2.1. Eddy viscosity from velocity field

At the core of the theory is the kinetic energy cascade rate, which equals the kinetic energy dissipation rate for homogeneous and isotropic turbulence (Kolmogorov Reference Kolmogorov1941a , Reference Kolmogorovb ) as well as small- ${\textit{Pr}}$ convection (Bhattacharya, Verma & Samtaney Reference Bhattacharya, Verma and Samtaney2021). The kinetic energy dissipation rate is given by

(2.1) \begin{equation} \epsilon (\boldsymbol{x},t)=\frac {\nu }{2}\sum _{i,j} \left ( \frac {\partial u_i}{\partial x_{\!j}} + \frac {\partial u_{\!j}}{\partial x_i} \right )^2, \end{equation}

where $u_i$ are the components of the turbulent velocity field. Note that in RBC, $\boldsymbol{u}=\boldsymbol{u}'$ . Based on (2.1), one can define the following coarse-grained dissipation field:

(2.2) \begin{equation} \epsilon _r(\boldsymbol{x},t)=\frac {1}{B_r} \int _{B_r} \epsilon (\boldsymbol{x}+\boldsymbol{y},t) \, \textrm{d}^3 y, \end{equation}

where $B_r$ is a sphere of radius $r$ centred at $\boldsymbol{x}$ . The coarse-graining scale $r$ varies between the viscous and outer scales of the inertial range of turbulence, which will be Kolmogorov scale $\eta$ and $H$ . The logarithm of the new energy dissipation field $\epsilon _r(\boldsymbol{x},t)$ is assumed to be normally distributed. Let us associate a characteristic velocity ${\mathcal U}_r=(r \epsilon _r)^{1/3}$ and a characteristic time ${\mathcal T}_r=(r^2/\epsilon _r)^{1/3}$ with the scale $r$ , which is taken in the inertial subrange (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997). Furthermore, we define the following dimensionless relative velocities (Kolmogorov Reference Kolmogorov1962):

(2.3) \begin{equation} \boldsymbol{w}(\boldsymbol{\xi },t)= \frac {\boldsymbol{u}(\boldsymbol{x}+\boldsymbol{\xi }r, \, t+\tau {\mathcal T}_r) - \boldsymbol{u}(\boldsymbol{x},t)}{{\mathcal U}_r}. \end{equation}

In the case of assumed statistical stationarity, the temporal arguments can be neglected for the subsequent discussion. The focus is on spatial scales which are locally isotropic. Thus, we set $\boldsymbol{\xi }=(1,0,0)$ . A longitudinal velocity increment along this direction follows then with the definition (2.1) to

(2.4) \begin{equation} \delta u (r) = u_x(\boldsymbol{x} + \boldsymbol{\xi } r, t) - u_x (\boldsymbol{x},t) = w_x(1,0,0,0)(r \epsilon _r)^{1/3}. \end{equation}

A scale-resolved eddy viscosity in the inertial subrange is then given by

(2.5) \begin{equation} \nu _e(r) \approx r \delta u(r) = w_x(1,0,0,0)\,r^{4/3}\epsilon _r^{1/3}. \end{equation}

In the following, we simply write $w_x$ and drop the argument vector. The distribution of the fluctuating prefactor is unknown and has to be determined from DNS. This was done in Iyer, Sreenivasan & Yeung (Reference Iyer, Sreenivasan and Yeung2017) for high-Reynolds-number data for isotropic box turbulence. The probability density function (PDF) of $w_x$ was found to obey super-Gaussian tails. Furthermore, $\langle w_x \rangle = 0$ and $\langle w_x^3 \rangle = -4/5$ in correspondence with the $4/5$ th law (Kolmogorov Reference Kolmogorov1941b ). The ansatz (2.5) for $\nu _e(r)$ is consistent because

(2.6) \begin{equation} \nu _e(\eta )=\lim _{r \rightarrow \eta } \nu _e(r)=\nu \quad \mbox{since} \quad \nu _e(\eta )=\eta \delta u (\eta ) = Re_\eta \nu , \end{equation}

with Kolmogorov scale $\eta =(\nu ^3/\langle \epsilon \rangle )^{1/4}$ and $Re_\eta =1$ (Yakhot Reference Yakhot2006). On the other hand, when $r \rightarrow l$ with $l$ being the size of the largest eddy, we have

(2.7) \begin{equation} \nu _e(l)=\lim _{r \rightarrow l} \nu _e(r)=l \delta u(l) \sim l u, \end{equation}

where $u$ is the characteristic velocity of the large-scale eddies. Noting that the kinetic energy dissipation rate $\langle \epsilon \rangle$ is of the order of $O(u^3/l)$ , (2.7) can be further manipulated using scaling analysis as follows:

(2.8) \begin{equation} \nu _e(l) \sim l u \sim \frac {l}{u^3} \, u^4 \sim \frac {u^4}{\langle \epsilon \rangle } \sim \frac {k_u^2}{\langle \epsilon \rangle }, \end{equation}

where $k_u=\langle u^2 \rangle /2$ is the turbulent kinetic energy. This is precisely the $k_u$ – $\epsilon$ type ansatz for $\nu _e$ (Pope Reference Pope2000), used widely for computing the turbulent viscosity. The quantity $\nu _e(r)$ is a highly fluctuating field due to the fluctuations of $\epsilon _r$ and $w_x$ . An effective eddy viscosity at scale $r$ in the inertial range follows in the present framework by an ensemble average

(2.9) \begin{equation} \langle \nu _e(r) \rangle = \big\langle w_x \epsilon _r^{1/3} \big\rangle r^{4/3}. \end{equation}

This moment is to be determined from the simulation data for $\eta \ll r \ll H$ , with $H$ being the (scale) height of the turbulent convection layer.

2.2. Eddy diffusivity from temperature field

In the next step, we extend this framework to the temperature field $T$ , inspired by the work of Stolovitzky et al. (Reference Stolovitzky, Kailasnath and Sreenivasan1995). The procedure is a bit less straightforward compared with that for the eddy viscosity. The scalar field at hand is the fluctuating temperature field which is given by

(2.10) \begin{equation} T'(\boldsymbol{x},t)=T(\boldsymbol{x},t)-\langle T(z) \rangle , \end{equation}

where $\langle T(z) \rangle$ is the area- and time-averaged temperature. The corresponding thermal dissipation rate is given by

(2.11) \begin{equation} \chi (\boldsymbol{x},t) = \kappa\!\left ( \frac {\partial T'}{\partial x_{\!j}} \right )^2. \end{equation}

Similar to (2.2), we define the coarse-grained thermal dissipation field:

(2.12) \begin{equation} \chi _r(\boldsymbol{x},t)=\frac {1}{B_r} \int _{B_r} \chi (\boldsymbol{x}+\boldsymbol{y},t) \, \textrm{d}^3y. \end{equation}

The coarse-graining scale $r$ varies again between the diffusive and outer scales of the inertial-convective range of scalar turbulence, which will be the Corrsin scale $\eta _C$ and $H$ (see § 3 for the definition). Following Stolovitzky et al. (Reference Stolovitzky, Kailasnath and Sreenivasan1995), the increment of temperature fluctuations is given by

(2.13) \begin{equation} \delta T' (r) = T'(\boldsymbol{x} + \boldsymbol{\xi }r,t) - T'(\boldsymbol{x},t) = w_T(1,0,0,0) \frac {(r \chi _r)^{1/2}}{(r \epsilon _r)^{1/6}}. \end{equation}

In (2.13), $w_T$ is the dimensionless relative temperature. Thus, we get the following expression for the eddy diffusivity, cf. (2.5):

(2.14) \begin{equation} \kappa _e(r) \approx \frac {[\delta T'(r) \, \delta u (r)]^2}{\chi _r}=w_x^2(1,0,0,0) w_T^2 (1,0,0,0)\, r^{4/3} \epsilon _r^{1/3}. \end{equation}

Again, when $r \rightarrow l$ , we have

(2.15) \begin{equation} \kappa _e(l)=\lim _{r \rightarrow l} \kappa _e(r) = \frac {[\delta T'(l) \delta u (l)]^2}{\chi _l} \sim \frac {\theta ^2 u^2}{\chi _l} \sim \frac {\langle T'^2 \rangle u^2}{\langle \chi \rangle } \sim \frac {k_T k_u}{\langle \chi \rangle }, \end{equation}

where $\theta \sim \sqrt {\langle {T^\prime} ^{2} \rangle }$ is the characteristic temperature of the large-scale eddies and $\chi_l$ is the large-scale thermal dissipation rate. We note again that (2.15) is consistent with the $k_u$ – $\epsilon$ type ansatz for an eddy diffusivity, $\kappa _e \sim k_u k_T/\langle \chi \rangle$ with the thermal variance $k_T=\langle T^{\prime 2} \rangle /2$ .

The effective eddy diffusivity at scale $r$ follows

(2.16) \begin{equation} \langle \kappa _e(r) \rangle = \big\langle w_x^2 w_T^2 \epsilon _r^{1/3}\big\rangle r^{4/3}. \end{equation}

We end up with the following expression for the scale-dependent turbulent Prandtl number:

(2.17) \begin{equation} {\textit{Pr}}_t(r) = \frac {\big\langle \nu _e(r) \big\rangle}{\big\langle \kappa _e (r)\big\rangle} = \frac {\big\langle w_x \epsilon _r^{1/3} \big\rangle }{\big\langle w_x^2 w_T^2 \epsilon _r^{1/3}\big\rangle }. \end{equation}

We note that the thermal dissipation rate disappears in this derivation for the eddy diffusivity and turbulent Prandtl number. Further, the statistics of the dimensionless $w_T$ will depend on the molecular Prandtl number. So, it can be expected that the ${\textit{Pr}}$ -dependence of ${\textit{Pr}}_t$ enters dominantly via this property. In this paper, we investigate how these definitions work for a very-low-Prandtl-number fluid.

5.1. Test of flux-gradient ansatz

In this section, we discuss the behaviour of scale-dependent eddy viscosity, eddy diffusivity and the turbulent Prandtl number. The eddy viscosity and diffusivity are frequently estimated using the flux-gradient relations, see e.g. Wilcox (Reference Wilcox1993). Accordingly, the eddy viscosity $\nu _e$ is estimated by dividing the Reynolds stress $-\langle u_i^{\prime} u_{\!j}^{\prime} \rangle$ by the time-averaged velocity gradient $\partial \langle u_i\rangle /\partial x_{\!j}$ , and the eddy diffusivity $\kappa _e$ is estimated by dividing the turbulent heat flux $-\langle u_{\!j}^{\prime}T' \rangle$ by the time-averaged temperature gradient $\partial \langle T\rangle /\partial x_{\!j}$ , as discussed in § 1 (see (1.2)).

However, in RBC there is an absence of a mean flow (Samuel et al. Reference Samuel, Bode, Scheel, Sreenivasan and Schumacher2024); mean velocity and temperature gradients in the bulk are practically zero or indefinite. This aspect is exhibited in figure 4( $a$ ), where we plot the turbulent momentum flux $\langle u_x^{\prime} u_z^{\prime} \rangle _{x,t}$ versus the mean velocity gradient $\partial \langle u_x \rangle _{x,t}/\partial z$ , and in figure 4( $b$ ), where we plot the turbulent heat flux $\langle u_z^{\prime} T' \rangle _{x,t}$ versus the mean temperature gradient $\partial \langle T \rangle _{x,t}/\partial z$ . Here, $\langle \boldsymbol{\cdot }\rangle _{x,t}$ represents averaging over time $t$ and the $x$ -direction, and the quantities are computed in the horizontal midplane. The phase points in figure 4( $a$ ) are distributed homogeneously in all the four quadrants, implying that the turbulent viscosity $\nu _e = -\langle u_x^{\prime} u_z^{\prime} \rangle _{x,t}/(\partial \langle u_x \rangle _{x,t}/\partial z)$ frequently changes sign. It must be noted that a similar analysis on turbulent momentum flux was carried out by Pandey et al. (Reference Pandey, Krasnov, Schumacher, Samtaney and Sreenivasan2022a ) in a slender RBC cell for which the turbulent flow is highly constrained; here, we reinforce their observations by showing the same results for RBC in horizontally extended cells. In figure 4( $b$ ), the data points are distributed mostly in the second quadrant, implying that the turbulent diffusivity $\kappa _e = -\langle u_z^{\prime} T' \rangle _{x,t}/(\partial \langle T \rangle _{x,t}/\partial z)$ is largely positive. This is expected since $u_z^{\prime}$ and $T'$ are positively correlated (Verma, Kumar & Pandey Reference Verma, Kumar and Pandey2017), and the temperature decreases weakly with $z$ in the bulk region for small Prandtl numbers (Pandey et al. Reference Pandey, Schumacher and Sreenivasan2021, Reference Pandey, Krasnov, Schumacher, Samtaney and Sreenivasan2022a , Reference Pandey, Krasnov, Sreenivasan and Schumacherb ). However, some of the data points lie in the first quadrant, resulting in several instances of positive temperature gradient and hence negative turbulent diffusivity. Thus, computing the eddy quantities using the flux-gradient method becomes highly challenging for RBC, and therefore we resort to the alternative approach of refined similarity hypothesis for computing these quantities.

5.2. Application of self-similarity framework

Having verified the approximate log–normal distributions of $\epsilon _r$ and $\chi _r$ , we proceed to compute the eddy viscosity and eddy diffusivity using (2.5) and (2.14), respectively, for scales $r\in [\eta , H/2]$ for ${\textit{Ra}}=10^6$ and $10^7$ , and for $r \in [\eta ,H]$ for ${\textit{Ra}}=10^5$ . Once again, the aforementioned quantities are computed using the data in the bulk region ( $5 \leqslant x/H \leqslant 20$ and $5 \leqslant y/H \leqslant 20$ ) on the horizontal midplane and are averaged over all time frames. It must be noted that we get similar results on computing the eddy quantities in other planes close to the horizontal midplane (see Appendix B). Here, we remark that for $r\gt H/2$ , the expressions for the eddy viscosity and diffusivity given by (2.5) and (2.14) lose accuracy, due to the effects of the walls; thus, we restrict distances to $r\leqslant H/2$ . However, for ${\textit{Ra}}=10^5$ , the Corrsin length scale itself is $\eta _C\approx 0.5$ (as shown in table 1). Thus, we had to go up to a distance of $r\sim H$ in order to capture an adequate range of length scales for which $\kappa _e\gt \kappa$ .

Figure 5. Plots of eddy viscosity, eddy diffusivity and scale-resolved turbulent Prandtl number versus the normalised length scale $r/\eta$ for ${\textit{Ra}}=10^5$ (red lines), ${\textit{Ra}}=10^6$ (green lines) and ${\textit{Ra}}=10^7$ (blue lines) in decreasing order of thickness. (a) Normalised eddy viscosity $\nu _e/\nu$ and eddy diffusivity $\kappa _e/\kappa$ . (b) Magnified plot of $\nu _e/\nu$ which fits closely with $r^{4/3}$ curve. (c) Magnified plot of $\kappa _e/\kappa$ along with the corresponding best-fit curves. (d) Scale-dependent turbulent Prandtl number ${\textit{Pr}}_t$ . In (a) and (b), the vertical dotted line represents $r/\eta =40$ , which marks the lower end of the inertial subrange. In (d), the dotted lines correspond to the regime where $\kappa _e\lt \kappa$ , and the solid lines correspond to $\kappa _e \geqslant \kappa$ . The inset in (d) exhibits the plots of ${\textit{Pr}}_t$ versus the length scale $r$ for $0.25 \leqslant r \leqslant 1$ .

We normalise the eddy viscosity and diffusivity with their respective molecular counterparts and plot these quantities versus $r/\eta$ in figure 5( $a$ ). It must be noted that magnitudes of the eddy diffusivity which are smaller than their molecular counterparts do not make sense as $\kappa _e \lt \kappa$ effectively means the presence of temperature filaments which are smaller than the Corrsin scale. Therefore, we exhibit only those regimes where $\kappa _e \geqslant \kappa$ . Since we consider ${\textit{Pr}} =0.001 \ll 1$ and moderate Rayleigh numbers, the range of length scales for which $\kappa _e \geqslant \kappa$ is small, pertaining to $r \geqslant 0.8H$ , $0.4H$ and $0.3H$ for ${\textit{Ra}}=10^5$ , $10^6$ and $10^7$ , respectively. Hence, only a very small portion of $\kappa _e$ curves is exhibited in figure 5( $a$ ) and there is no overlapping regime of $r/\eta$ corresponding to $\kappa _e \geqslant \kappa$ for the three Rayleigh numbers.

Figure 5(a) shows several interesting features. Both eddy viscosity and diffusivity increase with the length scale, which is expected. The curves of the normalised eddy viscosity collapse very well for all the Rayleigh numbers, implying that the normalised eddy viscosity depends only on the normalised length scale, and not on ${\textit{Ra}}$ . However, the normalised eddy diffusivity curves do not collapse, suggesting that these quantities do not depend on the normalised length scale alone, and there is an additional ${\textit{Ra}}$ -dependence. We have verified that a rescaling with $\eta _C$ rather than $\eta$ did not lead to a collapse of the data. The trend of the curves for $\kappa _e$ implies that for a particular $r/\eta$ , the normalised eddy diffusivity decreases with the increase of ${\textit{Ra}}$ . However, since there is no overlapping regime of $r/\eta$ corresponding to $\kappa _e\gt \kappa$ , the ${\textit{Ra}}$ -dependence of the normalised eddy diffusivity remains inconclusive.

Figure 5(b) reveals that the eddy viscosity curves fit well with $\nu _e \sim r^{4/3}$ for $r \gg \eta$ , which is consistent with the effective eddy viscosity relation given by (2.9). As mentioned in § 2.1, this scaling relation is expected to hold in the inertial subrange, which can be determined by investigating the kinetic energy spectrum $E(k)$ (Verma Reference Verma2019). The kinetic energy spectrum is defined as

(5.1) \begin{equation} E(k) = \sum _{k \leqslant |\boldsymbol{k}'| \lt k+1} \frac {1}{2} \big|\hat {\boldsymbol{u}}\big(\boldsymbol{k}'^2\big)\big|, \end{equation}

where $k$ is the wavenumber and $\hat {\boldsymbol{u}}$ is the Fourier transform of the velocity field. In this case, we consider two-dimensional three-component (2D-3C) energy spectrum, with $|\boldsymbol{k}|=(k_x^2+k_y^2)^{1/2}$ and $|\hat {\boldsymbol{u}}|=(\hat {u}_x^2+\hat {u}_y^2+\hat {u}_z^2)^{1/2}$ , where the subscripts $x$ , $y$ and $z$ represent the components along their respective directions. Pandey et al. (Reference Pandey, Krasnov, Schumacher, Samtaney and Sreenivasan2022b ) showed that $E(k)$ in low- ${\textit{Pr}}$ mesoscale convection follows the Kolmogorov scaling, i.e. $E(k) \sim k^{-5/3}$ (Kolmogorov Reference Kolmogorov1941b ). In figure 6, we display the normalised kinetic energy spectra in the midplane by plotting $ (k\eta )^{5/3} \, E(k\eta ) (\langle \epsilon \rangle \nu ^5)^{-1/4}$ against the normalised wavenumber $k\eta$ . A plateau – which becomes wider with increasing ${\textit{Ra}}$ – can be detected for all the Rayleigh numbers. The right-hand side end of the plateau is observed at $k\eta \approx 0.15$ for all the Rayleigh numbers, which provides an estimate for the lower end of the inertial subrange. This corresponds to $r/\eta \approx 40$ in figures 5(a) and 5(b), which is indicated by a vertical dotted line. It is clear that the scaling $\nu _e \sim r^{4/3}$ in figure 5(b) is observed in the inertial subrange, which extends up to $r/\eta = 850, 1550$ and $3150$ for ${\textit{Ra}} = 10^5, 10^6$ and $10^7$ , respectively.

Figure 6. Normalised kinetic energy spectra versus the normalised wavenumber $k\eta$ in the midplane exhibit an inertial subrange, which broadens with increasing ${\textit{Ra}}$ . The end of the plateau region at $k\eta \approx 0.15$ corresponds to the lower limit of the inertial subrange at $r/\eta \approx 40$ . The horizontal line indicates the Kolmogorov constant $K_K \approx 1.6$ .

As exhibited in figure 5( $c$ ), the best-fit curves for eddy diffusivity follow $\kappa _e \sim r^{3.9}$ for ${\textit{Ra}}=10^6$ and $10^7$ , and $\kappa _e \sim r^{3.4}$ for ${\textit{Ra}}=10^5$ . Comparing these fits with the derived expression for effective thermal diffusivity given by (2.14), we infer that the prefactor $\langle w_x^2 w_T^2 \epsilon _r^{1/3} \rangle$ has a strong dependence on the length scale $r$ . This shows that the scaling which we developed in § 2.2 is not detected for the present data. We also remark that the observed difference in the exponents between ${\textit{Ra}}=10^5$ and ${\textit{Ra}}\gt 10^5$ might be due to the fact that the length scales analysed for ${\textit{Ra}}=10^5$ are well above $0.5H$ and the effects of boundaries become relevant. It remains to be seen how the results change when higher Rayleigh numbers at this low ${\textit{Pr}}$ are accessible.

To gain further insights, we look at the thermal energy spectra $E_T(k)$ in the midplane, which is defined as

(5.2) \begin{equation} E_T(k) = \pi k |\hat {T}(k)|^2 \, , \end{equation}

where $\hat {T}(k)$ is the Fourier transform of the temperature field in the midplane; see Pandey et al. (Reference Pandey, Krasnov, Sreenivasan and Schumacher2022b ) for further details. The spectra $E_T(k)$ are displayed in figure 7(a), where the maximum wavenumbers are restricted up to twice the value corresponding to the Corrsin scale. The Corrsin wavenumber $2\pi /\eta _C$ is indicated by dashed vertical lines. There is a prominent maximum in $E_T(k)$ around $k \approx 2$ , beyond which a rapid decay of the spectrum is observed. The maxima correspond to the characteristic scales of the turbulent superstructures (Pandey et al. Reference Pandey, Scheel and Schumacher2018), and the rapid decline thereafter is consistent with a highly diffusive temperature field observed in very-low- ${\textit{Pr}}$ convection, see Pandey et al. (Reference Pandey, Krasnov, Sreenivasan and Schumacher2022b ) for visualisations of the temperature field for ${\textit{Pr}} = 10^{-3}$ .

It has been proposed that in the inertial-convective subrange, the thermal energy spectrum scales as (Corrsin Reference Corrsin1951; Sreenivasan Reference Sreenivasan1996)

(5.3) \begin{equation} E_T(k) = K_C \langle \epsilon \rangle ^{-1/3} \langle \chi \rangle k^{-5/3} \, , \end{equation}

which in the normalised form reads

(5.4) \begin{equation} E_T(k\eta ) = K_C (\langle \epsilon \rangle ^{-3} \nu ^5)^{1/4} \langle \chi \rangle (k\eta )^{-5/3}. \end{equation}

Here, $K_C$ is the Corrsin–Obukhov constant (Sreenivasan Reference Sreenivasan1996). In figure 7(b), we show the normalised thermal energy spectrum and observe that the inertial-convective subrange, where the spectrum exhibits a plateau, hardly exists. Only for ${\textit{Ra}} = 10^7$ , a narrow plateau region could be detected for $k\eta \in [0.001 - 0.003]$ , the upper end of which corresponds roughly to $r/\eta \approx 2000$ . This scale is beyond the maximum $r/\eta$ exhibited in figure 5(c,d).

Figure 7. (a) Thermal energy spectra in the midplane decay rapidly beyond the prominent maximum that corresponds to the characteristic scale of superstructures. Dashed vertical lines indicate the Corrsin wavenumber $2 \pi /\eta _C$ . (b) Normalised spectra do not exhibit an inertial-convective subrange, except for ${\textit{Ra}} = 10^7$ , where a narrow plateau region is detected for $0.001 \leqslant k\eta \leqslant 0.003$ .

Since the eddy diffusivity increases more steeply with $r$ compared with the eddy viscosity, the scale-dependent turbulent Prandtl number decreases with increasing $r$ . This is exhibited in figure 5( $d$ ), where we plot ${\textit{Pr}}_t$ versus the normalised length scale $r/\eta$ , focusing on the regions corresponding to $\kappa _e \gt \kappa$ . To this end, the curve segments of ${\textit{Pr}}_t(r)$ are plotted as solid lines for $\kappa _e \geqslant \kappa$ . Although ${\textit{Pr}}_t$ physically does not exist for $\kappa _e \lt \kappa$ , the plots of ${\textit{Pr}}_t$ are still shown in this regime as dotted lines. This is done primarily to understand the trend of ${\textit{Pr}}_t$ had the regime corresponding to $\kappa _e \geqslant \kappa$ been wider. Figure 5( $d$ ) shows that ${\textit{Pr}}_t$ takes values between 1 and 10, which is consistent with earlier studies on low- ${\textit{Pr}}$ flows (Reynolds Reference Reynolds1975; Jischa & Rieke Reference Jischa and Rieke1979; Bricteux et al. Reference Bricteux, Duponcheel, Winckelmans, Tiselj and Bartosiewicz2012; Pandey et al. Reference Pandey, Schumacher and Sreenivasan2021, Reference Pandey, Krasnov, Schumacher, Samtaney and Sreenivasan2022a ). This is unlike the case for moderate- and large- ${\textit{Pr}}$ fluids where ${\textit{Pr}}_t$ is less than unity. Thus, the scale-resolved turbulent Prandtl number is by almost four orders of magnitude higher than its molecular counterpart for the present case at very low ${\textit{Pr}}$ .

Like the eddy diffusivity, the turbulent Prandtl number shows additional dependence on ${\textit{Ra}}$ apart from $r/\eta$ . Again, a conclusive ${\textit{Ra}}$ -dependence of ${\textit{Pr}}_t$ cannot be obtained because of the lack of an overlapping regime corresponding to $\kappa _e \geqslant \kappa$ . However, going by the trend of the turbulent Prandtl number as shown by dotted lines, it can be suggested that for a particular $r/\eta$ , ${\textit{Pr}}_t$ increases with increasing ${\textit{Ra}}$ . This is consistent with the fact that the normalised eddy diffusivity decreases with an increase in ${\textit{Ra}}$ as explained earlier. However, on a particular length scale $r$ , ${\textit{Pr}}_t$ decreases with the increase of ${\textit{Ra}}$ ; this is exhibited in the inset of figure 5(d) where we plot ${\textit{Pr}}_t$ versus $r$ . This behaviour of ${\textit{Pr}}_t$ is conclusive for ${\textit{Ra}}=10^6$ and $10^7$ , where there is a small overlapping regime of $r$ corresponding to $\kappa _e \geqslant \kappa$ . The trend is also consistent with results of Pandey et al. (Reference Pandey, Schumacher and Sreenivasan2021, Reference Pandey, Krasnov, Schumacher, Samtaney and Sreenivasan2022a ), who computed ${\textit{Pr}}_t$ using the $k_u$ – $\epsilon$ approach. In a future work, we plan to extend our studies to a wider range of Rayleigh and Prandtl numbers to examine the universality of the observed trends of ${\textit{Pr}}_t$ in the present work.