3.1. Space–time correlation of scalar fluctuations

The two-point space–time correlation $R_{\phi \phi }(\delta /D, \tau U/D)$ represents the statistical relationship between scalar field values separated by a spatial separation $z$ and a time separation $\tau$ . It is defined as

(3.1) \begin{equation} R_{\phi \phi }(\delta /D, \tau U/D) = \frac {\langle \phi ^\prime (z+\delta /D, t+\tau U/D)\, \phi ^\prime (z, t)\rangle }{\phi _{\textit{rms}}(z)\, \phi _{\textit{rms}}(z+\delta /D)}, \end{equation}

where $\phi ^\prime (z)$ is a scalar concentration fluctuation at spatial position $z$ , and $\phi _{\textit{rms}}$ denotes the RMS of $\phi$ . Here, $\phi ^\prime = \phi - \langle \phi \rangle$ , with $\langle \phi \rangle$ the time-averaged scalar concentration. In addition, $\delta$ and $\tau$ are the space separation along the streamwise direction and time separation, respectively, and $\left \langle {\cdot }\right \rangle$ denotes the ensemble averaging operation. The distance separation $\delta$ is taken along the streamwise direction $z$ at a radial position. Examples of the space–time correlations are taken at $r/D=0.3$ within a streamwise range $3\leqslant z/D \leqslant 4$ . Contours of the correlation $R_{\phi \phi }$ for Schmidt numbers $\textit{Sc} = 10^{-3}$ , $1$ and $10$ are shown in figures 4(b–d). In addition, the space–time correlation of velocity fluctuations $R_{uu}(\tau ,\delta )$ is shown in contours of figure 4(a) for comparison. In this figure, the iso-correlation lines of $R_{uu}$ exhibit elliptical shapes, agreeing with the elliptical approximation framework of He et al. (Reference He and Zhang2017). Similar observations are also evidenced in experiments of turbulent pipe flows (Li et al. Reference Mohaghar, Dasi and Webster2025) and turbulent channel flows (Kim & Moin Reference Kim and Moin1989; He & Zhang Reference He and Tong2006). Regarding the scalar correlation $R_{\phi \phi }$ , for $\textit{Sc}=10^{-3}$ , the iso-correlation lines also display clear elliptical shapes. For $\textit{Sc}=1$ , the elliptical iso-correlation curves turn out to be narrower, and they are further narrower for $\textit{Sc}=10$ . As the time separation $\tau$ increases, the $R_{uu}$ curves have smaller peaks, and the widths of the curves are roughly unchanged. Similar observations can be found in $R_{\phi \phi }$ for each of the Schmidt numbers. Among the different Schmidt numbers, $R_{\phi \phi }$ curves are narrower as $\textit{Sc}$ increases. This implies that as the Schmidt number increases, the scalar fluctuations decorrelate more rapidly. As the Schmidt number increases, the scalar diffuses to the surroundings more slowly; the concentration conserved structures are not correlated in space and time. If the Schmidt number is assumed to approach infinity, then the scalar concentration field is assumed to be uncorrelated, as considered in the Kraichnan model (Kraichnan Reference Lavertu and Mydlarski1968; Falkovich et al. Reference Falkovich, Gawȩdzki and Vergassola2001; Yang et al. Reference Yao, Rezaeiravesh, Schlatter and Hussain2023).

Figure 4.

(a) Space–time correlation of velocity fluctuations $R_{uu}(\tau ,\delta )$ . (b–d) Space–time correlations $R_{\phi \phi }(\tau ,\delta )$ for (b) $\textit{Sc} = 10^{-3}$ , (c) $\textit{Sc} = 1$ and (d) $\textit{Sc} = 10$ . The black lines denote lines of iso-correlation (a) $R_{\phi \phi }=0.96$ and $0.98$ , (b–d) $R_{\phi \phi }=0.5$ and $0.6$ . All correlations are obtained at $r/D = 0.3$ and $3\leqslant z/D\leqslant 4$ .

The correlation contours differ from the one in the Rayleigh–Bérnard convection (He & Zhang Reference He and Tong2006; Zhou et al. Reference Zimmerman, Antonia, Djenidi, Philip and Klewicki2011), where the contours display circular-like shapes in the centre of the convection cell given by the lack of mean advection, and elliptical shapes from the large-scale flow motion near the cell wall.

3.2. Convection velocity of scalar fluctuations

Figure 5.

The measured peak position $\delta _p/D$ as a function of $\tau U/D$ for the scalar and velocity field, corresponding to figure 4.

Figure 6.

Scalar convection speed at the streamwise $z/D = 3.5$ . (a) Radial profiles of the scalar convection speed $U_\phi /U_c$ at different Schmidt numbers $\textit{Sc}$ , where $U_c=\langle U(r=0) \rangle$ . (b) The scalar convection speed in wall unit $U^+_\phi$ along the distance from the pipe wall in wall unit $y^+$ . (c) The scalar convection speed normalised by the local mean velocity. (d) The contour of the ratio ${U}_\phi /{U}_u$ . (e) The ratio of the mean scalar convection speed $\overline {U}_\phi$ and the mean convection speed of velocity fluctuations $\overline {U}_u$ as a function of $\textit{Sc}$ .

The convection speed of the scalar field, $U_\phi$ , can be determined by analysing the space–time correlation of the scalar field, following established methods (Kim & Hussain Reference Kim and Hussain1993; He et al. Reference He, Jin and Yang2010; Zhou et al. Reference Zimmerman, Antonia, Djenidi, Philip and Klewicki2011; Hogg & Ahlers Reference Hogg and Ahlers2013). Specifically, $U_\phi$ is calculated from a position ( $\delta _p$ ) that maximises $R_{\phi \phi }$ for a given time separation ( $\tau U/D$ ), i.e. $\delta _p=\mathrm{max}(R_{\phi \phi }\,|\,\tau )$ . As shown in figure 5, the peak positions $\delta _p$ exhibit an approximately linear dependence on the time separation $\tau$ for all Schmidt numbers. The convection velocity $U_\phi$ is obtained from a linear fit to the relation $\delta _p = U_\phi \tau$ over the range $0 \leqslant \tau U/D \leqslant 0.08$ , where the linear behaviour is clearly observed. The same fitting range is consistently applied to all $\textit{Sc}$ cases and radial positions to ensure comparability. A similar linear trend is also observed for the velocity correlation $R_{uu}$ , consistent with previous studies (He et al. Reference He, Jin and Yang2010; Zhou et al. Reference Zimmerman, Antonia, Djenidi, Philip and Klewicki2011; Hogg & Ahlers Reference Hogg and Ahlers2013).

This process is repeated for each pipe radial position, giving the profiles of the convection speed shown in figure 6(a), while the corresponding ratio of the convection speed with the local mean velocity is shown in figure 6(c). The convection speed profile of the velocity fluctuations in the pipe flow is similar to that in the turbulent channel flow (Kim & Hussain Reference Kim and Hussain1993; Jiménez et al. Reference Jiménez, del Álamo and Flores2004; Flores & Jiménez Reference Flores and Jiménez2006; del Álamo & Jiménez Reference Corrsin2009; Geng et al. Reference Geng, He, Wang, Xu, Lozano-Durán and Wallace2015). For instance, close to the wall, the convection speed is approximately $0.55$ , or $U^+_\phi \approx 11.5$ for the distance from the wall $y^+\lt 10$ (see figures 6 a,b), although the Schmidt numbers are different among the studies. For the scalar mixing, at $\textit{Sc}=1$ , the $U_\phi$ profile nearly collapses with the velocity convection profile, which is also similar to $\langle U \rangle$ for $r/D \lesssim 0.48$ . When $\textit{Sc}\lt 1$ , e.g. $\textit{Sc}=10^{-3}$ , high molecular diffusivity increases the spreading of scalar structures, reducing their coherence and effective propagation speed, thus lowering $U_\phi$ . Conversely, for $\textit{Sc} \gt 1$ , reduced diffusivity preserves compact scalar structures, with their convection enhanced by the turbulent flow. These results challenge the assumption that $U_\phi$ equals the mean flow velocity, especially in flows with $\textit{Sc} \neq 1$ . Close to the wall, the scalar convection speed ranges from $U_\phi /U_c\approx 0.5$ to $U_\phi /U_c\approx 0.56$ for $\textit{Sc}$ from $10^{-3}$ to $10$ , and in wall units $11\lesssim U_\phi ^+ \lesssim 12$ for $10^{-3} \leqslant Sc \leqslant 10$ . This finding has implications that experimental data relying on Taylor’s frozen turbulence hypothesis needs close examination (Zaman & Hussain Reference Zhou, Li, Lu and Liu1981; Geng et al. Reference Geng, He, Wang, Xu, Lozano-Durán and Wallace2015), and suggests that $\textit{Sc}$ -dependent corrections may be required to accurately transform temporal data into spatial information, especially for the turbulence close to the wall.

To further examine the spatial dependence of the Schmidt number effect, figure 6(d) shows the contours of the ratio $U_\phi /U_u$ as a function of the radial position ( $r/D$ ) and $\textit{Sc}$ . It can be seen that the influence of $\textit{Sc}$ on the scalar convection speed extends over a broad region of the pipe cross-section. Although the magnitude of $U_\phi /U_u$ varies with $r/D$ , particularly near the wall, the overall trend of increasing $U_\phi$ relative to $U_u$ with the increase of $\textit{Sc}$ remains consistent across most radial locations. A shallow local minimum can be observed near $r/D \approx 0.45$ for $\textit{Sc}=10^{-3}$ to $1$ , indicating a slight reduction of $U_\phi /U_u$ in the outer region at low Schmidt numbers. For higher Schmidt numbers ( $\textit{Sc}=5$ and $10$ ), a mild bulge of $U_\phi /U_u$ appears in the region $r/D \approx 0.2{-}0.45$ , suggesting enhanced scalar convection relative to velocity fluctuations in this intermediate radial range. With the $U_\phi$ profile, the cross-sectional average of the scalar convection speed is obtained as

(3.2) \begin{equation} \overline {U}_\phi = \frac {\int _{0}^{D/2} 2 \pi U_\phi r\, \mathrm{d} r}{\int _{0}^{D/2} 2 \pi r\, \mathrm{d} r}=(8/D^2)\int _{0}^{D/2} U_\phi r\, \mathrm{d} r. \end{equation}

Similarly, with the $U_u$ profile, the mean velocity convection speed ( $\overline {U}_u$ ) can be obtained. The effect of the Schmidt number on the mean convection speed is summarised in figure 6(e). For $\textit{Sc}\lesssim 1$ , $\overline {U}_\phi$ increases mildly with $\textit{Sc}$ , approaching $\overline {U}_u$ at $\textit{Sc}\approx 1$ . As $\textit{Sc}$ further increases, $\overline {U}_\phi$ continues to increase, indicating that scalar fluctuations are convected slightly faster than velocity fluctuations. However, the dependence on $\textit{Sc}$ remains relatively weak over the present range, and is therefore interpreted as a qualitative trend rather than a well-defined scaling law. If the observed trend is assumed to persist, then as $\textit{Sc}$ approaches zero, $U_\phi$ approaches zero. In this asymptotic limit, the scalar concentration is extremely rapidly uniform in space, and correspondingly $R_{\phi \phi }$ tends to unity for any space and/or time separations. However, $U_\phi$ approaching zero seems to be physically infeasible, speculated from the observation that the smallest advection speed of the velocity field is not zero although the ensemble-averaged velocity is zero there. As $\textit{Sc} \rightarrow \infty$ , if the scaling persists, then $U_\phi$ approaches infinity, and this is physically infeasible; in such a circumstance, $R_{\phi \phi }$ would tends to be a delta function for any small space or time separation, and $U_\phi$ cannot be extracted from such a correlation. Based on these discussions, we cautiously conjecture that $U_\phi (Sc)$ with $\textit{Sc}$ may approach an upper-bound saturation. A hypothetical upper bound is ${U}_u(r=0)$ , corresponding to the maximum convection speed of velocity fluctuations in a wall-bounded turbulent flow from a statistical perspective. The scalar convection speed, inferred from scalar space–time correlations, should therefore be interpreted as a statistical signal velocity associated with the underlying turbulent velocity field. At high $\textit{Sc}$ , molecular diffusion is very weak, and thin scalar filaments can persist over longer times. These filaments are not transported as rigid frozen objects, but are continuously stretched, tilted and deformed by the turbulent velocity field while being slowly smoothed by molecular diffusion. Thus the measured $U_\phi$ indicates the combined effects of turbulent advection and filament deformation. As a result, the measured $U_\phi$ may become more strongly associated with relatively fast-moving turbulent motions. In this sense, ${U}_u(r=0)$ may serve as a plausible statistical reference for the upper limiting behaviour of $U_\phi$ .

Similarly, as $\textit{Sc}$ decreases towards zero, stronger molecular diffusion suppresses thin scalar filaments, and produces broader, smoother scalar structures. The scalar field then loses small-scale spatial coherence more rapidly, and the scalar convection speed may approach the velocity convection speed and even a lower-bound saturation conjecturally close to ${U}_u(r/D=0.5)$ , which corresponds to the minimum convection velocity of velocity fluctuations in the present flow. This interpretation is consistent with the fact that passive scalar transport is mediated by the velocity field through the advection–diffusion equation, while its apparent propagation also depends on diffusion and structural deformation. We emphasise that this interpretation should be regarded only as a tentative hypothesis based on the trends observed within the limited Schmidt number range considered here, rather than as a complete physical argument or a rigorously established asymptotic result. Since the present dataset is not sufficient to confirm the proposed saturation behaviour, further work is needed to verify this trend over a broader range of Schmidt numbers, and to investigate the detailed physical mechanisms responsible for it. A more conclusive assessment would require additional data, especially at more extreme $\textit{Sc}$ , together with velocity–scalar space–time correlation analysis, conditional structural analysis and further theoretical investigation.

Figure 7.

Scale-dependent convection velocity of scalar fluctuations at the streamwise $z/D = 3.5$ . Wavenumber-dependent scalar convection velocity $U_k^\phi$ as a function of streamwise wavenumber $k_z D$ at different radial locations, compared with the velocity-field convection velocity $U_k^u$ (symbols), for (a) $\textit{Sc} = 10^{-3}$ , (b) $\textit{Sc} = 1$ and (c) $\textit{Sc} = 10$ . Different line styles denote different radial positions. (d–f) Contours of the ratio $U_k^\phi /U_k^u$ as a function of Schmidt number $\textit{Sc}$ and wavenumber $k_z D$ at $r/D = 0$ , $0.2$ and $0.4$ , respectively. The results show that the $\textit{Sc}$ dependence of the convection velocity is strongly scale-dependent, with enhanced convection velocity at high $\textit{Sc}$ primarily associated with small-scale (high-wavenumber) scalar structures.

The dependence of the scalar convection velocity on $\textit{Sc}$ suggests that the relevant transport mechanism is scale selective. To quantify this effect, we analyse the convection velocity as a function of wavenumber using the space–time spectrum. Following del Álamo & Jiménez (Reference Corrsin2009) and Wu & He (Reference Wu and He2021a , Reference Yang, Pan and Heb ), the convection velocity associated with a given wavenumber is obtained from the space–time spectrum $E(k,f)$ , namely

(3.3) \begin{equation} U_k(k) = \frac {f_c(k)}{k}, \quad f_c(k) = \frac {\int f E(k,f)\,\mathrm{d}f}{E(k)}. \end{equation}

Applying this definition to the scalar spectrum ( $E_\phi$ ) and velocity spectrum ( $E_u$ ) gives the scale-dependent convection velocities $U_k^\phi$ and $U_k^u$ , respectively. In this framework, the convection velocity shown in figure 6(a) is recovered as a spectrum-weighted average, namely, $U = {\int U_k(k)\,k^2\,E(k)\,\mathrm{d}k}/{\int k^2\,E(k)\,\mathrm{d}k}$ . Figures 7(a–c) show the variation of $U_k^\phi$ with the streamwise wavenumber $k_z D$ at different radial locations. It is observed that the scalar convection velocity depends weakly on wavenumber at low $\textit{Sc}$ , while a clear scale dependence emerges as $\textit{Sc}$ increases. In particular, for $\textit{Sc}\gt 1$ , $U_k^\phi$ exceeds $U_k^u$ over a broad range of intermediate and high wavenumbers, indicating that small-scale scalar structures are convected faster than the corresponding velocity fluctuations. The scale-dependent behaviour is more clearly illustrated in figures 7(d–f), which show the ratio $U_k^\phi /U_k^u$ as a function of $\textit{Sc}$ and $k_z D$ . For increasing $\textit{Sc}$ , the enhancement of convection velocity is primarily concentrated at high wavenumbers, whereas large-scale motions remain comparatively less affected. This demonstrates that the $\textit{Sc}$ -dependence of the convection velocity is intrinsically linked to the spectral distribution of scalar fluctuations. Physically, at high Schmidt numbers, molecular diffusion is weak, allowing scalar fluctuations to persist at small scales (e.g. Buaria et al. Reference Brethouwer, Boersma, Pourquié and Nieuwstadt2021a , Reference Buaria, Clay, Sreenivasan and Yeungb ). These scalar structures may be statistically associated with relatively fast-moving turbulent motions, and thus impose a higher effective convection velocity. In contrast, at low $\textit{Sc}$ , diffusion suppresses small-scale scalar structures, leading to a dominance of larger, smoother scalar features that exhibit lower convection velocities. This scale-dependent interpretation provides an association to the observed $\textit{Sc}$ -dependent convection velocity, and agrees with the interpretation that the scalar transport characteristics are governed by the interplay between scalar diffusivity and the multiscale structure of turbulence. The present analysis does not directly identify the instantaneous velocity structures responsible for scalar transport; further investigation using velocity–scalar space–time cross-correlations or conditional structural analysis would be needed.

Figure 8.

The space–time correlation profiles $R_{uu}(\tau U/D=0)$ and $R_{\phi \phi }(\tau U/D=0)$ at $r/D=0.3$ at the streamwise $z/D = 3.5$ .

3.3. Taylor microscale

Another important quantity from the space–time correlation is the Taylor microscale. In turbulence, the Taylor microscale of velocity fluctuations, introduced by Taylor (Reference Tennekes and Lumley1935), characterises an intermediate length scale in turbulent flows that marks the transition from the inertial range, where viscosity has minimal impact, to the dissipation range, where viscous forces dominate and kinetic energy is converted to heat (Tennekes & Lumley Reference Villermaux1972; Pope Reference Sawford2000). Specifically, in the context of a scalar field, the scalar Taylor microscale $\lambda _\phi$ represents the intermediate scale where molecular diffusion begins to influence turbulent scalar filaments (Mohaghar et al. Reference Monin, Yaglom and Lumley2020) and dissipate the scalar variance at smaller scales. The profiles of $R_{\phi \phi }$ and $R_{uu}$ from figure 4 at $\tau =0$ are shown in figure 8. Although, as shown above, $U_\phi (r)$ nearly collapses with the convection speed profile of the velocity fluctuations at $\textit{Sc}=1$ , their correlation profiles are quite different in that $R_{\phi \phi }$ decays more rapidly along $\delta$ than $R_{uu}$ . For $\textit{Sc}\gt 1$ , $R_{\phi \phi }$ profiles are narrower, whereas $R_{\phi \phi }$ are wider for $\textit{Sc}\lt 1$ than $R_{\phi \phi }(Sc=1)$ .

The Taylor microscale can be obtained by fitting a parabola (the first three points in this study; Mohaghar et al. Reference Monin, Yaglom and Lumley2020) to the autocorrelation function $R_{\phi \phi }(\delta /D, \tau U/D)$ near $\delta /D = 0$ (Pope Reference Sawford2000; Talluru et al. Reference Tang, Antonia and Djenidi2019):

(3.4) \begin{equation} R_{\phi \phi }(\delta /D, \tau U/D=0) \approx 1-\frac {\delta ^2}{\lambda _\phi ^2}, \end{equation}

where $\lambda _\phi$ is the scalar Taylor microscale. Figure 8 presents the ensemble-averaged autocorrelation functions, where the area under these curves decreases as $\textit{Sc}$ increases, indicating that the scalar field becomes less correlated over space. This reduction in correlation length scales reflects the diminishing role of molecular diffusion at higher Schmidt numbers.

Figure 9.

The scalar Taylor microscale at the streamwise $z/D = 3.5$ . (a) Radial profiles of the scalar Taylor microscale $\lambda _\phi$ for three representative Schmidt numbers. (b) Scalar Taylor microscale $\lambda _\phi$ at five radial positions as a function of the Schmidt number, where the black filled diamonds denote the cross-sectional mean $\overline {\lambda }_\phi$ , and the dashed lines indicate scalings for reference. (c) Plots of $\lambda _\phi /\lambda _u$ as a function of the Schmidt number, where $\lambda _u$ is the velocity Taylor microscale, and the grey thick line denotes $\lambda _\phi /\lambda _u\approx 0.15$ . (d) The non-dimensional scalar dissipation rate $C_{\epsilon \phi }$ as a function of the Schmidt number.

The radial profiles of the scalar Taylor microscales are shown in figure 9(a). All three example profiles have their corresponding peaks at $r/D \approx 0.4$ . As the Schmidt number increases, the profile becomes flatter. The mean Taylor microscale $\overline {\lambda }_\phi = (8/D^2){\int _{0}^{D/2} \lambda _\phi r\, \mathrm{d} r}$ is plotted versus the Schmidt number $\textit{Sc}$ in figure 9(b) as the black filled diamonds. For $\textit{Sc}\leqslant 1$ , as $\textit{Sc}$ decreases, $\overline {\lambda }_\phi$ increases, approximately following a scaling close to $\textit{Sc}^{-0.03}$ ; however, for $\textit{Sc}\lt 10^{-1}$ , $\overline {\lambda }_\phi$ gradually deviates from this scaling. For $\textit{Sc}\gt 1$ , $\overline {\lambda }_\phi$ decreases and approaches an approximate scaling $\textit{Sc}^{-1/2}$ as $\textit{Sc}$ increases. To better examine the deviation from $\textit{Sc}^{-0.03}$ , we show $\lambda _\phi$ at five $r/D$ values in figure 9(b). For $\textit{Sc} \lesssim 1$ , $\lambda _\phi$ at $r/D=0$ increases as $\textit{Sc}$ decreases, approximating a scaling $\textit{Sc}^{-0.03}$ until $\textit{Sc} = 10^{-3}$ . However, at large $r/D$ , particularly $r/D=0.4$ , gradual deviation from the scaling $\textit{Sc}^{-0.03}$ can be observed. We speculate that this is probably due to the velocity Taylor microscale $\lambda _u$ (obtained from $R_{uu}(\delta ,\tau =0)\approx 1-\delta ^2/\lambda _u^2$ ), which quantifies the characteristic length scale, at which the fluid viscosity begins to significantly influence the turbulent eddies. (Note that $\lambda _u$ is a constant for each $\textit{Sc}$ case.) The figure 9(c) illustrates this examination, showing that when the ratio is $\lambda _\phi /\lambda _u\gtrsim 0.15$ (the grey thick line), the scaling begins to depart from the $\textit{Sc}^{-0.03}$ scaling. This implies that the small scalar structures ( $\textit{Sc}\lesssim 1$ ) have a fixed size to the background turbulence, before the diffusivity takes the dominant role. In other words, the turbulent velocity field governs the somewhat small characteristic scales of scalar variance (for $\textit{Sc}\lesssim 1$ ), rather than the scalar diffusivity itself. For $\textit{Sc} \geqslant 1$ , $\lambda _\phi$ exhibits an approximate scaling $\textit{Sc}^{-1/2}$ in a limited range of the Schmidt numbers. This observation is consistent with the dimension analysis of decaying isotropic scalar turbulence (George Reference George1992) and the prior study of homogeneous isotropic turbulence (Fox Reference Fox2003). This behaviour aligns with the decrease of Batchelor scale with the Schmidt number $\eta _B = \eta _K\, \textit{Sc}^{-1/2}$ . In contrast, for $\textit{Sc} \lt 1$ , enhanced diffusion dissipates small-scale fluctuations, shifting the scalar dissipation scale to larger lengths, and increasing $\lambda _\phi$ . To further interpret these trends, we follow the framework developed for homogeneous isotropic turbulence by Donzis et al. (Reference Donzis, Sreenivasan and Yeung2005), Yeung & Sreenivasan (Reference Zaman and Hussain2014) and Tang et al. (Reference Taylor2023). In this framework, the ratio between the scalar and velocity Taylor microscales can be expressed in terms of the time scale ratio ( $R_\phi$ ) (Tang et al. Reference Taylor2023), namely

(3.5) \begin{equation} \frac {\lambda _\phi }{\lambda _u} = \left ( \frac {6 R_\phi }{5\, Sc} \right )^{1/2}= \left ( \frac {2 C_\epsilon }{5\,Sc\,C_{\epsilon \phi }} \right )^{1/2}, \end{equation}

where $C_\epsilon = \epsilon L/u_{\textit{rms}}^3$ , $C_{\epsilon \phi } = \epsilon _\phi L/(u_{\textit{rms}} \langle \phi '^2 \rangle )$ , $L$ is the velocity integral length scale, $\epsilon$ is the kinetic energy dissipation rate, and $\epsilon _\phi$ is the scalar dissipation rate. Equation (3.5) shows that the Schmidt number dependence of $\lambda _\phi /\lambda _u$ is controlled not only by the explicit factor $\textit{Sc}^{-1/2}$ , but also by the ratio between the mechanical and scalar dissipation time scales. In the present study, the velocity field is fixed for all scalar cases, so that at a given radial position, $\lambda _u$ and $C_\epsilon$ remain unchanged. It is therefore natural to examine the Schmidt number dependence of $C_{\epsilon \phi }$ , shown in figure 9(d). Over the present range, $C_{\epsilon \phi }$ decreases approximately as $\textit{Sc}^{-0.9}$ for $\textit{Sc}\lt 1$ , whereas for $\textit{Sc}\gtrsim 1$ , its dependence on $\textit{Sc}$ becomes much weaker, approximately $\textit{Sc}^{-0.05}$ . Substituting these trends into (3.5) gives

(3.6) \begin{equation} \frac {\lambda _\phi }{\lambda _u} \sim \left ( Sc\, C_{\epsilon \phi } \right )^{-1/2} \sim \begin{cases} \textit{Sc}^{-0.05} & (Sc\lt 1), \\[3pt] \textit{Sc}^{-0.475} & (Sc\gt 1).\end{cases} \end{equation}

These indicate that the two approximate scalings of $\lambda _\phi$ in figure 9(b) are closely connected to the Schmidt number dependence of $C_{\epsilon \phi }$ . Equivalently, $C_{\epsilon \phi }$ may be interpreted as the scalar dissipation time scale normalised by the large-eddy turnover time. For $\textit{Sc}\lt 1$ , molecular diffusion acts rapidly, and the scalar dissipation time scale changes strongly with $\textit{Sc}$ , which compensates most of the explicit $\textit{Sc}^{-1/2}$ dependence in (3.5). For $\textit{Sc}\gt 1$ , the scalar dissipation time scale varies much more weakly with $\textit{Sc}$ , therefore the explicit Schmidt number dependence becomes dominant, yielding the near- $\textit{Sc}^{-1/2}$ scaling.

Given by the moderate Reynolds number considered in this study, the useful Schmidt numbers for observation of the scaling $\textit{Sc}^{-0.03}$ lie in a limited range, difficult to support the argument of the scaling. We would like to point out that we claim this scaling not quantitatively, but qualitatively, that this scaling is smaller in magnitude than that for $\textit{Sc}\gt 1$ . Other studies in much larger Reynolds numbers and/or over a wider range of Schmidt numbers are necessary to re-examine this observation. Regarding the scaling for $\textit{Sc}\gt 1$ , larger-Schmidt-number studies are clearly desirable to establish whether a robust asymptotic scaling exists. In the present range, the observed decrease of $\lambda _\phi (Sc)$ is consistent with expectations based on arguments from George (Reference George1992), Fox (Reference Fox2003), Donzis et al. (Reference Donzis, Sreenivasan and Yeung2005) and Tang et al. (Reference Taylor2023), but should be interpreted as a qualitative trend rather than a definitive $\textit{Sc}^{-1/2}$ scaling. A comparison with recent experimental measurements at $\textit{Sc}\approx 2000$ (Li et al. Reference Mohaghar, Dasi and Webster2025) shows that the scalar Taylor microscale at the pipe centre, $\lambda _\phi \approx 0.0035D$ , does not quantitatively follow the $\textit{Sc}^{-1/2}$ trend observed in the present simulations. In that experiment, the scalar is continuously released from a needle-like small tube at the pipe centre, resulting in a sustained and spatially inhomogeneous scalar concentration field. In contrast, the present simulations consider a different scalar initialisation and subsequent evolution. Since $\lambda _\phi$ depends on both scalar variance and dissipation, it is influenced by the large-scale transport history as well as local mixing processes. This discrepancy therefore highlights the sensitivity of $\lambda _\phi$ to flow configuration and scalar injection conditions, rather than invalidating the qualitative trend of decreasing $\lambda _\phi$ with increasing $\textit{Sc}$ . These considerations motivate the further analysis presented in the next section.