ishihara et al. Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007). This is not consistent with Kolmogorov scaling, as inferred from the NS equation (e.g. Antonia et al. Reference Antonia, Djenidi and Danaila2014), nor is it consistent with the support for this scaling provided by previous DNS studies of forced periodic box turbulence (Jimenez et al. Reference Jimenez, Wray, Saffman and Rogallo1993; Yeung & Zhou Reference Yeung and Zhou1997; Gotoh et al. Reference Gotoh, Fukayama and Nakano2002; Yeung et al. Reference Yeung, Donzis and Sreenivasan2005), for $Re_{\unicode[STIX]{x1D706}}$ up to 700. As overwhelmingly supported by experimental data – and also the DNS data of Jimenez et al. (Reference Jimenez, Wray, Saffman and Rogallo1993), Yeung & Zhou (Reference Yeung and Zhou1997), Gotoh et al. (Reference Gotoh, Fukayama and Nakano2002, see figure 12 of their paper) and Yeung et al. (Reference Yeung, Donzis and Sreenivasan2005, up to $Re_{\unicode[STIX]{x1D706}}\approx 700$ , see figure 1 of their paper) (note that $S_{3}$ is constant for $Re_{\unicode[STIX]{x1D706}}\approx 240{-}700$ whereas a proper ‘ $4/5$ ’ inertial range has yet to be established) – the FRN effect first disappears at the smallest scales and subsequently at increasingly larger scales as $Re_{\unicode[STIX]{x1D706}}$ increases. This is in contrast to the claim by Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007, Reference Ishihara, Gotoh and Kaneda2009) that the Reynolds-number effect has ‘nearly’ disappeared in the so-called scaling range (in Ishihara et al. (Reference Ishihara, Gotoh and Kaneda2009), the scaling range exponents, which are independent of $Re_{\unicode[STIX]{x1D706}}$ , are inferred from Gotoh et al.’s (Reference Gotoh, Fukayama and Nakano2002) data at $Re_{\unicode[STIX]{x1D706}}=460$ ) whereas $S_{3}$ and $S_{4}$ continue to increase. Further, Ishihara et al. (Reference Ishihara, Gotoh and Kaneda2009) conclude (their summary point number 3) that

and (their summary point number 5) that

These conclusions imply that K41 and K62 may coexist, a possibility that is difficult to understand, at least in the context of the laboratory measurements and previous and earlier DNS results. Our observations indicate that the FRN effect first disappears in the dissipative range and lingers on in the scaling range, i.e. larger scales (in the scaling range) are more likely to continue to be affected by $Re_{\unicode[STIX]{x1D706}}$ than smaller scales (in the dissipative range). Evidently, Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007) do not share this view. To our knowledge, a plausible mechanism, which explains how intermittency, as expressed by $F$ say, can continue to increase with $Re_{\unicode[STIX]{x1D706}}$ whilst the scaling range exponents are no longer affected by $Re_{\unicode[STIX]{x1D706}}$ (Ishihara et al. Reference Ishihara, Gotoh and Kaneda2009), has yet to be proposed.

Figure 5.

Dependence of $F$ on $Re_{\unicode[STIX]{x1D706}}$ in SFPBT: $\times$ , Kerr (Reference Kerr1985); ▾ and ▴ correspond to $k_{max}^{\ast }=2$ and $k_{max}^{\ast }=1$ , respectively, from Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007); ♦, Gotoh et al. (Reference Gotoh, Fukayama and Nakano2002); ●, Wang et al. (Reference Wang, Chen, Brasseur and Wyngaard1996). The inset shows all the data using log–log coordinates. Also shown in the inset is the K62 prediction: $F=0.91Re_{\unicode[STIX]{x1D706}}^{0.39}$ (red curve) (Gylfason et al. Reference Gylfason, Ayyalasomayajula and Warhaft2004) and $F=1.36Re_{\unicode[STIX]{x1D706}}^{0.31}$ (green curve) (Hill Reference Hill2002); both predictions were used by Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007) (see their figure 6).

Figure 6.

Dependence of $F$ on $Re_{\unicode[STIX]{x1D706}}$ in flow between counter-rotating disks: ○, Tabeling et al. (Reference Tabeling, Zocchi, Belin, Maurer and Willaime1996); ▫, Belin et al. (Reference Belin, Maurer, Tabeling and Willaime1997).

Tabeling et al. (Reference Tabeling, Zocchi, Belin, Maurer and Willaime1996) and Belin et al. (Reference Belin, Maurer, Tabeling and Willaime1997) reported various statistics for $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ in helium gas at low temperature in a cylindrical container bounded by two counter-rotating disks over an impressively large range of $Re_{\unicode[STIX]{x1D706}}$ : 150–5040 (Tabeling et al. Reference Tabeling, Zocchi, Belin, Maurer and Willaime1996) and 150–2300 (Belin et al. Reference Belin, Maurer, Tabeling and Willaime1997). In the context of this paper, notwithstanding previous reservations (Sreenivasan Reference Sreenivasan1995; Sreenivasan & Antonia Reference Sreenivasan and Antonia1997) concerning the characteristics of the sensor used in these experiments, these results are useful since the range of $Re_{\unicode[STIX]{x1D706}}$ is large enough to allow the assessment of the FRN effect on $F$ with minimal ambiguity. As far as we are aware, there has been no systematic examination of LI in the central region of this flow. However, the isotropic form of the transport equation for $\overline{(\unicode[STIX]{x1D6FF}u)^{2}}$ is verified reasonably well (Moisy, Tabeling & Willaime Reference Moisy, Tabeling and Willaime1999). Antonia et al. (Reference Antonia, Tang, Djenidi and Danaila2015) have already reported and discussed elsewhere the data for $S_{3}$ , where $S_{3}$ remains constant ( $\simeq -0.50$ ) over a range of $Re_{\unicode[STIX]{x1D706}}$ extending up to 2000. The data for $F$ of Tabeling et al. (Reference Tabeling, Zocchi, Belin, Maurer and Willaime1996) and Belin et al. (Reference Belin, Maurer, Tabeling and Willaime1997) are shown in figure 6 for $Re_{\unicode[STIX]{x1D706}}$ up to 3000. The figure presents a similar trend to that of the data in figure 4: $F$ increases with $Re_{\unicode[STIX]{x1D706}}$ for $Re_{\unicode[STIX]{x1D706}}$ up to approximately 700, and, allowing for the scatter, seems to be approximately constant beyond that value. There is a peak in $F$ in the vicinity of $Re_{\unicode[STIX]{x1D706}}\simeq 700$ , which was speculated (by the authors) to correspond to a transition, possibly a ‘mixing’ transition as described by Dimotakis (Reference Dimotakis2000) and also observed by Pearson & Krogstad (Reference Pearson and Krogstad2001) in passive grid turbulence, to a new state of turbulence (see the discussion of this transition in Tabeling & Willaime (Reference Tabeling and Willaime2002)).

Figure 7.

Dependence of $F$ on $Re_{\unicode[STIX]{x1D706}}$ in shearless grid turbulence: ▫, Batchelor & Townsend (Reference Batchelor and Townsend1947, Reference Batchelor and Townsend1949); ▿ (magenta), estimated from the data of Zhou & Antonia (Reference Zhou and Antonia2000) ( $l_{w}/\unicode[STIX]{x1D702}=0.6{-}2.5$ ); ▿ (blue), Tong & Warhaft (Reference Tong and Warhaft1994) ( $l_{w}/\unicode[STIX]{x1D702}=0.8{-}1.3$ ); ●, Mydlarski & Warhaft (Reference Mydlarski and Warhaft1996) ( $l_{w}/\unicode[STIX]{x1D702}=1.8{-}5.2$ ); ▴ corresponds to the data from small and large tunnels, respectively, without mean shear, reproduced from figure 3 of Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004); note that only data for $l_{w}/\unicode[STIX]{x1D702}\leqslant 1.75$ are shown, as recommended by Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004). The inset shows all the data on a log–log plot. Also shown in the inset is one measurement ( $F=10.8$ , ▾) of Kahalerras et al. (Reference Kahalerras, Malecot and Gagne1998) along the axis of the ONERA wind tunnel (similar to grid turbulence) for $Re_{\unicode[STIX]{x1D706}}=2500$ and the K62 prediction (red curve): $F=0.91Re_{\unicode[STIX]{x1D706}}^{0.39}$ (Gylfason et al. Reference Gylfason, Ayyalasomayajula and Warhaft2004).

Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004) examined the Reynolds-number dependence of $F$ in grid turbulence with and without mean shear ( $Re_{\unicode[STIX]{x1D706}}=100{-}1000$ ). Here, we only show the data in grid turbulence without a mean shear since the latter has a significant effect on LI (Kim & Antonia Reference Kim and Antonia1993; Shen & Warhaft Reference Shen and Warhaft2000) (see also Tang et al. (Reference Tang, Antonia, Danaila, Djenidi, Zhou and Zhou2016), who showed that the higher-order statistics depart more strongly from LI in sheared than in shearless grid turbulence); not surprisingly, the magnitude of $F$ is increased, at the same $Re_{\unicode[STIX]{x1D706}}$ , relative to that measured without shear; also, $F$ increases along distinctly different paths for the shearless and sheared cases. Gylfason et al.’s (Reference Gylfason, Ayyalasomayajula and Warhaft2004) data (their figure 3) for shearless grid turbulence are reported in figure 7; note that only data for $l_{w}/\unicode[STIX]{x1D702}\leqslant 1.75$ are shown, as recommended by Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004). Also shown in this figure are data for grid turbulence collected from other published studies. Two comments can be made with regard to figure 7.

1. (i) All the data for $F$ collapse reasonably well onto a single distribution, up to $Re_{\unicode[STIX]{x1D706}}\simeq 730$ . This distribution has the same behaviour as that observed in figure 4: for $Re_{\unicode[STIX]{x1D706}}\leqslant$ 600, the magnitude of $F$ increases with $Re_{\unicode[STIX]{x1D706}}$ ; for $Re_{\unicode[STIX]{x1D706}}\geqslant 600$ , $F$ eventually reaches a constant, which is about the same as that measured by Kahalerras et al. (Reference Kahalerras, Malecot and Gagne1998) at $Re_{\unicode[STIX]{x1D706}}=2500$ . This trend is similar to that observed in figures 4 and 6. On the contrary, the inset of figure 7 shows that shearless grid turbulence data ( $50<Re_{\unicode[STIX]{x1D706}}<160$ ) from other published studies and especially the data point at $Re_{\unicode[STIX]{x1D706}}=2500$ are not adequately described by the relation $F=0.91Re_{\unicode[STIX]{x1D706}}^{0.39}$ (K62). Further, as discussed in the context of figures 4 and 5, Gylfason et al.’s relation $F=0.91Re_{\unicode[STIX]{x1D706}}^{0.39}$ cannot represent the data in other flows.

2. (ii) The data show that $F$ approaches a constant whose magnitude is in approximate agreement with that in plane and circular jets when $Re_{\unicode[STIX]{x1D706}}>600$ (figure 4).

Figure 8.

(a) Dependence of $F$ on $Re_{\unicode[STIX]{x1D706}}$ reproduced (log–log plot) from figure 6 of Sreenivasan & Antonia (Reference Sreenivasan and Antonia1997) without identifying the data sources. A ‘red’ line has been drawn around the ASL data to distinguish these from the other (laboratory) data. (b) An enlargement of the range for (laboratory) data using linear axes. Note the presentation in (b) is completely consistent with that in figures 4–7.

Figure 8(a) reproduces the compilation of $F$ versus $Re_{\unicode[STIX]{x1D706}}$ by Sreenivasan & Antonia (Reference Sreenivasan and Antonia1997) (their figure 6); we have drawn a ‘red’ line around the symbols corresponding to the ASL data. In order to compare figure 8(a) with figures 3–7, an enlargement of the $Re_{\unicode[STIX]{x1D706}}$ range corresponding to the laboratory data in figure 8(a) is shown in figure 8(b) with linear scales instead of log scales (the vertical scale is identical to that used in figures 4–7). Antonia et al. (Reference Antonia, Djenidi, Danaila and Tang2017) carried out a similar comparison for $S_{3}$ and made the following points.

1. (i) Since the data affected by the FRN effect were included in the compilation for $S_{3}$ , many values of the exponent $\unicode[STIX]{x1D6FC}_{3}$ in the power-law relation $S_{3}\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{3}}$ (K62) have been proposed.

2. (ii) The ASL data should not be used to test K41 and K62 because they were collected at relatively small heights (often below 30 m) above the ground or ocean surface where the requirements for K41 and K62 are unlikely to be met. Similar comments can be made for $F$ on the basis of figures 3–8.

It can be seen from figure 8(b) that there is a fair amount of ‘apparent’ scatter in the laboratory data for $Re_{\unicode[STIX]{x1D706}}\leqslant 600$ . In fact, this scatter reflects, to a large extent, the systematic difference in how $F$ approaches a constant in different flows, e.g. figure 3. In similar fashion to $S_{3}$ (Antonia et al. Reference Antonia, Djenidi, Danaila and Tang2017), no attempt was made to distinguish between distributions of $F$ in different flows.

Almost all the previous examinations of $F$ versus $Re_{\unicode[STIX]{x1D706}}$ have included the high- $Re_{\unicode[STIX]{x1D706}}$ ASL data, e.g. figure 6 of Sreenivasan & Antonia (Reference Sreenivasan and Antonia1997) (or the present figure 8 a). The inclusion of the ASL data leads to a power-law dependence of $F$ on $Re_{\unicode[STIX]{x1D706}}$ (K62) that is inconsistent with the results in figures 4, 6 and 7. We have excluded the ASL data from figures 4, 6 and 7 since Djenidi et al. (Reference Djenidi, Antonia, Talluru and Abe2017c ) have provided several reasons why the ASL data should be discarded when testing K41 and K62. We discuss this further here, in some detail. The ASL data have been obtained in a flow region where the effects of the mean shear and proximity to the surface (which causes blockage effects to arise) cannot be ignored, notwithstanding the locally high value of $Re_{\unicode[STIX]{x1D706}}$ , especially since both these effects are known to induce strong departures from LI. In fact, a large percentage of the ASL data in figure 8(a) were obtained by Wyngaard & Tennekes (Reference Wyngaard and Tennekes1970) at three heights ( $z=5.66$ , 11.3 and 22.6 m) above the ground (no information for the stability conditions is given). Under ‘neutral’ conditions, the ASL should be comparable (at similar values of $z/\unicode[STIX]{x1D6FF}$ ) with the laboratory boundary layer provided the nature of the surface is comparable. It is difficult to estimate with confidence the thickness $\unicode[STIX]{x1D6FF}$ of the neutral ASL. Wyngaard (Reference Wyngaard2010) suggests that, under neutral conditions, $\unicode[STIX]{x1D6FF}$ may scale with $U_{\unicode[STIX]{x1D70F}}/f$ ( $U_{\unicode[STIX]{x1D70F}}$ is the friction velocity and $f$ is the Coriolis parameter) and estimates, for mid-latitudes, $\unicode[STIX]{x1D6FF}$ to be within the range 250 to 2500 m. This implies that the majority of the neutral ASL data for $F$ were taken in the region $z/\unicode[STIX]{x1D6FF}<0.1$ . It is not difficult to conclude that there is quite a lot of uncertainty in the estimation of $\unicode[STIX]{x1D6FF}$ , e.g. a value of $\unicode[STIX]{x1D6FF}$ smaller than 100 m was found by Metzger, McKeon & Holmes (Reference Metzger, McKeon and Holmes2007) based on experiments in Utah’s western desert for almost ‘exactly’ neutral conditions; the authors stressed, however, the difficulties associated in making measurements under those conditions. In any case, it is not clear if any of the published ASL data were obtained under ‘exactly’ neutral conditions. Certainly, it is unwise to assume that the measurements of $F$ in region $z/\unicode[STIX]{x1D6FF}<0.1$ have not been affected by the proximity to the surface.

Recent measurements (Djenidi et al. Reference Djenidi, Antonia, Talluru and Abe2017c ) in a laboratory boundary layer over smooth and rough walls indicate that the magnitude of $F$ can increase, perhaps by as much as a factor of 3, in the region $z/\unicode[STIX]{x1D6FF}<0.1$ . For the ASL data in figure 8(a) of Sreenivasan & Antonia (Reference Sreenivasan and Antonia1997), it is almost certain that $F$ was invariably estimated over a range of heights that would correspond to the inner region of the laboratory boundary layer. For example, for the nearly neutral ASL data measured by Gibson et al. (Reference Gibson, Stegen and Williams1970) above the ocean surface, $F$ is in the range 13 to 26 (the height is in the range 2.25 m to 12.25 m); this variation is typical of that found in the inner region of the laboratory boundary layer (Djenidi et al. Reference Djenidi, Antonia, Talluru and Abe2017c ). Further, one should also check that there are no major departures from LI in this range due to the probable effect of the mean shear. In the region $z/\unicode[STIX]{x1D6FF}<0.1$ , a larger departure from LI is expected due to the combined effect of an increase in mean shear and the presence of the surface. Clearly, the use of $F$ in the first few metres of the ASL has misled a vast majority of researchers into preferring K62 over K41. More importantly, when the ASL data are disregarded, the (laboratory) data in figure 8(b) are consistent with the trend displayed in figures 4, 6 and 7, providing further strong support that $F$ increases as $Re_{\unicode[STIX]{x1D706}}$ increases, eventually becoming constant when $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large. Note that the log–log scale in figure 8(a), together with the weighting of the ASL data, have ‘aided and abetted’ the support for a power-law dependence of $F$ on  $Re_{\unicode[STIX]{x1D706}}$ .