3.1. Scheme and procedure

In this section, we will compare the scaling relations derived in the previous section against the results from our DNS. Equations (1.1a,b) and (1.2) are solved numerically using the in-house open-source code ‘AFiD’, which is based on a second-order finite-difference scheme (van der Poel et al. Reference van der Poel, Ostilla-Mónico, Donners and Verzicco2015). The code has been extensively validated (Verzicco & Orlandi Reference Verzicco and Orlandi1996; Verzicco & Camussi Reference Verzicco and Camussi1997; Stevens, Verzicco & Lohse Reference Stevens, Verzicco and Lohse2010; Stevens, Lohse & Verzicco Reference Stevens, Lohse and Verzicco2011; Kooij et al. Reference Kooij, Botchev, Frederix, Geurts, Horn, Lohse, van der Poel, Shishkina, Stevens and Verzicco2018). We impose periodic boundary conditions in the horizontal directions and no-slip boundary conditions at the top and bottom walls. For most simulations, we use domains of aspect ratios $\varGamma _x=8$ and $\varGamma _y=4$. We also performed CRB simulations with $\varGamma _x=48$ and $\varGamma _y=24$ for $Ra=10^7$, $Pr=1$ to study large-scale flow structures. Due to the need for high resolution at large $Ra$, the RB simulations for $Ra=10^{10}$, $Pr=1$ and $Ra=10^{11}$, $Pr=1$ were performed in domains of aspect ratios $\varGamma _x=\varGamma _y=4$, while the RB simulation for $Ra=10^{12}$, $Pr=1$ was performed in domain of aspect ratios $\varGamma _x=\varGamma _y=2$. For the CRB simulations, the wall velocities $(-U_w)$ and $U_w$ were imposed as Dirichlet boundary conditions on the bottom and top walls, respectively. This is done for numerical reasons (Bernardini et al. Reference Bernardini, Pirozzoli, Quadrio and Orlandi2013) and does not affect the analysis of the results. The equivalent velocity fields of the CRB system with the bottom wall at rest and the top wall at $2U_w$ can be obtained by a simple Galilean transformation, i.e. by adding $U_w$ to the numerically obtained flow field. For the PRB simulations, the volume forcing term $\varPi$ is computed at each time step to ensure a constant mass flow rate (Quadrio et al. Reference Quadrio, Frohnapfel and Hasegawa2016).

We use a uniform discretisation in the horizontal, periodic directions and a non-uniform grid in the wall-normal direction, in which we employ higher grid resolution in the BLs next to the walls. The thermal BL was ensured to be sufficiently resolved according to the resolution requirements put forward by Shishkina et al. (Reference Shishkina, Stevens, Grossmann and Lohse2010). The near-wall resolution is comparable to that of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014); Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014); Lee & Moser (Reference Lee and Moser2018) to ensure that the kinetic BL is sufficiently resolved. The simulations were run for a long enough physical time for the standard deviation of $Nu$ to converge within approximately 1 % of its mean value. In our previous work (Yerragolam et al. Reference Yerragolam, Verzicco, Lohse and Stevens2022b), we verified that the $Nu$ and $Re_{\tau }$ obtained from a domain with $\varGamma _x=8$ and $\varGamma _y=4$ shows a difference of less than 1 % from the $Nu$ obtained from a domain with $\varGamma _x=48$ and $\varGamma _y=24$. This observation is also supported by the fact that $Nu$ for the RB system converges at an approximate aspect ratio $\varGamma _x=\varGamma _y=4$ (Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018).

Since many of the scaling relations in § 2.4 rely on the value of the wind Reynolds number $Re_R$, obtaining an estimate for $Re_R$ from the numerical simulations of pure RB is necessary. The value of $Re_R$ can be estimated in two possible ways. The first estimate can be obtained by using ${Re_{max} = U_{max}H/\nu }$ where $U_{max}$ is the maximum value of the root mean squared (r.m.s.) horizontal velocity profile ${u_R(z) \equiv \sqrt {\langle u_x^2 + u_y^2\rangle _{A,t}}}$ at a height $z = \delta$ as shown in figure 2(a). The second estimate can be obtained by using the global r.m.s. velocity $Re_{rms} = U_{rms}H/\nu$ with $U_{rms} \equiv \sqrt {\langle u_x^2 + u_y^2 + u_z^2\rangle _{V,t}}$. In figure 2(b) and 2(c), we can see that these estimates are almost identical, providing strong evidence that the LSC rolls driving the wind at the wall also provide the dominant contribution to the mean kinetic energy in RB convection. In all the results discussed henceforth, we adopt $Re_{rms}$ as an estimate for $Re_R$. Unlike in Couette or Poiseuille flow, the mean shear stress at the wall is zero in RB convection. However, we can use the mean gradient of the r.m.s. horizontal velocity $\langle \partial _z u_R(z)\rangle _{W,t}$ to calculate the friction coefficient $C_R$ associated with the large-scale circulation. Here, $\langle \cdots \rangle _{W,t}$ indicates time averaging over the surface of the walls. The variation of $C_R$ with $Re_R$ is discussed separately in § 3.5.

Figure 2.

(a) The r.m.s. horizontal velocity profile $u_R(z)$ normalised with the free-fall velocity for RB system with $Ra=10^7$, $Pr=1.0$ is shown as a function of wall-normal height $z$, indicating the maximum value $U_{max}$ occurring at a wall-normal height $\delta$. (b) Value of $Re_{max}$ obtained from $U_{max}$ is plotted against the globally averaged r.m.s. velocity $Re_R$ of the RB system. The dashed grey line indicates $Re_{max} = Re_{rms}$, showing that these two estimates are virtually identical. (c) $Re_{max}$ compensated with $Re_{rms}$ plotted against $Re_{rms}$, once again emphasising that the two estimates are equivalent within the 5 decades of $Ra$ and nearly 3 decades of $Re_{rms}$ investigated here. The values of $Ra$ are shown on the top for reference. This figure is also available at https://multimediafluids.uam.elogim.com/S0022112024008723/JFM-Notebooks/files/figure2.

3.2. Global response parameters

We now validate the scaling relations for $Nu$ and $C_S$ derived in § 2.4. Within the parameter range of $Ra$, $Pr$ and $Re$ simulated, we already observe multiple transitions. As shear forcing is increased, we undergo transition from the buoyancy-dominated regime to the shear-dominated regime. For low shear forcing, $\epsilon _u \sim \epsilon _{u,bulk}$ with $\lambda _\theta < \lambda _u$ whereas for high shear forcing, $\epsilon _u \sim \epsilon _{u,BL}$ with $\lambda _\theta /\lambda _u \sim Pr^{1/2}$ (Yerragolam et al. Reference Yerragolam, Stevens, Verzicco, Lohse and Shishkina2022a). Additionally, at low shear forcing, we observe a laminar Prandtl–Blasius-type BL which undergoes a transition into a turbulent one at high shear.

In the buoyancy-dominated regime, we assume the presence of Prandtl–Blasius-type kinetic boundary layer with the friction coefficient $C_L$ given by (2.18). For the parameter range of our simulations, the relevant convection regime is $II_u$, so combining the relevant relation from table 2 with (2.18), we arrive at

(3.1)\begin{equation} \frac{Nu}{Nu_R} \sim \left(\frac{Re_T}{Re_L}\right)^{-1/5}. \end{equation}

In the buoyancy-dominated regimes, we also take (2.19) for the friction coefficient $C_S$ associated with the imposed shear. In order to further simplify these equations, we approximate $Re_L \approx Re_R$ in the buoyancy-dominated regime, which is justified by the weak variation of $Re_L$ with increasing $Re_S$ in (table 2). With this assumption, we can rewrite (2.19) as

(3.2)\begin{equation} C_S \sim \sqrt{Re_R}/Re_S, \end{equation}

and use (2.5) to rewrite (3.1) as

(3.3)\begin{equation} Nu/Nu_R \sim \left( \sqrt{1+\left(Re_S^2/Re_R^2\right)} \right)^{-1/5}. \end{equation}

These equations show good agreement with the numerical data plotted in figures 3(a)–3(d) for the buoyancy-dominated regime. Note that (3.1) and (3.3) do not explicitly state the dependence of $Nu_R(Ra, Pr)$ or $Re_R(Ra, Pr)$. There is no pure scaling exponent for $Nu_R(Ra)$ in pure RB for regime $II_u$. The values of $Nu_R$ and $Re_R$ used for figures 3(b) and 3(d) are obtained from numerical simulations of pure RB. The present extension to the GL-theory assumes that the values of $Nu_R$ and $Re_R$ are known a priori, and only attempts to provide scaling relations for the normalised quantities $Nu/Nu_R (Re_S/Re_R)$ and $C_S/C_R (Re_S/Re_R)$ in the buoyancy-dominated regime.

Figure 3.

(a) Value of $Nu/Pr^{1/2}$ plotted against $Re_S$. The black solid line indicates (3.5), the dashed lines indicate (2.20) and the dotted lines indicate (3.3). (b) Value of $Nu/Nu_R$ plotted against $Re_S/Re_R$. Dotted lines indicate (3.3). (c) Value of $C_S$ plotted against $Re_S$. The black solid line indicates (3.4) and coloured dashed lines indicate (2.19). (d) Value of $C_S$ normalised with $C_R$ plotted against $Re_S/Re_R$. Coloured dashed lines indicate (3.4) and black dashed line indicates (2.19). The data for PRB are indicated with plus markers and the data for CRB are indicated with dot markers. This figure is also available at https://multimediafluids.uam.elogim.com/S0022112024008723/JFM-Notebooks/files/figure3.

For the shear-dominated regime, we observe that the BL becomes turbulent. In this case, (2.19) is no longer valid. Instead, the relation between $C_T$ and $Re_T$ is given by (2.22). Note that, in the limiting case of very high-shear forcing, $Re_T \approx Re_S$. Equation (2.22) can be rewritten as

(3.4)\begin{equation} \sqrt{\frac{2}{C_S}} = \frac{1}{k} \ln{ \left( Re_S \sqrt{\frac{C_S}{8}} \right) } + B, \end{equation}

and (2.20) can be rewritten using (2.7) as

(3.5)\begin{equation} Nu \sim Pr^{1/2} C_S Re_S, \end{equation}

which agrees well with the numerical data points in figures 3(c) and 3(d) at very high shear forcing. However, it is more useful to substitute the value of $C_T$ obtained from (2.22) into (2.20) and approximate $Re_T$ from (2.5) as

(3.6)\begin{equation} Re_T \approx \sqrt{Re_R^2 + Re_S^2}, \end{equation}

to obtain the dashed lines plotted in figure 3(a), which show better agreement for a larger range of $Re_S/Re_R$ in the shear-dominated regime. The approximation given by (3.6) is then validated in figure 4(a). The additional energy term $\gamma C_S Re_S^2$ from (2.4) that corresponds to the turbulent fluctuations arising from shear forcing is shown in figure 4(b) with the value of the prefactor $\gamma \approx 10.24$. Note that the contribution from the fluctuations is much smaller than the contribution from the mean streamwise velocity, thereby making (3.6) a good approximation.

Figure 4.

(a) Value of $Re_T/Re_R$ plotted against $Re_S/Re_R$. The black solid line indicates the relation (2.5). (b) Value of $(\sqrt {Re_T^2 - Re_S^2})/Re_R$ plotted against $\sqrt {C_S} Re_S/Re_R$. The black dashed line indicates the relation (2.4) with $\gamma \approx 10.24$. The data for PRB are indicated with plus markers and the data for CRB are indicated with dot markers. This figure is also available at https://multimediafluids.uam.elogim.com/S0022112024008723/JFM-Notebooks/files/figure4.

3.3. Large-scale convection rolls

Next, we confirm the theoretical assumptions on the LSC rolls made in § 2.2. The three-dimensional volume visualisation in figure 5 shows the streamlines associated with these LSC rolls and it can be seen that they are predominantly oriented in the spanwise direction. However, the large-scale temperature structures in figure 5 are seen to be aligned neither fully along the streamwise direction, nor fully along the spanwise direction but along a diagonal. The thermal plumes that comprise these large-scale structures experience the advective effects of both $U_L$ in the spanwise direction and $U_S$ in the streamwise direction. Therefore, the orientation of these large scale temperature flow structures in the $x\unicode{x2013}y$ plane is tilted along a diagonal whose slope is approximately given by $U_L/U_S$. This is made clearer in figure 6 through the visualisation of the non-dimensional time-averaged local shear stress $\boldsymbol {\tau _w^{\prime }} \equiv (\tau _x^{\prime },\tau _y^{\prime },0)$ computed at the bottom wall in the large aspect ratio CRB system by subtracting the wall-averaged streamwise shear stress in the following way:

(3.7a,b)\begin{equation} \tau_x^{\prime} = H U_{F}^{-1}\left\langle \partial_z u_x - \left\langle \partial_z u_x \right\rangle_{x,y,t}\right\rangle_{t}, \quad \tau_y^{\prime} = H U_{F}^{-1}\left\langle \partial_z u_y \right\rangle_{t}. \end{equation}

Figure 5.

Three-dimensional volume visualisation of the time-averaged reduced temperature field of a CRB system with $Ra=10^7$, $Pr=1.0$, $Re_w=1414$, $\varGamma _x=48$, $\varGamma _y=24$ with the averaging time being 100 free-fall time units. The green curves indicate the streamlines of the time-averaged velocity field $\langle \boldsymbol {u}\rangle _{t} - U_w \hat {x}$. The spanwise reorientation and streamwise sweeping of the plumes are evident. The zoomed inset shows the time-averaged two-dimensional visualisation of the temperature and velocity vectors in the $x\unicode{x2013}z$ plane. The plumes carry streamwise momentum along with temperature and are swept in the streamwise direction, causing a reduction in the heat transport.

Figure 6.

Visualisation of $\boldsymbol {\tau _w^{\prime }}$ for the CRB system with $Ra=10^7$ and $Pr=1$. Panels (c), (e) and (g) are in the buoyancy-dominated regime and panel (i) is in the shear-dominated regime. The colour bars indicate the magnitude of $\boldsymbol {\tau _w^{\prime }}$ while the black arrows in the magnified panels (b), (d), (f), (h) and (j) indicate the direction of $\boldsymbol {\tau _w^{\prime }}$. The black lines in (c–h) indicate the slope of $(Re_w$/$Re_R)^{-1}$. This figure is also available at https://multimediafluids.uam.elogim.com/S0022112024008723/JFM-Notebooks/files/figure6.

Figure 6(a) shows the wall shear for the RB system i.e. for $Re_w=0$ and the zoomed inset in figure 6(b) shows the vectors of $\boldsymbol {\tau _w^{\prime }}$. As expected, the LSC rolls are randomly oriented and no global alignment of $\boldsymbol {\tau _w^{\prime }}$ is observed. A visual inspection of figures 6(c), 6(e) and 6(g) reveals that the large-scale flow structures seem to be oriented along the diagonal whose slope is approximately given by $Re_R$/$Re_w$. However, figures 6(d), 6(f) and 6(h) show that $\boldsymbol {\tau _w^{\prime }}$ is primarily oriented along the spanwise direction in the transitional regime. Figure 6(i) shows the breakdown of the LSC rolls and the formation of large meandering flow structures (Hutchins & Marusic Reference Hutchins and Marusic2007; Blass et al. Reference Blass, Zhu, Verzicco, Lohse and Stevens2020, Reference Blass, Tabak, Verzicco, Stevens and Lohse2021) in the shear-dominated regime, while figure 6(j) shows that $\boldsymbol {\tau _w^{\prime }}$ is predominantly aligned in the streamwise direction in the shear-dominated regime.

For further confirmation of the changes in the LSC rolls, we study the probability distribution function $\phi (\alpha )$ of the angle $\alpha$ spanned by the horizontal velocity component fluctuations $\boldsymbol {u_h^\prime } \equiv (u_x^{\prime }, u_y^{\prime }, 0)$ with the streamwise direction $x$. In figures 7(a)–7(e) it can be seen that the behaviour of $\phi (\alpha )$ is qualitatively quite similar for the CRB and PRB systems. In the RB system, the LSC rolls are randomly oriented as shown by a uniform $\phi (\alpha )$ in figure 7(a). In the buoyancy-dominated regime, the LSC roll has a strong tendency to align in the spanwise directions in the transitional regime, as shown in figures 7(b) and 7(d). In the shear-dominated regime, the velocity fluctuations are predominantly aligned in the streamwise direction as in the case of turbulent Couette/Poiseuille flows, as seen in figures 7(c) and 7(e). At the thermal BL height, the symmetry of $\phi (\alpha )$ about $\alpha =\pm {\rm \pi}/2$ is strongly suggested for all three regimes by the data from the numerical simulations as shown in figures 7(f)–7(h), confirming the assumption made in (2.6). In figure 7(f), the small non-uniformity in $\phi (\alpha )$ is attributed to the numerical confinement experienced by the flow structures in domains of a smaller aspect ratio of $\varGamma _x=8$ and $\varGamma _y=4$. For the relatively unconfined case with $\varGamma _x=48$ and $\varGamma _y=24$, the probability distribution is nearly uniform for all values of $\alpha$.

Figure 7.

The probability distribution $\phi (\alpha )$ of the flow orientation angle $\alpha$ for all heights $0 \leq z/\lambda _{\theta } \leq Nu$ for $Ra=10^7$, $Pr=1$, $\varGamma _x=48$ and $\varGamma _y=24$ in (a) RB flow (i.e. with $Re_S=0)$, (b,d) buoyancy-dominated regime with $Re_S/Re_R\approx 2.11$ and (c,e) shear-dominated regime with $Re_S/Re_R\geq 10$. Panels (b,c) are for the CRB system while panels (d,e) are for the PRB system. The black dashed lines indicate the height of the thermal BL and the colour bar indicates the magnitude of the probability. The probability distribution $\phi (\alpha )$ at the thermal BL height plotted against angle $\alpha$ for various $Ra$ and $Pr$ in (f) RB flow, (g) buoyancy-dominated regime and (h) shear-dominated regime. The solid lines are the CRB system with $\varGamma _x=48$ and $\varGamma _y=24$, the dashed lines are for the CRB system with $\varGamma _x=8$ and $\varGamma _y=4$ and the dotted lines are for the PRB system with $\varGamma _x=8$ and $\varGamma _y=4$. This figure is also available at https://multimediafluids.uam.elogim.com/S0022112024008723/JFM-Notebooks/files/figure7.

3.4. Dissipation rates

We now investigate the bulk and BL contributions to the global kinetic and thermal dissipation rates described in § 2.1. We validate the rigorous relations given by (2.1) and (2.2) using the data obtained from numerical simulations as shown in figures 8(a) and 8(b), respectively. For the range of $Ra$ and $Pr$ studied in this work, as long as the flow is buoyancy dominated, the system is in the $II_u$ regime with $\epsilon _u \sim \epsilon _{u,bulk}$. With increasing shear forcing, the shear term of (2.2) increases the BL contribution of the kinetic dissipation due to the formation of streamwise velocity gradients close to the wall. For sufficiently strong shear, the kinetic dissipation will be dominated by the BL contribution with $\epsilon _u \sim \epsilon _{u,BL}$ as shown in 8(c). On the contrary, figure 8(d) shows that the thermal dissipation rate is dominated by the BL contribution for the entire range of $Re_S$ considered in this study but the contribution reduces noticeably towards higher $Re_S$. For extremely strong shear, a possibility of a transition towards bulk dominance in thermal dissipation cannot be ruled out.

Figure 8.

(a) Ratio $(H^2 \epsilon _\theta \kappa ^{-1} \varDelta ^2 )/Nu$ showing good agreement with (2.1) indicated using the solid black line, and (b) ratio $( H^4 \epsilon _u/ \nu ^3 ) / ( (Nu - 1) Ra Pr^{-2} + C_S Re_S^3)$ plotted against $Re_S$ showing good agreement with (2.2) of the manuscript shown using the solid black line. (c) Boundary layer contribution to the global kinetic dissipation rate $\epsilon _{u,BL}$ and (d) the BL contribution to the global thermal dissipation rate $\epsilon _{\theta,BL}$. The circle markers are for the CRB system, while plus markers are for the PRB system. This figure is also available at https://multimediafluids.uam.elogim.com/S0022112024008723/JFM-Notebooks/files/figure8.

3.5. Friction coefficient in Rayleigh–Bénard flow

So far, we have considered only the laminar Prandtl–Blasius-type kinetic BLs in the buoyancy-dominated regime with the BL only becoming turbulent in highly shear-dominated regime. However, we now consider the possibility of turbulent kinetic BLs even in pure RB flow which corresponds to the so-called ‘ultimate’ regime in RB flow (Kraichnan Reference Kraichnan1962; Grossmann & Lohse Reference Grossmann and Lohse2011; Roche Reference Roche2020; Lohse & Shishkina Reference Lohse and Shishkina2023). While the existence of logarithmic temperature profiles has already been observed at high $Ra$ (Grossmann & Lohse Reference Grossmann and Lohse2012; Ahlers et al. Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012; Ahlers, Bodenschatz & He Reference Ahlers, Bodenschatz and He2014), it is yet to be seen if logarithmic behaviour given by

(3.8)\begin{equation} u_R^+ (z^+) = \frac{1}{k} \log (z^+) + B, \end{equation}

with

(3.9a–d)\begin{equation} u_R^+ (z^+) = u_R(z) /u_\tau, \quad z^+= \frac{z u_\tau}{2 \nu}, \quad u_R(z) \equiv \sqrt{\left\langle u_x^2 + u_y^2\right\rangle_{A,t}}, \quad u_\tau = \sqrt{\nu \left\langle\partial_z u_R\right\rangle_{W,t}}, \end{equation}

can be observed in the velocity profiles.

At the highest thermal forcing studied in this work with $Ra=10^{12}$, we start to also observe some hints of what could possibly be the onset of a log layer (see figure 9a), although it cannot be conclusively confirmed with the currently available data. Assuming that such a log layer could exist, an estimate of the modified von Kármán constant $k$ is obtained from the inflection point of the diagnostic function plotted in figure 9(b), giving $k \approx 1$. Correspondingly, the intercept $B \approx 5.2$ is found by fitting the data, as shown in figure 9(a).

Figure 9.

(a) Value of $u_R^+$ plotted against $z^+$, showing the onset of a log layer with the grey dashed line. The velocity profile for $Ra=10^{12}$, $Pr=1.0$ is indicated in magenta. (b) Diagnostic function plotted against $z^+$ with the inflection point and the corresponding von Kármán constant indicated. (c) Value of $\delta ^+$ plotted against $Re_R$ shows good agreement with $\delta ^+ = (1/2) Re_R^{1/2}$, indicated with the dashed grey line. (d) Value of $C_R$ plotted against $Re_\delta$ showing a collapse without any prefactors. The solid grey line indicates $C_R \sim 4Re_\delta ^{-2/3}$ and the dashed grey line indicates the modified Prandtl (Reference Prandtl1932) friction law given by (3.10a,b). (e) Value of $C_R/a$ plotted against $Re_R$, showing the Prandtl–Blasius scaling (2.18) at low $Re_R$ with the solid grey line and modified Prandtl (Reference Prandtl1932) friction law given by (3.10a,b) at high $Re_R$ with the dashed grey line. (f) Value of $C_\epsilon /b$ given by (3.11b) plotted against $Re_R$. The solid grey line indicates $C_\epsilon \sim Re_R^{-1/2}$ for the regime with $\epsilon _u \sim \epsilon _{u,BL}$, the dashed grey line indicates the modified Prandtl (Reference Prandtl1932) friction law (3.10a,b). At higher $Re_R$, $\epsilon _u \sim \epsilon _{u,bulk}$, leading to $C_\epsilon$ becoming independent of $Re_R$, as indicated with the dash-dotted and dotted grey lines. This figure is also available at https://multimediafluids.uam.elogim.com/S0022112024008723/JFM-Notebooks/files/figure9.

In the presence of such a logarithmic layer, we can now hypothesise about a relation for $C_R$ which is analogous to relation (3.4) for $C_S$ and (2.22) for $C_T$ by stating that in the limit of highly turbulent BL, $u_R^+=U_R$ at $z^+=\delta ^+$, where $\delta$ is the wall-normal distance to the peak velocity shown in figure 4. This gives us

(3.10a,b)\begin{equation} \sqrt{\frac{2}{C_R}} = \frac{1}{k} \ln{ \left( Re_\delta \sqrt{\frac{C_R}{8}} \right) } + B, \quad Re_\delta = U_R \delta / \nu .\end{equation}

If applicable to high $Ra$ RB flow, the general form of this equation could suggest universality in the behaviour of wall-bounded flows even though the values of $k$ and $B$ might be different from those observed for pipe or channel flows. By plotting $\delta ^+$ against $Re_R$, we find a very good fit with the scaling $\delta ^+ \sim Re_R^{1/2}$, as shown in figure 9(c). At present, we can only provide this scaling empirically because more data at extremely high $Ra$ are needed to understand the dynamics of the turbulent BL which is difficult due to the high computational expense of such numerical simulations. Plotting $C_R$ against $Re_\delta$ in 9(d), we find a nice collapse of the data for different aspect ratios and geometries. At low $Re_R$, $C_R$ seems to scale as $Re_\delta ^{-2/3}$, which can be obtained from the empirical scaling $\delta ^+ \sim Re_R^{1/2}$ shown in figure 9(c) and the assumption of a Prandtl–Blasius-type scaling of $C_R \sim Re_R^{-1/2}$. At high $Re_R$, we see good qualitative agreement between the data and the modified Prandtl (Reference Prandtl1932) friction law given by (3.10a,b).

We can further investigate the behaviour of $C_R$ by computing

(3.11a)$$\begin{gather} C_R \equiv a Re_R^{-1/2}, \end{gather}$$

(3.11b)$$\begin{gather} C_\epsilon \equiv b (Nu - 1) Ra Pr^{-2} Re_R^{-3}, \quad \begin{cases} C_\epsilon \sim C_R & \text{if} \epsilon_u \sim \epsilon_{u,BL} \\ C_\epsilon = \text{const.} & \text{if} \epsilon_u \sim \epsilon_{u,bulk}, \end{cases} \end{gather}$$

where $a$ and $b$ are prefactors that are obtained by fitting the data from figures 9(e) and 9(f), accounting for the effects of aspect ratio and geometry. Note that, when the kinetic dissipation of the RB system is dominated by the contribution from the BL, $C_\epsilon \sim C_R$ in (3.11b). The data points at lower $Re_R$ in figure 9(e) are observed to follow (2.18), which is consistent with the Prandtl–Blasius scaling. For higher $Re_R$, corresponding to $Ra>10^{10}$, $C_R$ shows better agreement with the Prandtl (Reference Prandtl1932) friction law given by (3.10a,b). Figure 9(f) shows a very similar behaviour to figure 9(e) but, in addition, we observe that $C_\epsilon$ becomes independent of $Re_R$ at very high values of $Re_R$. Although this does not reflect the true dependence of $C_R$ on $Re_R$, this apparent dependence is expected because the kinetic dissipation of the RB system undergoes a transition from being dominated by the BL to being dominated by the bulk (Lohse Reference Lohse1994). It should also be noted that this transition occurs at higher $Re_R$ for the more confined cylindrical RB simulations because more kinetic driving is required to overcome the viscous dissipation in the additional BLs on the side walls that are not present in the periodic box RB simulations (Ahlers et al. Reference Ahlers2022).