2. Dual scaling for the transport equation for $\overline {{{(\delta \theta )}^2}}$

The transport equation for $\overline {{{(\delta \theta )}^2}}$ in statistically stationary homogeneous isotropic turbulence under a uniform mean scalar gradient in the $x$ direction ( ${{\partial \Theta }}/{{\partial {x}}}$ ) is given by (Danaila & Mydlarski Reference Danaila and Mydlarski2001)

(2.1) \begin{equation} -\frac {2\,{{\rm d}\Theta /{\rm d} x}}{{{r^2}}}\int _0^r {s^2}\,\overline { \delta \theta\, \delta u}{\rm d}s - \overline {\delta u\,{{(\delta \theta )}^2}} + 2\kappa \frac {\partial }{{\partial r}}\overline {{{(\delta \theta )}^2}} = \frac {4}{3}{\overline \varepsilon _\theta }r, \end{equation}

where $s$ is a dummy variable. The second and third terms on the left-hand side of (2.1) represent the scalar variance transfer and molecular diffusion, respectively. The first term is a production term due to the mean temperature gradient.

We first examine the scaling based on $\theta _B$ , $u_K$ and $\eta$ . When the effect of the production term, i.e. the first term in (2.1), is neglected, the similarity of (2.1) is readily satisfied by

(2.2) \begin{align}&\,\,\, \frac {{\overline {{{(\delta \theta )}^2}} }}{{\theta _B^2}} = f_1\left ( {\frac {r}{{{\eta }}}} \right ), \end{align}

(2.3) \begin{align}& \frac {{\overline {{{\delta u\,{(\delta \theta )}^2}}} }}{{u_K \theta _B^2}} = g_1\left ( {\frac {r}{{{\eta }}}} \right ), \end{align}

when the $\theta _B$ , $u_K$ , $\eta$ scaling applies. This was discussed in Lemay, Djenidi & Antonia (Reference Lemay, Djenidi and Antonia2020) (see also Djenidi et al. Reference Djenidi, Antonia and Tang2022). Equation (2.1) can be rewritten as

(2.4) \begin{eqnarray} \frac {4}{3}\frac {r}{{{\eta }}} = \frac {{2\kappa \theta _B^2}}{{\overline \varepsilon _\theta \eta ^2}}\,f'_1\left ( {\frac {r}{{{\eta }}}} \right ) - \frac {{u_K\theta _B^2}}{{\overline \varepsilon _\theta {\eta }}}\,g_1\left ( {\frac {r}{{{\eta }}}} \right ) \end{eqnarray}

(here, the prime signifies a derivative with respect to $r/\eta$ ), so that $ {{\kappa \theta _B^2}}/{{\overline \varepsilon _\theta \eta ^2}}$ and ${{u_K\theta _B^2}}/{{\overline \varepsilon _\theta {\eta }}}$ (or ${{u_K\theta _B^2}}/{{\overline \varepsilon _\theta {\eta }}}$ and ${{u_K}/{\eta }}{Sc\,\nu }$ ) are constants when $Sc\approx 1$ . The scaling based on $\theta _B$ , $u_K$ and $\eta$ is a possible solution since

(2.5) \begin{eqnarray} \frac {{\kappa \theta _B^2}}{{\overline \varepsilon _\theta \eta ^2}}=1 \quad \text{and} \quad \frac {{u_K}{\eta }}{Sc\nu }=\frac {1}{Sc}. \end{eqnarray}

Figure 1.

Dependence of $S^*_\theta$ on $Re_\lambda$ in the stationary homogeneous isotropic turbulence with a uniform mean scalar gradient at $Sc=1$ ; they are estimated from the DNS data of Yeung et al. (Reference Yeung, Xu and Sreenivasan2002) (circles), Gotoh & Watanabe (Reference Gotoh and Watanabe2012) (triangles) and Gauding (Reference Gauding2014) (squares).

We next examine the scaling based on $\theta '$ , $u'$ and $L$ . When the production term, i.e. the first term in (2.1), is retained and the molecular diffusion term in (2.1) is neglected, one can show that the similarity of (2.1) is satisfied by

(2.6) \begin{align}&\,\,\, \frac {{\overline {{{(\delta \theta )}^2}} }}{\overline {\theta ^2}}= f_2\left ( {\frac {r}{{{L}}}} \right ), \end{align}

(2.7) \begin{align}& \frac {{\overline {{{\delta u\,{(\delta \theta )}^2}}} }}{{\overline {u^2}^{1/2}\, \overline {\theta ^2}}} = g_2\left ( {\frac {r}{{{L}}}} \right ), \end{align}

when the $\theta '$ , $u'$ , $L$ scaling applies. Indeed, under this scaling, (2.1) can be rewritten as

(2.8) \begin{eqnarray} \frac {4}{3}\frac {r}{{{L}}} = - \frac {{\overline {u^2}^{1/2}\, \overline {\theta ^2}}}{{\overline \varepsilon {L}}}\,g_2\left ( {\frac {r}{{{L}}}} \right ) -\frac {2S_\theta }{{{(r/L)^2}}}\int _0^{r/L} {(s/L)^2}\frac {\overline { \delta \theta\, \delta u}}{\overline {u^2}^{1/2}\, \overline {\theta ^2}^{1/2}}\, {\rm d}(s/L), \end{eqnarray}

where $S_\theta = {{{{\overline {{\theta ^2}} }^{1/2}}\,{{\overline {u^2} }^{1/2}}({{{{\rm d}\Theta }}/{{{\rm d}{x}}}}}})/{{{{\bar \varepsilon }_\theta }}}$ is the non-dimensional scalar gradient. Since the coefficient of the left-hand side term of (2.8) is constant, the coefficients of the first and second terms on the right-hand side must also be constant for similarity to be satisfied. Data of Yeung et al. (Reference Yeung, Xu and Sreenivasan2002), Gotoh & Watanabe (Reference Gotoh and Watanabe2012) and Gauding (Reference Gauding2014) (see figure 1) were used to estimate these two coefficients. We found that the first coefficient $C_{\varepsilon \theta }^{-1}$ , where $C_{\varepsilon \theta }= {{\overline \varepsilon {L}} }/{{\overline {u^2}^{1/2}\, \overline {\theta ^2}}}$ , is approximately constant when $Re_\lambda \gt 100$ and $Sc\approx 1$ (e.g. Donzis et al. Reference Donzis, Sreenivasan and Yeung2005; Buaria et al. Reference Buaria, Clay, Sreenivasan and Yeung2021). The second coefficient $S_\theta$ is also constant, with value approximately 2.

The above considerations indicate that similarity based on $\theta _B$ , $u_K$ and $\eta$ can satisfy (2.1) when $r$ is sufficiently small. Equally, (2.1) admits a similarity solution based on $\theta '$ , $u'$ and $L$ when $r$ is sufficiently large, which shows that both $\overline {{{(\delta \theta )}^2}}$ and $\overline {{{\delta u\,{(\delta \theta )}^2}}}$ satisfy the dual scaling as $Re_\lambda \to \infty$ . Although not shown here, it is worth mentioning that the same conclusions can be drawn when the above dual scaling is applied to a slightly more general transport equation for $\overline {{{(\delta \theta )}^2}}$ , e.g. (8) of Gauding et al. (Reference Gauding, Danaila and Varea2017), than (2.1). In the next section, we will assess the validity of the dual scaling in the context of $\overline {{{{(\delta \theta )}^2}}}$ , and the range of $r/\eta$ or $r/L$ over which it applies.

Figure 2.

Effect of $Re_{\lambda }$ on the distributions of $\overline {{(\delta \theta )}^2}$ in statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient at $Sc=1$ , normalised by (a) $\theta _B$ , $\eta$ , and (b) ( $\theta '$ , $L$ ). Data are from Gauding (Reference Gauding2014) and Gauding et al. (Reference Gauding, Danaila and Varea2017).

3. Dual scaling of $\overline {{{(\delta \theta )}^2}}$

Using the data of Gauding (Reference Gauding2014) and Gauding et al. (Reference Gauding, Danaila and Varea2017), we show in figure 2 the distributions of $\overline {{{(\delta \theta )}^2}}$ normalised by $\theta _B$ , $\eta$ and $\theta '$ , $L$ , respectively, in statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient for $Re_\lambda$ ranging from 88 to 529, and $Sc=1$ . One can observe that as $Re_\lambda$ increases, the scaling based on $\theta '$ , $L$ extends to increasingly smaller values of $r/L$ , while the scaling based on $\theta _B$ , $\eta$ extends to increasingly larger values of $r/\eta$ . These observations suggest that both scalings should eventually overlap as $Re_\lambda$ increases indefinitely. It is therefore reasonable to assume that similarity solutions of the form

(3.1) \begin{align}& \frac {{\overline {{{(\delta \theta )}^2}} }}{{\theta _B^2}} = {f_1}\left ( {\frac {r}{\eta }} \right ), \end{align}

(3.2) \begin{align}& \frac {{\overline {{{(\delta \theta )}^2}} }}{{\overline {{\theta ^2}} }} = {f_2}\left ( {\frac {r}{L}} \right ), \end{align}

Figure 3.

Local slope $LS={\rm d} \log (\overline {{{(\delta \theta )}^2}})/{\rm d} (\log r)$ , corresponding to the data in figure 2.

may be observed for a given $Sc$ and infinitely large $Re_\lambda$ . Figure 2 suggests that the ranges of $r/\eta$ and $r/L$ over which (3.1) and (3.2) are valid increase with increasing $Re_\lambda$ , with the expectation that these ranges eventually overlap over the inertial range as $Re_\lambda \to \infty$ . It is important to recall that that we are dealing with the same distribution of $\overline {{{(\delta \theta )}^2}}$ , after subjecting it to two different types of normalisation. Accordingly, matching (3.1) with (3.2) requires that

(3.3) \begin{eqnarray} \theta _B^2\,{f_1}\left ( {\frac {r}{\eta }} \right ) = \overline {{\theta ^2}}\, {f_2}\left ( {\frac {r}{L}} \right ). \end{eqnarray}

If one assumes that ${f_1}(r/\eta )$ exhibits a power-law form in the inertial range, namely

(3.4) \begin{eqnarray} {f_1}\left ( {\frac {r}{\eta }} \right ) =C {\left ( {\frac {r}{\eta }} \right )^\alpha }, \end{eqnarray}

where $C$ is a constant, then

(3.5) \begin{eqnarray} {f_2}\left ( {\frac {r}{L}} \right )=C \frac {{\theta _B^2}}{{\overline {{\theta ^2}} }}{\left ( {\frac {L}{\eta }} \right )^\alpha }{\left ( {\frac {r}{L}} \right )^\alpha } = C\frac {{{C_{\varepsilon \theta }}}}{{{C_\varepsilon }}}\frac {{{{15}^{1/2}}}}{{{{{Re} }_\lambda }}}{\left ( {\frac {{{{C_\varepsilon }\, Re} _\lambda ^{3/2}}}{{{{15}^{3/4}}}}} \right )^\alpha }{\left ( {\frac {r}{L}} \right )^\alpha }. \end{eqnarray}

To arrive at (3.5), we used ${{u_K^2}}/{{\overline {{u^2}} }} = {{{{15}^{1/2}}}}/{Re_\lambda }$ and ${L}/{\eta } = C_\varepsilon\, {Re_\lambda ^{3/2}}/{{{{15}^{3/4}}}}$ . The plots in figure 2 support the argument that (3.4) and (3.5) will eventually and simultaneously be satisfied for significant ranges of $r/\eta$ and $r/L$ . Since $C_\varepsilon$ and $C_{\varepsilon \theta }$ are approximately constant for $Re_\lambda \gtrsim 200$ (see the inset of figure 2 b), the collapse based on $\theta '$ , $L$ becomes more convincing for $Re_\lambda \gtrsim 200$ . While an overlap region has not emerged yet in the distributions of figure 2, on can expect that as $Re_\lambda$ continues to increase, a region of overlap between the two different normalisations will emerge. Eventually, when $Re_\lambda \to \infty$ , an inertial range should be established rigorously where the two sets of scales become interchangeable. It is of interest to note that the formulation (3.5), or equivalently

(3.6) \begin{eqnarray} {f_2}\left ( {\frac {r}{L}} \right )=C_1\,Re_\lambda ^{3/2\alpha - 1}{\left ( {\frac {r}{L}} \right )^\alpha }, \end{eqnarray}

where $C_1={15}^{1/2-3\alpha /4} C C_{\varepsilon \theta }C_{\varepsilon }^{\alpha -1}$ , is independent of $Re_\lambda$ when $3/2\alpha -1=0$ , i.e. when $\alpha =2/3$ , which corresponds to the Kolmogorov–Obukhov–Corrsin prediction. Although not reported here, it is easy to show that applying a plausibility analysis (see Djenidi, Antonia & Tang Reference Djenidi, Antonia and Tang2023) to both (3.5) and (3.6), one can show that $\alpha = 2/3$ is the only possibility that does not violate the plausibility constraints.

Figure 3 shows the corresponding local slope $LS={\rm d} \log (\overline {{{(\delta \theta )}^2}})/{\rm d} (\log r)$ of the distributions shown in figure 2. We can observe that there is no well-defined plateau for all the data, even at $Re_\lambda \sim 500$ , underlining the importance of the finite Reynolds number effect in the context of establishing the inertial range. The same feature has been observed for the velocity structure functions in grid turbulence at higher $Re_\lambda$ ( $ \sim 1500$ ) (e.g. Tang et al. Reference Tang, Antonia and Djenidi2023). This indicates that there is no well-established overlap region (or inertial range) for the $Re_\lambda$ range considered. We note, however, that the larger $Re_\lambda$ , the closer $LS$ is to $2/3$ at scales $r/\eta \sim 200$ . The same feature can be observed for the data of Gotoh et al. (Reference Gotoh, Watanabe and Suzuki2011) and Shete et al. (Reference Shete, Boucher, Riley and de Bruyn Kops2022) at comparable $Re_\lambda$ ( $=174{-}633$ ) and $Sc=1$ (figure 4).

Figure 4.

Distributions of $\overline {{(\delta \theta )}^2} /\theta _B^2$ and corresponding local slope $LS(r/\eta )$ at $Sc=1$ in statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient. These are calculated based on the spectra of Gotoh et al. (Reference Gotoh, Watanabe and Suzuki2011) ( $Re_\lambda =174{-}468$ ) and Shete et al. (Reference Shete, Boucher, Riley and de Bruyn Kops2022) ( $Re_\lambda =633$ ); note that the distribution at $Re_\lambda =586$ of Gotoh et al. (Reference Gotoh, Watanabe and Suzuki2011) is not shown for clarity.

Figure 5.

Local slope $LS={\rm d} \log (\overline {{{(\delta \theta )}^2}})/{\rm d} (\log r)$ at $r/\eta =200$ . Black and blue symbols correspond to the data in figures 4 and 3, respectively. Note that the data for only $Re_\lambda \gtrsim 184$ in figure 3 are shown since this range of $Re_\lambda$ is large enough so that the scale $r=200\eta$ is not close to $L$ . The black and blue curves are the fits to the data based on (3.7) with $C_2=250$ and 155, respectively. The dotted horizontal line corresponds to the value $2/3$ .

In order to quantify, albeit approximately, the value of $Re_\lambda$ required for the $2/3$ power law to be established, we use the following Reynolds number dependent expression of $LS(r/\eta )$ at $r/\eta =200$ :

(3.7) \begin{equation} LS(r/\eta =200)=2/3-\frac {C_2}{{Re_\lambda ^{1.25}} }. \end{equation}

The constants $C_2$ (250 or 155) and exponent 1.25 are selected so that (3.7) fits the data in figures 3 and 4 reasonably well, as seen in figure 5. The data at $Re_\lambda =88$ and 119 in figure 3 are not shown since one cannot expect to observe a scaling range at such a relatively low $Re_\lambda$ . One can see that at least $Re_\lambda \sim 10^4$ is required to reach the $2/3$ power-law scaling. This value of $Re_\lambda$ is currently beyond the reach of the experiments in the laboratory and DNS. One expects that the value should be much larger in the absence of a mean scalar gradient. Interestingly, the way the $2/3$ power-law scaling is approached is different for the data of Gotoh et al. (Reference Gotoh, Watanabe and Suzuki2011) and Shete et al. (Reference Shete, Boucher, Riley and de Bruyn Kops2022) (blue symbols) and those of Gauding et al. (Reference Gauding, Danaila and Varea2017) (black symbols). A possible explanation is that the large-scale forcing for velocity and/or scalar fields varies between different studies, affecting the magnitude of the last term in (2.1), leading to different behaviour of $\overline {{{(\delta \theta )}^2}}$ at moderate scales; this merits further investigation.

Figure 6.

Distributions of $\overline {{(\delta \theta )}^2} /\theta _B^2$ and corresponding local slope $LS$ at $Sc\approx 0.7$ (blue and red curves) in statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient. They are calculated based on the spectra of Shete et al. (Reference Shete, Boucher, Riley and de Bruyn Kops2022) ( $Re_\lambda =633$ ) and Gotoh & Watanabe (Reference Gotoh and Watanabe2015) ( $Re_\lambda =805$ ). For comparison, also shown is the distribution of Gotoh & Watanabe (Reference Gotoh and Watanabe2015) with $Re_\lambda =805$ in stationary box turbulence with the scalar being excited by a random external force at low wavenumbers (black curve).

We conclude this section by examining (see figure 6) the distributions of $\overline {{{(\delta \theta )}^2}} /\theta _B^2$ and their local slope at $Re_\lambda =633$ (blue curves; Shete et al. Reference Shete, Boucher, Riley and de Bruyn Kops2022) and $Re_\lambda =805$ (red curves; Gotoh & Watanabe Reference Gotoh and Watanabe2015), and $Sc=0.7$ , in stationary forced periodic box turbulence with a uniform mean scalar gradient. The increase in $Re_\lambda$ is reflected in the distributions $\overline {{{(\delta \theta )}^2}} /\theta _B^2$ as well as in the local slopes beyond the dissipative range (i.e. $r/\eta \gtrsim 20$ ). Gotoh & Watanabe (Reference Gotoh and Watanabe2015) argue that the power-law exponent for $\overline {{{(\delta \theta )}^2}}$ in the range $80 \lesssim r/\eta \lesssim 400$ is approximately 0.642. However, they also argue that the power-law exponent for stationary box turbulence with the scalar being excited by a random external force at low wavenumbers is given by $0.639 +0.057\log (r/r_x)$ , where $r_x=70\eta$ , which led them to conclude that the ‘scaling exponents of passive scalars are not universal’. Included in figure 6 is the distribution of Gotoh & Watanabe (Reference Gotoh and Watanabe2015) at $Re_\lambda =805$ for stationary box turbulence with the scalar being excited by a random external force at low wavenumbers (black curve). Comparison between the distributions at the same $Re_\lambda =805$ shows clear differences between the cases with and without scalar gradient. The addition of a scalar gradient appears to accentuate the large-scale effects on the small scales, as illustrated on $LS(r/\eta )$ . This is indeed observed in figure 2 of Gotoh & Watanabe (Reference Gotoh and Watanabe2015), which shows that the contributions of the scalar gradient at $r/L\approx 0.145$ ( $r/\eta \approx 200$ ) are larger for the statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient than with the scalar being excited by a random external force at low wavenumbers. Consequently, one expects the emergence of an inertial-convective range to be delayed in the former case; this is indeed observed in figure 6. Evidently, it would be desirable to further investigate the finite Reynolds number effect on the behaviour of $\overline {{{(\delta \theta )}^2}}$ and its local slope at moderate scales by further increasing $Re_\lambda$ .