2. Transport equations for the fourth-, eighth- and twelfth-order velocity derivative moments

The generalized transport equation for $\overline {{({{\partial {u_i}}}/{{\partial {x_k}}})({{\partial {u_j}}}/{{\partial {x_k}}})}}$ is given by (e.g. Gerolymos & Vallet Reference Gerolymos and Vallet2016)

(2.1)\begin{align} &\frac{\partial }{{\partial t}}\overline {\left( {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_j}}}{{\partial {x_k}}}} \right)} \!+\! {\bar{U}_l}\,\frac{\partial }{{\partial {x_l}}}\overline {\left( {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_j}}}{{\partial {x_k}}}} \right)} \!+ \frac{\partial }{{\partial {x_l}}}\overline {{u_l}\left( {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_j}}}{{\partial {x_k}}}} \right)} \!-\! \nu\, \frac{{{\partial ^{\rm{2}}}}}{{\partial {x^2_l}}}\overline {\left( {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_j}}}{{\partial {x_k}}}} \right)}\,\, \overline{\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_j}}}{{\partial {x_l}}}}\, \frac{{\partial {{\bar{U}}_l}}}{{\partial {x_k}}}\nonumber\\ &\qquad +\overline{\frac{{\partial {u_j}}}{{\partial {x_k}}}\,\frac{{\partial {u_i}}}{{\partial {x_l}}}}\, \frac{{\partial {{\bar{U}}_l}}}{{\partial {x_k}}} + \overline {\frac{{\partial {u_i}}}{{\partial {x_k}}}\, \frac{{\partial {u_l}}}{{\partial {x_k}}}}\,\frac{{\partial {{\bar{U}}_j}}}{{\partial {x_l}}} + \overline {\frac{{\partial {u_j}}}{{\partial {x_k}}}\,\frac{{\partial {u_l}}}{{\partial {x_k}}}}\, \frac{{\partial {{\bar{U}}_i}}}{{\partial {x_l}}} \nonumber\\ &\quad={-} \overline {{u_l}\,\frac{{\partial {u_i}}}{{\partial {x_k}}}}\, \frac{{{\partial ^2}{{\bar{U}}_j}}}{{\partial {x_k}\,\partial {x_l}}} - \overline{{u_l}\,\frac{{\partial {u_j}}}{{\partial {x_k}}}}\,\frac{{{\partial ^2}{{\bar{U}}_i}}}{{\partial {x_k}\,\partial {x_l}}}-\overline {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_l}}}{{\partial {x_k}}}\,\frac{{\partial {u_j}}}{{\partial {x_l}}}} - \overline {\frac{{\partial {u_j}}}{{\partial {x_k}}}\,\frac{{\partial {u_l}}}{{\partial {x_k}}}\,\frac{{\partial {u_i}}}{{\partial {x_l}}}} \nonumber\\ &\qquad - \overline {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{{\partial ^2}p}}{{\partial {x_k}\,\partial {x_j}}}} - \overline {\frac{{\partial {u_j}}}{{\partial {x_k}}}\,\frac{{{\partial ^2}p}}{{\partial {x_k}\,\partial {x_i}}}} - {{2}}\nu\,\overline {\frac{{{\partial ^2}{u_i}}}{{\partial {x_k}\,\partial {x_l}}}\,\frac{{{\partial ^2}{u_j}}}{{\partial {x_k}\,\partial {x_l}}}}. \end{align}

Note that unless otherwise stated, only the repeated index $l$ indicates summation hereafter. Other indices such as $i$, $j$, $k$, $m$, $n$, ${t}$ and $q$ can be made equal to either 1, 2 or 3. In this paper, we focus only on the statistically stationary turbulence under a uniform mean velocity gradient in the $x_2$ direction (i.e. $\partial {\bar {U}_1}/\partial {x_2} \ne 0$). Therefore, all terms on the first and third lines of (2.1) are ignored. We thus have

(2.2)\begin{align} &\overline {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_j}}}{{\partial {x_l}}}}\,\frac{{\partial {{\bar{U}}_l}}}{{\partial {x_k}}}+ \overline {\frac{{\partial {u_j}}}{{\partial {x_k}}}\,\frac{{\partial {u_i}}}{{\partial {x_l}}}}\,\frac{{\partial {{\bar{U}}_l}}}{{\partial {x_k}}} + \overline {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_l}}}{{\partial {x_k}}}}\, \frac{{\partial {{\bar{U}}_j}}}{{\partial {x_l}}} + \overline {\frac{{\partial {u_j}}}{{\partial {x_k}}}\,\frac{{\partial {u_l}}}{{\partial {x_k}}}}\,\frac{{\partial {{\bar{U}}_i}}}{{\partial {x_l}}} \nonumber\\ &\quad ={-} \overline {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{\partial {u_l}}}{{\partial {x_k}}}\,\frac{{\partial {u_j}}}{{\partial {x_l}}}} - \overline {\frac{{\partial {u_j}}}{{\partial {x_k}}}\,\frac{{\partial {u_l}}}{{\partial {x_k}}}\,\frac{{\partial {u_i}}}{{\partial {x_l}}}} - \overline {\frac{{\partial {u_i}}}{{\partial {x_k}}}\,\frac{{{\partial ^2}p}}{{\partial {x_k}\,\partial {x_j}}}} - \overline {\frac{{\partial {u_j}}}{{\partial {x_k}}}\,\frac{{{\partial ^2}p}}{{\partial {x_k}\,\partial {x_i}}}}\nonumber\\ &\qquad - {\rm{2}}\nu\,\overline {\frac{{{\partial ^2}{u_i}}}{{\partial {x_k}\,\partial {x_l}}}\,\frac{{{\partial ^2}{u_j}}}{{\partial {x_k}\,\partial {x_l}}}}. \end{align}

The four terms on the first line are the (large-scale) production terms associated with the mean shear, which will be simplified after taking the values for $i, j, k$ (see below). The five terms on the second line are small-scale terms. Since $i$, $j$ and $k$ can assume values of 1, 2 and 3, (2.2) represents in fact 27 independent equations. In order to quantify the effect of large-scale forcing caused by the mean shear on the degree of isotropy of the small scales, we should focus on the equations that satisfy the following conditions: (i) regardless of whether LI is satisfied, the large-scale production terms are non-zero; (ii) the corresponding small-scale terms are zero when LI is assumed. The degree of local anisotropy for small-scale quantities can thus be quantified by the magnitudes of the large-scale production terms. All large-scale production terms that satisfy condition (i) are discussed in Appendix A. Here, we focus on only the equation for $i = 1$, $j = 2$ and $k = 1$ since it will lead to a ratio of transverse-to-streamwise velocity derivative variances, i.e.

(2.3)\begin{equation} {R_2} = \frac{{\overline {{{\left( {\partial {u_2}/\partial {x_1}} \right)}^2}} }}{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }}\approx2 \end{equation}

in the production term; this will be discussed later in the context of (2.5). In this case, (2.2) becomes

(2.4)\begin{align} \overline {\frac{{\partial {u_2}}}{{\partial {x_1}}}\,\frac{{\partial {u_2}}}{{\partial {x_1}}}}\,\frac{{\partial {{\bar{U}}_1}}}{{\partial {x_2}}} &={-} \overline {\frac{{\partial {u_1}}}{{\partial {x_1}}}\,\frac{{\partial {u_l}}}{{\partial {x_1}}}\,\frac{{\partial {u_2}}}{{\partial {x_l}}}} - \overline {\frac{{\partial {u_2}}}{{\partial {x_1}}}\,\frac{{\partial {u_l}}}{{\partial {x_1}}}\,\frac{{\partial {u_1}}}{{\partial {x_l}}}} - \overline {\frac{{\partial {u_1}}}{{\partial {x_1}}}\,\frac{{{\partial ^2}p}}{{\partial {x_1}\,\partial {x_2}}}} - \overline {\frac{{\partial {u_2}}}{{\partial {x_1}}}\,\frac{{{\partial ^2}p}}{{\partial {x_1}\,\partial {x_1}}}} \nonumber\\ &\quad - {{2}}\nu\,\overline {\frac{{{\partial ^2}{u_1}}}{{\partial {x_1}\,\partial {x_l}}}\,\frac{{{\partial ^2}{u_2}}}{{\partial {x_1}\,\partial {x_l}}}}. \end{align}

Note that the production term, i.e. the term on the first line in (2.4), is non-zero and thus condition (i) is satisfied. Further, when LI is assumed, all combinations of the indices (i.e. $i = 1$, $j = 2$ and $k = 1$) for the small-scale quantities in (2.4) (i.e. all terms on the second line) that involve the Kronecker delta $\delta _{12}$ will be zero. Namely, condition (ii) mentioned above is satisfied. We now normalize the large-scale production terms and the corresponding small-scale quantities in (2.4). Here, we normalize (2.4) using $\overline {{{( {\partial {u_1}/\partial {x_1}} )}^2}}$, which is equivalent to using $\overline {{{( {\partial {u_1}/\partial {x_2}} )}^2}}$ since, as shown below, the ratio $\overline {{{\left ( {\partial {u_1}/\partial {x_1}} \right )}^2}} /\overline {{{\left ( {\partial {u_1}/\partial {x_2}} \right )}^2}}$ is close to its isotropic value of 0.5 in this flow, i.e.

(2.5)\begin{align} \frac{{\overline {\dfrac{{\partial {u_2}}}{{\partial {x_1}}}\,\dfrac{{\partial {u_2}}}{{\partial {x_1}}}}\,\dfrac{{\partial {{\bar{U}}_1}}}{{\partial {x_2}}}}}{{{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }^{3/2}}}} &= \frac{{\overline {{{\left( {\partial {u_2}/\partial {x_1}} \right)}^2}} }}{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }}\,\frac{{\partial {{\bar{U}}_1}/\partial {x_2}}}{{{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }^{1/2}}}}\nonumber\\ &= {R_2}\,\frac{{15\nu\,\overline {{u_1^2}}\,\partial {{\bar{U}}_1}/\partial {x_2}\,{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }^{1/2}}}}{{15\nu\,\overline {{u_1^2}}\,\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }}\nonumber\\ &= {R_2}\,\frac{{15\nu\,\overline {{u_1^2}}\,\partial {{\bar{U}}_1}/\partial {x_2}\,{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }^{1/2}}}}{{\overline {{u_1^2}} {{\bar{\varepsilon}}_{iso}}}}\nonumber\\ &= {R_2}\,\frac{{{S^*}15\nu\,{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }^{1/2}}}}{{\overline {{u_1^2}} }} = \frac{{15{R_2}{S^*}}}{{{Re_\lambda }}}={A_{1}}, \end{align}

where ${\bar {\varepsilon }}_{iso}={15\nu \,\overline {{{( {\partial {u_1}/\partial {x_1}} )}^2}} }$ is the isotropic mean turbulent energy dissipation rate, $S^*$ is the non-dimensional shear parameter

(2.6)\begin{equation} {S^*} = \frac{{\overline {u_1^2}\,\dfrac{{\partial {{\bar{U}}_1}}}{{\partial {x_2}}}}}{{\bar{\varepsilon}_{iso} }}, \end{equation}

and the parameter ${A_{1}}$ is defined as

(2.7)\begin{align} {A_1} = \dfrac{{ - \overline {\dfrac{{\partial {u_1}}}{{\partial {x_1}}}\,\dfrac{{\partial {u_l}}}{{\partial {x_1}}}\,\dfrac{{\partial {u_2}}}{{\partial {x_l}}}} - \overline {\dfrac{{\partial {u_2}}}{{\partial {x_1}}}\,\dfrac{{\partial {u_l}}}{{\partial {x_1}}}\,\dfrac{{\partial {u_1}}}{{\partial {x_l}}}} - \overline {\dfrac{{\partial {u_1}}}{{\partial {x_1}}}\,\dfrac{{{\partial ^2}p}}{{\partial {x_1}\,\partial {x_2}}}} - \overline {\dfrac{{\partial {u_2}}}{{\partial {x_1}}}\,\dfrac{{{\partial ^2}p}}{{\partial {x_1}\,\partial {x_1}}}} - {{2}}\nu\,\overline {\dfrac{{{\partial ^2}{u_1}}}{{\partial {x_1}\,\partial {x_l}}}\,\dfrac{{{\partial ^2}{u_2}}}{{\partial {x_1}\,\partial {x_l}}}} }}{{{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }^{3/2}}}}. \end{align}

If LI is satisfied, then $A_{1}=0$; all combinations of the indices in ${A_1}$ will lead to the Kronecker delta $\delta _{12}$ under the assumption of LI. As an example, the third term in the numerator of ${A_1}$ can be written as

(2.8)\begin{equation} \overline {\frac{{\partial {u_1}}}{{\partial {x_1}}}\,\frac{{{\partial ^2}p}}{{\partial {x_1}\,\partial {x_2}}}} = \frac{3}{4}\left( {{\delta _{11}}{\delta _{12}} + {\delta _{12}}{\delta _{11}} - \frac{2}{3}\,{\delta _{11}}{\delta _{12}}} \right)\overline {\frac{{\partial {u_1}}}{{\partial {x_1}}}\,\frac{{{\partial ^2}p}}{{\partial {x_1}\,\partial {x_1}}}} = 0 \end{equation}

after using the isotropic form of a single-point fourth-order velocity derivative tensor $\overline {({{\partial {u_i}}}/{{\partial {x_j}}})({{{\partial ^2}p}}/{{\partial {x_k}\,\partial {x_m}}})}$ (e.g. Antonia & Browne Reference Antonia and Browne1983; Wyngaard Reference Wyngaard2010). Equation (2.5) relates the normalized mean shear parameter $S^*$ to a measure of local anisotropy quantified by $A_{1}$. We note that the experimental data of Shen & Warhaft (Reference Shen and Warhaft2002) indicate that $R_2=2.07$ and 1.87 at $Re_\lambda = 193$ and 974, respectively, in statistically stationary turbulent shear flows (the isotropic value of $R_2$ is 2); the corresponding values of $S^*$ are 5.27 and 6.11. For Ferchichi & Tavoularis (Reference Ferchichi and Tavoularis2000), $\overline {{{\left ( {\partial {u_1}/\partial {x_1}} \right )}^2}} /\overline {{{\left ( {\partial {u_1}/\partial {x_2}} \right )}^2}} \approx 0.5$ at $Re_\lambda = 140\unicode{x2013}660$, while for Shen & Warhaft (Reference Shen and Warhaft2002), $\overline {{{\left ( {\partial {u_1}/\partial {x_1}} \right )}^2}} /\overline {{{\left ( {\partial {u_1}/\partial {x_2}} \right )}^2}} =0.49$ and 0.52 at $Re_\lambda = 193$ and 974, respectively. Finally, the ratio of $\bar {\varepsilon }_{iso}$ to the mean turbulent energy dissipation rate $\bar {\varepsilon }$ can be written as

(2.9)\begin{equation} {{\bar \varepsilon }_{iso}}/\bar \varepsilon = \frac{{15\nu \displaystyle\int_0^\infty {k_1^2\,{\phi _{u_1}}({k_1})\,{\rm d}{k_1}} }}{{\nu \left[ \displaystyle{\int_0^\infty {{\phi _{{\omega _1}}}({k_1})} \,{\rm d}{k_1} + \int_0^\infty {{\phi _{{\omega _2}}}({k_1})} \,{\rm d}{k_1} + \int_0^\infty {{\phi _{{\omega _3}}}({k_1})}\, {\rm d}{k_1}} \right]}}, \end{equation}

where ${\phi _{u_1}}({k_1})$ and ${{\phi _{{\omega _i}}}({k_1})}$ are the spectra of $u_1$ and $\omega _i$, respectively. Using the spectra of Sekimoto, Dong & Jiménez (Reference Sekimoto, Dong and Jiménez2016) (which can be downloaded from the website https://torroja.dmt.upm.es/turbdata/HST/stats/spectra), we can estimate that ${{\bar \varepsilon }_{iso}}/\bar \varepsilon \approx 0.93$ and 0.98 at $Re_\lambda \approx 100$ and 250, respectively. That is, the departure of ${{\bar \varepsilon }_{iso}}/\bar \varepsilon$ from the isotropic value is small. Since $R_2$ does not depart significantly from the isotropic value, (2.5) indicates that for a given $S^*$, the approach towards LI follows $Re_\lambda ^{-1}$. This is consistent with the conventional scaling of Lumley (Reference Lumley1967) using dimensional arguments. We should stress that (2.5) is an exact relation derived from the Navier–Stokes equations for a statistically stationary turbulence with a uniform mean velocity gradient, and is thus more rigorous than the derivations of Lumley (Reference Lumley1967).

We now extend the above analysis to higher-order moments. We first derive the transport equations for $\overline {{ ({{\partial {u_i}}}/{{\partial {x_k}}})({{\partial {u_j}}}/{{\partial {x_k}}})({{\partial {u_m}}}/{{\partial {x_k}}})({{\partial {u_n}}}{{\partial {x_k}}} ) } }$; the detailed derivation can be found in Appendix B. Using the same assumptions associated with (2.5), we obtain

(2.10)\begin{equation} \frac{15R_4S^*}{Re_\lambda } = A_2, \end{equation}

where

(2.11)\begin{equation} R_4 = \frac{{\overline {{{\left( {\partial {u_2}/\partial {x_1}} \right)}^4}} }}{{{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }^2}}} \end{equation}

and

(2.12)\begin{align} {A_2} &= \left[ { - 3\overline {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^2}\frac{{\partial {u_1}}}{{\partial {x_1}}}\,\frac{{\partial {u_l}}}{{\partial {x_1}}}\,\frac{{\partial {u_2}}}{{\partial {x_l}}}} - \overline {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^3}\frac{{\partial {u_l}}}{{\partial {x_1}}}\,\frac{{\partial {u_1}}}{{\partial {x_l}}}} } \right.\nonumber\\ &\quad - 3\,\overline {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^2}\frac{{\partial {u_1}}}{{\partial {x_1}}}\,\frac{\partial }{{\partial {x_1}}}\left( {\frac{{\partial p}}{{\partial {x_2}}}} \right)} - \overline {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^3}\frac{\partial }{{\partial {x_1}}}\left( {\frac{{\partial p}}{{\partial {x_1}}}} \right)}\nonumber\\ &\quad \left. \left. {}- 3\nu\,\overline {\frac{{{\partial ^2}{u_2}}}{{\partial {x_1}\,\partial {x_l}}}\,\frac{\partial }{{\partial {x_l}}}\left( {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^2}\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right)} - \nu\,\overline {\frac{{{\partial ^2}{u_1}}}{{\partial {x_1}\,\partial {x_l}}}\,\frac{\partial }{{\partial {x_l}}}{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^3}} \right]\right/ {\overline {{{\left( {\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right)}^2}} }^{5/2}. \end{align}

Similarly to (2.5), (2.10) relates $S^*$, the fourth-order moment $R_4$ and the normalized fifth-order moment ${A_2}$. Note that LI requires ${A_2}$ to be zero since all combinations of the indices in ${A_2}$ will lead to the Kronecker delta $\delta _{12}$, which is zero. Using a procedure similar to that in Appendix B, we can derive the transport equations for $\overline {{ ({{\partial {u_i}}}/{{\partial {x_k}}})({{\partial {u_j}}}/{{\partial {x_k}}})({{\partial {u_m}}}/{{\partial {x_k}}})({{\partial {u_n}}}/{{\partial {x_k}}})({{\partial {u_{t}}}}/{{\partial {x_k}}})({{\partial {u_q}}}/{{\partial {x_k}}}) } }$; the detailed derivation can be found in Appendix C, which relates the mean shear $S^*$, the normalized sixth-order moment

(2.13)\begin{equation} R_6=\frac{{\overline {{{\left( {\partial {u_2}/\partial {x_1}} \right)}^6}} }}{{{{\overline {{{\left( {\partial {u_1}/\partial {x_1}} \right)}^2}} }^3}}}, \end{equation}

and the normalized seventh-order moment

(2.14)\begin{align} {A_3} &= \left[ { - 5\overline {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^4}\frac{{\partial {u_1}}}{{\partial {x_1}}}\,\frac{{\partial {u_l}}}{{\partial {x_1}}}\,\frac{{\partial {u_2}}}{{\partial {x_l}}}} - \overline {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^4}\frac{{\partial {u_2}}}{{\partial {x_1}}}\,\frac{{\partial {u_l}}}{{\partial {x_1}}}\,\frac{{\partial {u_1}}}{{\partial {x_l}}}} } \right.\nonumber\\ &\quad - 5\,\overline {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^4}\frac{{\partial {u_1}}}{{\partial {x_1}}}\,\frac{\partial }{{\partial {x_1}}}\left( {\frac{{\partial p}}{{\partial {x_2}}}} \right)} - \overline {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^5}\frac{\partial }{{\partial {x_1}}}\left( {\frac{{\partial p}}{{\partial {x_1}}}} \right)}\nonumber\\ &\quad \left. \left. {}- 5\nu\,\overline {\frac{{{\partial ^2}{u_2}}}{{\partial {x_1}\,\partial {x_l}}}\,\frac{\partial }{{\partial {x_l}}}\left( {{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^4}\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right)} - \nu\, \overline {\frac{{{\partial ^2}{u_1}}}{{\partial {x_1}\,\partial {x_l}}}\frac{\partial }{{\partial {x_l}}}{{\left( {\frac{{\partial {u_2}}}{{\partial {x_1}}}} \right)}^5}} \right] \right/ {\overline {{{\left( {\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right)}^2}}^{7/2}}, \end{align}

which is also zero when LI is satisfied since all combinations of the indices in ${A_3}$ will lead to the Kronecker delta $\delta _{12}$. That is,

(2.15)\begin{equation} \frac{15R_6 S^*}{Re_\lambda} = A_3. \end{equation}

Equations (2.5), (2.10) and (2.15) are of the form

(2.16)\begin{equation} 15R_{2n}\,\frac{{{S^*}}}{{{Re_\lambda }}} = A_n \quad {\rm with} \ n=1, 2\ {\rm and}\ 3, \end{equation}

where $A_n$ quantifies the degree of local anisotropy. It is clear that $A_n$ is controlled by $S^*$, $Re_\lambda$ and the normalized even moment ($R_{2n}$). For a given $S^*$, the available data in this flow indicate that $R_2\approx 2$ (Shen & Warhaft Reference Shen and Warhaft2002), $R_4\sim Re_\lambda ^{0.25}$ and $R_6\sim Re_\lambda ^{0.7}$ (Shen & Warhaft Reference Shen and Warhaft2000). Consequently, (2.16) indicates that the approach to LI will be slowest for ${A_3}$, followed by ${A_2}$, and then $A_{1}$. We recall that the odd-order moments of transverse velocity derivatives ($M_{2n+1}$) are frequently used to test LI in this flow. It is thus interesting to examine whether $M_{2n+1}$ will follow the prediction of (2.16). On one hand, the available data for $S^*$ show that there is very large scatter between different investigations over a large range of $Re_\lambda$ in the statistically stationary turbulence (see, for example, figure 6 of Schumacher et al. Reference Schumacher, Sreenivasan and Yeung2003). On the other hand, we note that $R_4$ and $R_6$ increase as $Re_\lambda$ increases up to $Re_\lambda \approx 1000$ (e.g. Ferchichi & Tavoularis Reference Ferchichi and Tavoularis2000; Shen & Warhaft Reference Shen and Warhaft2000; Warhaft & Shen Reference Warhaft and Shen2001; Schumacher et al. Reference Schumacher, Sreenivasan and Yeung2003). It is therefore more meaningful to examine the effects of the normalized moment $R_{2n}$ on $M_{2n+1}$ over a relatively large range of $Re_\lambda$. The available data collected by Schumacher et al. (Reference Schumacher, Sreenivasan and Yeung2003) that satisfy this condition are the experimental data of Shen & Warhaft (Reference Shen and Warhaft2000) and Ferchichi & Tavoularis (Reference Ferchichi and Tavoularis2000) over a significant range of $Re_\lambda$ ($100 \lesssim Re_\lambda \lesssim 1000$). We next reappraise these data in detail.

Figures 1(a), 2(a) and 3(a) show the dependence on $Re_\lambda$ of $S^*/Re_\lambda$, $15R_4S^*/Re_\lambda$ and $15R_6S^*/Re_\lambda$. Due to the large scatter in these figures, we use parallel lines to encompass as much data as possible. Figures 1(b), 2(b) and 3(b) show the distributions of $M_3$, $M_5$ and $M_7$ versus $Re_\lambda$ for the same data sets. Also shown in figures 1(b), 2(b) and 3(b) are the distributions (green lines) with the same power-law exponents as in figures 1(a), 2(a) and 3(a), and the predictions of Shen & Warhaft (Reference Shen and Warhaft2000): $M_3= 8.72 Re_\lambda ^{-0.52}$, $M_5\sim Re_\lambda ^{-0.02}$ and $M_7\sim Re_\lambda ^{0.63}$ (red lines). The same scales for the horizontal axis and two decades for the vertical axis are used in figures 1–3. It can be seen from figure 1(a) that $S^*/Re_\lambda$ follows approximately the relation $\sim Re_\lambda ^{-1}$. Since experimental data of Shen & Warhaft (Reference Shen and Warhaft2002) show that $R_2=2.07$ and 1.87 at $Re_\lambda = 193$ and 974, respectively, we can obtain $S^*/Re_\lambda ={A_{1}}/15R_2\approx {A_{1}}/30$ when taking $R_2 \approx 2$. We can observe in figure 1(b) that $M_3$ follows the relation $\sim Re_\lambda ^{-1}$ for $Re_\lambda \gtrsim 200$. In contrast, the distribution for $8.72 Re_\lambda ^{-0.52}$ proposed by Shen & Warhaft (Reference Shen and Warhaft2000) does not follow the data well. For simplicity, the distribution for $S^*/Re_\lambda$ described by the black line ($6.14 Re_\lambda ^{-1}$) in figure 1(a) has been used to plot the distributions of $15R_4S^*/Re_\lambda$ and $15R_6S^*/Re_\lambda$ in figures 2(a) and 3(a). Figures 2(a) and 3(a) show that $15R_4S^*/Re_\lambda$ ($=A_2$) and $15R_6S^*/Re_\lambda$ ($=A_3$) decrease as ${\sim } Re_\lambda ^{-0.75}$ and ${\sim } Re_\lambda ^{-0.3}$, respectively. The results in figures 1(a), 2(a) and 3(a) indicate that both the mean shear $S^*$ and the even moments of transverse velocity derivatives ($R_2$, $R_4$ and $R_6$) affect significantly the rate at which LI is approached in the context of ${A_1}$, ${A_2}$ and $A_{3}$. In particular, since $R_2\approx 2$ (Shen & Warhaft Reference Shen and Warhaft2002), $R_4\sim Re_\lambda ^{0.25}$ and $R_6\sim Re_\lambda ^{0.7}$ (Shen & Warhaft Reference Shen and Warhaft2000), the larger the order, the slower the rate at which this approach is reached. However, despite the large scatter, $M_5$ is approximately constant, and $M_7$ increases as $Re_\lambda$; they do not follow the behaviour of ${A_2}$ and $A_{3}$. The reason for the difference in behaviour between $M_7$ and $A_3$, $M_5$ and $A_2$, deserves further investigation. Our analytical derivations indicate that the small-scale quantities ${A_1}$, ${A_2}$ and $A_{3}$ will tend towards isotropy with increasing $Re_\lambda$ provided that the product of the normalized even-order moments (i.e. $R_2$, $R_4$ and $R_6$) and $S^*$ does not increase as rapidly as $Re_\lambda$. However, other small-scale quantities, such as $M_7$, may not follow the same trend, or even exhibit an opposite trend. We recall that Shen & Warhaft (Reference Shen and Warhaft2000) used an active grid to achieve high $Re_\lambda$. The high turbulence levels generated by the active grid may potentially produce large-scale anisotropy that may induce an additional effect on the behaviour of $M_7$. Evidently, it would be desirable to examine the behaviour of $M_5$ and $M_7$ at higher $Re_\lambda$.

Figure 1.

Dependence on $Re_\lambda$ of (a) $S^*/Re_\lambda$ and (b) $M_3$, in the statistically stationary turbulence. Note that $S^*/Re_\lambda ={A_{1}}/15R_2$, see (2.16). Blue circles correspond to Ferchichi & Tavoularis (Reference Ferchichi and Tavoularis2000), and black diamonds correspond to Shen & Warhaft (Reference Shen and Warhaft2000). Green lines indicate ${\sim }Re_\lambda ^{-1}$, the black curve indicates $6.14 Re_\lambda ^{-1}$, and the red line indicates $8.72 Re_\lambda ^{-0.52}$ (Shen & Warhaft Reference Shen and Warhaft2000).

Figure 2.

(a) Dependence of $15R_4S^*/Re_\lambda$ in the statistically stationary turbulence. (b) Dependence of $M_5$ on $Re_\lambda$. Blue circles correspond to Ferchichi & Tavoularis (Reference Ferchichi and Tavoularis2000), and black diamonds correspond to Shen & Warhaft Reference Shen and Warhaft2000). Green lines indicate ${\sim }Re_\lambda ^{-0.75}$, and the red line indicates ${\sim } Re_\lambda ^{-0.02}$ (Shen & Warhaft Reference Shen and Warhaft2000).

Figure 3.

(a) Dependence of $15R_6S^*/Re_\lambda$ on $Re_\lambda$ in the statistically stationary turbulence (Shen & Warhaft Reference Shen and Warhaft2000). (b) Dependence of $M_7$ on $Re_\lambda$ (Shen & Warhaft Reference Shen and Warhaft2000). Green lines indicate ${\sim }Re_\lambda ^{-0.3}$, and the red line indicates ${\sim } Re_\lambda ^{0.63}$ (Shen & Warhaft Reference Shen and Warhaft2000).