5.1. Validation of equation (2.9)

Here, we test (2.9) against measurements at $h^{+}=180$ . A perfect validation of (2.9) requires that the diffusion term (left-hand side) balances the sum $S_{1,1}+2G/R_{{\it\lambda}}$ exactly. On the centreline of the channel, it can be seen from figure 1 that $-S_{1,1}$ is about 0.36. Following the procedure shown in the above section, we estimate that $2G/R_{{\it\lambda}}$ is about 0.60 at this location. Figure 10 shows the distribution of $\overline{u_{2}^{+}{\it\epsilon}_{1}^{+}}$ , with respect to $x_{2}^{+}$ , obtained with the one-component vorticity probe, where three different measurements of $u_{1}$ are made simultaneously with $u_{2}$ at each $x_{2}$ location. Each point shown in figure 10 is an average of the three values. The magnitude of $\overline{u_{2}^{+}{\it\epsilon}_{1}^{+}}$ increases rapidly in the inner layer, reaching a maximum at about $x_{2}^{+}=50$ , and then decreases slowly farther away from the wall before approaching zero on the centreline of the channel. The term on the left-hand side of (2.9) is obtained by applying a least-squares linear fit to the distribution of $\overline{u_{2}^{+}{\it\epsilon}_{1}^{+}}$ over the range $x_{2}^{+}=110{-}180$ , and then normalizing the slope of the fitted line with $\overline{{\it\epsilon}}_{iso}$ on the channel centreline and ${\it\nu}$ . Its magnitude is thus found to be equal to 0.25, as shown in table 1, where estimates of $S_{1,1}$ , $2G/R_{{\it\lambda}}$ , and $S_{1,1}+2G/R_{{\it\lambda}}$ are also included. The conclusion we draw from this table is that (2.9) is satisfied adequately on the centreline of the channel.

Figure 10.

Distribution of $\overline{u_{2}^{+}{\it\epsilon}_{1}^{+}}$ in the $x_{2}^{+}$ direction at $h^{+}=180$ . Solid line is a linear fit over the range $x_{2}^{+}=110{-}180$ .

Table 1.

Check of equation (2.9) at $h^{+}=180$ using data from a one-component vorticity probe.

The vorticity budget is given by (Abe et al. Reference Abe, Antonia and Kawamura2009)

where the mean strain rate $\unicode[STIX]{x1D61A}_{ij}$ and the strain-rate fluctuations $\unicode[STIX]{x1D634}_{ij}$ are given by

The DNS vorticity budget data (Antonia & Kim Reference Antonia and Kim1994) at $h^{+}=180$ and 395 and (Abe et al. Reference Abe, Antonia and Kawamura2009) at $h^{+}=180$ , 395 and 640 have been used to assess the magnitude of the ratios $Dif/D$ and $P/D$ in the outer part of the channel. In figure 11, the ratios shown up to $x_{2}^{+}\simeq 180$ are for $h^{+}=180$ . For $200\leqslant x_{2}^{+}\leqslant 400$ , the ratios are for $h^{+}=395$ ; and for $x_{2}^{+}\geqslant 450$ , the ratios are for $h^{+}=640$ . For both DNS data sets at $h^{+}=180$ , there is a tendency for $|Dif/D|$ to increase and $|P/D|$ to decrease as the centreline is approached; this is more marked for the experimental data than for the DNS data. At the centreline, $|Dif/D|\simeq 0.21$ for the DNS data of Abe et al. (Reference Abe, Antonia and Kawamura2009) compared to 0.40 for the experiment ( $|Dif/D|\simeq 0.25$ for Antonia & Kim Reference Antonia and Kim1994). For $h^{+}=395$ , the variation with $x_{2}^{+}$ is small in the outer region, and $|Dif/D|$ is only 0.15 at the centreline for both DNS data sets. $|Dif/D|$ becomes even smaller at $h^{+}=640$ ( $|Dif/D|=0.12$ on the centreline). In contrast, $|P/D|$ contributes significantly to the vorticity budget with increasing $h^{+}$ . Figure 11 shows a decreasing trend for $|Dif/D|$ with increasing $h^{+}$ . This is consistent with, at sufficiently high Reynolds number, $|Dif/D|$ becoming negligible, i.e. $|P/D|\approx 1$ . The same should be true for the isotropic budget (2.9) and will be addressed in § 5.2.

Figure 11.

Vorticity budget of DNS data (Antonia & Kim Reference Antonia and Kim1994) at $h^{+}=180$ and 395 (solid curves) and (Abe et al. Reference Abe, Antonia and Kawamura2009) at $h^{+}=180$ , 395 and 640 (dashed curves) in the outer layer. Note that each term is normalized by the dissipation term; three tagged points are the ratios $Dif/D$ at the channel centreline for Abe et al. (Reference Abe, Antonia and Kawamura2009). Symbols are the isotropic budget (2.9) of experimental data at $h^{+}=180$ : ▫, diffusion term; ○, production term ( $S_{1,1}$ ); ♢, dissipation term ( $2G/R_{{\it\lambda}}$ ); note that each is divided by the dissipation term.

5.2. Scaling of the transport equation

After applying dimensional arguments to the kinetic energy production term, Bernardini, Pirozolli & Orlandi (Reference Bernardini, Pirozolli and Orlandi2014) argued that, for the region close the centreline of the channel, while $v^{\prime }$ , $w^{\prime }$ and $u^{\prime }v^{\prime }$ scale with $U_{{\it\tau}}$ , $u^{\prime }$ scales with $\sqrt{U_{{\it\tau}}U_{cl}}$ . Here we extend their argument to the transport equation for $\overline{{\it\epsilon}}_{iso}$ along the channel centreline. Since the turbulent kinetic energy production term $P$ should scale as $P\sim U_{{\it\tau}}^{2}U_{cl}/h$ , $\overline{{\it\epsilon}}_{iso}$ should thus have the same scaling as $P$ ,

The diffusion of ${\it\epsilon}_{1}$ by $u_{2}$ should scale as

where ${\it\tau}=h/U_{{\it\tau}}$ is the typical eddy turnover time in the outer region of the channel (Jimenez, Hoyas & Mizuno Reference Jimenez, Hoyas and Mizuno2009). Hence, it follows that

The term on the left of (2.9) can therefore be rewritten as follows:

where

Figure 12 confirms the Reynolds number independence of $C_{1}$ as well as of the ratios $\sqrt{\overline{u^{2}}/(U_{{\it\tau}}U_{cl})}$ and $\overline{{\it\epsilon}}_{iso}h/(U_{{\it\tau}}^{2}U_{cl})$ , thus indicating that (5.6) is reasonably well supported by the present experimental data. Therefore, (2.9) can be rewritten in the form

where $C\,({\approx}9.7)$ estimated at $h^{+}=180$ should be independent of $R_{{\it\lambda}}$ . Arguably, the first term in (2.9) approaches zero at sufficiently large $R_{{\it\lambda}}$ . This is consistent with $C/R_{{\it\lambda}}$ approaching zero at infinite $R_{{\it\lambda}}(\lim _{R_{{\it\lambda}}\rightarrow \infty }(C/R_{{\it\lambda}})=0)$ . The main objective in deriving (5.7) is to assess the way $S_{1,1}+2G/R_{{\it\lambda}}$ approaches zero along the centreline of a fully developed channel flow as $R_{{\it\lambda}}$ increases and compare it with how $S_{1,1}+2G/R_{{\it\lambda}}$ approaches zero in other flows, e.g. decaying HIT, the axis of a turbulent round jet, and SFPBT.

Figure 12.

Variation of $C_{1}$ with $h^{+}$ on the channel centreline (▫); $\sqrt{\overline{u^{2}}/(U_{{\it\tau}}U_{cl})}$ (▵); and the normalized dissipation rate $\overline{{\it\epsilon}}_{iso}h/(U_{{\it\tau}}^{2}U_{cl})$ (▿). Black symbols, present data; red, Moser et al. (Reference Moser, Kim and Mansour1999); blue, Abe et al. (Reference Abe, Antonia and Kawamura2009); green, Bernardini et al. (Reference Bernardini, Pirozolli and Orlandi2014); the horizontal lines indicate the mean values of the present data.

The statement that $S_{1,1}+2G/R_{{\it\lambda}}$ approaches zero as the Reynolds number increases may be obtained via several different arguments.

We first consider that (5.4) applies, with ${\it\tau}$ the characteristic time of the turbulent diffusion of ${\it\epsilon}_{1}$ along $x_{2}$ . Therefore, the left-hand side of (2.9) (hereafter denoted $LSE$ ) scales as

Together with ${\it\tau}\sim \mathscr{L}\big/\mathscr{U}$ (where $\mathscr{L}$ and $\mathscr{U}$ are the length and velocity scales) and the normalized energy dissipation rate

$LSE$ becomes, after some straightforward calculations

Since $C_{{\it\epsilon}}$ tends to a constant (which may depend on IBC) for $R_{{\it\lambda}}\geqslant 100$ (e.g. Abe & Antonia Reference Abe and Antonia2011), then $LSE\rightarrow 0$ as $Re_{\mathscr{L}}$ increases. If $\mathscr{L}$ is the integral scale of the flow, then $R_{{\it\lambda}}\sim Re_{\mathscr{L}}^{1/2}$ and the right-hand side of (5.7) is retrieved.

A similar development should hold for any other flow, with the restriction that local isotropy holds. The particular path along which $C_{{\it\epsilon}}$ tends to a constant is, again, flow specific, but it is commonly accepted that a constant value (though IBC dependent) is reached for sufficiently high Reynolds numbers (see for example Sreenivasan Reference Sreenivasan1998; Burattini, Lavoie & Antonia Reference Burattini, Lavoie and Antonia2005; Vassilicos Reference Vassilicos2015). In the near-field region downstream of the fractal grids as well as regular grids, $C_{{\it\epsilon}}$ is not constant with $x$ but may be approximated by a ratio $Re_{I}/Re_{\mathscr{L}}$ , where $Re_{I}$ is a global/inlet Reynolds number (see for example Valente & Vassilicos Reference Valente and Vassilicos2012; Hearst & Lavoie Reference Hearst and Lavoie2014; Vassilicos Reference Vassilicos2015). In this case, (5.10) would lead to $LSE\sim Re_{I}^{-1/2}$ rather than $LSE\sim R_{{\it\lambda}}^{-1}$ , as indicated by (5.7). In the far field of an axisymmetric jet, included in figures 14 and 15, $R_{{\it\lambda}}$ is constant along the axis. This region of the flow is exactly self-preserving and $C_{{\it\epsilon}}$ like $R_{{\it\lambda}}$ is constant with respect to $x$ .

A different way to reach the same outcome is to explicitly suppose self-preservation of the functions involved in $LSE$ , viz.

and

where $f_{v{\it\epsilon}}$ and $f_{{\it\epsilon}}$ are self-preserving functions which may depend on IBC. It follows that $LSE$ scales as $Re_{\mathscr{L}}^{-1/2}$ . This demonstration circumvents the explicit dependence on $C_{{\it\epsilon}}$ , and requires self-preservation. Note that the latter assumption is, however, implicit when writing equation (5.4).

A more general viewpoint may be provided in flows characterized by large-scale production, e.g. shear flows. In this case, with the one-point energy budget being given by $\bar{{\it\epsilon}}=2\overline{uv}(\partial \overline{U}/\partial y)$ , $LSE$ can be written as

Now,

and (5.13) becomes

Similarly, any flow, either decaying or forced, may be tackled, with the same qualitative final result. The constants in front of term $R_{{\it\lambda}}^{-1}$ are flow and IBC dependent, as the precise form of $LSE$ changes from flow to flow.

We have thus reinforced the claim that $S_{1,1}+2G/R_{{\it\lambda}}$ tends to zero for sufficiently high Reynolds numbers (independently of the exact definition of the Reynolds number). Moreover, as already emphasized, because $G/R_{{\it\lambda}}$ becomes constant at $R_{{\it\lambda}}\simeq 70$ , then $S_{1,1}$ becomes constant for large Reynolds numbers, which is analogous to the normalized dissipation $C_{{\it\epsilon}}$ becoming constant at a sufficiently high Reynolds number.

Figure 13.

Dependence of $2G/R_{{\it\lambda}}$ on $R_{{\it\lambda}}$ in different flows. Channel centreline: ○, present data; grid turbulence: ▫, Zhou & Antonia (Reference Zhou and Antonia2000) and ▿, Larssen & Devenport (Reference Larssen and Devenport2011); SFPBT: △, Jimenez et al. (Reference Jimenez, Wray, Saffman and Rogallo1993); pipe axis: ♢, Rosenberg et al. (Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013).

Figure 14.

Dependence of $-S_{1,1}$ on $R_{{\it\lambda}}$ in different flows. (a) Channel centreline: ○, present data; ♢, Abe et al. (Reference Abe, Antonia and Kawamura2009); *, Vreman & Kuerten (Reference Vreman and Kuerten2014b ); pipe axis: ✩, Antonia & Pearson (Reference Antonia and Pearson2000a ); SFPBT: ▵, Jimenez et al. (Reference Jimenez, Wray, Saffman and Rogallo1993);  $\times$ , Kerr (Reference Kerr1985); ▹, Wang et al. (Reference Wang, Chen, Brasseur and Wyngaard1996); green curve, Qian (Reference Qian1994); EDQNM simulation of decaying HIT: black curve, Meldi & Sagaut (Reference Meldi and Sagaut2013). Red curve is inferred from $S_{1,1}=C/R_{{\it\lambda}}-2G/R_{{\it\lambda}}$ by assuming $2G/R_{{\it\lambda}}=\text{constant}$ ( ${\approx}0.52$ for $R_{{\it\lambda}}>70$ ) on the channel centreline (similarly, the blue curve in (b) is inferred from the same equation along the jet axis). (b) Grid turbulence: ▫, Zhou & Antonia (Reference Zhou and Antonia2000); ○, Mydlarski & Warhaft (Reference Mydlarski and Warhaft1996); jet axis: ●, Mi, Xu & Zhou (Reference Mi, Xu and Zhou2013); ▪, Lefeuvre, Djenidi & Antonia (Reference Lefeuvre, Djenidi and Antonia2015); ▵, P. Burattini (private communication);  $+$ , Kahalerras, Malecot & Gagne (Reference Kahalerras, Malecot and Gagne1998); ▿, Friehe et al. (Reference Friehe, Van Atta and Gibson1971). Some data in (a) are reproduced in (b) for the purpose of comparison.

5.3. Validation of equation (5.7)

Estimates of $2G/R_{{\it\lambda}}$ and $S_{1,1}$ for the present flow are shown in figures 13 and 14 by following the procedure outlined in § 4. Also included in figure 13 are estimates inferred from experimental and DNS data in other flows, such as SFPBT, along the axis of a pipe, and grid turbulence. As noted in the Introduction, there is abundant evidence to indicate that the ratio $G/R_{{\it\lambda}}$ approaches a constant relatively rapidly with increasing $R_{{\it\lambda}}$ , e.g. Lee et al. (Reference Lee, Djenidi, Antonia and Danaila2013). This is consistent with figure 13 which shows that, for all the flows considered here, $2G/R_{{\it\lambda}}$ is very nearly constant for $R_{{\it\lambda}}\geqslant 70$ . For $R_{{\it\lambda}}<70$ , $2G/R_{{\it\lambda}}$ appears to be flow dependent. For example, the approach to a constant value is slow in grid turbulence and rapid in SFPBT. For the present flow along the centreline of the channel, the approach is between grid turbulence and SFPBT. It should be noted that the constancy of $2G/R_{{\it\lambda}}$ is not an artefact of the extrapolation because the present extrapolated spectra collapse well with all other non-extrapolated spectra (see figure 9 d) in the dissipative range in accordance with Kolmogorov scaling. In his analysis, Champagne (Reference Champagne1978) used a ninth-order least-squares polynomial fit to extrapolate their spectrum and reported a value of 0.64 for $2G/R_{{\it\lambda}}$ (not shown in figure 13) on the centreline of the far field of a plane wake at $R_{{\it\lambda}}=182$ ( $x/d=299$ ). It is, however, likely that this polynomial fit strongly depends on the accuracy of the spectrum in the high-wavenumber range where the extrapolation is applied. The inadequacy of the spatial resolution ( ${\it\eta}/l_{w}=0.81$ ) tends to amplify the errors caused by the polynomial fit, thus yielding overestimated values of $2G/R_{{\it\lambda}}$ . Using either the spectral model of Pope (see § 4) or DNS spectra to extrapolate the spectrum of Champagne (Reference Champagne1978), we obtained a value of about 0.52, which is consistent with figure 13.

Figure 15.

Dependence of $S_{1,1}+2G/R_{{\it\lambda}}$ on $R_{{\it\lambda}}$ in different flows. Symbols are the experimental and numerical data. Channel centreline: ○, present data; grid turbulence: ▫, Zhou & Antonia (Reference Zhou and Antonia2000); SFPBT: ▵, Jimenez et al. (Reference Jimenez, Wray, Saffman and Rogallo1993); jet axis: ▪, Lefeuvre et al. (Reference Lefeuvre, Djenidi and Antonia2015). The four curves (solid, dashed, dash-dotted and dotted curves), inferred from (5.7) ( $C$ is flow dependent), correspond to grid turbulence, the centreline of a channel flow, the axis of a round jet, and SFPBT, respectively.

In contrast to figure 13, the distributions of $S_{1,1}$ (figure 14 a,b) exhibit significant differences between different flows. Figure 14(a) shows the data, both DNS and measurements, for $-S_{1,1}$ on the centreline of the channel flow as well as measurements on the centreline of a pipe, SFPBT and eddy-damped quasi-normal Markovian (EDQNM) results for decaying homogeneous and isotropic turbulence (Meldi & Sagaut Reference Meldi and Sagaut2013). Figure 14(b) compares $-S_{1,1}$ between the present measurements in the channel flow and measurements in other turbulent flows (e.g. grid turbulence and jet flows). As expected, the magnitude of $S_{1,1}$ is virtually constant in SFPBT; the value of 0.52 is in good agreement with the analytical prediction of Qian (Reference Qian1994). Along the axis of the jet flow, the magnitude of $S_{1,1}$ is somewhat smaller than that in SFPBT at small $R_{{\it\lambda}}$ but approaches quickly the value of 0.52 as $R_{{\it\lambda}}$ increases. The data shown include the measurements of Mi et al. (Reference Mi, Xu and Zhou2013), Lefeuvre et al. (Reference Lefeuvre, Djenidi and Antonia2015) with $d_{w}=1.24~{\rm\mu}\text{m}$ , P. Burattini (private communication), Friehe et al. (Reference Friehe, Van Atta and Gibson1971), and Kahalerras et al. (Reference Kahalerras, Malecot and Gagne1998). Except for Mi et al. (Reference Mi, Xu and Zhou2013), all the values of $-S_{1,1}$ were obtained by assuming that $\partial /\partial x=-(1/U_{c})(\partial /\partial t)$ , where $U_{c}\equiv U$ is the instantaneous velocity along $x$ . Kahalerras et al. (Reference Kahalerras, Malecot and Gagne1998) showed that this is preferable to the use of Taylor’s hypothesis ( $U_{c}\equiv \overline{U}$ ) which is inadequate when the local turbulence intensity is high (typically 25–30 %). The large value of $-S_{1,1}$ (0.68) reported by Champagne (Reference Champagne1978) on the jet axis is not included in figure 14 as we suspect that it has been adversely affected by corrections for Taylor’s hypothesis. Note that the values of Mi et al. (Reference Mi, Xu and Zhou2013) may also be overestimated since they also used Taylor’s hypothesis ( $U_{c}\equiv \overline{U}$ ), although they adopted an algorithm of high-order spectral-like stencils to calculate the velocity gradient with high accuracy. For the grid turbulence data, $-S_{1,1}$ first decreases slightly at low $R_{{\it\lambda}}$ and then, notwithstanding the scatter for the active grid turbulence data ( $90<R_{{\it\lambda}}<600$ , Mydlarski & Warhaft Reference Mydlarski and Warhaft1996), increases towards a constant. It should be noted that the EDQNM data which extend to $R_{{\it\lambda}}=10^{5}$ show that $-S_{1,1}$ remains constant when $R_{{\it\lambda}}\geqslant 1000$ . This is also observed (not shown here) in the EDQNM results of Bos et al. (Reference Bos, Chevillard, Scott and Rubinstein2012).

The smallest values of $-S_{1,1}$ are on the axis of the pipe (Antonia & Pearson Reference Antonia and Pearson2000a ). These values increase slightly in the range $60<R_{{\it\lambda}}<180$ and are roughly 40 % smaller than in SFPBT for the same Reynolds number range. The magnitude of $S_{1,1}$ on the centreline of the present channel flow is actually slightly larger than on the pipe axis. On the channel centreline, the magnitude of $S_{1,1}$ is smaller for our experiments than for the DNS data of Abe et al. (Reference Abe, Antonia and Kawamura2009) or Vreman & Kuerten (Reference Vreman and Kuerten2014b ). We suspect that this reflects differences in IBCs between simulations and experiments which lead to different magnitudes of the diffusion term and hence different values of $C$ in (5.7).

The main feature of figure 14 is that the magnitude of $S_{1,1}$ at a fixed $R_{{\it\lambda}}$ and its variation with $R_{{\it\lambda}}$ are flow dependent. The error bars on our data indicate clearly that the variation of $-S_{1,1}$ between different flows (at a fixed Reynolds number) exceeds the experimental uncertainty associated with the measurement of $-S_{1,1}$ in any particular flow. Indeed, Thiesset et al. (Reference Thiesset, Antonia and Djenidi2014) pointed out that since the imbalance between $S_{1,1}$ and $G$ depends on the type of large-scale forcing, the evolution of $S_{1,1}$ with $R_{{\it\lambda}}$ may differ from flow to flow. This is convincingly supported by figure 14. Most importantly, the distributions of $S_{1,1}$ , inferred from $S_{1,1}=C/R_{{\it\lambda}}-2G/R_{{\it\lambda}}$ after assuming that $2G/R_{{\it\lambda}}=\text{constant}$ ( ${\approx}0.52$ for $R_{{\it\lambda}}>70$ ) on the channel centreline and along the jet axis, respectively, are reasonably well supported by the experimental data.

Since $S_{1,1}$ and $2G/R_{{\it\lambda}}$ have now been measured independently, we can test (5.7). Figure 15 shows $S_{1,1}+2G/R_{{\it\lambda}}$ versus $R_{{\it\lambda}}$ in different flows, where $S_{1,1}$ and $2G/R_{{\it\lambda}}$ were also estimated independently. In this figure, the four curves (solid, dashed, dash-dotted and dotted curves), inferred from (5.7), where $C$ is a flow-dependent constant, correspond to grid turbulence with $n=1.1$ , the centreline of a fully developed channel flow, the axis of a round jet, and SFPBT, respectively. By definition, $C$ is zero for SFPBT. Symbols are used for the experimental and numerical data. A number of comments can be made based on figure 15. (i) Due to the fundamental constraint of the mean transport equation for $\overline{{\it\epsilon}}_{iso}$ , $S_{1,1}+2G/R_{{\it\lambda}}$ approaches its stationary state ( $S_{1,1}+2G/R_{{\it\lambda}}=0$ ) along different paths in different flows (four curves). The approach is slower along the centreline of a channel flow than along the axis of a round jet. Evidently, and not surprising, the approach is fastest for SFPBT. (ii) For grid turbulence, the centreline of the channel flow, and SFPBT, the approaches are reasonably well supported by experimental and numerical data despite the scatter. (iii) Since $2G/R_{{\it\lambda}}$ approaches a constant relatively rapidly with increasing $R_{{\it\lambda}}$ (in figure 13, $2G/R_{{\it\lambda}}\approx 0.52\pm 0.1$ for $R_{{\it\lambda}}\geqslant 70$ ), the different distributions of $S_{1,1}+2G/R_{{\it\lambda}}$ (or $S_{1,1}$ ) in different flows at $R_{{\it\lambda}}\geqslant 70$ are mainly due to the type of large-scale forcing (the term $C/R_{{\it\lambda}}$ in (5.7)), which differs from flow to flow. (iv) Since the term $C/R_{{\it\lambda}}$ should become negligible at sufficiently large $R_{{\it\lambda}}$ , $-S_{1,1}$ cannot grow unboundedly as $R_{{\it\lambda}}$ increases because $2G/R_{{\it\lambda}}$ becomes constant (e.g. figure 13). This would be consistent with K41 and the prediction of Qian (Reference Qian1994). It would also be consistent with the predictions of Tennekes (Reference Tennekes1968) based on a vortex tube model and the heuristic model of Saffman (Reference Saffman1970). It is not, however, consistent with K62 or the predictions from log-normal and fractal models (e.g. Frisch, Sulem & Nelkin Reference Frisch, Sulem and Nelkin1978; Van Atta & Antonia Reference Van Atta and Antonia1980; Sreenivasan & Antonia Reference Sreenivasan and Antonia1997). Strictly, the prediction from K62, like K41, should have been tested at very large $R_{{\it\lambda}}$ , and under conditions for which LI applies. Unfortunately, this has not been the case since laboratory data, typically obtained in the range $40<R_{{\it\lambda}}<1000$ , were used. For these data, the finite Reynolds number effect which is intimately linked to the dependence of $C$ on the flow, cannot be ignored. Nor can the departure from LI be completely dismissed. High- $R_{{\it\lambda}}$ atmospheric surface layer (ASL) data have also been used in conjunction with laboratory data for testing K62. Indeed, the ASL data have been instrumental in the context of providing support for K62. Since we cannot be certain that the ASL data, which were mostly obtained in relatively close proximity to the ground or ocean surface, have not been adversely affected by this latter effect and since there is no information about the mean-square vorticity budget, the most prudent course of action would be to avoid using these data when testing K62.