2. Theoretical considerations and discussions

The dynamical equation for the scalar fluctuation $\theta$ is given by Corrsin (Reference Corrsin1952) as

(2.1)\begin{equation} \frac{{\partial \theta }}{{\partial t}} + {\bar{U}_j}\frac{{\partial \theta }}{{\partial {x_j}}} + {u_j}\frac{{\partial \varTheta}}{{\partial {x_j}}} + {u_j}\frac{{\partial \theta }}{{\partial {x_j}}} - \overline{{u_j}\frac{{\partial \theta }}{{\partial {x_j}}}} = \kappa \frac{\partial }{{\partial {x_j}}}\frac{{\partial \theta }}{{\partial {x_j}}}, \end{equation}

where $\varTheta$ is the mean scalar; $\bar {U}_i$ is the mean velocity in the $x_i$ direction. Sreenivasan & Tavoularis (Reference Sreenivasan and Tavoularis1980) derived a transport equation for $\overline {({\partial \theta /\partial {x_1}})^3}$ in a steady, homogeneous shear flow with uniform mean velocity and scalar gradients in the $x_2$ direction (${{\partial \bar {U}_1}}/{{\partial {x_2}}}={\rm const.}$ and ${{\partial \varTheta }}/{{\partial {x_2}}}={\rm const.}$). Using a similar procedure as Sreenivasan & Tavoularis (Reference Sreenivasan and Tavoularis1980), we first differentiate (2.1) with respect to $x_{\alpha }$ (${\alpha }=1,2,3$), then multiply the resulting equation by $({\partial \theta /\partial {x_{\alpha }}})^n$ and finally take the average to obtain

(2.2)\begin{align} &\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{\partial }{{\partial {x_{\alpha}}}}\frac{{\partial \theta }}{{\partial t}}} + \frac{{\partial {\bar{U}_j}}}{{\partial {x_{\alpha}}}}\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{{\partial \theta }}{{\partial {x_j}}}} + {\bar{U}_j}\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{\partial }{{\partial {x_{\alpha}}}}\frac{{\partial \theta }}{{\partial {x_j}}}} \nonumber\\ &\quad +\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{{\partial {u_j}}}{{\partial {x_{\alpha}}}}} \frac{{\partial \varTheta}}{{\partial {x_j}}} + \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}{u_j}} \frac{\partial }{{\partial {x_{\alpha}}}}\frac{{\partial \varTheta}}{{\partial {x_j}}} + \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{{\partial {u_j}}}{{\partial {x_{\alpha}}}}\frac{{\partial \theta }}{{\partial {x_j}}}} \nonumber\\ &\quad +\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}{u_j}\frac{\partial }{{\partial {x_{\alpha}}}}\frac{{\partial \theta }}{{\partial {x_j}}}} - \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}} \frac{\partial }{{\partial {x_{\alpha}}}}\overline{{u_j}\frac{{\partial \theta }}{{\partial {x_j}}}} = \kappa \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{\partial }{{\partial {x_{\alpha}}}}\frac{\partial }{{\partial {x_j}}}\frac{{\partial \theta }}{{\partial {x_j}}}} . \end{align}

Since we consider only the statistically stationary case, we have

(2.3)\begin{equation} \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{\partial }{{\partial {x_{\alpha}}}}\frac{{\partial \theta }}{{\partial t}}} = \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{\partial }{{\partial t}}\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} = \frac{1}{n+1}\frac{\partial }{{\partial t}}\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^{n+1}}}=0.\end{equation}

In this paper, we focus on the passive scalar field convected by statistically stationary homogeneous isotropic turbulence under a uniform mean scalar gradient in the $x_1$ direction (${{\partial \varTheta }}/{{\partial {x_1}}}$). Therefore, the last two terms on the first line and the second term on the second line of (2.2) can be ignored. Further, since the flow is homogeneous, we have

(2.4)$$\begin{gather} \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}{u_j}\frac{\partial }{{\partial {x_{\alpha}}}}\frac{{\partial \theta }}{{\partial {x_j}}}} = \frac{1}{{n + 1}}\frac{\partial }{{\partial {x_j}}}\overline{{u_j}{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^{n + 1}}} = 0, \end{gather}$$

(2.5)$$\begin{gather}\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}} \frac{\partial }{{\partial {x_{\alpha}}}}\overline{{u_j}\frac{{\partial \theta }}{{\partial {x_j}}}} = \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}} \frac{\partial }{{\partial {x_{\alpha}}}}\frac{{\partial \overline{{u_j}\theta } }}{{\partial {x_j}}} = 0. \end{gather}$$

Finally, (2.2) reduces to

(2.6)\begin{equation} \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{{\partial {u_1}}}{{\partial {x_{\alpha}}}}} \frac{{\partial \varTheta}}{{\partial {x_1}}} ={-} \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{{\partial {u_j}}}{{\partial {x_{\alpha}}}}\frac{{\partial \theta }}{{\partial {x_j}}}} + \kappa \overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_{\alpha}}}}} \right)}^n}\frac{\partial }{{\partial {x_{\alpha}}}}\frac{\partial }{{\partial {x_j}}}\frac{{\partial \theta }}{{\partial {x_j}}}} . \end{equation}

The left-hand side term is the (large-scale) mean scalar-gradient production. The two terms on the right side are the small-scale terms, which can be interpreted as representing the production and destruction of $\overline {{{({{{\partial \theta }}/{{\partial {x_{\alpha }}}}} )}^{n + 1}}}$ respectively. Note that, when ${{\partial \varTheta }}/{{\partial {x_1}}}=0$ and $n=1$, (2.6) is the stationary form of the equation for $\overline {{{({{{\partial \theta }}/{{\partial {x_{\alpha }}}}})}^{2}}}$ first written by Corrsin (Reference Corrsin1953). The equation, which contained both mean velocity and mean scalar-gradient terms, was tested by Abe, Antonia & Kawamura (Reference Abe, Antonia and Kawamura2009) in a turbulent channel flow at different Reynolds numbers. Wyngaard (Reference Wyngaard1971) considered (2.6) with $n=1$. He interpreted the first term on the right side of (2.6) as the production rate of $\overline {{{({{{\partial \theta }}/{{\partial {x_{\alpha }}}}})}^{2}}}$ due to the stretching of the temperature field by the turbulent strain field, which is balanced by the molecular smoothing of the gradient temperature field. Since (2.6) contains both large- and small-scale terms, it can be used to quantify the effect of the large-scale forcing associated with the mean scalar gradient on small-scale quantities. When $n$ is even and ${\alpha }=2$ or 3, the first term in (2.6) is zero if local isotropy is satisfied since all combinations of the indices in $\overline {{{({{{\partial \theta }}/{{\partial {x_{\alpha }}}}})}^n}({{\partial {u_1}}}/{{\partial {x_{\alpha }}}})}$ will lead to the Kronecker delta $\delta _{1{\alpha }}$ (${\alpha }=2$ or 3) under the assumption of local isotropy. As an example, the first term in (2.6) for ${\alpha }=2$ and $n=2$ can be written as

(2.7)\begin{equation} \overline{\frac{{\partial {u_1}}}{{\partial {x_2}}}\frac{{\partial \theta }}{{\partial {x_2}}}\frac{{\partial \theta }}{{\partial {x_2}}}} \frac{{\partial \varTheta }}{{\partial {x_1}}} = \frac{3}{4}\left( {{\delta _{12}}{\delta _{22}} + {\delta _{12}}{\delta _{22}} - \frac{2}{3}{\delta _{12}}{\delta _{22}}} \right)\overline{\frac{{\partial {u_1}}}{{\partial {x_1}}}{{\left( {\frac{{\partial \theta }}{{\partial {x_1}}}} \right)}^2}} \frac{{\partial \varTheta }}{{\partial {x_1}}} = 0, \end{equation}

after using the isotropic form of a single-point fourth-order velocity derivative tensor $\overline {({{\partial {u_i}}}/{{\partial {x_j}}})({{\partial \theta }}/{{\partial {x_n}}})({{\partial \theta }}/{{\partial {x_m}}})}$ (e.g. Antonia & Browne Reference Antonia and Browne1983; Wyngaard Reference Wyngaard2010). When $n$ is odd and ${\alpha }=2$ or 3, the first term in (2.6) is also zero if local isotropy is satisfied since all combinations of the indices in $\overline {{{( {{{\partial \theta }}/{{\partial {x_{\alpha }}}}})}^n}({{\partial {u_1}}}/{{\partial {x_{\alpha }}}})}$ will lead to the Kronecker delta $\delta _{1{\alpha }}$ (${\alpha }=2$ or 3) or the permutation symbol $\epsilon _{1{\alpha \alpha }}$ or $\epsilon _{{\alpha \alpha \alpha }}$ under the assumption of local isotropy. Therefore, in (2.6), we only consider the subscript ${\alpha }=1$, which represents the direction of the mean scalar gradient. After normalization by ${\overline {{{({\partial \theta /\partial {x_1}})}^2}} ^{(n + 1)/2}}{\overline {{{({\partial {u_1}/\partial {x_1}})}^2}} ^{1/2}}$, the first term in (2.6) becomes

(2.8) \begin{equation} \left.\begin{array}{c@{}} \dfrac{{\overline{{{\left( {\dfrac{{\partial \theta }}{{\partial {x_1}}}} \right)}^n}\dfrac{{\partial {u_1}}}{{\partial {x_1}}}} \dfrac{{\partial \varTheta }}{{\partial {x_1}}}}}{{{{\overline{{{( {\partial \theta /\partial {x_1}} )}^2}} }^{(n + 1)/2}}{{\overline{{{( {\partial {u_1}/\partial {x_1}} )}^2}} }^{1/2}}}} = \dfrac{{\overline{{{\left( {\dfrac{{\partial \theta }}{{\partial {x_1}}}} \right)}^n}\dfrac{{\partial {u_1}}}{{\partial {x_1}}}} \dfrac{{\partial \varTheta }}{{\partial {x_1}}}}}{{{{\overline{{{( {\partial \theta /\partial {x_1}} )}^2}} }^{n/2}}{{\overline{{{( {\partial {u_1}/\partial {x_1}} )}^2}} }^{1/2}}{{\overline{{{( {\partial \theta /\partial {x_1}} )}^2}} }^{1/2}}}}\\ \dfrac{{{R_n}\dfrac{{\partial \varTheta }}{{\partial {x_1}}}}}{{{{\overline{{{( {\partial \theta /\partial {x_1}} )}^2}} }^{1/2}}}} = \dfrac{{{R_n}6\kappa {{\overline{{\theta ^2}} }^{1/2}}{{\overline{u_1^2} }^{1/2}}\dfrac{{\partial \varTheta }}{{\partial {x_1}}}{{\overline{{{\left( {\dfrac{{\partial \theta }}{{\partial {x_1}}}} \right)}^2}} }^{1/2}}}}{{{{\overline{{\theta ^2}} }^{1/2}}{{\overline{u_1^2} }^{1/2}}6\kappa \overline{{{( {\partial \theta /\partial {x_1}} )}^2}}}} = \dfrac{{6{R_n}S_\theta ^*}}{{Pe_{{\lambda_\theta}}}} \end{array} \right\},\end{equation}

where $Pe_{{\lambda _\theta }}$ is defined in (1.2); $S_\theta ^*$ is defined as

(2.9)\begin{equation} S_\theta^* = \frac{{{{\overline{{\theta ^2}}}^{1/2}}{{\overline{u_1^2} }^{1/2}}\dfrac{{\partial \varTheta }}{{\partial {x_1}}}}}{{{{\bar \varepsilon }_\theta }}}, \end{equation}

with ${{\bar {\varepsilon }_\theta } = 6\kappa \overline {{{({{{\partial \theta }}/{{\partial {x_1}}}})}^2}}}$; for convenience, we refer to $S_\theta ^*$ as the non-dimensional scalar gradient. Here, ${R_n}$ is the normalized correlation involving the streamwise derivative of $\theta$ and $u_1$, defined as

(2.10)\begin{equation} {R_n} = \frac{{\overline{{{\left( {\dfrac{{\partial \theta }}{{\partial {x_1}}}} \right)}^n}\dfrac{{\partial {u_1}}}{{\partial {x_1}}}} }}{{{{\overline{{{({\partial \theta /\partial {x_1}} )}^2}} }^{n/2}}{{\overline{{{({\partial {u_1}/\partial {x_1}})}^2}} }^{1/2}}}}. \end{equation}

Finally, after normalization by ${\overline {{{( {\partial \theta /\partial {x_1}} )}^2}} ^{(n + 1)/2}}{\overline {{{({\partial {u_1}/\partial {x_1}} )}^2}} ^{1/2}}$, (2.6) for ${\alpha }=1$ can be rewritten as

(2.11)\begin{equation} \frac{{6{R_n}S_\theta ^*}}{{Pe_{\lambda_\theta} }} = {A_{n}},\end{equation}

where $A_{n}$ is defined as

(2.12)\begin{align} {A_n} = \frac{{ - \overline{{{\left( {\dfrac{{\partial \theta }}{{\partial {x_1}}}} \right)}^n}\dfrac{{\partial {u_j}}}{{\partial {x_1}}}\dfrac{{\partial \theta }}{{\partial {x_j}}}} + \kappa \overline{{{\left( {\dfrac{{\partial \theta }}{{\partial {x_1}}}} \right)}^n}\dfrac{\partial}{{\partial {x_1}}}\dfrac{\partial }{{\partial {x_j}}}\dfrac{{\partial \theta }}{{\partial {x_j}}}}}}{{{{\overline{{{({\partial \theta /\partial {x_1}} )}^2}} }^{(n + 1)/2}}{{\overline{{{({\partial {u_1}/\partial {x_1}})}^2}} }^{1/2}}}}. \end{align}

The numerator of $A_n$ is simply the right-hand side of (2.6) with $\alpha =1$. In that respect, $A_n$ represents the normalized sum of the production and destruction of $\overline {{{( {{{\partial \theta }}/{{\partial {x_{1}}}}} )}^{n + 1}}}$. Some remarks are warranted on the use of $\overline {{{( {{{\partial \theta }}/{{\partial {x_1}}}} )}^2}}$, which is one of the three components in the full scalar dissipation rate defined as $2 \kappa \overline {{{( {{{\partial \theta }}/{{\partial {x_i}}}} )}^2}}$, in equations (2.11) and (2.9). The values of $\overline {{{( {{{\partial \theta }}/{{\partial {x_i}}}} )}^2}}$ along the directions perpendicular to the mean scalar gradient, i.e. the $x_2$ and $x_3$ directions, should be equal because of the symmetry. On the other hand, Yeung et al. (Reference Yeung, Xu and Sreenivasan2002) showed that the ratio of parallel-to-perpendicular scalar-gradient variances (i.e. $\overline {{{({\partial \theta /\partial {x_1}})}^2}} /\overline {{{({\partial \theta /\partial {x_2}})}^2}}$ and $\overline {{{( {\partial \theta /\partial {x_1}} )}^2}} /\overline {{{({\partial \theta /\partial {x_3}})}^2}}$) is close to unity (their table III or figure 11a). In particular, this ratio is equal to 1.05 at $Sc=1$ and $Re_\lambda =140$ and 240, respectively (their table III). This implies that local isotropy is satisfied adequately in the context of the ratio of parallel-to-perpendicular scalar-gradient variances in this flow and justifies the use of $\overline {{{( {{{\partial \theta }}/{{\partial {x_1}}}} )}^2}}$ in (2.11) and (2.9). It is worth mentioning that local isotropy does not require ${R_2}$, ${R_4}$ and ${R_6}$ in (2.11) to be zero. For example, ${R_2}$, the mixed velocity-scalar derivative skewness, represents the production of ${\bar {\varepsilon }_\theta }$ generated by stretching of the scalar field as a result of the turbulent strain rate. Further, for $n=2,4,6$ and when local isotropy is satisfied, any combination of the indices in ${A_2}, {A_4}$ and ${A_6}$ will lead to a term involving the permutation symbols $\epsilon _{111}$, $\epsilon _{11j}$ or $\epsilon _{1jj}$, which are zero. Based on the above analysis, it can be concluded from (2.11) that, due to the presence of the mean shear, ${A_2}, {A_4}$ and ${A_6}$ may not be 0 at finite $Pe_{\lambda _\theta }$ and thus the flow is anisotropic in the context of ${A_2}, {A_4}$ and ${A_6}$. Therefore, (2.11) provides a relation between the mean scalar gradient and the local anisotropy. Namely, since, regardless of whether local isotropy is satisfied, the large-scale production term (the term on left-hand side of (2.11)) is non-zero and the corresponding small-scale terms (${A_2}, {A_4}$ and ${A_6}$ in (2.11)) are zero when local isotropy is assumed, we can use (2.11) to quantify the effect of the large-scale forcing caused by the mean shear on the degree of isotropy of the small scales. Specifically, the relation shows how the level of local anisotropy depends on the magnitudes of ${S^*_\theta }$, $Pe_{\lambda _\theta }$ and ${R_n}$ for $n=2,4$ and 6. If ${R_n}{S^*_\theta }$ for $n=2,4, 6$ does not increase as rapidly as $Pe_{\lambda _\theta }$, ${A_2}, {A_4}$ and ${A_6}$ should then go to zero when $Pe_{\lambda _\theta }$ is sufficiently large, irrespective of the magnitude of ${S^*_\theta }$. In particular, for a given ${S^*_\theta }$, if ${R_n}$ for $n=2,4,6$ are constant, $A_n$ will behave as $Pe_{\lambda _\theta }^{-1}$ in statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient. Figure 1(a) shows the distributions of $S^*_\theta$ and $R_2$ in statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient at $Re_\lambda =38$. We can observe that the magnitude of $R_2$ decreases slightly as $Sc$ increases for $Sc\lesssim 10$ and appears to be approximately constant for $Sc\gtrsim 10$. Similarly, the magnitude of $S^*_\theta$ increases as $Sc$ increases for $Sc\lesssim 10$ and also becomes approximately constant for $Sc\gtrsim 10$. We thus can conclude that $A_2$ should behave as $Pe_{\lambda _\theta }^{-1}$ for $Sc\gtrsim 10$. It is worth mentioning that the values of $R_4$ and $R_6$ in this flow are not available. However, $R_2$, $R_4$ and $R_6$ can be related to the normalized high-order moments of scalar derivatives via the following Cauchy–Schwarz inequalities:

(2.13a)$$\begin{gather} \overline{\left| {{{\left( {\frac{{\partial \theta }}{{\partial {x_1}}}} \right)}^2}\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right|} \le {\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_1}}}} \right)}^4}} ^{1/2}}{\overline{{{\left( {\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right)}^2}} ^{1/2}} \Rightarrow | {{R_2}} | \le \frac{{{{\overline{{{( {\partial \theta /\partial {x_1}} )}^4}} }^{1/2}}}}{{\overline{{{( {\partial \theta /\partial {x_1}} )}^2}} }} = F_{{\parallel} 4}^{1/2}, \end{gather}$$

(2.13b)$$\begin{gather}\overline{\left| {{{\left( {\frac{{\partial \theta }}{{\partial {x_1}}}} \right)}^4}} \right|\frac{{\partial {u_1}}}{{\partial {x_1}}}} \le {\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_1}}}} \right)}^8}} ^{1/2}}{\overline{{{\left( {\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right)}^2}} ^{1/2}} \Rightarrow | {{R_4}} | \le \frac{{{{\overline{{{( {\partial \theta /\partial {x_1}} )}^8}} }^{1/2}}}}{{{{\overline{{{( {\partial \theta /\partial {x_1}} )}^2}} }^2}}} = F_{{\parallel} 8}^{1/2}, \end{gather}$$

(2.13c)$$\begin{gather}\overline{\left| {{{\left( {\frac{{\partial \theta }}{{\partial {x_1}}}} \right)}^6}\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right|} \le {\overline{{{\left( {\frac{{\partial \theta }}{{\partial {x_1}}}} \right)}^{12}}} ^{1/2}}{\overline{{{\left( {\frac{{\partial {u_1}}}{{\partial {x_1}}}} \right)}^2}} ^{1/2}} \Rightarrow | {{R_6}} | \le \frac{{{{\overline{{{( {\partial \theta /\partial {x_1}} )}^{12}}} }^{1/2}}}}{{{{\overline{{{( {\partial \theta /\partial {x_1}} )}^2}} }^3}}} = F_{{\parallel} 12}^{1/2}, \end{gather}$$

where $F_{\parallel 4}$, $F_{\parallel 8}$ and $F_{\parallel 12}$ are the normalized 4th, 8th and 12th moments of scalar derivatives along the direction parallel to the mean gradient. The inequalities in (2.13) imply that the magnitudes of $R_2$, $R_4$ and $R_6$ should not exceed those of $F_{\parallel 4}$, $F_{\parallel 8}$ and $F_{\parallel 12}$. Figure 11(c) of Yeung et al. (Reference Yeung, Xu and Sreenivasan2002) ($Sc=0.25\unicode{x2013} 64$) shows that the flatness of scalar derivatives along directions parallel ($F_{\parallel 4}$) and perpendicular ($F_{ \bot 4}$) to the mean gradient are approximately constant for $Sc>10$ at $Re_\lambda =38$. Also, at a higher $Re_\lambda (=140)$, figure 8 of Buaria et al. (Reference Buaria, Clay, Sreenivasan and Yeung2021a) ($Sc=1\unicode{x2013} 512$) shows that the normalized $4$th, $6$th and $8$th moments of scalar derivatives along directions parallel ($F_{\parallel 4}$, $F_{\parallel 6}$ and $F_{\parallel 8}$) and perpendicular ($F_{\bot 4}$, $F_{\bot 6}$ and $F_{\bot 8}$) to the mean gradient approach each other and become independent of $Sc$ for $Sc\gtrsim 8$. Based on this information, for the normalized moments of the scalar derivatives and the inequalities in (2.13), it seems reasonable to assume that $R_4$ and $R_6$ become independent of $Sc$, like $R_2$ (figure 1a), for $Sc>10$. In this situation, we can conclude that $A_4$ and $A_6$, like $A_2$, should behave as $Pe_{\lambda _\theta }^{-1}$ for $Sc\gtrsim 10$.

Figure 1.

(a) Dependence of ${S^*_\theta }$ (${\bigtriangledown}$, blue) and $R_2$ ($\bullet$, red) on $Sc$ for $Re_\lambda =38$. (b) Dependence of $|S_3|$ ($\bullet$, blue), $|S_5|$ ($\blacktriangledown$, red) and $|S_7|$ (${\blacklozenge}$) at $Re_\lambda =38$. Lines: ${\sim }Pe_{{\lambda _\theta }}^{-1}$. The horizontal arrows in (a) and (b) indicate the region in which $R_2\approx const$ and ${S^*_\theta }\approx const$. To plot (a) and (b), we have used the DNS data of Yeung et al. (Reference Yeung, Xu and Sreenivasan2002); see text for details.

We recall that $|{S_{3}}|$, $|{S_{5}}|$ and $|{S_{7}}|$ are frequently used to test local isotropy of a passive scalar. In this context, it is interesting to examine whether $|{S_{3}}|$, $|{S_{5}}|$ and $|{S_{7}}|$ will follow the prediction (2.11), equivalently whether $|{S_{3}}|$, $|{S_{5}}|$ and $|{S_{7}}|$ will follow ${A_n}\sim Pe_{\lambda _\theta }^{-1}$ for $n=2,4,6$ for $Sc\gtrsim 10$. Although ${A_n}$ and $|{S_m}|$ are different small-scale quantities, they have some features in common. For example, they both are strongly affected by the uniform mean scalar gradient; they both quantify the degree of local anisotropy of the passive scalar. Figure 1(b) shows the distributions of $|{S_{3}}|$, $|{S_{5}}|$ and $|{S_{7}}|$ vs $Pe_{\lambda _\theta }$, using the data of Yeung et al. (Reference Yeung, Xu and Sreenivasan2002). In order to obtain the values of $Pe_{\lambda _\theta }$, we have used the following relation:

(2.14)\begin{equation} \frac{{P{e_{{\lambda _\theta }}}}}{{R{e_\lambda }}} = {\left( {\frac{{6Sc{R_\theta }}}{5}} \right)^{1/2}} ,\end{equation}

where the time scale ratio $R_\theta$ can be rewritten as

(2.15)\begin{equation} {R_\theta } = \frac{{\overline{{\theta ^2}} /{{\bar \varepsilon }_\theta }}}{{\overline{{q^2}} /\bar \varepsilon }} = \frac{{\overline{{\theta ^2}} \bar \varepsilon }}{{3\overline{u_1^2} {{\bar \varepsilon }_\theta }}} = \frac{{\dfrac{{\bar \varepsilon L}}{{{{\overline{u_1^2} }^{3/2}}}}}}{{\dfrac{{3{{\bar \varepsilon }_\theta }L}}{{{{\overline{u_1^2} }^{1/2}}\overline{{\theta ^2}} }}}} = \frac{{{C_\varepsilon }}}{{3{C_{\varepsilon \theta }}}}, \end{equation}

where $L$ is the integral length scale. The parameters $C_\varepsilon$ and $C_{\varepsilon \theta }$ at $Re_\lambda$=38 and $Sc=0.25\unicode{x2013} 64$ are calculated from table 1 of Donzis et al. (Reference Donzis, Sreenivasan and Yeung2005). Note that, at $Pe_{\lambda _\theta }=121$ (or equivalently $Sc=16$), there are two values of $|S_m|$ at each $m$ (=3,5,7) (see figure 12 of Yeung et al. Reference Yeung, Xu and Sreenivasan2002), corresponding to two simulations at different spatial resolutions, i.e. $k_{max}\eta _ B=1.48$ and 2.95 (see table I of Yeung et al. (Reference Yeung, Xu and Sreenivasan2002); here, $k_{max}$ is the resolved highest wavenumber and $\eta _ B$ is the Batchelor length scale). In figure 1(b), the values of $|S_m|$ at $k_{max}\eta _ B=1.48$ are not shown since small values of $k_{max}\eta _ B$ would result in an underestimation of the magnitude of $|S_m|$; for example, this can be observed from table III of Yeung et al. (Reference Yeung, Xu and Sreenivasan2002) in the context of $|S_3|$ and $|S_4|$ at $Sc=16$. We can observe from figure 1(b) that $|S_3|$, $|S_5|$ and $|S_7|$ exhibit a tendency to behave as $Pe_{\lambda _\theta }^{-1}$ for $Sc>10$ (or equivalently ${Pe_{{\lambda _\theta }}}>91$). Namely, the flow approaches local isotropy as $Pe_{\lambda _\theta }^{-1}$ in the context of $|S_3|$, $|S_5|$ and $|S_7|$.

We now examine whether the behaviours of $|S_3|$, $|S_5|$ and $|S_7|$ follow the behaviours of ${A_2}, {A_4}$ and ${A_6}$ (or the prediction (2.11)) at $Re_\lambda =140$. Buaria et al. (Reference Buaria, Clay, Sreenivasan and Yeung2021b) show that $C_{\varepsilon \theta }$ decreases as $Sc$ increases from 1 to 512 at $Re_\lambda =140$ (see their figure 2, which was digitized to replot $|S_3|$, $|S_5|$ and $|S_7|$ vs $Pe_{\lambda _\theta }$). Figure 2 shows the distributions of $|S_3|$, $|S_5|$ and $|S_7|$ of Buaria et al. (Reference Buaria, Clay, Sreenivasan and Yeung2021a) as a function of $Pe_{\lambda _\theta }$. To replot the data we used (2.14) and (2.15) with $C_\varepsilon =0.46$ (estimated using data in table 1 of Donzis et al. (Reference Donzis, Sreenivasan and Yeung2005) at $Re_\lambda =140$) and the values of $C_{\varepsilon \theta }$, the distributions of $|S_3|$, $|S_5|$ and $|S_7|$ vs $Sc$ for the data of Buaria et al. (Reference Buaria, Clay, Sreenivasan and Yeung2021a) are replotted in figure 2 as a function of $Pe_{\lambda _\theta }$. Figure 2 shows that $|S_3|$, $|S_5|$ and $|S_7|$ follow approximately the behaviour $Pe_{\lambda _\theta }^{-1}$ for $Sc>10$ (or equivalently ${P{e_{{\lambda _\theta }}}}>257$). Namely, the flow approaches local isotropy as $Pe_{\lambda _\theta }^{-1}$ in the context of $|S_3|$, $|S_5|$ and $|S_7|$. We can observe that the behaviour of $|S_3|$, $|S_5|$ and $|S_7|$ for $Sc>10$ in figures 1(b) and 2 is consistent with the prediction in (2.11).

Figure 2.

Dependence of $|S_3|$ ($\bullet$, blue), $|S_5|$ ($\blacktriangledown$, red) and $|S_7|$ (${\blacklozenge}$) on $Pe_{\lambda _\theta }$ for $Re_\lambda =140$. Lines: ${\sim }{P{e_{{\lambda _\theta }}}}^{-1}$. The horizontal arrow indicates the region where $Sc>10$, $S_\theta ^*$ and $|R_2|$ are approximately constant and the normalized $4$th, $6$th and $8$th moments of scalar derivatives along directions parallel and perpendicular to the mean gradient are also approximately constant. The symbols correspond to the DNS data of Buaria et al. (Reference Buaria, Clay, Sreenivasan and Yeung2021a).