3.1. Thermal resistance at the walls

The temperature drop at the wall $\Delta T_w$ in RBC occurs across the thermal boundary layer thickness, which is defined as $\delta _T = \Delta T_w/ ({{\textrm d}\overline {T}}{/}{{\textrm d}z}) \big \rvert _w$ , where $({{\textrm d}\overline {T}}{/}{{\textrm d}z}) \big \rvert _w$ is the mean temperature gradient at the wall. Thus, ${\delta _T}/{H} = {(\Delta T_w}/{\Delta T)} Nu^{-1}$ . For turbulent convection in a regular RBC cell with a nearly isothermal core, $\Delta T_w = \Delta T/2$ and ${\delta _T}/{H} = ({1}/{2}) {Nu}^{-1}$ . However, for a slender RBC cell, we need to account for the drop in the tube part. Using (2.5), the thermal boundary layer thickness for slender RBC can be written as

(3.1) \begin{equation} \frac {\delta _T}{H} = \frac {1}{2} \left (1-\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}}\right ) Nu^{-1}. \end{equation}

To obtain the resistance at the walls, we use the correlations for heat flux in classical RBC ( $\varGamma \gtrsim 1$ ). Instead of Nusselt number $\textit{Nu}$ , however, we use an alternative parameter as proposed by Theerthan & Arakeri (Reference Theerthan and Arakeri2000) to non-dimensionalise the heat flux, which is more convenient and meaningful:

(3.2) \begin{equation} C_{q_w} = \frac {q''}{k \Delta T_w} \left (\frac {g \beta }{\nu \alpha } \Delta T_w\right )^{-1/3} = Ra_{\delta _T}^{-1/3}. \end{equation}

This proposal is an outcome of the observation that in turbulent natural convection, $\textit{Nu}$  scales roughly as $Ra^{1/3}$ , making the heat flux nearly independent of the length scale. By assuming the heat flux to be dependent only on $\Delta T_w$ , the temperature difference across the wall boundary layer, and the fluid properties, dimensional analysis gives the above expression for $C_{q_w}$ . It turns out that $C_{q_w} = Ra_{\delta _T}^{-1/3}$ , where $Ra_{\delta _T} = {g \beta \Delta T_w \delta _T^3}/(\nu \alpha )$ is the Rayleigh number based on conduction-layer thickness $\delta _T$ . The advantage of such a representation is the fact that unlike $\textit{Nu}$ which changes by orders of magnitude with changing $Ra$ , the value of $C_{q_w}$ lies between 0.1 to 0.3 not just for RBC, but for various configurations (see Arakeri Reference Arakeri2012; Arakeri, Kumar & Mahapatra Reference Arakeri, Kumar and Mahapatra2024) involving natural convection in the presence of a wall. The heat flux $q''= k ({{\textrm d}\overline {T}}/{{\textrm d}z}) \big \rvert _w = k {\Delta T_w}/{\delta _T}$ in terms of $C_{q_w}$ is thus given by

(3.3) \begin{equation} q'' = C_{q_w} k \left (\frac {g \beta }{\nu \alpha }\right )^{1/3} \Delta T_w^{4/3}. \end{equation}

In ‘standard’ RBC ( $\varGamma \gtrsim 1$ ) since $\Delta T_w \approx \Delta T/2$ , using (3.3) we get an expression for $C_{q_w}$ in terms of $\textit{Nu}$ and $Ra$ as $C_{q_w} = 2^{4/3} Nu Ra^{-1/3}$ . Using the usual representation of scaling for RBC, $Nu = CRa^n Pr^p$ , we can obtain $C_{q_w} = 2^{4/3} C Ra^{n-1/3} Pr^p$ . For the classical scaling regime ( $n=1/3$ ), we get $C_{q_w} = 2^{4/3} C Pr^p$ ; $C_{q_w}$ is only a function of $Pr$ and not $Ra$ . For example, the scaling proposed by Globe & Dropkin (Reference Globe and Dropkin1959), $Nu\,{=}\,0.069 Ra^{1/3} Pr^{0.074}$ gives $C_{q_w} = 0.1739 Pr^{0.074}$ . Note that $C_{q_w}$ is not just a compensated Nusselt number, it is a different and more suitable non-dimensional measure of heat flux than $\textit{Nu}$ . The physical significance of $C_{q_w}$ is the fact that $C_{q_w} = \textrm{const.}$ implies that for different $Ra$ , the conduction layer thickness $\delta _T$ adjusts itself to compensate for the temperature difference across it ( $\Delta T_w$ ) to keep $Ra_{\delta _T}$ a constant, which fits in with the argument of Malkus (Reference Malkus1954) and forms the basis for the models for turbulent RBC by Howard (Reference Howard and Görtler1966) and Theerthan & Arakeri (Reference Theerthan and Arakeri1998). For more discussion on $C_{q_w}$ and its advantages, the reader is referred to Theerthan & Arakeri (Reference Theerthan and Arakeri2000), Puthenveettil & Arakeri (Reference Puthenveettil and Arakeri2005), Arakeri (Reference Arakeri2012) and Arakeri et al. (Reference Arakeri, Kumar and Mahapatra2024).

For free convection over heated horizontal flat plates, at high enough Rayleigh numbers $Nu = 0.15 Ra_{w}^{1/3}$ for $8 \times 10^6 \lt Ra_w \lt 1.6 \times 10^9$ (Lloyd & Moran Reference Lloyd and Moran1974) with the length scale equal to the width of the plate $d$ and temperature scale $\Delta T_w$ as the temperature difference between the plate and the ambient. From this correlation, we get $C_{q_w} = 0.15$ . In both the above examples, the $Nu{-}Ra$ scaling laws follow the classical $1/3$ power law which gives a constant value (with a weak function of $Pr$ ) for $C_{q_w}$ . For a non-classical scaling, for example, $Nu=0.15Ra^{0.29}$ for $Pr=0.7$ in a RBC container of $\varGamma =1$ (Scheel & Schumacher Reference Scheel and Schumacher2017), we obtain $C_{q_w}=0.378 Ra^{-0.0433}$ , a weak function of $Ra$ .

We next check the values of $C_{q_w}$ obtained in the slender RBC cell using the $\textit{Nu}$ versus $Ra$ data presented in Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ). For a slender RBC cell, using (2.5) in (3.3) the expression for $C_{q_w}$ becomes

(3.4) \begin{equation} C_{q_w} = Ra_{\delta _T}^{-1/3} = 2^{4/3} Nu Ra^{-1/3} \left (1-\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}}\right )^{-4/3}. \end{equation}

Figure 2. (a) Variation of the dimensionless wall heat transfer coefficient $C_{q_w}$ with $Ra_d=Ra\varGamma ^3$ in a slender RBC cell, calculated using (3.4) for Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) ( $\varGamma =0.1$ and $Pr=1$ ). Red curve shows $C_{q_w}$ calculated from the Grossmann–Lohse (GL) theory (Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013) for regular RBC with $\varGamma =1$ , $Pr=1$ and black dotted line is the corresponding least-squares fit given by $C_{q_w}=0.1328+1.235Ra_d^{-0.18}$ . Blue dashed line shows a constant $C_{q_w}=0.1739$ approximation from Globe & Dropkin (Reference Globe and Dropkin1959). (b) Scaling of gradient-based dimensionless quantities $Nu_g$ versus $Ra_g$ in the tube region of Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ), calculated using (3.5) and (3.6) and compared with the two regimes of scaling in Pawar & Arakeri (Reference Pawar and Arakeri2016). In both panels, the data from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) are calculated from Nusselt number averaged at the top/bottom plates $\textit{Nu}$ (black markers) and that averaged in the whole volume $Nu_v$ (light-green markers).

The values of $C_{q_w}$ so calculated for slender RBC with $\varGamma =0.1$ and $Pr=1$ from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data are shown in figure 2(a) and plotted against $Ra_d=Ra\varGamma ^3$ . Two versions of $C_{q_w}$ are shown: one calculated from Nusselt number averaged over the top/bottom walls ( $\textit{Nu}$ ) and other averaged in the whole domain ( $Nu_v$ ). To compare this with that of a regular RBC, we show for a cell with $\varGamma =1$ and $Pr=1$ , the $C_{q_w}$ prediction from the GL theory with the updated prefactors (Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). A least-squares curve fit to the GL curve gives $C_{q_w}=0.1328+1.235Ra_d^{-0.18}$ for $Pr=1$ . Ideally, we should be using $Ra_w$ instead of $Ra_d$ for comparing the $C_{q_w}$ values for regular and slender RBC cases, because $\Delta T_w$ is the forcing at the wall and not $\Delta T$ as is assumed in the figure. However, $Ra_d=2Ra_w (1- ({{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}} )^{-1}$ and for the cases we are considering, the values of ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ are small, and since $C_{q_w}$ is a very weak function of $Ra_w$ , the plot of $C_{q_w}$ versus $Ra_w$ closely resembles that of $C_{q_w}$ versus $Ra_d$ . It can be seen that both versions (from $\textit{Nu}$ and $Nu_v$ ) of the Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data follow the prediction of GL theory very closely, validating our hypothesis that the heat transfer mechanism at the walls of both regular and slender RBC cells is the same. Also shown is the constant approximation of $C_{q_w}=0.1739$ based on the Globe & Dropkin (Reference Globe and Dropkin1959) fit for reference. The main conclusion is that heat transfer at the walls of slender RBC is conveniently obtained by using the non-dimensional heat transfer measure $C_{q_w}$ , whose value is nearly constant.

3.2. Thermal resistance in the tube part

Convection in the tube part of the slender RBC cell, which we term as TC, has a very different character compared with the convection near the walls. At high enough Rayleigh numbers the flow is axially homogeneous and the turbulence is entirely buoyancy-driven (Cholemari & Arakeri Reference Cholemari and Arakeri2009).

A conventional definition of Rayleigh number for the tube part would be given by $Ra_{\textit{tub}} = {g \beta \Delta T_{\textit{tub}} H^3}/(\nu \alpha )$ , where $T_{\textit{tub}}$ is the difference between the knee temperatures at the tube ends. However, since the convection is axially homogeneous and driven by a linear temperature gradient, it is more appropriate to use what we call the gradient-based Rayleigh number (Arakeri et al. Reference Arakeri, Avila, Dada and Tovar2000), based on the vertical temperature gradient in the tube and the tube diameter: $Ra_g = {g \beta \Delta T_d d^3}/(\nu \alpha ) = {g \beta ({{\textrm d}\overline {T}}/{{\textrm d}z}) d^4}/(\nu \alpha )$ , where $\Delta T_d = ({{\textrm d} \overline {T}}/{{\textrm d}z}) d$ is the temperature difference in the tube part over a diameter height. Essentially, away from the top and bottom ends, the convection is driven by the local temperature gradient and the length scale is the tube diameter (Cholemari & Arakeri Reference Cholemari and Arakeri2005, Reference Cholemari and Arakeri2009). The relation between $Ra_g$ and $Ra$ in slender RBC is obtained using (2.3) and (2.4) as

(3.5) \begin{equation} Ra_g = Ra \left ( \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} \right ) \varGamma ^4. \end{equation}

Again, for the tube part of slender RBC, a gradient-based Nusselt number can be defined as $Nu_g = {q'' d}/{k \Delta T_d} = {q''}/(k ({{\textrm d}\overline {T}}/{{\textrm d}z}))$ which depends only on the local temperature gradient in the tube. Taking the ratio $Nu_g/Nu$ and using (2.4), we have

(3.6) \begin{equation} Nu_g = {Nu}{\left (\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}}\right )}^{-1}. \end{equation}

A general scaling relation for heat flux in the tube region may be written as

(3.7) \begin{equation} Nu_g = C_{\textit{tub}} Ra_g^a Pr^b. \end{equation}

Several regimes, depending on $Ra_g$ and $Pr$ , exist in TC. Some have been documented for high $Pr$ in Arakeri et al. (Reference Arakeri, Avila, Dada and Tovar2000). At high enough $Ra_g$ , the convection becomes turbulent with no mean flow. For the turbulent flow itself, two regimes have been identified (Pawar & Arakeri Reference Pawar and Arakeri2016). Combining all the experimental and numerical data that were reported until then at different Prandtl numbers, Pawar & Arakeri (Reference Pawar and Arakeri2016) argued that the correlations for the two turbulent regimes are best given in terms of the gradient-based Grashof number $\textit{Gr}_g = {g \beta ({{\textrm d}\overline {T}}/{{\textrm d}z}) d^4}/{\nu ^2} = Ra_g/Pr$ , and proposed for the two regimes the following relations:

(3.8a) \begin{align} Nu_g &= C_{0.3} \textit{Gr}_g^{0.3} Pr = C_{0.3} Ra_g^{0.3} Pr^{0.7}\quad \textrm{for}\ \textit{Gr}_{g0} \lt \textit{Gr}_g \lt \textit{Gr}_{gc}, \end{align}

(3.8b) \begin{align} Nu_g &= C_{0.5} \textit{Gr}_g^{0.5} Pr = C_{0.5} Ra_g^{0.5} Pr^{0.5}\quad \textrm{for}\ \textit{Gr}_g \gt \textit{Gr}_{gc},\\[-18pt]\nonumber\end{align}

where $\textit{Gr}_{gc} \simeq 1.6 \times 10^5$ is the critical gradient Grashof number where the flow transitions from 0.3 to 0.5 power regime and $ \textit{Gr}_{g0}$ is a possible unknown lower critical limit below which the scaling is unexplored. The empirically determined values for the prefactors are $C_{0.3}=8.3 \pm 3$ and $C_{0.5}=0.75 \pm 0.1$ . The scaling exponents in (3.7) take the values $a=0.3$ , $b=0.7$ for the 0.3 regime ( $\textit{Gr}_g \lt \textit{Gr}_{gc}$ ) and $a=0.5$ , $b=0.5$ for the 0.5 regime ( $\textit{Gr}_g \gt \textit{Gr}_{gc}$ ).

The 0.5 regime ( $\textit{Gr}_g \gt \textit{Gr}_{gc}$ ) corresponds to the ‘ultimate regime’ where heat flux becomes independent of the viscosity and thermal diffusivity of the fluid, and is easily and unambiguously achievable with ordinary fluids at laboratory scales in TC as compared with RBC. For the 0.3 regime ( $\textit{Gr}_g \lt \textit{Gr}_{gc}$ ), it was shown (Pawar & Arakeri Reference Pawar and Arakeri2016) that viscous effects start affecting the flux, and we obtain a ‘viscous turbulent’ regime in TC. It is important to note that the nature of the correlations is very different from those for regular RBC, in particular the dependence on $Pr$ is strong.

The plot of $Nu_g$ versus $Ra_g$ (figure 2b ) shows that the data from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) closely follow the TC correlations ((3.8a ), (3.8b )), including the flow transitioning from 0.3 to 0.5 regime at the critical gradient Grashof number $\textit{Gr}_{gc}$ . Equations (3.5) and (3.6) were used to calculate $Nu_g$ and $Ra_g$ from the simulation data (from both versions of Nusselt number, $\textit{Nu}$ and $Nu_v$ ). The close match between data and prediction validates our hypothesis that the flow in the tube part of the slender RBC is very similar to or the same as that in TC.

The values of all the non-dimensional parameters calculated for the simulations of Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) at $Pr=1$ and $\varGamma =0.1$ are given in table 2. The equations used for ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ , $Ra_w$ , $C_{q_w}$ , $Ra_g$ and $Nu_g$ have been discussed in §§ 2 and 3. The calculation of Reynolds number $Re_d$ and the mean plume spacing $\lambda _p/d$ are discussed later in the paper.

Table 2. Summary of different non-dimensional parameters at different $Ra$ calculated for Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data ( $Pr=1$ , $\varGamma =0.1$ ). The subscripts $s$ and $v$ are used to represent quantities calculated from the Nusselt number data of the reference evaluated at the top/bottom plates $(Nu)$ and evaluated in the domain volume $(Nu_v)$ , respectively.

4.1. The generalised $\textit{Nu}$ – $Ra$ correlation for slender RBC

Substituting (4.1) and (4.4) to (2.2), we get

(4.5) \begin{equation} \Delta T = 2 K_w^{-3/4} (q'')^{3/4} + K_{\textit{tub}}^\frac {-1}{1+a} (q'')^\frac {1}{1+a} \varGamma ^\frac {-4a}{1+a} H^\frac {1-3a}{1+a}. \end{equation}

We can non-dimensionalise this equation by multiplying throughout by $ ({g \beta }/{\nu \alpha }) H^3$ :

(4.6) \begin{equation} \frac {g \beta \Delta T H^3}{\nu \alpha } = 2 \left (\frac {g \beta }{\nu \alpha }\right ) K_w^{-3/4} (q'')^{3/4} H^3+ \left (\frac {g \beta }{\nu \alpha }\right ) K_{\textit{tub}}^\frac {-1}{1+a} (q'')^\frac {1}{1+a} \varGamma ^\frac {-4a}{1+a} H^\frac {1-3a}{1+a} H^3. \end{equation}

Substituting for $K_w$ and $K_{\textit{tub}}$ and rearranging terms:

(4.7) \begin{align} \frac {g \beta \Delta T H^3}{\nu \alpha } &= 2 C_{q_w}^{-3/4} \left (\frac {g \beta \Delta T H^3}{\nu \alpha }\right )^{3/4} \left (\frac {q'' H}{k \Delta T}\right )^{3/4} \nonumber \\ &\quad+C_{\textit{tub}}^{\frac {-1}{1+a}} \left (\frac {g \beta \Delta T H^3}{\nu \alpha }\right )^{\frac {1}{1+a}} \left (\frac {q'' H}{k \Delta T}\right )^{\frac {1}{1+a}} \left (\frac {\nu }{\alpha }\right )^{\frac {-b}{1+a}} \varGamma ^{\frac {-4a}{1+a}}. \end{align}

Thus, we have the generalised $\textit{Nu}$ – $Ra$ correlation for slender RBC cells as

(4.8) \begin{equation} 2 C_{q_w}^{-3/4} Nu^{3/4} Ra^{3/4} + C_{\textit{tub}}^\frac {-1}{1+a} Nu^\frac {1}{1+a} Ra^\frac {1}{1+a} Pr^\frac {-b}{1+a} \varGamma ^\frac {-4a}{1+a} = Ra. \end{equation}

Instead of the familiar power-law form, we have an implicit equation between $\textit{Nu}$ , $Ra$ , $Pr$ and $\varGamma$ . The first term on the left-hand side corresponds to the wall part and second term corresponds to the tube part. Due to the implicit nature of the equation, the $\textit{Nu}$ value has to be calculated iteratively. The values of $C_{\textit{tub}}$ , $a$ and $b$ depend on the regime of TC scaling in the tube part of the slender RBC. For the two regimes for the tube part mentioned in (3.8), (4.8) becomes

(4.9a) \begin{align} 2 C_{q_w}^{-3/4} Nu^{3/4} Ra^{3/4} + C_{0.3}^{-10/13} Nu^{10/13} Ra^{10/13} Pr^{-7/13} \varGamma ^{-12/13} &= Ra\quad \nonumber\\ \textrm { for } \textit{Gr}_{g0}\lt \textit{Gr}_g\lt \textit{Gr}_{gc}, \end{align}

(4.9b) \begin{align} 2 C_{q_w}^{-3/4} Nu^{3/4} Ra^{3/4} + C_{0.5}^{-2/3} Nu^{2/3} Ra^{2/3} Pr^{-1/3} \varGamma ^{-4/3} &= Ra\nonumber\\\textrm { for } \textit{Gr}_g\gt \textit{Gr}_{gc}, \end{align}

where $C_{\textit{tub}}=C_{0.3}$ , $a=0.3$ , $b=0.7$ for the 0.3 regime and $C_{\textit{tub}}=C_{0.5}$ , $a=b=0.5$ for the 0.5 regime. Depending on whether the gradient-based Grashof number $\textit{Gr}_g$ in the tube part of the slender RBC cell is below or above the critical value $\textit{Gr}_{gc} = 1.6 \times 10^5$ , one has to solve either (4.9a ) or (4.9b ) to obtain $\textit{Nu}$ at a particular $Ra$ , $Pr$ and $\varGamma$ . In order to determine the regime into which the convection in the tube part falls, we need to know the value of $\textit{Gr}_g$ , which is unknown a priori. Hence, we need to find the critical Grashof number $\textit{Gr}_c$ (or the critical Rayleigh number $Ra_c$ ) based on the total temperature difference and cell height when the convection regime in the tube part changes from the 0.3 to 0.5 regime. The procedure to determine $\textit{Gr}_c$ or $Ra_c$ is given in Appendix A.

4.2. Comparison of model predictions with data

Next, we explore the validity of our model by comparing it against some of the recent studies involving convection in slender cells. We use only those data from the different studies that satisfy the restrictions on $Ra$ placed by our model. We will see that in several cases, the flow was not ‘turbulent’ in the tube or at the wall or both, and thus did not satisfy the conditions required by the model.

The variation of $\textit{Nu}$ (compensated by $Ra^{1/3}$ ) with $Ra$ for a slender RBC cell of aspect ratio $\varGamma =0.1$ and $Pr=1$ obtained using (4.9a ) and (4.9b ) is shown in figure 3(a). The Nusselt number data from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) (both versions, $\textit{Nu}$ and $Nu_v$ along with the corresponding error bars) and the data-fit proposed by them are shown, along with the LES results of Samuel et al. (Reference Samuel, Samtaney and Verma2022) for the same Rayleigh numbers and conditions ( $\varGamma =0.1$ , $Pr=1$ ). We note here that the wall heat flux values from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) may not be fully converged for $Ra\gt 10^{13}$ , but it it is unlikely that the converged values will be significantly different. The differences in the $\textit{Nu}$ and $Nu_v$ data, as well as between the two studies, highlight the challenges in simulations at these extreme Rayleigh numbers. The blue and red curves correspond to the predictions using the current model with a constant ( $C_{q_w}=0.1739$ from the Globe & Dropkin (Reference Globe and Dropkin1959) fit) and variable ( $C_{q_w}=0.1328+1.235Ra_d^{-0.18}$ from GL theory) approximations for $C_{q_w}$ , respectively. For this configuration, the value of the critical Rayleigh number where the regime changes in the tube part is determined to be $Ra_c \approx 5.3 \times 10^9$ (see Appendix A). Below the critical Rayleigh number (the exact value of which depends very weakly on the choice of $C_{q_w}\!$ ), (4.9a ) corresponding to the 0.3 regime is used, and above which (4.9b ) corresponding to the 0.5 power regime is used. Also shown are some other well-known correlations in the literature for RBC, none of which fit well with the slender RBC data. The classical scaling relation given by Globe & Dropkin (Reference Globe and Dropkin1959) and the fit proposed by Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) appear as horizontal lines in the compensated plot. We may mention that the predictions using the classical RBC correlations become worse as the cells become taller, as $\varGamma$ reduces. It can be seen that while our model with the constant $C_{q_w}$ approximation does not faithfully reproduce the given trend of data points, the model with variable $C_{q_w}$ approximation follows the data remarkably well. It should be noted that even with the constant $C_{q_w}=0.1739$ approximation, the error in the estimate is less that $20\,\%$ . In fact, a $C_{q_w}=0.15$ approximation (from the horizontal flat plate) gives a better prediction at higher $Ra$ , although the departure is higher at lower $Ra$ . However, given that the agreement is best captured with the variable $C_{q_w}$ , we choose to use this option further in our analysis. It is evident that the change of regime from 0.3 to 0.5 power law in the tube is responsible for the sharp change in the slope at the critical Rayleigh number.

Figure 3. (a) Compensated $\textit{Nu}$ versus $Ra$ for slender RBC using the current model (4.9) at $\varGamma =0.1$ and $Pr=1$ with constant $C_{q_w} = 0.1739$ (blue solid line) and variable $C_{q_w} = 0.1328+1.235Ra_d^{-0.18}$ (red solid line). Prediction is compared with the Nusselt number data from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) averaged at the top/bottom plates $\textit{Nu}$ (black markers), averaged in the whole volume $Nu_v$ (light-green markers) and the data-fit proposed by them (black dashed line). Data from Samuel et al. (Reference Samuel, Samtaney and Verma2022) (orange markers), scaling proposed by Shraiman & Siggia (Reference Shraiman and Siggia1990) (dashed green line), Globe & Dropkin (Reference Globe and Dropkin1959) (dashed pink line) and the estimate from GL theory with updated prefactors from Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013) (dashed brown line) are shown. (b) Compensated $\textit{Nu}$ versus $Ra$ at lower Rayleigh numbers for different $\varGamma$ (shown in different colours). $Pr=1$ for $\varGamma =0.2$ and $Pr=4.38$ for all other $\varGamma$ values. Solid lines show current model estimate and dashed lines show the Ahlers et al. (Reference Ahlers2022) model. Markers show data from DNS of Zhang (Reference Zhang2019) (open circles) and experiments of Zhang & Xia (Reference Zhang and Xia2023b ) (open triangles) at $Pr=4.38$ ; and no-tilt case of Zwirner & Shishkina (Reference Zwirner and Shishkina2018) (filled diamonds) at $Pr=1$ . Dotted lines showing GL theory for each $Pr$ are almost overlapped. For the current model estimates, $C_{q_w} = 0.1328+1.235Ra_d^{-0.18}$ is used, and each truncated at a lower limit of $\textit{Gr}_g=5{\times }10^3$ .

In figure 3(b), we show the results of DNS and experimental studies from Zhang (Reference Zhang2019) and Zhang & Xia (Reference Zhang and Xia2023b ) for different aspect ratios in the range $0.05 \leqslant \varGamma \leqslant 0.16$ at relatively low Rayleigh numbers and $Pr=4.38$ . Data from the ‘non-tilted cases’ of Zwirner & Shishkina (Reference Zwirner and Shishkina2018) at $\varGamma =0.2$ and $Pr=1$ are also shown. Zhang & Xia (Reference Zhang and Xia2023b ) found delayed onset of convection with decreasing aspect ratio, in line with the predictions of Shishkina (Reference Shishkina2021) and Ahlers et al. (Reference Ahlers2022). Further to the onset, they observed a steep rise in $\textit{Nu}$ , followed by an enhanced heat transfer in slender cells compared with the $\varGamma =1$ (base) case. An estimate using our model for each of the cases is shown in figure 3(b), each trimmed at a lower limit for $\textit{Gr}_g$ at $\textit{Gr}_{g0}=5{\times }10^3$ (see (3.8) and the discussion in § 4.5). That is why the curves corresponding to our model appear to start mid-way in the plot. In particular, the regime of validity of our model starts far from the onset, after the compensated $\textit{Nu}$ starts to decrease for each case. Ahlers et al. (Reference Ahlers2022) showed that results from various studies with different $\varGamma$ collapse and lie close to the GL theory estimate, when the Rayleigh number is re-scaled as $Ra_{\ell } = Ra(1+1.49 \varGamma ^{-2})^{-3/2}$ . The modified GL theory estimate for each $\varGamma$ , based on this re-scaling, is shown as the Ahlers model in figure 3(b), along with the actual GL theory (Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013) estimate for $\varGamma =1$ . It is worth mentioning that there are no other models that account for the $\varGamma$ dependence of $\textit{Nu}$ at such low aspect ratios. It can be seen that within its regime of applicability, our model follows the data from various aspect ratios closely, and that they lie closer to the data than the predictions of the Ahlers model.

Figure 4. (a) Plots of $\textit{Nu}$ versus $\varGamma$ for slender RBC using the current model at $Pr=4.38$ for different $Ra$ , compared with data for cylindrical domain (circular markers) and square-base domain (open square markers) from Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021). Here $C_{q_w} = 0.1328+1.235Ra_d^{-0.18}$ was used for current model. The prediction by the Ahlers et al. (Reference Ahlers2022) model is shown with dashed lines. Dot-dashed lines show iso-lines of $\textit{Gr}_g=1{\times }10^3$ (grey) and $\textit{Gr}_g=5{\times }10^3$ (black), the lower limit of the current model applicability. (b) Plots of $\textit{Nu}$ versus $Pr$ for four different values of $Ra$ at $\varGamma =0.1$ compared with the DNS data of Pandey & Sreenivasan (Reference Pandey and Sreenivasan2021) (circular markers). Solid lines show current model estimate, dashed lines show GL estimate ( $\varGamma =1$ ) and dotted lines show the Ahlers et al. (Reference Ahlers2022) model estimate at the same $Ra$ values. Current model is trimmed at a lower bound of $\textit{Gr}_g=5{\times }10^3$ shown by black dot-dashed curve. The value of $C_{q_w}$ for each case was obtained from GL theory at corresponding $Ra_d$ and $Pr$ (see discussion).

Figure 4(a) shows the variation of $\textit{Nu}$ as a function of aspect ratio for different Rayleigh numbers at $Pr=4.38$ from the DNS studies of Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021). Data with both cylindrical and square-base domains are shown. The authors reported enhanced heat transfer for small-aspect-ratio cells, and found that there was an optimal aspect ratio $\varGamma _{{opt}}$ that maximised the Nusselt number. The $\textit{Nu}$ augmentation and $\varGamma _{{opt}}$ were found to be a function of Rayleigh number and the shape of the container, as can be seen in the figure. Also shown are the $\textit{Nu}$ predictions of the Ahlers et al. (Reference Ahlers2022) model and the current model for each $Ra$ . Current model is extended to a lower limit of $\textit{Gr}_g=1{\times }10^3$ , although we do not know if the 0.3 scaling regime in the tube part is applicable below $\textit{Gr}_g=5{\times }10^3$ . Iso-lines of both these $\textit{Gr}_g$ limits are also shown. We have also extended the model for aspect ratios to $\varGamma =1$ , as the tube resistance becomes negligible and the flux will be controlled entirely by the wall convection. It can be observed that the current model prediction is very close to that of the data even beyond its regime of applicability. In fact, our estimates lie somewhere in between the data for cylindrical and square-base domains. The model predicts a mild maximum for $\textit{Nu}$ and that this $\varGamma _{{opt}}$ is a function of $Ra$ , though the value of this $\varGamma _{{opt}}$ is different from that found by Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021) and the reason is also likely to be different. The model predictions are close to the data in the range where it is applicable, $\textit{Gr}_g \gt 5{\times }10^3$ , and reasonably close even beyond the range to $\textit{Gr}_g = 1{\times }10^3$ . In general, the predictions are better than that of the Ahlers et al. (Reference Ahlers2022) model, which predicts monotonically increasing $\textit{Nu}$ with decreasing $\varGamma$ . At higher aspect ratios ( $0.2 \leqslant \varGamma \leqslant 1$ ) the predictions from our model and the Ahlers model converge. As mentioned above, as $\varGamma$ approaches unity, the relative contribution of the tube part approaches 0, and the overall model prediction is still accurate.

In figure 4(b), the variation of $\textit{Nu}$ with $Pr$ at different Rayleigh numbers is shown for an aspect ratio of $\varGamma =0.1$ from the simulations of Pandey & Sreenivasan (Reference Pandey and Sreenivasan2021). The authors report that for $Pr \geqslant 1$ and high $Ra$ , the heat transfer in slender cells is very similar to that in regular cells with $\varGamma \sim 1$ . At lower $Pr$ , they found that $\textit{Nu}$ values in slender cells are much lower than those in wider cells, with steep variation of $\textit{Nu}$ with $Pr$ especially for the low- $Ra$ cases. The expected behaviour of wide cells ( $\varGamma \gtrsim 1$ ) is shown in the figure by the GL theory for each $Ra$ , along with the modified GL prediction adjusted for $\varGamma$ dependence using the Ahlers et al. (Reference Ahlers2022) model. The current model estimate for each $Ra$ is also shown, once again trimmed at a lower limit of $\textit{Gr}_g=5{\times }10^3$ . The values of $C_{q_w}$ used for the current model were obtained as functions of $Pr$ from curve fits to GL theory at $Ra_d$ values of $10^5$ , $10^6$ , $10^7$ and $3{\times }10^7$ as $0.2616Pr^{0.01238}+0.03251{\times }0.9189^{Pr}$ , $0.2498Pr^{-0.03176}-0.01478Pr^{-0.9368}$ , $0.2183Pr^{-0.03471}-0.01787Pr^{-0.7077}$ and $0.2761Pr^{0.05626}-0.08877Pr^{0.1557}$ corresponding to $Ra$ of $10^8$ , $10^9$ , $10^{10}$ and $3{\times }10^{10}$ , respectively (although a $Pr$ -independent $C_{q_w} = 0.1328+1.235Ra_d^{-0.18}$ also gives reasonable estimates). It can be seen that our model follows the data points remarkably well, except for the lowest $Ra$ of $10^8$ . In particular, the steep change in $\textit{Nu}$ at low $Pr$ values is also well captured. This behaviour is due to the strong $Pr$ dependence in the $\textit{Nu}$ correlations for the tube part. In regular RBC, the Prandtl number effect for $\textit{Nu}$ scaling is not significant (low or negligible values of exponent $p$ in the usual power-law form $Nu = C Ra^{n} Pr^{p}$ ), as can be seen by the flat profiles of GL curves. However, in TC, the $Pr$ dependency is very strong, in both the 0.3 and 0.5 regimes (see (3.8)). Thus, in slender RBC, especially at low $Ra$ and $Pr$ values, the tube part plays a dominant role. It should be noted that the TC scaling we adapted from Pawar & Arakeri (Reference Pawar and Arakeri2016) has used data that existed in TC only for $Pr \geqslant 1$ ; however, we see that the capability of the model well surpasses its expected applicability regime (§ 4.5), even giving accurate estimates for $Pr$ as low as 0.1.

4.3. Temperature drop in the tube part

One of the main objectives of this paper is to uniquely determine the non-dimensional temperature gradient ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ in the tube part which is also equal to the ratio of temperature drop in the tube to the total temperature difference $\Delta T_{\textit{tub}}/\Delta T$ (2.4) as a function of the total forcing $Ra$ and the aspect ratio of the cell $\varGamma$ . For turbulent convection in a regular RBC setup, as mentioned above, the core is isothermal due to the efficient mixing by LSC. For slender RBC, we get a linear temperature profile in the tube part, away from the ends, which adds to the overall thermal resistance. This was indeed one of the assumptions we made to derive the generalised $\textit{Nu}$ – $Ra$ correlation in the previous section. We can get the value of this linear temperature gradient in the core using our model. Using (3.5) and (3.6) to substitute for $\textit{Nu}$ and $Ra$ with $Nu_g$ and $Ra_g$ in the generalised $\textit{Nu}$ – $Ra$ correlation (4.8), we have

(4.10) \begin{align}\nonumber\\[-24pt]2 C_{q_w}^{-3/4} Nu_g^{3/4} Ra_g^{3/4} \varGamma + C_{\textit{tub}}^\frac {-1}{1+a} Nu_g^\frac {1}{1+a} Ra_g^\frac {1}{1+a} Pr^\frac {-b}{1+a} = Ra_g \left (\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}}\right )^{-1}. \end{align}

Now, using the scaling for TC (3.7) and writing $Nu_g$ in terms of $Ra_g$ and simplifying, we get

(4.11) \begin{equation} \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} = \left (1 + 2 C_{q_w}^{-3/4} C_{\textit{tub}}^{3/4} Ra_g^{\frac {3a-1}{4}} Pr^{\frac {3b}{4}} \varGamma \right )^{-1}. \end{equation}

This relation gives ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ at a particular $Ra_g$ . To rewrite this equation in terms of $Ra$ , we use (3.5):

(4.12) \begin{equation} 2 C_{q_w}^{-3/4} C_{\textit{tub}}^{3/4} Ra^{\frac {3a-1}{4}} Pr^{\frac {3b}{4}} \varGamma ^{3a} \left (\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}}\right )^{\frac {3(1+a)}{4}} + \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} = 1. \end{equation}

Similar to the generalised $\textit{Nu}$ – $Ra$ correlation, we have an implicit relation for the non-dimensionalised temperature gradient in the tube part in terms of $Ra$ , $Pr$ and $\varGamma$ . The values of $a$ and $b$ depend on whether the convection in the tube belongs to the 0.5 (ultimate) or 0.3 (viscous turbulent) regime. For the two regimes of TC, the above relation becomes

(4.13a) \begin{align} 2 C_{q_w}^{-3/4} C_{0.3}^{3/4} Ra^{-1/40} Pr^{21/40} \varGamma ^{9/10} \left (\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}}\right )^{39/40} + \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} &= 1\quad \textrm { for } \textit{Gr}_{g0}\lt \textit{Gr}_g\lt \textit{Gr}_{gc}, \end{align}

(4.13b) \begin{align} 2 C_{q_w}^{-3/4} C_{0.5}^{3/4} Ra^{1/8} Pr^{3/8} \varGamma ^{3/2} \left (\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}}\right )^{9/8} + \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} &= 1 \quad \textrm { for } \textit{Gr}_g\gt \textit{Gr}_{gc}. \end{align}

The variation of non-dimensional temperature gradient for a slender RBC cell of $\varGamma =0.1$ and $Pr=1$ estimated using the current model (4.13a,b ) is shown in figure 5(a) for the two different choices of $C_{q_w}$ . The data from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) for the same parameter set are also shown for comparison. It can be seen that while the model with variable $C_{q_w}$ approximations shows excellent match with the data, even the constant $C_{q_w}$ approximation gives very reasonable estimates, especially at high Rayleigh numbers. Below the critical Rayleigh number for regime change in the tube part, (4.13a ) is used, while for above (4.13b ) is used. The change in regime can be seen as a sudden change in the slope at the critical Rayleigh numbers corresponding to both the $C_{q_w}$ choices. It is also evident that in the 0.5 regime, ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ decreases more rapidly compared with that in the 0.3 regime, with increasing $Ra$ . Since ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ is also the fraction of the total temperature in the tube, it can be seen that for the current configuration ( $\varGamma =0.1$ and $Pr=1$ ), around $10\%$ to $30\, \%$ of the temperature drop is in the tube, depending on $Ra$ . The temperature drop in the tube part becomes smaller with increasing $Ra$ , suggesting lower thermal resistance values in the tube part compared with the wall part. As we show in § 4.6, the fraction of temperature drop in the tube depends strongly on aspect ratio and $Pr$ . For $Pr = 1$ , $\varGamma = 0.01$ , the ratio can be as much as $80\, \%$ , and for high $Pr$ it becomes negligible.

Figure 5. (a) Non-dimensional temperature gradient ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ (equal to the relative temperature drop in the tube $\Delta T_{\textit{tub}}/\Delta T$ ) in the tube part of slender RBC versus $Ra$ for $\varGamma =0.1$ and $Pr=1$ calculated using the current model (4.13), compared with Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data (black circular markers). Estimates using both a constant $C_{q_w} = 0.1739$ (blue solid line) and variable $C_{q_w} = 0.1328+1.235Ra_d^{-0.18}$ (red solid line) are shown. The regime changes from 0.3 to 0.5 power scaling at the critical Rayleigh number $Ra_c \approx 5.3{\times }10^9$ with a sharp change in slope. (b) Plot of ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ as a function of $Pr$ using the current model (solid lines) at two different $Ra$ values for $\varGamma =0.1$ , compared with the simulation results of Pandey & Sreenivasan (Reference Pandey and Sreenivasan2021) (circular markers).

In figure 5(b), we show the variation of ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ with Prandtl number for two different Rayleigh numbers, $Ra=10^8$ and $10^{10}$ , for an aspect ratio $\varGamma =0.1$ predicted using the current model with a lower limit of $\textit{Gr}_g=5{\times }10^3$ . The data from the simulations of Pandey & Sreenivasan (Reference Pandey and Sreenivasan2021) are shown for comparison. Again, it is seen that the model predicts the variation of temperature gradient in the tube quite well, even for Prandtl number as low as $10^{-2}$ , showing the beyond-the-expected capability of the model. As shown by Pandey & Sreenivasan (Reference Pandey and Sreenivasan2021), the temperature gradient and hence the relative temperature drop in the tube part is as high as 1 at low Prandtl numbers, and reduces to very low values with increasing $Pr$ . As discussed above, the strong Prandtl number dependence of scaling in TC results in this behaviour. It is worth mentioning that to the best of our knowledge, there are no correlations in the literature that predict the temperature gradient or drop in the core for slender RBC, and hence for the figures discussed above, we do not have any correlations or data to compare with; except the DNS data from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) and Pandey & Sreenivasan (Reference Pandey and Sreenivasan2021).

An alternative expression for the non-dimensional temperature gradient in the tube can be obtained if the value of $\textit{Nu}$ is known a priori, say from experiments. For this, we note that the ratio of temperature drop in the tube to the total temperature difference is the same as the ratio between the second term of the left-hand side and the right-hand-side term of the generalised $Nu{-}Ra$ correlation (4.8). That is,

(4.14) \begin{equation} \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} = \frac {\Delta T_{\textit{tub}}}{\Delta T} = \frac {C_{\textit{tub}}^\frac {-1}{1+a} Nu^\frac {1}{1+a} Ra^\frac {1}{1+a} Pr^\frac {-b}{1+a} \varGamma ^\frac {-4a}{1+a}}{Ra} = C_{\textit{tub}}^\frac {-1}{1+a} Nu^\frac {1}{1+a} Ra^\frac {-a}{1+a} Pr^\frac {-b}{1+a} \varGamma ^\frac {-4a}{1+a}. \end{equation}

For the two regimes of TC scaling, the above relation becomes

(4.15a) \begin{align} \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} &= C_{0.3}^{-10/13} Nu^{10/13} Ra^{-3/13} Pr^{-7/13} \varGamma ^{-12/13}\quad \textrm { for } \quad \textit{Gr}_{g0}\lt \textit{Gr}_g\lt \textit{Gr}_{gc},\\[-10pt]\nonumber \end{align}

(4.15b) \begin{align} \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} &= C_{0.5}^{-2/3} Nu^{2/3} Ra^{-1/3} Pr^{-1/3} \varGamma ^{-4/3}\quad \textrm { for }\quad \textit{Gr}_g\gt \textit{Gr}_{gc}. \end{align}

Thus, we have explicit relations for ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ which might be useful if the value of $\textit{Nu}$ is known beforehand.

The entire procedure described in the previous sections to characterise the heat flux and temperature gradient for a slender RBC configuration (i.e. given $Ra$ , $Pr$ and $\varGamma$ ) can be summarised as follows. The first step is to plot the $Ra$ – $\varGamma$ parameter space for the particular $Pr$ , and determine whether the given configuration falls inside the region of applicability (ROA) of the current model (explained in § 4.5). If it does, then the next step is to establish whether it belongs to the 0.3 or the 0.5 regime of TC by finding the critical Rayleigh number $Ra_c = \textit{Gr}_c Pr$ of transition between the two regimes, corresponding to $\textit{Gr}_{gc}=1.6 \times 10^5$ (see Appendix A for details). The next step is to select an expression for $C_{q_w}$ at the particular $Pr$ for wall convection, using any existing scaling relations such as GL theory, as we did in § 3.1. Now, $\textit{Nu}$ can be estimated by solving (4.9a ) or (4.9b ), and ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ by solving (4.13a ) or (4.13b ), depending on the scaling regime.

4.4. Critical Rayleigh numbers and transition to ultimate regime

There are different critical Rayleigh numbers that may be linked to transition between the various convection regimes in slender RBC, and $Ra_c$ is only one among them. The first is the the critical Rayleigh number for onset of convection $Ra_{ons}$ . The aspect ratio dependency of $Ra_{ons}$ is discussed in detail by Shishkina (Reference Shishkina2021), Ahlers et al. (Reference Ahlers2022) and Zhang & Xia (Reference Zhang and Xia2023b ), and we do not discuss this aspect here. Next is $Ra_0$ , marking the transition of convection in the tube part from an (yet) unknown regime to the 0.3 regime, marked by $\textit{Gr}_g = \textit{Gr}_{g0}$ (see § 4.5). We do not have enough information about the TC scaling and hence slender RBC scaling below this limit. The next is $Ra_c$ ( $=\textit{Gr}_c Pr$ ) that we discussed in the previous section, that demarcates the 0.3 (viscous turbulent) and 0.5 (ultimate) regimes in the tube part of the convection. And lastly, $Ra_u^*$ , which gives the value of $Ra$ above which the ultimate regime of convection is obtained in the whole RBC cell, both the tube and wall parts.

Figure 6. (a) Critical Rayleigh number $Ra_c$ of transition between the 0.3 and 0.5 regimes in the tube part, as a function of aspect ratio $\varGamma$ for different $Pr$ . (b) Critical Rayleigh number $Ra_u^*$ of transition to the ultimate regime in the entire cell as a function of aspect ratio $\varGamma$ for $Pr=1$ estimated using the current model, such that a critical shear Reynolds number $Re_s$ is achieved. Two estimates with $Re_s = 420$ (red markers) and with $Re_s = 300$ (green markers) are shown. Red and green solid lines are curve fits to these data; grey line is the extrapolation of the estimate by Bodenschatz et al. (Reference Bodenschatz, He, van Gils and Ahlers2015); blue line the estimate by Ahlers et al. (Reference Ahlers2022).

The variation of the critical Rayleigh number $Ra_c$ (see Appendix A) with aspect ratio is shown in a log–log plot in figure 6(a) for different Prandtl numbers. Only $\varGamma$ values up to 0.2 are shown since above this limit, the correlations used for the tube part may not be valid. From the figure, it can be seen that the value of $Ra_c$ increases considerably with decreasing aspect ratio for all Prandtl numbers. When the aspect ratio decreases by two orders of magnitude, the critical Rayleigh number increases by seven orders of magnitude. Thus, for highly slender cells, one cannot expect the 0.5 or the ultimate regime scaling in the tube part until very high Rayleigh numbers, as taller cells lead to lower values of $\textit{Gr}_g$ for the same $Gr$  ((A4a ) and (A4b )). Variations for four different Prandtl numbers are shown. It can be seen that $Ra_c$ increases considerably with $Pr$ , and high- $Pr$ fluids require extremely high Rayleigh numbers for this transition.

As mentioned above, $Ra\gt Ra_c$ only implies that the convection in the tube part alone has reached ultimate-regime-type scaling; however, the boundary layer at the walls may not have transitioned to turbulence and the scaling of the global heat flux $\textit{Nu}$ for the entire cell would not have reached the ultimate regime. Although the existence of such scaling is still debated, it has been proposed that the cross-over to ultimate scaling is achieved when the shear Reynolds number $Re_s=\delta _vu/\nu$ crosses a critical limit of $Re_s\,{\approx}\,420$ (Grossmann & Lohse Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2002; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009), $u$ being the mean wind velocity of the LSC and $\delta _v$ the velocity boundary layer thickness (discussed in detail in § 6.4). For slender RBC, we may estimate the critical Rayleigh number for this transition $Ra_u^*$ by finding the $Ra$ at which $Re_s$ crosses the value of 420. For this, we define the shear Reynolds number as $Re_s=\delta _v u_{\textit{rms}}/\nu$ , where, in the absence of LSC, shear is induced on the boundary layer by the fluctuating velocity from the tube part $u_{\textit{rms}} \sim w_m$ . Thus, $Re_s\,{=}\,(\delta _v/d) (u_{\textit{rms}}d/\nu ) = 0.3655 Re_d^{0.5}$ ( $Re_d$ is the Reynolds number based on tube width, see § 5), where we used a Blasius-type scaling for the velocity boundary layer $\delta _v/d \sim Re_d^{-0.5}$ and a prefactor of 0.3655 obtained from a curve fit to Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data (see § 6.4). For a slender RBC cell with $\varGamma =0.1$ and $Pr=1$ , we use our model (5.7) and (4.9) and find that the critical $Ra$ for $Re_s=420$ is $Ra_u^*=4.8{\times }10^{17}$ . For lower aspect ratios, the $Ra_u^*$ estimates are higher, and plotted against $\varGamma$ in figure 6(b). A curve fit to our model prediction gives $Ra_u^*=2.2{\times }10^{14}\times \varGamma ^{-3.24}$ . Also shown is an extrapolation of the fit given by Bodenschatz et al. (Reference Bodenschatz, He, van Gils and Ahlers2015), based on their experiments at higher aspect ratios ( $\varGamma =0.33$ , $0.5$ and $1$ ), and that proposed by Ahlers et al. (Reference Ahlers2022). It can be observed that our exponent $B=-3.24$ in the scaling $Ra_u^*\sim \varGamma ^{B}$ is close to the value of $B=-3.26$ by the former and to the exponent $B=-3$ of Roche (Reference Roche2020) and Ahlers et al. (Reference Ahlers2022) for $\varGamma \ll 1$ , based on theoretical arguments. It is interesting that the different approaches give similar aspect ratio dependencies.

It should be noticed, however, that our estimate of $Ra_u^*$ is solely based on the assumption that ultimate regime transition in the entire cell is triggered at $Re_s=420$ . In particular, this approach is valid when the turbulent boundary layer is triggered by the shear of the wind above. The value of 420 is usually used for regular RBC, where the steady wind of LSC is expected to be dominant, even at high $Ra$ . For slender RBC, the time-averaged flow above the boundary layers at high $Ra$ is very weak, as is the case for high-aspect-ratio ( $\varGamma =4$ ) periodic domains reported recently (Samuel et al. Reference Samuel, Bode, Scheel, Sreenivasan and Schumacher2024). For these cases, the transition to the (potential) ultimate regime could be triggered by a possible destruction of the kinetic boundary layer by turbulent fluctuations (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). If that is the case, the transition could be triggered at even low values of $Re_s$ (based on root mean square (r.m.s.) of fluctuations), since unsteady boundary layers are known to have a lower threshold for transition to turbulence. Hence here we show an estimate of $Ra_u^*$ using $Re_s=300$ as well, in addition to that of 420. This again gives a very close exponent of $B=-3.26$ , although the prefactor has reduced. This estimate lies even closer to the estimates of Bodenschatz et al. (Reference Bodenschatz, He, van Gils and Ahlers2015) and Ahlers et al. (Reference Ahlers2022). It should be reiterated that this approach of finding $Ra_u^*$ based on a critical shear Reynolds number does not guarantee the existence of the ultimate regime beyond this limit, which for slender RBC has not yet been observed. However, this approach gives an estimate of a lower bound for full ultimate transition in slender RBC cells. In summary, both the critical Rayleigh numbers $Ra_c$ for transition to the ultimate regime in the tube part and $Ra_u^*$ for transition to the ultimate regime of the entire domain are strong functions of aspect ratio, and the more slender the geometry, the higher is the Rayleigh number needed for transition. The above calculation is for $Pr =1$ . It is expected that the value of $Ra_u^*$ will depend on $Pr$ , and may be easily calculated using the same approach.

4.5. Parameter space

Next we look at the boundaries of the $Ra$ – $\varGamma$ – $Pr$ parameter space within which our model is valid. One is the lower limit of applicability of the 0.3 regime in TC, $\textit{Gr}_{g0}$  (3.8a ), below which no correlations are available for $Nu_g$ . Based on study reported by Pawar & Arakeri (Reference Pawar and Arakeri2016) we arrive at $\textit{Gr}_{g0}=5 \times 10^3$ , and that is why in figures discussed above, the curves do not span the entire abscissa range, and appear to start mid-way. Secondly, the scaling relations proposed by Pawar & Arakeri (Reference Pawar and Arakeri2016) were based on experiments and simulations with $Pr\gt 1$ only, and so does the expected validity of our correlations. The prefactor $C_{\textit{tub}}$ and exponents $a$ and $b$ of the general scaling relation for TC (3.7) might have to be modified when studies that encompass a broader range of $\textit{Gr}_g$ and $Pr$ emerge. A third limitation arises at the wall, where the assumed $C_{q_w}$ scaling is expected to be valid only for $Ra_w$ values greater than around $10^5$ , i.e. when the wall convection is turbulent and at least one plume or more is present at the wall, i.e. $\lambda _p/d\lt 1$ , where $\lambda _p$ is the average plume spacing. Plume spacing predicted by the model of Theerthan & Arakeri (Reference Theerthan and Arakeri1998) and validated by experiments is discussed in detail in § 6.2. At lower $Ra_w$ values, a modified scaling for $C_{q_w}$ needs to be established. And lastly, the domain has to be ‘slender’ enough for the TC scaling relations to be used. Aspect ratio $\varGamma \leqslant 0.2$ is a reasonable limit so that fully developed, axially homogeneous TC region is present away from the walls. For example, from the vertical profiles of temperature and velocity fluctuations in Zhang & Xia (Reference Zhang and Xia2023b ; see their figure 5b), axial invariability was observed for $\varGamma \leqslant 0.16$ , but not for $\varGamma =0.3$ . This criterion, however, is not very stringent, since the contribution of the tube part in our model diminishes as $\varGamma$ approaches 1, and the overall scaling controlled by the wall convection approaches that of regular RBC given by the (modified) GL theory, as we discussed earlier (figure 4a ).

Figure 7. Parameter space of $Ra$ versus $\varGamma$ for (a) $Pr=1$ and (b) $Pr=4.38$ . Dashed lines are iso-lines of $Ra_{ons}$ representing the onset of convection (based on Shishkina (Reference Shishkina2021)), $\textit{Gr}_g=\textit{Gr}_{g0}$ , $\varGamma =0.2$ , $\lambda _p/d = 1$ , $\textit{Gr}_g=\textit{Gr}_{gc}$ and $Ra_u^*|_{Re_s=420}$ . Shaded/coloured regions show the expected Region of Applicability (ROA) of the current model. Yellow region: 0.3 (viscous turbulent) regime in the tube part; pink region: 0.5 (ultimate) regime in tube part only; blue region: ultimate regime in the full cell. Inset for (a) shows the same figure in an extended range of $\varGamma$ . Markers indicate the parameter values from the various studies.

In figures 7(a) and 7(b), we sketch out the $Ra$ – $\varGamma$ parameter space for $Pr=1$ and $Pr=4.38$ , respectively, along with the parameter ranges covered by some of the low-aspect-ratio studies with which we compared our model in the previous sections. The iso-lines of $\textit{Gr}_{g0} = 5{\times }10^3$ ; $\varGamma =0.2$ ; and $\lambda _p/d = 1$ are shown, which form the expected boundaries within which our model assumptions are valid. This region that we call the ROA of the current model is highlighted using coloured regions. The iso-line of $\textit{Gr}_{gc} = 1.6{\times }10^5$ splits the ROA into the 0.3 regime (yellow-shaded area) and the 0.5 regime (pink-shaded area) of convection in the tube part. For $Pr=1$ , we have also shown the iso-line for $Ra_u^*$ based on $Re_s=420$ beyond which the entire cell may transition to the ultimate regime (blue-shaded area in the inset). The lines of $\textit{Gr}_{g0}$ and $\lambda _p/d=1$ are very close to each other and have a very similar slope, and depending on $Pr$ , either one of these can be the deciding criterion. While the simulations of Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) lie deep into the ROA of our model, most of the cases from other studies, Zhang (Reference Zhang2019), Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021), Zhang & Xia (Reference Zhang and Xia2023b ) and Zwirner & Shishkina (Reference Zwirner and Shishkina2018), lie outside or are just inside the boundaries of the ROA; only a few of the points lie inside our ROA. The cases of Pandey & Sreenivasan (Reference Pandey and Sreenivasan2021) are not shown here due to their varying $Pr$ . In the previous sections, we showed that despite the fact that the majority of these studies lie outside the ROA, the model was able to reasonably well predict the $\textit{Nu}$ and temperature gradients for these cases in a wide range of $Pr$ (as low as 0.1) and aspect ratios ( $\varGamma \to 1$ ). Thus, the validity of the model seems to extend beyond its ROA. However, $\textit{Gr}_{g0}$ seems to be a stricter criterion, as below this (yet unknown) value, sharp changes in the kind of TC scalings are expected, evident from the low- $Ra$ data points (near the onset of convection) of Zhang & Xia (Reference Zhang and Xia2023b ) in figure 3(b). At such low $\textit{Gr}_g$ values, one may start seeing temperature inversions and locked-flow sublayers as shown in the temperature profiles of Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021) (see figure 5e of their paper and figure S2 in their supplementary material), which depart from the assumed temperature profile of our model (figure 1c ). It is interesting to note that when a power-law-type scaling is fitted on the $\textit{Gr}_g = \textrm{const.}$ lines, i.e. $\textit{Gr}_{g0}$ and $\textit{Gr}_{gc}$ , both the corresponding critical Rayleigh numbers $Ra_0$ and $Ra_c$ have aspect ratio dependencies close to $\varGamma ^{-3.8}$ (for $Pr=1$ ) and $\varGamma ^{-3.6}$ (for $Pr=4.38$ ). These exponents are very close to the $\varGamma ^{-4}$ lines given by Zhang & Xia (Reference Zhang and Xia2023b ) which demarcate the different regimes in their simulations (see figure 7c of their paper). As for the ultimate regime in the entire cell, it can be seen that all the existing studies, including Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ), lie well below the mark needed to cross $Ra_u^*=Ra|_{Re_s=420}$ .

Figure 8. Compensated $\textit{Nu}$ using the current model (4.9): (a) versus $Ra$ for different aspect ratios ( $\varGamma$ ) at $Pr=1$ , (b) versus $\varGamma$ for different $Ra$ at $Pr=1$ and (c) versus $Ra$ for different $Pr$ at $\varGamma =0.1$ . In (b), solid lines show current model and dashed lines show the Ahlers et al. (Reference Ahlers2022) model. The inset in (b) shows aspect ratio re-scaled as $\varGamma _s = \varGamma ^{3/2} Ra^{1/8} Pr^{3/8}$ , and the curves for high $Ra$ collapsing onto a universal curve, given by $0.0265 [ 1 + \tanh ( 1.18 \log _{10} (\varGamma _s) + 0.75 ) ]$ . Each curve is terminated at a lower bound of $\textit{Gr}_g=5{\times }10^3$ .

4.6. Effect of aspect ratio and Prandtl number

In this section, we explore the effect of varying aspect ratio and Prandtl number on the Nusselt number and temperature gradient in the tube part for slender RBC. The compensated plot of $\textit{Nu}$ with $Ra$ estimated using the current model for different aspect ratios is shown in figure 8(a) for $Pr=1$ using $C_{q_w} = 0.1328+1.235Ra_d^{-0.18}$ . Here $\textit{Nu}$ decreases with reducing aspect ratios at all Rayleigh numbers due to the increasing resistance in the tube part resulting in the reduction in overall heat flux. For the Rayleigh number range considered here, there is a reduction of at least 4–5 times in $\textit{Nu}$ as $\varGamma$ decreases from 0.1 to 0.01. The change in slope in each curve corresponds to the regime change in the tube part from 0.3 to 0.5. The delayed onset of the 0.5 regime due to the higher critical Rayleigh number is also evident with decreasing aspect ratio. When plotted against aspect ratio for different Rayleigh numbers (at $Pr=1$ ) in figure 8(b), it is clear that there exists an optimum aspect ratio $\varGamma _{{opt}}$ at which $\textit{Nu}$ is maximum, for the lower Rayleigh numbers. This was shown earlier for $Pr=4.38$ and lower $Ra$ values while comparing the model with the Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021) results and the associated discussion in figure 4(a). At higher Rayleigh numbers, beyond $Ra \sim 10^{13}$ , we do not find the existence of such augmented $\textit{Nu}$ or optimum aspect ratio: $\textit{Nu}$ seems to decrease monotonically with reducing $\varGamma$ . Another interesting observation from figure 8(a,b) is that for $Pr=1$ , above an aspect ratio of $\varGamma =0.1$ , the compensated Nusselt number $NuRa^{-1/3}$ in the 0.5 regime decreases with increasing $Ra$ , while below this limit, it increases with increasing $Ra$ . At  $\varGamma =0.1$ , it appears that $NuRa^{-1/3}$ is almost independent of $Ra$ , and is approximately 0.05. At higher Prandtl numbers (not shown here) this observation is consistent, although $\varGamma$ at which this inversion occurs seems to decrease with $Pr$ .

At lower aspect ratios, the $NuRa^{-1/3}$ curves seem to decline rapidly. At very low aspect ratios ( $\varGamma \sim 10^{-3}$ ), the curves flatten out to exceedingly low values, the value of which depends on $Ra$ . For lower Rayleigh numbers, with reducing $\varGamma$ , the curves terminate before reaching very small values because of the limit on $\textit{Gr}_g$ reaching $\textit{Gr}_{g0}$ , beyond which they cross-over to an unknown regime below the 0.3 regime. The family of these curves for different $Ra$ seem to show similarity to an S-shaped curve. When the aspect ratio $\varGamma$ is re-scaled as $\varGamma _s=\varGamma ^{3/2}Ra^{1/8}Pr^{3/8}$ , we find that the curves at different $Ra$ collapse to a single universal curve, especially for higher Rayleigh numbers (inset of figure 8b ). A least-squares curve fit provides an empirical expression for this universal curve as $NuRa^{-1/3} \approx 0.0265[1+\tanh (1.18\log _{10}(\varGamma _s)+0.75)]$ . Although we show the collapse only for $Pr\,{=}\,1$ here, we have verified that this re-scaling is valid for different Prandtl numbers also ( $Pr=0.1$ – $600$ ). Note that the expression for the scaled aspect ratio $\varGamma _s$ is the same as the coefficients in the first term of (4.13b ), corresponding to the 0.5 regime. That is why the curves start to deviate slightly once they enter the 0.3 regime. We emphasise that the above correlation is valid only for high Rayleigh numbers, particularly for $Ra \gt 10^{13}$ . For lower $Ra$ values, the $NuRa^{-1/3}$ curves start to deviate, as seen in figure 8(b).

The (compensated) $Nu{-}Ra$ plot at different (most commonly encountered) Prandtl numbers for an aspect ratio of $\varGamma =0.1$ is shown in figure 8(c). Note that we have used $C_{q_w} = 0.1328+1.235Ra_d^{-0.18}$ for $Pr=$ 0.7, 1 and 7 as the $C_{q_w}$ versus $Ra_d$ curves obtained from the GL theory were very similar for these three cases. For $Pr=0.1$ , we obtained $C_{q_w} = 0.1387+14.55Ra_d^{-0.44}$ and for $Pr=600$ , we obtained $C_{q_w} = 0.1372+4.1Ra_d^{-0.287}$ from curve fits to the respective GL solutions, as discussed in § 3. There is a slight decrease in $\textit{Nu}$ with decrease in $Pr$ from 600 to 7. As we saw above (figure 4 b and the associated discussion), $\textit{Nu}$ decreases steeply with decreasing $Pr$ at low Rayleigh and Prandtl numbers. The effect of $Pr$ on the heat flux mainly comes from the tube part scaling.

Figure 9. Non-dimensional temperature gradient ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ or the fraction of temperature drop in the tube part $\Delta T_{\textit{tub}}/\Delta T$ using the current model (4.13): (a) versus $Ra$ for different aspect ratios ( $\varGamma$ ) at $Pr=1$ , (b) versus $\varGamma$ for different $Ra$ at $Pr=1$ and (c) versus $Ra$ for different $Pr$ at $\varGamma =0.1$ . The inset in (b) shows that as the aspect ratio is re-scaled as $\varGamma _s = \varGamma ^{3/2} Ra^{1/8} Pr^{3/8}$ , the curves collapse onto a universal curve given by $0.5 [ 1 - \tanh ( 1.09 \log _{10} (\varGamma _s) + 0.88 ) ]$ .

The variation of non-dimensional temperature gradient ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ (or the fraction of temperature drop $\Delta T_{\textit{tub}}/\Delta T$ ) in the tube part with the total $Ra$ for different aspect ratios estimated using (4.13) is shown in figure 9(a), for $Pr=1$ . As the aspect ratio $\varGamma$ decreases, the increased resistance results in a higher relative temperature drop $\Delta T_{\textit{tub}}/\Delta T$ in the tube part. With a ten-fold decrease in the aspect ratio from 0.1 to 0.01, the relative drop in the tube part increases from about $10\%{-}20\, \%$ to around $70\, \%$ for the Rayleigh number range considered here. The variation of ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ with $\varGamma$ for different Rayleigh numbers at $Pr=1$ is shown in figure 9(b). The observations are same: relative temperature drop in the tube increases with decreasing $\varGamma$ , and the variation is pronounced for lower Rayleigh numbers. Note that at $\varGamma = 0.1$ , nothing remarkable happens for ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ , in contrast to the nearly constant behaviour of $NuRa^{-1/3}$ (figure 8b ). The value of ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ increases gradually with reducing aspect ratio until $\varGamma \sim 0.2$ , and then increases rapidly with further reduction in $\varGamma$ for all $Ra$ . At very low aspect ratios, however, the curves tend to flatten out such that the majority of the temperature drop is in the tube part. Once again, when the aspect ratio is re-scaled as $\varGamma _s=\varGamma ^{3/2}Ra^{1/8}Pr^{3/8}$ , the curves for different $Ra$ collapse onto a single universal curve, even for low Rayleigh numbers (figure 9 b inset). A curve fit to this universal curve gives ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}} \approx 0.5[1-\tanh (1.09\log _{10}(\varGamma _s)+0.88)]$ , which is valid for different Prandtl numbers as well (not shown). A close inspection shows departure from the fit when the curves enter the 0.3 regime, although the deviations are extremely small. The collapse is good even at lower $Ra$ , unlike for $NuRa^{-1/3}$ . The existence of such universal curves for ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ and $NuRa^{-1/3}$ is remarkable, in our opinion. One can very easily get reasonable estimates of the Nusselt number and the temperature gradient, without even solving (4.9) and (4.13), the proposed scaling laws for slender RBC.

The variation of ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ with $Ra$ for different Prandtl numbers at an aspect ratio of $\varGamma =0.1$ shows the increasing tube resistance with decrease in $Pr$ (figure 9c ). At very high Prandtl numbers, the temperature gradient in the tube is negligible, and the core is near isothermal, as in the cases with $\varGamma \gtrsim 1$ . This behaviour of ${{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}}$ was already shown when comparing our model with the results of Pandey & Sreenivasan (Reference Pandey and Sreenivasan2021) in § 4.3. We reiterate that $C_{q_w}$ is a very weak function of $Pr$ , meaning that heat transfer in regular RBC is only weakly dependent on $Pr$ for $Pr \gtrsim 1$ and the effect of $Pr$ on $\textit{Nu}$ in slender RBC is chiefly due to the effect of the tube part, where the $Pr$ dependence is strong. Note that the $Pr = 600$ case has been included in the figures because it corresponds to where the convection experiments are conducted with salt (NaCl) mixed in water to create a density difference. A possible way to do slender RBC experiments with salt is to use permeable membranes as the top and bottom walls, as was done in Puthenveettil & Arakeri (Reference Puthenveettil and Arakeri2005) for studying high- $Ra$ , high- $Pr$ , regular Rayleigh–Bénard-type convection.

6.1. Temperature variation at the walls

For slender RBC, thermal boundary layer thickness based on the slope method corresponds to the intersection point between the slope of mean temperature at the wall and the linear temperature in the tube, giving (3.1). For detailed information on this and other methods of defining thermal boundary layer thickness, the reader is referred to Scheel, Kim & White (Reference Scheel, Kim and White2012) and the discussion in Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009). The near-wall variation of the non-dimensional mean temperature $-(\overline {T}(z)-T_0)/\Delta T_w$ along the vertical direction for different $Ra$ of Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) is plotted in figure 12. Here, $\overline {T}(z)=\left \langle T \right \rangle _{A,t}$ , with $T_0=T(z=0)$ . The vertical distance is non-dimensionalised by the boundary layer thickness $\delta _T$ using (6.5) in figure 12(a) and by the wall scale $Z_w$ using (6.4) in figure 12(b). It can be seen that in both the figures, the plots collapse well for different Rayleigh numbers. It can also be noticed that the wall length scale $Z_w$ is also a valid scale near the wall, and most of the variation in mean temperature happens within $Z_w \leqslant 15$ , an observation consistent with the literature on conventional RBC (see Theerthan & Arakeri Reference Theerthan and Arakeri1998).

Figure 12. Near-wall profiles of mean temperature of Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data along the vertical direction. Temperature is non-dimensionalised by $\Delta T_w$ and $z$ non-dimensionalised by (a) $\delta _T$ and (b) $Z_w$ .

6.2. Plume spacing

Plumes are the near-wall coherent structures that primarily transport heat from the conduction to the convection layer (Theerthan & Arakeri Reference Theerthan and Arakeri1998; Chilla & Schumacher Reference Chillà and Schumacher2012). Very close to the wall, they appear as sheets and further away they may form into mushroom-type structures. They originate as fragments of the thermal boundary layer which detach due to boundary layer instability and move into the bulk followed by broadening due to diffusion and mixing. Higher Rayleigh numbers increase the number of these plumes. The LSC itself is driven partly by merging of groups of localised plumes (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). However, the large-scale wind affects the plume motion also, often aligning them at high enough $Ra$ , as can be seen from the planform views. Based on free-convection experiments on RBC and heated horizontal surfaces, Theerthan & Arakeri (Reference Theerthan and Arakeri1998) observed the dominant near-wall structures to be a network of randomly spaced and oriented line plumes that are moving about continuously on the surface. The authors proposed a model in which these plumes are fed locally by laminar or viscous free-convection boundary layers either side of each plume. Based on stability arguments they obtained the average spacing of the plumes to be given by $Ra_{\lambda _p}^{1/3} = \lambda _p/Z_w = 52 Pr^{-0.012}$ , where $Ra_{\lambda _p} = {g \beta \Delta T_w \lambda _p^3}/(\nu \alpha )$ is the Rayleigh number based on average plume spacing $\lambda _p$ and $Z_w$ is the wall length scale given in (6.1). The model has been extended to high Pr, $Ra_{\lambda _p}^{1/3} \approx 92$ (Puthenveettil & Arakeri Reference Puthenveettil and Arakeri2005). It has been shown (Theerthan & Arakeri Reference Theerthan and Arakeri2000; Puthenveettil & Arakeri Reference Puthenveettil and Arakeri2005; Puthenveettil et al. Reference Puthenveettil, Gunasegarane, Agrawal, Schmeling, Bosbach and Arakeri2011) that the model predictions of plume spacing agree well with what has been obtained in experiments. For slender RBC, at the top and bottom walls, we may express the plume spacing non-dimensionalised with the width of the plate $d$ by invoking the definitions of $Ra_{\lambda _p}$ , $Ra_w$ and (2.6) as

(6.7) \begin{equation} \frac {\lambda _p}{d} = 52 Ra_w^{-1/3} Pr^{-0.012} = 65.5 Ra^{-1/3} \left ( 1-\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} \right )^{-1/3} \varGamma ^{-1} Pr^{-0.012}. \end{equation}

Figure 13. Temperature field at the mid-boundary layer height $\delta _T/2$ showing the plume structure at different $Ra$ : (a) $Ra=10^{11}$ ; (b) $Ra=10^{13}$ ; and (c) $Ra=10^{15}$ (reproduced from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) with permission). The black horizontal bar below each case shows a scale of $52 Z_w$ , the expected mean plume spacing. The width of each image corresponds to the tube diameter $d$ .

For the simulation cases of Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) at $Pr=1$ , the values of $Ra_w$ and plume spacing $\lambda _p/d$ estimated using (6.7) are given for the various $Ra$ in table 2. The last row, $d/\lambda _p$ , gives the approximate mean number of plumes contained within the plate width. With increasing $Ra$ , the plume spacing obviously decreases and a greater number of finer and finer plumes are contained within the plate width. For the lowest- $Ra$ case, notice that $\lambda _p/d \gt 1$ which means that there is hardly one plume to be expected at any given time. This may be considered as another lower limit for the validity of our model with respect to the convection at the walls, as the value of $C_{q_w}$ might start deviating from our estimate. Thus, from the point of view of the convection at the walls, the applicability of the model is when $\lambda _p/d \lesssim 1$ , or equivalently, $Ra_w \gtrsim 10^5$ . Contour images of the temperature field near the bottom boundary layer for a few cases from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) are reproduced in figure 13. Based on the temperature threshold criteria, these structures can be identified as plumes rising from the bottom surface. An estimate of the mean plume spacing given by $52 Z_w/d = 52 Ra_w^{-1/3}$ (6.7) is shown below each case for comparison. It can be observed that the structures are, in fact, separated roughly by a magnitude of $\lambda _p \approx 52 Z_w$ at least for the high- $Ra$ cases, thus proving the validity of the plume model (6.7) at the walls for slender RBC.

6.3. Velocity variation at the walls

As discussed in § 5, in contrast to the LSC of regular-sized RBC, the large-scale structure in slender RBC is of helical nature, which scales with the mixing length velocity $w_m$ of TC. In the tube part, this results in an axially homogeneous turbulence, meaning the statistical quantities are homogeneous along the vertical axis. Near the top and bottom ends of the domain, however, we expect a transition, where this large-scale flow communicates with the near-wall plumes and the (top and bottom) boundary layers. Thus, above the edge of the boundary layer, one should expect velocity fluctuations of the order of the mixing length velocity $w_m$ , and it replaces the LSC velocity scale for RBC with $\varGamma \gtrsim 1$ . This is shown in figure 14 where the vertical profiles of the r.m.s. horizontal velocity, $u_{x,\textit{rms}}$ , near the walls are shown for different $Ra$ from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data. The data are averaged in both horizontal plane and time, and the data corresponding to the top and bottom walls have been combined. The velocity is non-dimensionalised using the free-fall velocity $U_{\!f}$ in figure 14(a) and using the mixing length scale $w_m$ in figures 14(b) and 14(c). The vertical distance is non-dimensionalised by the velocity boundary layer thickness $\delta _v$ in figures 14(a) and 14(b) and by the thermal boundary layer thickness $\delta _T$ in figure 14(c). Here, a Blasius profile is assumed to find the velocity boundary layer thickness (Sun, Cheung & Xia Reference Sun, Cheung and Xia2008; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009) $\delta _v/d = a Re_d^{-1/2}, a=0.3655$ . We discuss more about estimating the velocity boundary layer thickness in the next section. It can be seen that scaling the velocities with $w_m$ results in better data collapse and is obviously a better choice for non-dimensionalising the horizontal velocity fluctuations than $U_{\!f}$ . It can be also be observed that while scaling with the thermal boundary layer thickness $\delta _T$ is not very promising (or with the wall length scale $Z_w$ which is not shown here), scaling with the velocity boundary layer thickness $\delta _v$ results in a reasonable collapse of $u_{x,\textit{rms}}/w_m$ data for different $Ra$ , except for slight deviations at the highest $Ra$ . Thus, it is evident that fluctuations of the horizontal velocity $u_{x,\textit{rms}}$ grow from 0 at the wall to $w_m$ in the tube part in a distance of around one boundary layer thickness.

Figure 14. Near-wall profiles of r.m.s. of horizontal velocity fluctuations $u_{x,\textit{rms}}$ from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data with different choices of non-dimensionalisation for velocity and vertical distance: (a) $u_{x,\textit{rms}}/U_{\!f}$ versus $z/\delta _v$ , (b)  $u_{x,\textit{rms}}/w_m$ versus $z/\delta _v$ and (c) $u_{x,\textit{rms}}/w_m$ versus $z/\delta _T$ .

Figure 15. Near-wall profiles of r.m.s. of vertical velocity fluctuations $u_{z,\textit{rms}}$ from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data with different choices of non-dimensionalisation for velocity and vertical distance: (a) $u_{z,\textit{rms}}/U_{\!f}$ versus $z/\delta _T$ , (b)  $u_{z,\textit{rms}}/U_w$ versus $z/\delta _T$ and (c) $u_{z,\textit{rms}}/U_w$ versus $z/\delta _v$ .

However, the r.m.s. of vertical velocity fluctuations, $u_{z,\textit{rms}}$ , show a different behaviour near the walls. The variations of $u_{z,\textit{rms}}/U_{\!f}$ with $z/\delta _T$ and $u_{z,\textit{rms}}/U_w$ with both $z/\delta _T$ and $z/\delta _v$ for Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data are shown in figure 15 for different $Ra$ . While $U_{\!f}$ is not a good choice for scaling the vertical component of velocity, a reasonable collapse results when the wall velocity scale $U_w$  (6.1) is used instead. That $U_w$ is the relevant scale for the vertical velocity is understandable because most of the contribution comes from the plumes. The model proposed by Theerthan & Arakeri (Reference Theerthan and Arakeri1998) is based on a unit cell of a line plume fed by a boundary layer on either side. It was shown there that in conventional RBC also, data for $u_{z,\textit{rms}}$ scale well with $U_w$ . It can be noticed that the data collapse better when the distance is scaled with the thermal boundary layer thickness (or the wall length scale $Z_w$ not shown here) than when scaled with velocity boundary layer thickness. This could be due to the fact that vertical velocities scale with the extent of penetration of plumes into the bulk, which again scales with $\delta _T$ or $Z_w$ . Thus, we see that while the vertical velocity fluctuations near the wall scale with $\delta _T$ or $Z_w$ and $U_w$ , the horizontal velocity fluctuations scale with $\delta _v$ and the outer velocity scale $w_m$ , produced by the TC. It is worth mentioning that while moving from the wall to the tube region, the vertical velocity fluctuations grow at a much slower rate than the horizontal components, and the scale reaches the bulk velocity $w_m$ not after one boundary layer thickness, but only around half-a-diameter distance from the wall.

6.4. Velocity boundary layer

Similar to the thermal boundary layer, the kinetic or the velocity boundary layer may be defined in many ways. One can define this based on either the mean velocity profile $\delta _v^{M}$ or the (r.m.s. of) the fluctuation velocity profile $\delta _v^{\sigma }$ . It could be based on local time-averaged profiles or both area- and time-averaged profiles. It could also be based on the components of velocity or the velocity magnitude itself. Again, the boundary layer thickness may be defined to be based on the slope of the vertical profile at the wall $\delta _v^{sl}$ , or the distance of the location of the maxima from the wall $\delta _v^{max}$ .

The scale of the velocity boundary layer thickness is usually represented as a power law of the form $\delta _v/H \sim Re^{\xi }$ or $\delta _v/H \sim Ra^{\eta }$ . A simple approximation of $\xi =-1/2$ resulting from laminar boundary layer of Prandtl–Blasius theory has been successful in predicting the mean velocity boundary layer thickness. This approximation was central to the derivation of the GL theory (Grossmann & Lohse 2000, Reference Grossmann and Lohse2004) and is justified due to the low values of shear Reynolds number $Re_s=\delta _vu/\nu$ usually encountered. Above a critical limit of $Re_s \approx 420$ , it is believed that the laminar scaling approximations break down and the velocity boundary layers become fully turbulent. Even below this limit when the laminar boundary layer (Blasius) scaling is valid, the mean velocity profile itself departs from the classical Blasius velocity profile due to the continuous chaotic and time-dependent behaviour owing to the continuous detachment of thermal plumes (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). Note that since $Re \sim Ra^{1/2}$ , an approximation of $\xi \approx -1/2$ implies that $\eta \approx -1/4$ . From numerical simulations of RBC with $\varGamma =1$ and $Pr=1$ , Verzicco & Camussi (Reference Verzicco and Camussi1999) gave a fit $\delta _v^{sl,M}{/}H=0.95Ra^{-0.23}$ for the area-averaged boundary layer thickness based on the mean horizontal velocity. Sun et al. (Reference Sun, Cheung and Xia2008) performed high-resolution measurements in RBC experiments with water ( $Pr\,{=}\,4.3$ ) in a rectangular cell of $\varGamma =1$ in the range $10^9 \leqslant Ra \leqslant 10^{10}$ and proposed $\delta _v{/}H\,{\sim}\,Ra^{-0.27}$ and $\delta _v{/}H\,{\sim}\, Re^{-0.50}$ from independently determined Reynolds number, which was consistent with the Prandtl–Blasius scaling. In their case, the boundary layer thickness was estimated from the mean horizontal velocity profiles at the centre of the plate using both the slope and maxima methods, both yielding similar scaling. Numerical simulations of Scheel et al. (Reference Scheel, Kim and White2012) for $10^5 \leqslant Ra \leqslant 10^9$ and $Pr=0.4,0.7$ in a cylindrical cell of $\varGamma =1$ yield $\delta _v^{sl,M}/H\,{\sim}\,Ra^{-0.18}$ and $\delta _v^{max,\sigma}/H\,{\sim}\,Ra^{-0.25}$ , from the mean and r.m.s. horizontal velocity profiles, respectively. Later, owing to the inhomogeneity of the boundary layer dynamics across the area of the plates, Scheel & Schumacher (Reference Scheel and Schumacher2014) computed fully local inner and outer scales for the boundary layer thickness and showed that the conventional boundary layer thickness definition based on the slope method lay in between the averaged values of these scales. For a detailed discussion on various definitions of boundary layer thickness and comparison of numerical and experimental data, we refer the reader to Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009), Scheel et al. (Reference Scheel, Kim and White2012) and Scheel & Schumacher (Reference Scheel and Schumacher2014).

For slender RBC, as shown in § 6.3, the horizontal velocity near the walls which scales with the mixing length velocity ( $u^{\prime}_x \sim w_m$ ) acts as the streamwise ‘wind’ that interacts with the plumes. We also saw that the vertical velocity fluctuations near the walls scale with the wall velocity ( $u^{\prime}_z \sim U_w$ ). We take the approach that it is the plumes that dictate the growth of the boundary layer, not viscosity (Chandra Reference Chandra2000). We assume that the rate of boundary layer growth along the ‘mean flow’ direction is proportional to the vertical velocity fluctuations, i.e. ${\textrm d}\delta/ {\textrm d}t \sim u^{\prime}_z$ , the time derivative being taken in the convective sense. The convective time scale is $\tau =d{/}w_m$ . Thus, $\delta _v (w_m{/}d) \sim U_w$ , or $\delta _v{/}d \sim U_w{/}w_m$ . Using (6.1) and (5.1), and noticing that $\delta _v/H = (\delta _v/d)\varGamma$ , we have

(6.8) \begin{equation} \frac {\delta _v}{H} \sim \frac {U_w}{w_m} \varGamma = 2^{-1/3} Ra^{-1/6} \left ( 1-\frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}} \right )^{1/3} \frac {{\textrm d}\widetilde {T}}{{\textrm d}\tilde {z}}^{-1/2}. \end{equation}

Notice that as the temperature drop in the tube part diminishes, i.e. $ {{\textrm d}\widetilde {T}}/{{\textrm d}\tilde {z}} \rightarrow 0$ , the above relation implies that $\eta \approx -0.17$ . Boundary layer thickness obtained using this relation is compared with Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data for slender RBC ( $\varGamma =0.1$ ) and a few other scaling laws in the literature (for $\varGamma \gtrsim 1$ ) in figure 16 for various $Ra$ . For extracting the boundary layer thickness from the data, we have used both the methods: based on slope ( $\delta _v^{sl,\sigma }$ ) and based on the location of the maxima ( $\delta _v^{max,\sigma }$ ). Both of these are for the profiles of the r.m.s. horizontal velocity fluctuation, as apparently there is no mean flow or wind in the case of slender RBC. It can be observed that the curve for the boundary layer thickness based on the slope method from the data has a trend very similar to that of the Blasius profile $\delta _v/H = a Re_d^{-1/2} \varGamma$ . A curve fit to the data from the slope method gives a prefactor of $a = 0.3655$ for the Blasius scaling. Most of the aforementioned scalings from the literature have a similar trend as they have an $\eta$ value close to $-0.25$ ; however, all of them overpredict the data. It also needs to be mentioned that the correlations cited from the literature correspond to the boundary layer associated with mean flow due to the LSC, which would be present only in the $\varGamma \sim 1$ cells. However, it is evident that the boundary layer thickness based on location of the local maxima, $\delta _v^{max,\sigma }$ , has a very different trend, and it follows our prediction (6.8) remarkably well with a prefactor of $0.1313$ (obtained from curve fitting). Thus, even though there is no mean flow, the velocity boundary layer thickness estimate using the slope method is well captured by Blasius-like scaling or other predictions in the literature with $\eta \approx -0.25$ , whereas the estimate using the maxima method is well captured by the boundary layer thickness determined by the plume-based model (6.8). The reason for the difference in the two scalings is not clear. From the plume structure (figure 13), it is evident that unlike in RBC with $\varGamma \sim 1$ , there is no alignment of the plumes in slender RBC suggesting an absence of a persistent large-scale flow. The boundary layer thickness associated with the fluctuating velocity would be due to the fluctuating wind caused by the tube-convection eddies.

Figure 16. Velocity boundary layer thickness $\delta _v$ obtained from the data of Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) for different $Ra$ . Boundary layer thickness estimates, calculated both using the methods of slope $\delta _v^{sl,\sigma }$ (triangle markers) and maxima $\delta _v^{max,\sigma }$ (circular markers) are shown. Red solid line shows the estimate from the current plume-based model $\delta _v/d \sim U_w/w_m$ (6.8) with an appropriate prefactor of 0.1313. Blue solid line is the Blasius scaling with an appropriate prefactor; other solid lines are various estimates from the literature. The superscripts ‘max’ and ‘sl ’ stand for the boundary layer thickness definition based on the slope and location of maxima methods, respectively, for the horizontal velocity profile. Similarly, superscripts ‘M’ and ‘ $\sigma$ ’ stand for boundary layer definition based on the mean and r.m.s., respectively, of the horizontal velocity profile.

6.5. Velocity variation near the sidewalls

Although not as important as the top and bottom walls, sidewalls do play an important role in the flow dynamics in RBC, and a predominant one in TC. Ideally, sidewalls are insulated; however, the non-zero conductivities and heat capacities of the sidewalls create additional complexities and the need for sidewall corrections while calculating the convective fluxes in experiments (Roche et al. Reference Roche, Castaing, Chabaud, Hébral and Sommeria2001; Verzicco Reference Verzicco2002). In fact, performing experiments on slender RBC would be even more challenging, due to large surface area of sidewalls in comparison with the top and bottom walls. For TC and consequently in the tube part of slender RBC, however, the role of the sidewalls is more passive due to the absence of a time-average velocity in the tube at high $Ra_g$ when the flow is turbulent. Away from the sidewalls in the bulk region (towards the centre of the tube), the r.m.s. velocities scale with the mixing length velocity $w_m$ as discussed in § 5. From the radial velocity profile of r.m.s. velocity fluctuations, Cholemari & Arakeri (Reference Cholemari and Arakeri2009) showed that as the sidewall is approached, the lateral (wall-normal) component starts diminishing early and monotonically to zero at the wall due to kinematic blocking effect. The flow near the wall is akin to shear-free turbulence near a wall, with splat and anti-splat events. Unlike the lateral velocity, the axial (vertical) velocity fluctuation profile peaks near the wall and diminishes rather quickly to zero at the walls due to viscous effects and the no-slip boundary condition. It can be shown that the extent of this viscous affected region may be given by a length scale $l_v \sim \sqrt {\nu \tau }$ , where $\tau = d/w_m$ is the eddy turnover time, and can be expressed as $l_v/d \sim \textit{Gr}_g^{-1/4}$ (Pawar & Arakeri Reference Pawar and Arakeri2016). In figure 17(a), the variation in the radial direction of r.m.s. of total velocity fluctuations $u_{\textit{rms}}$ at the mid-height are shown for Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data. Only the near-wall region is shown and the radius $r$ is non-dimensionalised with diameter $d$ of the tube and velocity by the free-fall velocity $U_{\!f}$ . At the right end, $r/d = r_0/d = 0.5$ , where $r_0$ is the radius of the tube. The dashed vertical lines show the approximate location of the viscous affected region calculated as $l_v/d=\textit{Gr}_g^{-1/4}$ from the wall on the right side. It can be seen that most of the variation of the velocity profile is within this length scale for each case. The same data are re-plotted in figure 17(b) with the velocity non-dimensionalised with the mixing length velocity $w_m$ and the distance by the viscous length scale $l_v$ , with the coordinate shifted such that left end corresponds to the sidewall. It can be seen that with this choice of scaling for distance and velocity, the data for different $Ra$ collapse onto each other within a small error margin, except for the smallest $Ra$ . The viscous affected, boundary layer-like region falls within a distance of $r/l_v \approx 2$ from the sidewall for all the cases. Note that we have used the total velocity fluctuation profile here and not the vertical velocity which would have shown a peak at the edge of the viscous affected region where vertical velocity fluctuations are maximum. The total velocity r.m.s. increases monotonically from 0 at the wall and reaches the mixing length velocity within a few $l_v$ lengths. In fact, the reason for change in the scaling regime from 0.3 to 0.5 in TC can be traced back to the extent of this viscous affected region (Pawar & Arakeri Reference Pawar and Arakeri2016). For the 0.3 scaling regime, the viscous region extends to a significant portion of the tube as in the lowest- $Ra$ case $Ra=10^8$ in figure 17. Note that for this case, the entire width of the tube is around 3.5 $l_v$ , meaning that the entire tube is affected by viscosity. For the 0.5 regime, the extent of the viscous region is very small and negligible compared with the tube width, meaning the flow in the bulk is unaffected by diffusive effects resulting in the ultimate ( $1/2$ -power) scaling.

Figure 17. Profiles of r.m.s. of total velocity fluctuations $u_{\textit{rms}}$ near the sidewall for Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020a ) data in the radial direction: (a) $u_{\textit{rms}}/U_{\!f}$ versus $r/d$ and (b) $u_{\textit{rms}}/w_m$ versus $(r_0-r)/l_v$ . Here, $r_0=d/2$ is the radius of the tube and $l_v = \sqrt {\nu \tau } = \sqrt {\nu d/w_m}$ . Data are sampled at mid-height $z/H =0.5$ and averaged in the azimuthal direction and time. Dashed vertical lines in (a) show the location of $l_v$ measured from the wall (right end) for various $Ra$ .