2. Dual scaling in the context of the Kármán–Howarth equation

We consider first the applicability of the inner (Kolmogorov scales, $\eta$ and $u_K$) and outer ($u'$ and $L$) scaling in the context of the Kármán–Howarth equation which governs the transport of $\overline {{(\delta u)}^2}$:

(2.1)\begin{equation} \frac{4}{5} \bar{\epsilon} r ={-} \overline{{(\delta u)}^3} + 6\nu \frac{\partial }{{\partial r}}\overline{{(\delta u)}^2} + I_u(r), \end{equation}

where $I_u(r)$ is the term which reflects the contribution from the large scales to the transport of $\overline {{(\delta u)}^2}$ in homogeneous and isotropic turbulence (e.g. Danaila et al. Reference Danaila, Anselmet, Zhou and Antonia1999; Hill Reference Hill2001). We will focus on decaying grid turbulence, partly because this flow can satisfy isotropy, at least for small scales, to a reasonable approximation and also because moderately large values of $Re_\lambda$ can be achieved, e.g. by using an active grid (e.g. Mydlarski & Warhaft Reference Mydlarski and Warhaft1996, Reference Mydlarski and Warhaft1998; Gylfason, Ayyalasomayajula & Warhaft Reference Gylfason, Ayyalasomayajula and Warhaft2004) or by varying the density of the fluid (e.g. Bodenschatz et al. Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014; Sinhuber et al. Reference Sinhuber, Bodenschatz and Bewley2015, Reference Sinhuber, Bewley and Bodenschatz2017; Kaminsky et al. Reference Kaminsky, Birnir, Bewley and Sinhuber2020). The departure from local isotropy at all scales can be quantified by the ratio between calculated and measured second-order spanwise velocity structure functions, i.e. $\overline {{{(\delta w)}^2}}_{iso}/\overline {{{(\delta w)}^2}}$, where $\overline {{{(\delta w)}^2}}_{iso}$ is given by (Monin & Yaglom Reference Monin and Yaglom2007)

(2.2)\begin{equation} \overline{{{(\delta w)}^2}}_{iso} = \left( {1 + \frac{r}{2}\frac{{\rm d}}{{{\rm d}r}}} \right)\overline{{{(\delta u)}^2}}. \end{equation}

At $r = \lambda$, a separation which resides near the lower end of the scaling range (Tang et al. Reference Tang, Antonia, Djenidi, Danaila and Zhou2017), the departure from local isotropy is relatively small ($\lesssim 10\,\%$, see figure 10 of Lavoie, Djenidi & Antonia (Reference Lavoie, Djenidi and Antonia2007) at $Re_\lambda =36\unicode{x2013}43$ in grid turbulence with and without a secondary contraction). Further, it was shown that the ratio $\overline {{{(\delta w)}^2}}_{iso}/\overline {{{(\delta w)}^2}}$ satisfies isotropy within approximately $10\,\%$ at all separations when a secondary contraction is introduced. This result is consistent with that of Comte-Bellot & Corrsin (Reference Comte-Bellot and Corrsin1966) and Comte-Bellot & Corrsin (Reference Comte-Bellot and Corrsin1971) who used a contraction to improve the isotropy of grid turbulence, with $u'$ being nearly equal to $v'$. It is worth mentioning that since the magnitude of $Re_\lambda$ in the present paper is much larger than that of Lavoie et al. (Reference Lavoie, Djenidi and Antonia2007), we thus expect a further improved local isotropy, compared with that of Lavoie et al. (Reference Lavoie, Djenidi and Antonia2007). Also, since isotropy requires homogeneity, and isotropy at all scales is approximately satisfied in grid turbulence, the weak spatial decay along $x$, the mean flow direction, should not affect the behaviour of $\overline {{{(\delta u)}^n}}$. In other words, the influence of any small inhomogeneity in the $x$ direction on the behaviour of $\overline {{{(\delta u)}^n}}$ should be negligible. Further, it is well established, based on the KH equation, that the effect of the spatial decay along the $x$ direction in the scaling range weakens as $Re_\lambda$ increases (Antonia & Burattini Reference Antonia and Burattini2006; Tang et al. Reference Tang, Antonia, Djenidi, Danaila and Zhou2017; Antonia et al. Reference Antonia, Tang, Djenidi and Zhou2019). A further attractive feature of grid turbulence is that the energy budget is relatively simple and can hence be measured fairly accurately thus yielding a reliable estimation of $\bar {\varepsilon }$ (e.g. Antonia, Zhou & Zhu Reference Antonia, Zhou and Zhu1998).

The two main items of interest here are the conditions for which inner (or Kolmogorov) scaling ($u_K$, $\eta$) and outer scaling ($u'$, $L$) satisfy the similarity of (2.1).

(i) When the effect of the large scale term $I_u(r)$ is neglected, the similarity of (2.1) is readily satisfied by

(2.3)$$\begin{gather} \frac{{\overline{{{(\delta u)}^2}} }}{{u_0^2}} = f\left( {\frac{r}{{{l_0}}}} \right), \end{gather}$$

(2.4)$$\begin{gather}\frac{{\overline{{{(\delta u)}^3}} }}{{u_0^3}} = g\left( {\frac{r}{{{l_0}}}} \right), \end{gather}$$

when $u_0 \equiv u_K$ and $l_0 \equiv \eta$, viz. the ($u_K$, $\eta$) scaling applies. This was discussed in some detail by Antonia, Djenidi & Danaila (Reference Antonia, Djenidi and Danaila2014). Equation (2.1) can be rewritten as

(2.5)\begin{equation} \frac{4}{5}\frac{r}{{{l_0}}} = \frac{{6\nu u_0^2}}{{\bar{\epsilon} l_0^2}}f' - \frac{{u_0^3}}{{\bar{\epsilon} {l_0}}}g\left( {\frac{r}{{{l_0}}}} \right) \end{equation}

(here, the prime signifies a derivative with respect to $r/l_0$), so that $\bar {\varepsilon } {l_0}/u_0^3$ and $\bar {\varepsilon } l_0^2/\nu u_0^2$ (or $\bar {\varepsilon } {l_0}/u_0^3$, ${u_0}{l_0}/\nu$) are constants. Clearly, ($u_K$, $\eta$) is a possible solution since

(2.6a,b)\begin{equation} \frac{\bar{\varepsilon} {\eta}}{u_K^3}=1 \quad {\rm and} \quad \frac{{u_K}{\eta}}{\nu}=1. \end{equation}

Scaling based on ($u'$, $L$) is also possible provided

(2.7a,b)\begin{equation} \frac{\bar{\epsilon} {L}}{u'^3}=C_\epsilon={\rm constant} \quad {\rm and} \quad \frac{{u'}{L}}{\nu}={\rm constant}. \end{equation}

The first requirement in (2.7a,b) has been validated in several different flows (e.g. Sreenivasan Reference Sreenivasan1984; Burattini, Lavoie & Antonia Reference Burattini, Lavoie and Antonia2005b; Ishihara, Gotoh & Kaneda Reference Ishihara, Gotoh and Kaneda2009; Mi, Xu & Zhou Reference Mi, Xu and Zhou2013; Vassilicos Reference Vassilicos2015; McComb et al. Reference McComb, Berera, Yoffe and Linkmann2015), at least when $Re_\lambda$ is sufficiently large. The second can, to our knowledge, hold only in the far field of a circular jet (Antonia, Satyaprakash & Hussain Reference Antonia, Satyaprakash and Hussain1980; Burattini, Antonia & Danaila Reference Burattini, Antonia and Danaila2005a) when $Re_\lambda$ is finite. It should however be satisfied in decaying grid turbulence when $Re_\lambda \to \infty$ (e.g. George Reference George1992; Speziale & Bernard Reference Speziale and Bernard1992; Antonia et al. Reference Antonia, Smalley, Zhou, Anselmet and Danaila2003). The above observations lead to the conclusion that the scaling ($u_K$, $\eta$) should be effective regardless of the flow since the two dimensionless parameters in (2.6a,b) are indeed universal (with a value of 1). Nevertheless, the neglect of the large-scale term (2.5) for values of $r/\eta$ that extend into the IR is unlikely to be fully justifiable unless $Re_\lambda \to \infty$. Neither of the parameters in (2.7a,b) is likely to be flow independent; indeed, there is adequate evidence (e.g. Sreenivasan Reference Sreenivasan1984; Burattini et al. Reference Burattini, Lavoie and Antonia2005b; Mi et al. Reference Mi, Xu and Zhou2013; Vassilicos Reference Vassilicos2015) that the magnitude of $C_\epsilon$ depends on the flow.

(ii) $I_u(r)$ is now retained and the viscous term in (2.1) is neglected. The focus now is on the larger scales of motion. Equation (2.1) can be rewritten as

(2.8)\begin{equation} \frac{4}{5}\frac{r}{{{l_0}}} ={-} \frac{{u_0^3}}{{\bar{\varepsilon} {l_0}}}g\left( {\frac{r}{{{l_0}}}} \right) + \frac{{{I_u}(r)}}{{\bar{\varepsilon} {l_0}}}. \end{equation}

We concentrate on grid turbulence, for which

(2.9)\begin{equation} I_u(r)={-} \frac{3}{{{r^4}}}\int_0^r {{s^4}}\left[ U\frac{{\partial \overline{(\delta u)^2} }}{{\partial x}}\right]\,{\rm d}s, \end{equation}

where $s$ is a dummy variable, identifiable with the separation along $x$, and $U$ is the (constant) mean velocity in the $x$ direction (e.g. Danaila et al. Reference Danaila, Anselmet, Zhou and Antonia1999; Antonia et al. Reference Antonia, Zhou, Danaila and Anselmet2000; Danaila, Anselmet & Antonia Reference Danaila, Anselmet and Antonia2002). Using (2.9), ${I_u}/\bar {\varepsilon } {l_0}$ becomes

(2.10)\begin{equation} \frac{{3}}{{{{(r/{l_0})}^4}}}\left( {\int_0^{r/{l_0}} {{{\left( {\frac{s}{{{l_0}}}} \right)}^4}f\left( {\frac{r}{{{l_0}}}} \right)\frac{{{\rm d}s}}{{{l_0}}}} } \right)\left[ {\frac{U}{{\bar{\varepsilon} }}\frac{{\partial u_0^2}}{{\partial x}} - \frac{{Uu_0^2}}{{\bar{\varepsilon} }}{{\left( {\frac{r}{{{l_0}}}} \right)}^2}\frac{f'}{{{l_0}}}\frac{{{\rm d}{l_0}}}{{{\rm d}\kern0.06em x}}} \right]. \end{equation}

The expression within square brackets must be constant for similarity to be satisfied. The first term of this expression is indeed constant since $\bar \epsilon =-\tfrac {3}{2}U({{\partial {{u_0^2}} }}/{{\partial x}})$ is the turbulent energy budget (when $u_0 \equiv u'$). The second term is also constant with the ($u'$, $L$) scaling since $\bar {\varepsilon } {L}/u'^3$ is constant and ${({1}/{{{u'}}})({{{\rm d}{L}}}/{{{\rm d}\kern0.06em x}})}$ is also constant as both $u'$ and ${\rm d}L/{{\rm d}\kern0.06em x}$ vary as $x^{n/2}$ when the energy decays according to a power-law, viz. $\overline {u^2}\sim x^{n}$ and $\bar {\epsilon }\sim x^{n-1}$; the latter two relations, together with $C_\varepsilon =\bar {\epsilon } L/u'^3={\rm const}$, lead to $L \sim x^{n/2+1}$. It is worth mentioning that $U$ in the second term within the square brackets does not affect the above analysis and the corresponding results since we can replace $U/{{\rm d}\kern0.06em x}$ with $1/{\rm d}t$ in grid turbulence. The previous similarity requirements were previously outlined by Kármán & Von Howarth (Reference Von Kármán and Howarth1938) and Townsend (Reference Townsend1976). We conclude that the similarity of (2.8), i.e. the transport equation of ${\overline {{{(\delta u)}^2}} }$ when $r$ is sufficiently large to allow the viscous term to be neglected, is satisfied by the ($u'$, $L$) scaling. We should note however that the ($u_K$, $\eta$) scaling can also satisfy similarity of (2.8) when $n=-1$ (in this case, both ${({1}/{{\bar {\varepsilon } }})({{\partial u_K^2}}/{{\partial x}})}$ and ${({1}/{{u_K }})({{\partial \eta }}/{{\partial x}})}$ are constant). This rate of decay was shown by Speziale & Bernard (Reference Speziale and Bernard1992) to be the asymptotically consistent high-Reynolds-number solution. The condition ${{u'}{L}}/{\nu }={\rm constant}$ in (2.7a,b) is immediately satisfied in this case so that the scales ($u'$, $L$), just like ($u_K$, $\eta$), would satisfy complete similarity (or self-preservation), i.e. similarity for all scales of motion. The Taylor microscale $\lambda$ should then be a valid replacement for $L$ since $Re_\lambda$ should no longer depend on $x$ and $\lambda$ should be proportional to $L$. The general self-preservation analysis of Djenidi & Antonia (Reference Djenidi and Antonia2015) confirmed the dependence of $n$ on the initial conditions and showed how this dependence is related to the initial length and velocity scales. These authors suggested that it may be possible to have initial conditions that lead to $n=-1$ with a finite $Re_\lambda$. However, $Re_\lambda$ must remain constant throughout the decay.

The available grid turbulence data in the literature indicate that $n$ is typically smaller than $-1$, usually with a value ranging between $-1.1$ and $-1.5$ (e.g. George Reference George1992; Lavoie et al. Reference Lavoie, Djenidi and Antonia2007; Lee et al. Reference Lee, Djenidi, Antonia and Danaila2013; Sinhuber et al. Reference Sinhuber, Bodenschatz and Bewley2015). In particular, Sinhuber et al. (Reference Sinhuber, Bodenschatz and Bewley2015) showed that $n$ is independent of the Reynolds number, with a value of $n = - 1.18 \pm 0.02$, over a wide range of Reynolds numbers; this value is close to the prediction of Saffman (Reference Saffman1967) ($n =-1.2$). Lavoie et al. (Reference Lavoie, Djenidi and Antonia2007) showed experimentally that even when the turbulence is nearly perfectly isotropic, $n$ is not equal to $-1$. Our interest is to assess the validity of the dual scaling and the range of $r/l_0$ over which it applies. The scaling based on ($u_K$, $\eta$) is expected to extend to increasingly larger values of $r/\eta$ as $Re_\lambda$ increases. Correspondingly, the scaling based on ($u'$, $L$) is expected to extend to increasingly smaller values of $r/L$ as $Re_\lambda$ increases. Since both scalings must eventually overlap as $Re_\lambda \to \infty$, the overlap region should include the inertial range. Regardless of the scaling used, this overlap region should satisfy similarity and hence be independent of $Re_\lambda$. All of the above analysis is focused on the grid turbulence where the influence of large scales on $-\overline {{{(\delta u)}^3}}$ can be quantified by the $I_u(r)$ term in (2.9). If one considers different flows with different large-scale forcing, the influence of the large-scale motion is likely to differ from that in grid turbulence. Accordingly, in those situations, the flow may locally reflect the large-scale anisotropy at finite $Re_\lambda$ imposed by the forcing. For example, when anisotropic forcing is used in direct numerical simulations of box turbulence at $Re_{q\lambda }=20\unicode{x2013}32$ ($=\overline {u_iu_i}^{1/2}\lambda /\nu$), it induces small-scale anisotropy which is traced to the dynamics of non-local triads with one low wavenumber in the directly forced wavenumber band (Yeung & Brasseur Reference Yeung and Brasseur1991). Also, Kurien & Sreenivasan (Reference Kurien and Sreenivasan2000) observed that the structure functions of order two to six exhibit a strong anisotropy at moderate scales in the atmospheric surface layer at $Re_\lambda =870\unicode{x2013}2100$.

3. Basic flow parameters in grid turbulence

The grid turbulence data we used for our study were digitized from figures presented by Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) and Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) ($Re_\lambda =110\unicode{x2013}1620$). Since all the experimental data reported here have already been published, the reader can find detailed descriptions of the experimental conditions and measurement techniques in the original papers. Here we only describe and discuss the basic flow parameters associated with those two datasets and explain the treatment we have had to apply to these data.

Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) estimated $\bar {\varepsilon }$ from $\overline {(\delta u)^3}$ by assuming $\bar {\varepsilon }=\max (-5/4\overline {(\delta u)^3}/r)$, i.e. assuming that (1.2) is tenable. This would result in $\bar {\varepsilon }$ being underestimated since the maximum value of $\overline {(\delta u)^3}/ \bar {\varepsilon } r$ has not yet reached the value of 4/5 for the available data in grid turbulence (see for example figure 1a of Antonia et al. (Reference Antonia, Tang, Djenidi and Zhou2019)). Indeed, their magnitudes of $C_\epsilon$ ($\equiv \bar {\varepsilon } L/u'^3=0.58$, 0.71, 0.83, 0.77, 0.77 and 0.75 at $Re_\lambda=150$, 400, 549, 730, 1370 and 1620, respectively, Bodenschatz et al. Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) are smaller than those of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) ($C_\varepsilon=1.06$, 1.12, 0.91, 0.91 and 0.81 at $Re_\lambda=110$, 264, 508, 1000 and 1450, respectively), in which $\bar {\varepsilon }$ was estimated from the turbulent energy budget $\bar {\epsilon }=-\tfrac {3}{2}U({{\partial {{u'^2}} }}/{{\partial x}})$. Also, the spectra of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) in the dissipative range do not collapse, see figure 1(b), which is reproduced from their figure 30. To obtain a more accurate estimation of $\bar {\epsilon }$, the spectral chart method of Djenidi & Antonia (Reference Djenidi and Antonia2012) has been applied to the spectra of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) (their figure 30, or figure 1(b) in the present paper). The ‘new’ estimates of $\bar {\varepsilon }$ are denoted by $\bar {\varepsilon }_{spec}$. This method is based on the observation that there is collapse in the upper part of the dissipative range not only for grid turbulence, but also for many other turbulent flows; see for example figure 9 of Saddoughi & Veeravalli (Reference Saddoughi and Veeravalli1994), figure 6.14 of Pope (Reference Pope2000), figure 3 of Gotoh, Fukayama & Nakano (Reference Gotoh, Fukayama and Nakano2002), figure 5 of Larssen & Devenport (Reference Larssen and Devenport2011), figures 1–2 of Antonia et al. (Reference Antonia, Djenidi and Danaila2014) and figure 1(a) of Tang et al. (Reference Tang, Antonia, Djenidi and Zhou2020). Therefore, the values of $\bar {\varepsilon }_{spec}$ ensure that there is collapse in the upper part of the dissipative range, as illustrated in figure 1(a) where the distributions are normalized by $\bar {\varepsilon }_{spec}$ and $\nu$. For comparison, we have included the grid turbulence data of Larssen & Devenport (Reference Larssen and Devenport2011) at $Re_\lambda= 124$ and 1360. It is worth mentioning that all the data of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) were obtained in passive grid turbulence. However, the data of Larssen & Devenport (Reference Larssen and Devenport2011) at $Re_\lambda =124$ and 1362 were measured in passive and active grid turbulence, respectively. The distribution of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) is in almost perfect agreement with that of Larssen & Devenport (Reference Larssen and Devenport2011) at $Re_\lambda =124$ at all scales. This is not too surprising since both distributions were measured in passive grid turbulence at the same $Re_\lambda$. We can observe that there is a large spike at low wavenumbers (around $k_1\eta \approx 0.0002$) for the data of Larssen & Devenport (Reference Larssen and Devenport2011) at $Re_\lambda =1362$. This is due to the rotation of the active grid which ‘introduces additional energy into the spectra’ (Larssen & Devenport Reference Larssen and Devenport2011). It is plausible that this spike, which reflects the presence of a strong coherent motion, may affect the medium wavenumber range, i.e. $0.002\lesssim k_1\eta \lesssim 0.04$, over which the magnitude of the energy spectrum at $Re_\lambda =1362$ is larger than all others. We recall that Larssen & Devenport (Reference Larssen and Devenport2011) calculated $\bar {\varepsilon }$ by integrating the longitudinal dissipation spectrum, i.e. $\bar {\varepsilon } = 15\nu \int {k_1^2{\phi _u}({k_1})d} {k_1}$. Local isotropy requires that the integration of the Kolmogorov-normalized longitudinal dissipation spectrum is equal to 1/15 ($\equiv 0.0667$). Although not shown here, we should stress that the dissipation spectrum at $Re_\lambda =1362$ is in good agreement with all other distributions at all scales for $Re_\lambda \leq 701$, confirming the accuracy of the dissipation spectrum at $Re_\lambda =1362$. More importantly, the integration across all scales at $Re_\lambda =1362$ leads to 0.069, which is only 3.5 % larger than the isotropic value of 1/15. This further indicates that the magnitude of $\bar {\varepsilon }$ at $Re_\lambda =1362$ should be reasonably accurate. More importantly, there is reasonable agreement between the ‘$\bar {\varepsilon }_{spec}$’-normalized spectra and the spectra of Larssen & Devenport (Reference Larssen and Devenport2011) for $k_1\eta \gtrsim 0.3$ at all $Re_\lambda$. In contrast, as seen in figure 1(b), there is no collapse in the upper part of the dissipative range since $\bar {\epsilon }$ was underestimated by using the maximum value of $-\overline {(\delta u)^3}/ \bar {\varepsilon } r$ at finite $Re_\lambda$. In particular, as $Re_\lambda$ increases, the approach to a $-$5/3 scaling of the spectra, as reflected by the nearly horizontal distributions, is similar to that of Mydlarski & Warhaft (Reference Mydlarski and Warhaft1996). The magnitudes of $C_\varepsilon$, based on $\bar {\varepsilon }_{spec}$, are close to those of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) (figure 2), confirming that $\bar {\varepsilon }_{spec}$ should be very close to the true value $\bar {\varepsilon }$. To further confirm the magnitudes of $\bar {\varepsilon }_{spec}$, we have also included the grid turbulence data of Comte-Bellot & Corrsin (Reference Comte-Bellot and Corrsin1971) (${Re}_{\lambda }=61\unicode{x2013}72$, ) and Larssen & Devenport (Reference Larssen and Devenport2011) (${Re}_{\lambda }=101\unicode{x2013}1362$, )) (see figure 2). We can observe that the magnitudes of $C_\varepsilon$, based on $\bar {\varepsilon }_{spec}$, are also close to those of Comte-Bellot & Corrsin (Reference Comte-Bellot and Corrsin1971) and Larssen & Devenport (Reference Larssen and Devenport2011), especially for ${Re}_{\lambda }\lesssim 300$ and ${Re}_{\lambda }\gtrsim 900$; the scatter for the data of Larssen & Devenport (Reference Larssen and Devenport2011) in the range $300 \lesssim {Re}_{\lambda }\lesssim 900$ is relatively large. We recall that $C_\varepsilon$ in stationary forced periodic box turbulence follows a functional form (Donzis, Sreenivasan & Yeung Reference Donzis, Sreenivasan and Yeung2005):

(3.1)\begin{equation} {C_\varepsilon } = A(1 + \sqrt {1 + {{(B/{Re_\lambda })}^2}} ), \end{equation}

where $A=0.2$ and $B=92$ (see green curve in figure 2). When using (3.1) to fit the grid turbulence data in figure 2, we observe that the asymptotic value of ${C_\varepsilon }$ in grid turbulence is approximately 0.8, which corresponds to $A=0.4$. Here, $B$ is essentially constant which reflects the finite Reynolds number effect on ${C_\varepsilon }$ in this flow. We take $B=60$ which appears to fit all the grid turbulence data reasonably well. We finally compare the $\overline {(\delta u)^2}/u_K^2$ distributions, calculated from the Kolmogorov-normalized energy spectra (figure 1a) using the following relation (e.g. Dickey & Mellor Reference Dickey and Mellor1979; Monin & Yaglom Reference Monin and Yaglom2007):

(3.2)\begin{equation} \overline{{{(\delta {u})}^2}} /u_K^2 = 2\int_0^\infty {{\phi _{{u}}}(k_1\eta)} (1 - \cos (k_1r))\,{\rm d}(k_1\eta), \end{equation}

with the other grid turbulence data (Comte-Bellot & Corrsin Reference Comte-Bellot and Corrsin1971; Malecot Reference Malecot1998; Zhou & Antonia Reference Zhou and Antonia2000; Larssen & Devenport Reference Larssen and Devenport2011) (see figure 3). Evidently, the distributions of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) are quite close to the other grid turbulence data at all scales when ${Re_\lambda }$ is comparable and in the dissipative range, regardless of ${Re_\lambda }$. Based on the above analysis, we can conclude that the magnitudes of $\bar {\varepsilon }_{spec}$, obtained from the spectral chart method, are reasonably accurate and reliable. The values of $Re_\lambda$ are then re-estimated from the isotropic relation $\overline {u^2}/u_K^2=Re_\lambda /15^{1/2}$. The final values are 124, 379, 495, 701, 1310 and 1558, which are slightly smaller than the values of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014), i.e. $Re_\lambda =150, 400, 549, 730, 1370$ and 1620, respectively. For reference, we have also included the data for stationary forced periodic box turbulence. We can observe from figure 2 that $C_\varepsilon$ is close to 0.8 at large $Re_\lambda$ for the grid turbulence and to 0.4 for the stationary forced periodic box turbulence. This difference is consistent with the difference observed in $L/\eta$ since ${C_\varepsilon }$ can be written as

(3.3)\begin{equation} {C_\varepsilon } = \frac{{\bar{\varepsilon} L}}{{u'{^3}}} = \frac{{L/\eta }}{{u'{^3}/u_K^3}} = \frac{{L/\eta }}{{Re_\lambda ^{3/2}/{{15}^{3/4}}}}, \end{equation}

where the isotropic relation $u{'^2}/u_K^2 = {Re_\lambda }/{15^{1/2}}$ has been used. A plausible reason for the difference in the magnitude of ${C_\varepsilon }$ between grid and box turbulence is the difference in $L/\eta$ at any given $Re_\lambda$, which reflects, at least partially, the different types of large-scale forcing between these two flows. For example, $L/\eta =2522$ at $Re_\lambda =1300$ in box turbulence (Iyer, Sreenivasan & Yeung Reference Iyer, Sreenivasan and Yeung2020), whereas $L/\eta =5516$ at $Re_\lambda =1360$ in grid turbulence (Larssen & Devenport Reference Larssen and Devenport2011). Tsuji (Reference Tsuji2009) observed a difference of a factor of two in $L/\eta$ between the DNS data and experimental data in most other flows (see their figure 1b). This difference in $L/\eta$ is equivalent to the factor of two difference in ${C_\varepsilon }$ between grid turbulence and box turbulence.

Figure 1. (a) Kolmogorov normalized compensated spectra of $u$ in grid turbulence (Bodenschatz et al. Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) ($Re_\lambda=$ 110–1450), normalized by $\bar {\epsilon }_{spec}$ and $\nu$ (the spectral chart method). For comparison, we have included the grid turbulence data of Larssen & Devenport (Reference Larssen and Devenport2011) (dashed curves) at $Re_\lambda = 124$ and 1360. (b) Corresponding grid turbulence data of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) ($Re_\lambda = 110\unicode{x2013}1450$), normalized by $\bar {\epsilon }$, estimated from $\overline {(\delta u)^3}$ (see text), and $\nu$; note that to see the dissipative range more clearly, a smaller range $k_1\eta =0.01\unicode{x2013}1$ is used for the vertical axis.

Figure 2. Distributions of $C_\epsilon = \bar {\epsilon } L /u'^3$ versus ${Re}_{\lambda }$ in grid turbulence. ( red), (Kaminsky et al. Reference Kaminsky, Birnir, Bewley and Sinhuber2020); ( blue), re-estimated $C_\epsilon$, based on $\bar {\epsilon }_{spec}$ (the spectral chart method), versus $Re_\lambda$ for the data of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) (see text). For comparison, we have included the grid turbulence data of Comte-Bellot & Corrsin (Reference Comte-Bellot and Corrsin1971) () and Larssen & Devenport (Reference Larssen and Devenport2011) (). Black curve, (3.1) with $A=0.4$ and $B=60$. Also included are the data in stationary forced periodic box turbulence (($\bigcirc$ black), Wang et al. (Reference Wang, Chen, Brasseur and Wyngaard1996); ($\bigcirc$ blue), Ishihara & Kaneda (Reference Ishihara and Kaneda2002); (${\ast }$ black), Jimenez et al. (Reference Jimenez, Wray, Saffman and Rogallo1993); ( green), Ishihara et al. (Reference Ishihara, Gotoh and Kaneda2009); ($+$ black), Cao, Chen & Doolen (Reference Cao, Chen and Doolen1999); ( blue), Yeung & Zhou (Reference Yeung and Zhou1997); ( blue), Gotoh et al. Reference Gotoh, Fukayama and Nakano2002). Green curve, (3.1) with $A=0.2$ and $B=92$.

Figure 3. Distributions of $\overline {(\delta u)^2}/u_K^2$ versus $r/\eta$ in grid turbulence for the data of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) at ${Re_\lambda }=124$, 379, 495, 701, 1310 and 1558 (green curves). For comparison, other grid turbulence data are also shown; red curves, Zhou & Antonia (Reference Zhou and Antonia2000) (${Re_\lambda }=27$, 51, 62, 75, 83, 89 and 100); cyan curves, Comte-Bellot & Corrsin (Reference Comte-Bellot and Corrsin1971) (${Re_\lambda }=61$ and 65); black curves, Larssen & Devenport (Reference Larssen and Devenport2011) (${Re_\lambda }=124$ and 1360); blue curve, Malecot (Reference Malecot1998) (${Re_\lambda }=144$); pink curve, $(r/\eta )^2/15$. The arrow indicates the direction of increasing ${Re_\lambda }$. Note that the data of Comte-Bellot & Corrsin (Reference Comte-Bellot and Corrsin1971), Larssen & Devenport (Reference Larssen and Devenport2011) and Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) are calculated from their spectra, based on (3.2).

Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) have shown distributions of $\overline {(\delta u)^2}$, $\overline {|\delta u|^3}$, $\overline {(\delta u)^4}$, $\overline {(\delta u)^6}$ and $\overline {(\delta u)^8}$ in grid turbulence at $Re_\lambda = 110$, 264, 508, 1000 and 1450, respectively; the $\overline {(\delta u)^2}{/u_K^2}$ distributions are shown in figure 4(a) (green curves). Figures 1–5 of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) were digitized to obtain $\overline {{(\delta u)^2}}$ distributions. We then divided $\overline {{(\delta u)^2}}$ by $u_K^2$ ($u_K$ was calculated from $\eta$ and $\nu$ shown in their table 1). For comparison, also included are the other grid turbulence data, same as in figure 3. It can be observed that the distributions of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) strongly depart from the other distributions. This is because they actually plotted $\overline {{(\delta u)^n}}/u_K^n$ versus $r/(\eta M)$, instead of $\overline {{(\delta u)^n}}/u_K^n$ versus $r/\eta$. We recall that there is an overwhelming amount of data confirming that the $\overline {(\delta u)^2}/u_K^2$ distributions collapse in the dissipative range over a large range of ${Re_\lambda }$ in various flows (e.g. Pearson & Antonia Reference Pearson and Antonia2001; Gotoh et al. Reference Gotoh, Fukayama and Nakano2002; Ishihara et al. Reference Ishihara, Gotoh and Kaneda2009; McComb et al. Reference McComb, Yoffe, Linkmann and Berera2014). The same is observed in grid turbulence (figure 3). It is then expected that the distributions of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) should collapse with those reported in the literature, at least in the small-scale range, if one removes the factor $M$ from the ratio $r/(\eta M)$. Since the value of $M$ was not provided by Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020), we replotted their data with $r/(\eta M)$ multiplied by different values of $M$ so the distributions align with with other grid turbulence data (figure 4b). The values of $M$ are 29, 6.9, 2.96, 1 and 0.54 at $Re_\lambda =110$, 264, 508, 1000 and 1450, respectively; see also table 1. These values will be used to plot $\overline {(\delta u)^4}/u_K^4$ and $\overline {(\delta u)^6}/u_K^6$ versus $r/\eta$, which will be presented and discussed later. In other words, we do not use ‘$M$’ to correct $\eta$ (or equivalently $u_K$); instead, it is simply a means of forcing agreement with the other grid turbulence data in the dissipative range. As discussed above, the universality of this range is supported by an overwhelming amount of data over a large range of ${Re_\lambda }$ in various flows, as expected on the basis of the first similarity hypothesis of K41. Finally, we compare the distributions of $\overline {(\delta u)^4}/u_K^4$ at ${Re_\lambda }=110$ and 508 with those of Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004) at comparable ${Re_\lambda }$ (168 and 656) in grid turbulence (figure 5a). It is worth mentioning that $\overline {(\delta u)^4}/u_K^4$ leads to $15^{-2}F_4(r/\eta )^4$ as $r/\eta \to 1$. Also, we can observe (later from figure 16) that $F_4$ at ${Re_\lambda }=550$ and 656 are close to each other; the same can be observed for ${Re_\lambda }=110$ and 168. This implies that the distributions at ${Re_\lambda }=550$ and 656 (and also those at ${Re_\lambda }=110$ and 168) should be close to each other at small $r/\eta$. These two expectations are indeed observed in figure 5(a) in the separation range $r/\eta \lesssim 20\unicode{x2013}30$. Figure 5(a) further justifies our use of the numerical factor $M$ on the data of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020). As mentioned above, $\overline {(\delta u)^4}/u_K^4$ approaches $15^{-2}F_4(r/\eta )^4$ as $r/\eta \to 1$. Also, since $F_4$ increases systematically with increasing $Re_\lambda$ for ${Re_\lambda }\lesssim 500$ (see figure 16 later), the $\overline {(\delta u)^4}/u_K^4$ distribution at $Re_\lambda =264$ at small $r/\eta$ must lie between the distributions of $Re_\lambda=110$ and 508. Yet, we can observe from figure 5(b) that, when $M=6.9$, the distribution at $Re_\lambda =264$ is smaller than that at $Re_\lambda =110$; this is incorrect and should be rectified. The value $M=10$, which ensures that the distribution at $Re_\lambda =264$ at small $r/\eta$ lies between the distributions of $Re_\lambda = 110$ and 508, as illustrated in figure 5(b), is a reasonable value for plotting $\overline {(\delta u)^4}/u_K^4$ at $Re_\lambda =264$ (see figure 5b). Also, the data for $\overline {(\delta u)^8}$ are not presented in the present paper since their behaviour is similar to that of $\overline {(\delta u)^6}$. As mentioned above, Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) have also shown the $\overline {|\delta u|^3}$ distributions. In the present paper, we will only discuss the behaviour of $\overline {(\delta u)^3}$ using the data of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) in the same flow, instead of $\overline {|\delta u|^3}$, since there is an exact equation for the former quantity, as discussed in § 1. It is worth noting that the data mentioned above were measured by a hot-wire anemometry and Taylor's hypothesis was used to calculate $\overline {(\delta u)^n}$. It is generally accepted that this hypothesis is reliable when the mean rate of shear and turbulence intensity are small. Indeed, according to He, Jin & Yang (Reference He, Jin and Yang2017), in a flow such as grid turbulence where the mean shear rate is zero, the conversion from time to space can be written as

(3.4)\begin{equation} r = \sqrt {1 + (u'/U)^2} Ut. \end{equation}

For the VDTT grid turbulence data, the turbulence intensity $u'/U$ is only in the range $2.35\,\%\sim 3.7\,\%$ (see table II of Bodenschatz et al. Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014), and thus justifies the use of Taylor's hypothesis. Pope (Reference Pope2000) pointed out that ‘in grid turbulence with $u'/U\ll 1$, it [Taylor's hypothesis] is quite accurate.’

Figure 4. (a) Distributions of $\overline {(\delta u)^2}/u_K^2$ versus $r/\eta$ in grid turbulence for the data of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) at ${Re_\lambda }=110$, 264, 508, 1000 and 1450 (green curves). Panel (b) corresponds to the same data, after applying a numerical factor $M$ to their data (see text). For comparison, other grid turbulence data, same as in figure 3, are also shown in (a,b). Pink curve, $(r/\eta )^2/15$. The arrow indicates the direction of increasing ${Re_\lambda }$.

Figure 5. (a) Distributions of $\overline {(\delta u)^4}/u_K^4$ versus $r/\eta$ in grid turbulence (Kaminsky et al. Reference Kaminsky, Birnir, Bewley and Sinhuber2020) after applying a numerical factor $M(=29$ and 2.96 for $Re_\lambda=110$ and 508) to their data (see text). For comparison, the grid turbulence data of Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004) (without shear) at $Re_\lambda=168$ (pink curve) and 656 (cyan curve) are shown; they are reproduced from figure 5(c,d) of Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004). (b) Distributions of $\overline {(\delta u)^4}/u_K^4$ versus $r/\eta$ in grid turbulence (Kaminsky et al. Reference Kaminsky, Birnir, Bewley and Sinhuber2020) at $Re_\lambda =264$ after applying a numerical factor $M(=6.9$ and 10) to their data (see text). The dotted curve, $15^{-2}F_4(r/\eta )^4$ with $F_4=9.6$. Other curves are the same as (a).

Table 1. Values of $M$ used to plot $\overline {(\delta u)^2}/u_K^2$, $\overline {(\delta u)^4}/u_K^4$ and $\overline {(\delta u)^6}/u_K^6$ versus $r/\eta$ at each $Re_\lambda$ for the data of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020) (see text). Note that a value of 10, instead of 6.9, is used to plot $\overline {(\delta u)^4}/u_K^4$ at $Re_\lambda =264$ since its distribution should lie between the distributions of $Re_\lambda =110$ and 508 in the dissipative range (see figure 5).

4. Dual scaling and its consequences

We show in figures 6–9 the distributions of $\overline {(\delta u)^n}$ ($n=2,3,4,6$) normalized by ($u_K$, $\eta$) and ($u'$, $L$) in grid turbulence at $Re_\lambda = 110\unicode{x2013}1558$, using the data of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) (after reevaluating $\bar {\varepsilon }$, as discussed in § 3) and Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020). We can observe from figures 6–9 that the scaling based on ($u'$, $L$) extends to increasingly smaller values of $r/L$ as $Re_\lambda$ increases. Further, the scaling based on ($u_K$, $\eta$) extends to increasingly larger values of $r/\eta$ as $Re_\lambda$ increases. These results indicate that both scalings should eventually overlap at larger values of $Re_\lambda$. Another feature of figures 6–9 is that the range over which the scaling based on ($u'$, $L$) is tenable appears to be larger than that over which the scaling based on ($u_K$, $\eta$) holds over the present $Re_\lambda$ range, especially for $n>3$. One possible reason for this behaviour is that $\overline {(\delta u)^n}/u'^n$ as $r\to L$ approaches the appropriate Gaussian values, which are $Re_\lambda$-independent, while $\overline {(\delta u)^n}/u_K^n$ as $r\to \eta$ leads to the normalized moments of the velocity derivative ($F_n$), which are likely to be constant only at large $Re_\lambda$ for $n>3$ (note that $F_2=1$ by definition and $|F_3|\approx 0.53$ for $Re_\lambda \gtrsim 300$ (Antonia et al. Reference Antonia, Djenidi, Danaila and Tang2017)); the behaviour of $F_n$ will be discussed briefly later in the context of figure 16. However, as $r\to \infty$, we can estimate that, for $n=2, 4$ and 6, $\overline {(\delta u)^n}/u'^n$ approach the values of 2, 12 and 120, respectively, when (3.1), (3.7) and (3.10) of Pearson & Antonia (Reference Pearson and Antonia2001) are used. The $Re_\lambda$ dependence of the limiting values of $\overline {(\delta u)^n}/u_K^n$ ($n=2,3,4,6$) as $r\to \infty$ has been discussed by Pearson & Antonia (Reference Pearson and Antonia2001) (see their figure 6). They concluded that the assumption that $\delta u$ is ‘Gaussian when $r$ is large is, experimentally, almost impossible to disprove’.

Figure 6. Effect of $Re_\lambda$ on the distributions of $\overline {(\delta u)^2}$ in grid turbulence. Panels (a,b) are normalized by ($u_K$, $\eta$) and ($u'$, $L$), respectively. These data are reproduced from Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020).

Figure 7. Effect of $Re_\lambda$ on the distributions of $-\overline {(\delta u)^3}$ in grid turbulence. Panels (a,b) are normalized by ($u_K$, $\eta$) and ($u'$, $L$), respectively. These data are reproduced from Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014); note that for (a), $u_K$ and $\eta$ were re-evaluated as explained in the text.

Figure 8. Effect of $Re_\lambda$ on the distributions of $\overline {(\delta u)^4}$ in grid turbulence. Panels (a,b) are normalized by ($u_K$, $\eta$) and ($u'$, $L$), respectively. These data are reproduced from Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020).

Figure 9. Effect of $Re_\lambda$ on the distributions of $\overline {(\delta u)^6}$ in grid turbulence. Panels (a,b) are normalized by ($u_K$, $\eta$) and ($u'$, $L$), respectively. These data are reproduced from Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020).

The considerations in § 2 indicate that similarity based on ($u_K$, $\eta$) can satisfy the Kármán–Howarth equation even when $Re_\lambda$ is finite provided $r$ is sufficiently small. Equally, the Kármán–Howarth equation admits a similarity solution based on ($u'$, $L$) when $r$ is sufficiently large. It is therefore reasonable to assume that there are similarity solutions of the form

(4.1)$$\begin{gather} \frac{{\overline{{{(\delta u)}^2}} }}{{u_K^2}} = f_1\left( {\frac{r}{{{\eta}}}} \right), \end{gather}$$

(4.2)$$\begin{gather}\frac{{\overline{{{(\delta u)}^2}} }}{\overline{u^2}} = f_2\left( {\frac{r}{{{L}}}} \right), \end{gather}$$

for sufficiently small and large $r$, respectively. As $Re_\lambda$ increases, the range of $r/\eta$ and $r/L$ over which (4.1) and (4.2) are valid also increases (figure 6). The expectation that these ranges must eventually overlap over the inertial range is highly plausible (see Gamard and George's discussion of the overlap in the spectra of $u$, when normalized by ($u_K$, $\eta$) and ($u'$, $L$) Gamard & George Reference Gamard and George2000), especially as $Re_\lambda \to \infty$. Note that we are dealing with the same distributions of $\overline {{{(\delta u)}^2}}$, after subjecting it to two different types of normalization. Matching (4.1) with (4.2) requires that

(4.3)\begin{equation} u_K^2{f_1}\left( {\frac{r}{\eta }} \right) = \overline{{u^2}} {f_2}\left( {\frac{r}{L}} \right). \end{equation}

If it is further assumed that, in the inertial range, ${f_1}(r/\eta )$ exhibits a power-law form, viz.

(4.4)\begin{equation} {f_1}\left( {\frac{r}{\eta }} \right) \sim {\left( {\frac{r}{\eta }} \right)^\alpha }, \end{equation}

then

(4.5)\begin{equation} {f_2}\left( {\frac{r}{L}} \right) \sim \frac{{u_K^2}}{{\overline{{u^2}} }}{\left( {\frac{L}{\eta }} \right)^\alpha }{\left( {\frac{r}{L}} \right)^\alpha }. \end{equation}

For isotropic turbulence, ${{u_K^2}}/{{\overline {{u^2}} }} = {{{{15}^{1/2}}}}/{Re_\lambda }$ and ${L}/{\eta } = {Re_\lambda ^{3/2}}/{{{{15}^{3/4}}}}$ if $C_\epsilon =1$. The plots in figure 6 show that (4.1) and (4.2) are closely satisfied for significant ranges of $r/\eta$ and $r/L$. In particular, at the two largest values of $Re_\lambda$, the upper value of $r/\eta$ exceeds 50 whilst the lower value of $r/L$ lies below 0.01. This implies that an overlap region is starting to conform with the expected limits of the inertial range, i.e. $\eta \ll r \ll L$. The Reynolds number independence displayed by figure 6 for $r/\eta \lesssim 50$ and $r/L\gtrsim 0.01$ is expected to be approached in the overlap region. As discussed in §§ 1 and 2, the two scalings should become interchangeable as $Re_\lambda \to \infty$ so that the inertial range should also be independent of $Re_\lambda$. The formulation (4.5) or ${f_2}\sim Re_\lambda ^{3/2\alpha - 1}{(r/L)^\alpha }$ is independent of $Re_\lambda$ when $3/2\alpha -1=0$, i.e. when $\alpha =2/3$, which corresponds to the K41 prediction. This result was also derived by Lundgren (Reference Lundgren2002) from the Kármán–Howarth equation using matched asymptotic expansions when $Re_\lambda \to \infty$. A similar result was derived earlier by Gamard & George (Reference Gamard and George2000) in the context of the $u$ spectrum; the approach adopted in this paper is consistent with the asymptotic invariance principle and the methodology of near-asymptotics, introduced by George (Reference George1989, Reference George1994). The previous approach for $\overline {{{(\delta {u})}^2}}$ can be extended to $\overline {{{(\delta {u})}^n}}$ (see for example Djenidi et al. Reference Djenidi, Antonia and Tang2019).

Figures 7–9 indicate that the dual scaling applies to $\overline {{{(\delta {u})}^3}}$, $\overline {{{(\delta {u})}^4}}$ and $\overline {{{(\delta {u})}^6}}$. One would expect that, in general, the dual scaling will also apply to $\overline {{{(\delta {u})}^n}}$, if the probability distribution function (p.d.f.) of $\delta {u}$ complies with the two types of normalization. For simplicity, we next focus on a parametrized form of $\overline {{{(\delta {u})}^n}}$ (Dhruva Reference Dhruva2000; Kurien & Sreenivasan Reference Kurien and Sreenivasan2000), viz.

(4.6) \begin{equation} \dfrac{{\overline{{{(\delta u)}^n}} }}{{u_K^n}} = \dfrac{1}{{{{15}^{n/2}}}}{F_n}\dfrac{{{{\Big(\dfrac{r}{\eta }\Big)}^n}{{\Big({1 + {D_n}\Big(\dfrac{r}{L}\Big)} \Big)}^{2{C_n} - n}}}}{{{{\Big( {1 + {B_n}{{\Big(\dfrac{r}{\eta }\Big)}^2}}\Big)}^{{C_n}}}}}, \end{equation}

where $B_n$, $C_n$, $D_n$ and $F_n$ ($\equiv \overline {{{(\partial u/\partial x)}^n}} /{\overline {{{(\partial u/\partial x)}^2}} ^{n/2}}$) are constants. Note that, for $n>2$, $F_n$ is only expected to approach a constant value at increasingly larger values of $Re_\lambda$ as $n$ increases. An earlier version of (4.6), without the large-scale term, viz. ${{{\left ( {1 + {D_n}({r}/{L})} \right )}^{2{C_n} - n}}}$, was proposed by Stolovitzky, Sreenivasan & Juneja (Reference Stolovitzky, Sreenivasan and Juneja1993) who tested it against laboratory boundary layer data for a moderate Reynolds number ($Re_\lambda \approx 200$). The stated objective was to obtain reliable estimates of the scaling exponents (i.e. $C_n$). One can however query this objective given that the power-law behaviour of $\overline {{{(\delta {u})}^n}}$ is rigorous only when the inertial range is established, i.e. $Re_\lambda \to \infty$. As will be shown later, ‘$\zeta _n$’ for $\overline {{{(\delta {u})}^n}}$ cannot be defined even at the largest $Re_\lambda$ ($\approx 1500$) for the present grid turbulence data; this is not surprising since the 4/5 law has yet to be established (see figure 12a later). Dhruva (Reference Dhruva2000) found that (4.6) provided a good fit to atmospheric surface layer data for $\overline {{{(\delta {u})}^2}}$, $\overline {{{(\delta {u})}^4}}$ and $\overline {{{(\delta {u})}^6}}$ over the complete range of $r$. Our main interest is that (4.6) should represent as adequate a fit as possible to the experimental data for $\overline {{{(\delta {u})}^n}}$ irrespectively of $n$ and $Re_\lambda$. In particular, the constants in (4.6) should satisfy the asymptotic case ($Re_\lambda \to \infty$). This is really the context in which (4.6) was used by Antonia & Burattini (Reference Antonia and Burattini2006) and Antonia et al. (Reference Antonia, Tang, Djenidi and Zhou2019) with $n=2$. In other words, we shall not use (4.6) to estimate $C_n$ for different values of $Re_\lambda$, as was done, for example, by Stolovitzky et al. (Reference Stolovitzky, Sreenivasan and Juneja1993). When $Re_\lambda$ is finite, at least at $Re_\lambda \sim 10^3$, (4.6) does not reduce to a simple power-law form ($\sim r^\alpha$) in the scaling range in various turbulent flows such as the stationary forced periodic box turbulence and along the axes of axisymmetric and plane jets (Tang et al. Reference Tang, Antonia, Djenidi, Danaila and Zhou2017).

Equation (4.6) should be interpreted as a composite model for $\overline {{{(\delta {u})}^n}}$, which satisfies the dual scaling at small and large $r$. In particular, in the inertial range ($\eta \ll r \ll L$), (4.6) reduces to

(4.7)\begin{equation} \frac{{\overline{{{(\delta u)}^n}} }}{{u_K^n}} = \frac{1}{{{{15}^{n/2}}}}{F_n}B_n^{ - {C_n}}{\Big(\dfrac{r}{\eta }\Big)^{n - 2{C_n}}}. \end{equation}

Using the isotropic relations ${L/\eta } = {C_\epsilon }{15^{ - 3/4}}Re_\lambda ^{3/2}$ and $\overline {u^2}/u_K^2=Re_\lambda /15^{ 1/2}$, we have

(4.8)\begin{equation} \frac{{\overline{{{(\delta u)}^n}} }}{{{{\overline{{u^2}} }^{n/2}}}} = \frac{{{{15}^{n/4}}Re_\lambda ^{3/2(n - 2{C_n})}}}{{{{15}^{n/2}}Re_\lambda ^{n/2}}}{F_n}B_n^{ - {C_n}}{\left(\frac{r}{L}\right)^{n - 2{C_n}}}C_\varepsilon ^{^{n - 2{C_n}}}{15^{ - 3/4(n - 2{C_n})}}. \end{equation}

Note that (4.8) is the generalization of (4.5) to $\overline {{{(\delta {u})}^n}}$. It is also the starting point of nearly all intermittency models, viz. ${{\overline {{{(\delta u)}^n}} }}/{{{u'^n}}} \sim {({r}/{L})^{{\zeta _n}}}$, with a prefactor that is independent of $Re_\lambda$ but possibly dependent on the macrostructure of the flow. We underline that (4.8) is equivalent to (4.7) since only a transformation of variables has been applied. If (4.8) is $Re_\lambda$-independent, we immediately obtain

(4.9)\begin{equation} \tfrac{3}{2}(n - 2{C_n}) - n/2 = 0, \end{equation}

which leads to

(4.10)\begin{equation} C_n= n/3. \end{equation}

We will use this value of $C_n$ in (4.6) in the next section. Substituting $C_n= n/3$ into (4.8), we can obtain

(4.11)\begin{equation} \frac{{\overline{{{(\delta u)}^n}} }}{{{{\overline{{u^2}} }^{n/2}}}} = \frac{1}{{{{15}^{n/2}}}}{F_n}B_n^{ - {C_n}}{\left(\frac{r}{L}\right)^{n - 2{C_n}}}C_\varepsilon ^{n - 2C_n}. \end{equation}

Note that (4.7) and (4.11) are identical if $C_\varepsilon =1$. The use of $C_n= n/3$ in (4.6) is fully justifiable since (4.6) should comply with complete similarity, i.e. similarity at all values of $r/\eta$ (or $r/L$) as $Re_\lambda \to \infty$. For finite values of $Re_\lambda$, (4.6) should describe the incomplete similarity inherent in the dual scaling scheme (this will be illustrated in figures 18–21). For $n=2$, (4.6) was, as noted earlier, used previously and shown to describe adequately the dependence of $\overline {{{(\delta u)}^2}}$ on $Re_\lambda$ in various flows.

Figure 10. Distributions of $\overline {{{(\delta u)}^2}} /(\bar {\epsilon } r)^{2/3}$ based on (4.6) at ${Re_\lambda }=124$, 379, 495, 701, 1310 and 1558. Symbols with the same colour correspond to the data of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) shown in figure 3.

Figure 11. (a) Distributions of $\overline {{{(\delta u)}^2}} /(\bar {\epsilon } r)^{2/3}$ based on (4.6) at ${Re_\lambda }=110$, 264, 508, 1000, 1450, $10^4$ and $10^5$. Symbols with the same colour are the corresponding grid turbulence data, reproduced from figure 6(a). (b) Local slope ($LS_2= {\rm d}\log (\overline {{{(\delta {u})}^2}}) /{\rm d}\log r$). Dashed horizontal line, $2/3$.

Figure 12. (a) Distributions of $-\overline {{{(\delta u)}^3}} /\bar {\epsilon } r$ based on (4.6) at ${Re_\lambda }=117$, 352, 535, 679, 1323, 1548, $10^4$ and $10^5$. Symbols with the same colour are the corresponding grid turbulence data, reproduced from figure 7(a). (b) Local slope ($LS_3= {\rm d}\log (-\overline {{{(\delta {u})}^3}}) /{\rm d}\log r$). Dashed horizontal line, $1$.

Figure 13. (a) Distributions of $\overline {{{(\delta u)}^4}} /(\bar {\epsilon } r)^{4/3}$ based on (4.6) at ${Re_\lambda }=110$, 264, 508, 1000, 1450, $10^4$ and $10^5$. Symbols with the same colour are the corresponding grid turbulence data, reproduced from figure 8(a). (b) Local slope ($LS_4= {\rm d}\log (\overline {{{(\delta {u})}^4}}) /{\rm d}\log r$) corresponding to the distributions in (a). Dashed horizontal line, $4/3$.

Figure 14. (a) Distributions of $\overline {{{(\delta u)}^6}} /(\bar {\epsilon } r)^{6/3}$ based on (4.6) at ${Re_\lambda }=110$, 264, 508, 1000, 1450, $10^4$ and $10^5$. Symbols with the same colour are the corresponding grid turbulence data, reproduced from figure 9(a). (b) Local slope ($LS_6= {\rm d}\log (\overline {{{(\delta {u})}^6}}) /{\rm d}\log r$). Dashed horizontal line, $6/3$.

It is worth mentioning that, as $r\to \infty$, (4.6) is proportional to $Re_\lambda ^{n/2}$ after taking $C_n= n/3$ and using the relation ${L/\eta } = {C_\epsilon }{15^{ - 3/4}}Re_\lambda ^{3/2}$. This is consistent with the limiting values as $r\to \infty$ for $n=2$, 4 and 6 (see (3.1), (3.7) and (3.10) of Pearson & Antonia Reference Pearson and Antonia2001).

5. Comparison between (4.6) and experimental data

Figures 10, 11(a)–14(a) show a comparison between the experimental data in figures 3, 6(a)–9(a) and (4.6). Note that we have used $C_n= n/3$; the values for $B_n$, $C_\epsilon$, $D_n$ and $F_n$ are listed in tables 2 and 3. It is worth mentioning that $B_2=0.0056$ and $F_2=1$ lead to a Kolmogorov constant for ${\overline {{{(\delta u)}^2}} }$ of $A_2=$ 2.11. Also, $B_3=0.0114$ and $-F_3=0.53$ lead to a Kolmogorov constant for ${\overline {{{(\delta u)}^3}} }$ of $A_3=$ 4/5. The magnitudes of $F_3$, $F_4$ and $F_6$ used in (4.6) are consistent with the available data in grid turbulence and along the axis of the plane jet (see figure 16 later). The magnitude of $C_\epsilon\ (=0.8)$ is consistent with the VDTT data at large $Re_\lambda$ (figure 2). Here, $D_2$ and $D_3$ are selected so that (4.6) fits the data for $\overline {(\delta u)^2}/u_K^2$ and $\overline {(\delta u)^3}/u_K^3$ reasonably well at all $Re_\lambda$. The magnitudes of $D_4$ and $B_4$ are determined by trial and error until (4.6) fits the data for $\overline {(\delta u)^4}/u_K^4$ at all $Re_\lambda$; the magnitudes of $D_6$ and $B_6$ are determined in the same way. It should be pointed out that all the constants $B_n$, $C_n$, $D_n$ and $F_n$ are $Re_\lambda$-independent except for $F_4$ in the range $Re_\lambda <1000$ and for $F_6$ in the range $Re_\lambda <1450$. Here, $F_4=10$ for $Re_\lambda \geq 1000$ and $B_4=0.011$ lead to a Kolmogorov constant for ${\overline {{{(\delta u)}^4}} }$ of $A_4=18.18$. Also, $F_6=440$ for $Re_\lambda \geq 1450$ and $B_6=0.0178$ lead to a Kolmogorov constant for ${\overline {{{(\delta u)}^6}} }$ of $A_6=413.33$. In particular, we can obtain two ratios: $A_4/A_2^2=4.1$ and $A_6/A_2^3=43.7$. Their magnitudes are consistent with the predictions of Qian (Reference Qian2000) who used a non-Gaussian p.d.f. of $\delta u$ to obtain $A_4/A_2^2=4.1\unicode{x2013}5.4$ and $A_6/A_2^3=37.4\unicode{x2013}69.0$. Here, the main objective of using (4.6), with the above magnitudes for $B_n$, $C_n$, $D_n$ and $F_n$ in grid turbulence, is to describe the dependence on $Re_\lambda$ of the available experimental data for $\overline {(\delta u)^n}/u_K^n$ which allows us to extrapolate the $\overline {(\delta u)^n}/u_K^n$ distributions to infinitely large $Re_\lambda$. The local slope $LS_n$ ($= {\rm d}\log |\overline {{{(\delta {u})}^n}}| /{\rm d}\log r$), based on (4.6), is shown in figures 11(b)–14(b). Note that the local slope $LS_2$ of the distributions in figure 10 is similar to that in figure 11(b) and thus not shown. We can observe from these figures that the agreement between curves and symbols is satisfied approximately at almost all scales over a significant range of ${Re_\lambda }$, confirming the adequacy of the fit used to emulate the available data for ${\overline {{{(\delta u)}^2}} }$, ${\overline {{{(\delta u)}^3}} }$, ${\overline {{{(\delta u)}^4}} }$ and ${\overline {{{(\delta u)}^6}} }$. It should be pointed out that the agreement between the model and the data for some of the distributions at large $Re_\lambda$ and at moderate scales is not so good, as seen for example for the distributions of $\overline {{{(\delta u)}^2}} /(\bar {\epsilon } r)^{2/3}$ at ${Re_\lambda }= 1310$ and 1558 in the range $r/\eta \approx 20\unicode{x2013}100$, which correspond to the range $k_1\eta \approx 0.06\unicode{x2013}0.31$ in the energy spectra. We can indeed observe from figure 1(a) that the magnitude of the Kolmogorov-normalized energy spectra in the range $k_1\eta \approx 0.06\unicode{x2013}0.31$ at ${Re_\lambda }= 1310$ and 1558 is smaller than that for all the other distributions, including those of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014) at ${Re_\lambda }=124$, 379, 495 and 701 and Larssen & Devenport (Reference Larssen and Devenport2011) at ${Re_\lambda }=124$ and 1360. This corresponds to $\overline {{{(\delta u)}^2}} /(\bar {\epsilon } r)^{2/3}$ being underestimated at ${Re_\lambda }=$ 1310 and 1558 in the range $r/\eta \approx 20\unicode{x2013}100$ which is most likely due to the measurement uncertainty. We next use the Kármán–Howarth equation (2.1) to further examine the data of Bodenschatz et al. (Reference Bodenschatz, Bewley, Nobach, Sinhuber and Xu2014). The distributions of $-\overline {{{(\delta u)}^3}} /\bar {\epsilon } r$ in figure 15 are calculated using (2.1) and (4.6) with $n=2$ at ${Re_\lambda }=117$, 352, 535, 679, 1323 and 1548. We emphasize that using (4.6) with $n=2$ is equivalent to using of the experimental data for $\overline {{{(\delta u)}^2}} /(\bar {\epsilon } r)^{2/3}$ in figure 10; here we used the former since the calculation of the integral term in (2.1), i.e. $I_u(r)$ term, based on (4.6) with $n=2$ should be more accurate than that based on the discrete experimental data. Also shown in figure 15 are the grid turbulence data (symbols) at the same $Re_\lambda$, reproduced from figure 12(a). The agreement between curves and symbols in figure 15 is satisfactory. This agreement not only confirms the fit (i.e. (4.6) with $n=2$) used to emulate the available data for $\overline {{{(\delta u)}^2}} /(\bar {\epsilon } r)^{2/3}$ (figure 10), but also vindicates the choice of $M$ for $\overline {{{(\delta u)}^2}} /u_K^2$ in figure 4. Finally, figure 15 further indicates that the magnitude of the model at ${Re_\lambda }= 1310$ and 1558 in the range $r/\eta \approx 20\unicode{x2013}100$ being larger than that of the data most likely reflects a measurement uncertainty in the experimental data (figure 10). Correspondingly, the magnitude of the energy spectrum distributions in the range $k_1\eta \approx 0.06\unicode{x2013}0.31$ at ${Re_\lambda }= 1310$ and 1558 in figure 1(a) being smaller than that of all other distributions also most likely reflects a measurement uncertainty. Further, there is no well-defined plateau for all the experimental data (figures 10, 11a–14a), even at $Re_\lambda \sim 1500$, underlining the importance of the FRN effect in the context of establishing the inertial range. This effect can be also observed in the $LS_n$ distributions for all the experimental data; as an example, we have included in figure 13(b) the local slope $LS_4$ for the corresponding VDTT grid turbulence data of Kaminsky et al. (Reference Kaminsky, Birnir, Bewley and Sinhuber2020). In particular, $LS_3$ for $Re_\lambda =1558$ is far from exhibiting a plateau (figure 12b), implying that (1.2) is actually far from being satisfied by the VDTT grid turbulence data, which underlines the absence of an IR. Note that the calculated distributions indicate that a plateau is likely to emerge as $Re_\lambda$ increases to an infinitely large value. Both the VDTT grid turbulence data and $LS_n$ distributions based on (4.6) in figures 10, 11–14 show an evolution of ${\overline {{{(\delta u)}^n}} }$ and the corresponding local slopes with ${Re_\lambda }$ with a slow approach to $n/3$. Such a trend is not only consistent with all the analysis in the present paper, but also consistent with the plane jet data of Tang et al. (Reference Tang, Antonia, Djenidi, Danaila and Zhou2017). Finally, we should also stress that the local slopes $LS_n$ are not affected by the values of $\bar {\varepsilon }$. Obviously, they are not affected by the choice of the value of $M$ used in replotting the $\overline {(\delta u)^n}/u_K^n$ distributions.

Table 2. Values of $B_n$, $C_\epsilon$, $D_n$, $F_2$ and $F_3$ used in figures 10–21. Note that the magnitudes of $F_3$ are compared with data in grid turbulence and along the axis of the plane jet in figure 16.

Table 3. Values of $F_4$ and $F_6$ used in figures 13–17 and 20–21. Note that the magnitudes of $F_4$ and $F_6$ are compared with data in grid turbulence and along the axis of the plane jet in figure 16.

Figure 15. Analytical predictions of $-\overline {{{(\delta u)}^3}} /\bar {\epsilon } r$ (curves) based on (2.1) and (4.6) with $n=2$ at ${Re_\lambda }=117$, 352, 535, 679, 1323 and 1548. The details of how the integral term in (2.1), i.e. $I_u(r)$ term, was calculated are discussed elsewhere (see for example (4.9) of Tang et al. (Reference Tang, Antonia, Djenidi, Danaila and Zhou2017) where the power-law decay rate for $\overline {u^2}$, viz. $\overline {u^2}\sim x^{n}$ with $n\approx -1.2$ (Sinhuber et al. Reference Sinhuber, Bodenschatz and Bewley2015), was used here). Symbols with the same colour are the corresponding grid turbulence data, reproduced from figure 12(a). The dashed horizontal line in each plot corresponds to the value of 4/5.

Figures 11–14 indicate that $Re_\lambda$ may need to be of order $10^5$ before an inertial range is established unequivocally. To our knowledge, this should be the first time that plausible evidence has been presented for the approach to the IR based on values of $n$ extending to six. We next quantify, albeit approximately, the values of $Re_\lambda$ required for an inertial range to be established for $\overline {(\delta u)^n}$. Figure 17 shows the range of $r/\lambda$ over which the distributions, based on (4.6), of $\overline {{{(\delta {u})}^2}}/ {(\bar {\epsilon } r)^{ {2/3}}}$, $-\overline {{{(\delta {u})}^3}} /{\bar {\epsilon } r}$, $\overline {{{(\delta {u})}^4}} /{(\bar {\epsilon } r)^{ {4/3}}}$ and $\overline {{{(\delta {u})}^6}}/ {(\bar {\epsilon } r)^{ {6/3}}}$ depart from 2.11, 4/5, 18.16 and 413 (these values correspond to the plateaus at $Re_\lambda =10^5$ in figures 11(a)–14a), respectively, by no more than 2.5 %. It should be pointed out that another possibility would be to use $r=\lambda$ as the ‘base’ line for IR. This is consistent with Obligado & Vassilicos (Reference Obligado and Vassilicos2019) and Meldi & Vassilicos (Reference Meldi and Vassilicos2021) who showed, using the eddy damped quasinormal Markovian, that the interscale energy flux is closest to $\bar {\epsilon }$ when $r$ is close to $\lambda$ while the larger $r$, compared to $\lambda$, the larger the departure from equilibrium in which the interscale energy flux is equal to $\bar {\epsilon }$. This would result in slightly larger values of $Re_\lambda$ being required to establish the IR. We can observe from figure 17 that, when $n$ is even, the larger $n$ is being required, the slower is the rate at which $C_n = n/3$ is established convincingly. Interestingly, the $r/\lambda$ range for $\overline {{{(\delta {u})}^6}}/ {(\bar {\epsilon } r)^{ {6/3}}}$ is close to that for $-\overline {{{(\delta {u})}^3}} /{\bar {\epsilon } r}$. Finally, we can observe that $C_2 = 2/3$ is attained at a smaller $Re_\lambda$ than $C_3 = 1$. This is consistent with the observation of Antonia et al. (Reference Antonia, Tang, Djenidi and Zhou2019) where a similar model for $\overline {{{(\delta {u})}^2}}$ with a slightly different constant ($B_2=0.0061$) was used and $\overline {{{(\delta {u})}^3}} /{(\bar {\epsilon } r)}$ was calculated from the Kármán–Howarth equation.

Figure 16. The magnitudes (diamonds) of $|F_{3}|$, $F_{4}$ and $F_{6}$ used to plot figures 12–14. For reference, the data along the axis of the plane jet, reproduced from Antonia et al. (Reference Antonia, Djenidi, Danaila and Tang2017) (triangles), and in grid turbulence, reproduced from Gylfason et al. (Reference Gylfason, Ayyalasomayajula and Warhaft2004) (squares), are also shown; for the latter, only the laboratory shearless grid turbulence data are shown. The horizontal dashed lines indicate the mean values of each of the plotted quantities for the plane jet data.

Figure 17. The range of $r/\lambda$ over which $\overline {{{(\delta {u})}^2}} /{(\bar {\epsilon } r)^{{2/3}}}$ (blue line), $\overline {{{(\delta {u})}^3}} /{(\bar {\epsilon } r)}$ (red line), $\overline {{{(\delta {u})}^4}} /{(\bar {\epsilon } r)^{{4/3}}}$ (green line) and $\overline {{{(\delta {u})}^6}} /{(\bar {\epsilon } r)^{6/3}}$ (pink line) depart from 2.11, 4/5, 18.16 and 413 (these values correspond to the plateaus at $Re_\lambda =10^5$ in figures 11a–14a), respectively, by no more than 2.5 %; they are calculated based on (4.6), using $C_n= n/3$ and the constants for $B_n$, $C_\epsilon /D_n$ and $F_n$ in table 2. The locations of the vertical arrows give an approximate indication of the values of $Re_\lambda$ needed to attain an IR, of two decades in extent, for $\overline {{{(\delta {u})}^2}} /{(\bar {\epsilon } r)^{{2/3}}}$ (blue arrow), $\overline {{{(\delta {u})}^3}} /{(\bar {\epsilon } r)}$ (red arrow), $\overline {{{(\delta {u})}^4}} /{(\bar {\epsilon } r)^{{4/3}}}$ (green arrow) and $\overline {{{(\delta {u})}^6}} /{(\bar {\epsilon } r)^{6/3}}$ (pink arrow).

Figure 18. Distributions of $\overline {(\delta u)^2}$ at large $Re_\lambda$ in grid turbulence. Panels (a,b) are normalized by ($u_K$, $\eta$) and ($u'$, $L$), respectively.

Figure 19. Distributions of $-\overline {(\delta u)^3}$ at large $Re_\lambda$ in grid turbulence. Panels (a,b) are normalized by ($u_K$, $\eta$) and ($u'$, $L$), respectively.

Figure 20. Distributions of $\overline {(\delta u)^4}$ at large $Re_\lambda$ in grid turbulence. Panels (a,b) are normalized by ($u_K$, $\eta$) and ($u'$, $L$), respectively.

Figure 21. Distributions of $\overline {(\delta u)^6}$ at large $Re_\lambda$ in grid turbulence. Panels (a,b) are normalized by ($u_K$, $\eta$) and ($u'$, $L$), respectively.

It is of interest to examine the dual scaling using (4.6) at large $Re_\lambda$. Figures 18–21 show the distributions of $\overline {(\delta u)^n}$ ($n=2,3,4,6$) normalized by ($u_K$, $\eta$) and ($u'$, $L$) in grid turbulence at $Re_\lambda = 1600\unicode{x2013}10^5$. In conformity with the observation based on figures 6–9, the scaling based on ($u'$, $L$) extends to increasingly smaller values of $r/L$ while the scaling based on ($u_K$, $\eta$) extends to increasingly larger values of $r/\eta$ as $Re_\lambda$ increases. In particular, at the two largest values of $Re_\lambda$, the upper value of $r/\eta$ exceeds $10^4$ for all $\overline {(\delta u)^n}$ whilst the lower value of $r/L$ lies below $5\times 10^{-4}$. Taking $Re_\lambda = 2\times 10^4$ and using the isotropic relation ${L}/{\eta } = C_\epsilon ({Re_\lambda ^{3/2}}/{{{{15}^{3/4}}}})$ with $C_\epsilon =0.8$, we can estimate that the overlap region is given by $10^2\lesssim r/\eta \lesssim 10^4$. Namely, a distinct range of overlap between the ($u'$, $L$) and ($u_K$, $\eta$) scalings can be observed at $Re_\lambda =2\times 10^4$, suggesting the onset of an inertial range for $\overline {(\delta u)^n}$ ($n=2,3,4,6$) (see also figure 17). These results further confirm that the dual scaling applies to $\overline {(\delta u)^n}$ and an overlap range begins to emerge when $Re_\lambda >10^4$.