2.1. Classical regime by Malkus (Reference Malkus1954)

The first regime of convection, called classical, was proposed by Malkus (Reference Malkus1954) and Priestley (Reference Priestley1954). It has the merit of simplicity and predicts a scaling law $Nu_0 \sim Ra^{1/3}$, hence with an exponent 1/3 close to the exponents observed both in the experiments and the numerical simulations in the range of $Ra$ between $10^6$ and $10^{12}$. This regime of convection is entirely characterized by a constant Rayleigh number for each boundary layer as

(2.2)\begin{equation} \frac{g \alpha (\rm \Delta) T \delta_{T}^3}{2 \nu \kappa} = Ra^*. \end{equation}

Using (2.1) and (2.2), we obtain for the classical regime

(2.3)\begin{equation} Nu_0 = \left (\frac{Ra}{2^4 Ra^*}\right)^{1/3}. \end{equation}

2.2. Ultimate regime by Kraichnan (Reference Kraichnan1962)

For very large Rayleigh numbers, the thermal boundary layers observed in the case of the classical regime can be destabilized and Kraichnan (Reference Kraichnan1962) and Spiegel (Reference Spiegel1971) assumed that they could become similar to the velocity boundary layers observed in the case of a fully developed mean shear flow. This ultimate regime is then characterized by a constant but Prandtl-dependent Péclet number for each thermal boundary layer

(2.4)\begin{equation} \frac{v_0^* \delta_{T}}{\kappa} = Pe^*(Pr), \end{equation}

where $\kappa = \lambda / (\rho c_p)$ is the thermal diffusivity of the fluid. For small Prandtl numbers, the thickness of the viscous sublayer is smaller than $\delta _{T}$ leading to a constant Péclet number $Pe^* = Pe^*_{Pr \to 0}$. On the contrary, at moderate $Pr$ numbers, $Pe^*$ varies as $\sqrt {Pr}$ since $Pe^* = \sqrt {Pe^*_{Pr \to 0} Re_s Pr }$, where $Re_s$ is the characteristic Reynolds number for the top of the viscous sublayer (Kraichnan Reference Kraichnan1962). The new unknown parameter $v_0^*$ can be interpreted as a friction velocity and measures the root-mean-square value of velocity fluctuations at the edge of each boundary layer, similarly to the friction velocity defined in the case of a channel flow. Unlike the classical regime for which the characteristic Rayleigh number $Ra^*$ depends only on $(\rm \Delta) T$ and $\delta _{T}$, $Pe^*$ is linked to the convective flow in the bulk by the velocity fluctuations $v_0^*$. Thus, determining the Nusselt number for the ultimate regime requires additional assumptions and equations. The parameter $v_0^*$ is an increasing function of the large-scale mean velocity ($U_0$), also called as the wind turbulence. By analogy with what is well known for the channel flow, Kraichnan (Reference Kraichnan1962) assumed that $v_0^* \sim U_0 / \ln Re_0$, with $Re_0 = U_0 h / \nu$. In addition, the wind velocity is obtained by writing that the Richardson number in the bulk flow is of order 1, i.e. $Ri = g \alpha (\overline {w' T'}) h / U_0^3 \sim 1$. Using the definitions of $Re_0$, $Ra$ and $Nu_0$, this last equation yields

(2.5)\begin{equation} Re_0^3 \sim \frac{Ra Nu_0}{Pr^2}. \end{equation}

We can note that (2.5) is valid for the classical regime obtained by Malkus (Reference Malkus1954), the ultimate regime proposed by Kraichnan (Reference Kraichnan1962) and the two convection regimes (II and IV) of the Grossmann & Lohse (Reference Grossmann and Lohse2000) theory (see § 2.3). Using (2.1), (2.5) and $v_0^* \sim U_0 / \ln Re_0$, (2.4) becomes

(2.6)\begin{equation} Re_0^2 \ln (Re_0) \sim \frac{Ra}{Pr Pe^*}. \end{equation}

Then, using (2.6), (2.5) gives the Nusselt number for the ultimate regime

(2.7)\begin{equation} Nu_0 \sim \left (\frac{Pr \, Ra}{ [ Pe^* \ln (Re_0) ]^3 }\right)^{1/2}. \end{equation}

For small $Pr$ numbers (typically $Pr < Pe^*_{Pr \to 0} / Re_s$), $Nu_0 \sim Pr^{1/2} Ra^{1/2} / (\ln Re_0 )^{3/2}$ while for moderate $Pr$ numbers, $Nu_0 \sim Pr^{-1/4} Ra^{1/2} / [\ln (Re_0) ]^{3/2}$.

2.3. RB theory by Grossmann & Lohse (Reference Grossmann and Lohse2000)

Grossmann & Lohse (Reference Grossmann and Lohse2000) (GL) proposed a RB theory to describe with more precision the Rayleigh and Prandtl dependence of the Nusselt number. The kinetic energy and thermal dissipation rates which are defined respectively as $\epsilon _u = ({\nu }/{2}) \sum _{i,j} (\partial _j u_i + \partial _i u_j)^2$ and $\epsilon _T = \kappa \sum _{i} (\partial _i T)^2$ play a central role in GL theory. In steady state and averaging over the whole RB cell the two equations of conservation of the turbulent kinetic energy ($\frac {1}{2}\sum _i u_i^2$) and of the square of the temperature give the following two exact relations:

(2.8)\begin{gather} \langle\epsilon_u\rangle = \frac{\nu^3}{h^4} \frac{(Nu_0 - 1) Ra}{Pr^{2}} , \end{gather}

(2.9)\begin{gather}\langle \epsilon_T \rangle = \kappa \left (\frac{(\rm \Delta) T}{h} \right )^2 Nu_0. \end{gather}

The key idea of the GL theory is to split both mean dissipation rates into two contributions each, one from the bulk (Bu) and one from the boundary layers (BLs) as

(2.10)\begin{gather} \langle\epsilon_u\rangle = \langle\epsilon_u\rangle_{Bu} + \langle\epsilon_u\rangle_{BL}, \end{gather}

(2.11)\begin{gather}\langle \epsilon_T \rangle = \langle \epsilon_T \rangle_{Bu} + \langle \epsilon_T \rangle_{BL} , \end{gather}

where

(2.12a,b)\begin{equation} \langle\epsilon_u\rangle_{Bu} = \frac{1}{h} \int_{\delta_u}^{h-\delta_u} \overline{\epsilon_u} (z)\,\mathrm{d} z\quad \text{and}\quad \langle \epsilon_T \rangle_{Bu} = \frac{1}{h} \int_{\delta_{T}}^{h-\delta_{T}} \overline{\epsilon_T} (z)\,\mathrm{d} z \end{equation}

are, respectively, the viscous and thermal dissipation taking place in the bulk flow. Whereas the viscous and thermal dissipation taking place in the boundary layers can be written as

(2.13a,b)\begin{equation} \langle\epsilon_u\rangle_{BL} = \frac{2}{h} \int_{0}^{\delta_u} \overline{\epsilon_u} (z)\,\mathrm{d} z\quad \text{and}\quad \langle \epsilon_T \rangle_{BL} = \frac{2}{h} \int_{0}^{\delta_{T}} \overline{\epsilon_T} (z)\,\mathrm{d} z. \end{equation}

In (2.12a,b) and (2.13a,b), the kinetic energy and thermal dissipation rates are first averaged over a horizontal cross-section giving $\overline {\epsilon _u}$ and $\overline {\epsilon _T}$, respectively. The thickness of the thermal BLs ($\delta _T$) is given by (2.1) while a Blasius-type layer is assumed for the viscous BLs, with a thickness of

(2.14)\begin{equation} \frac{\delta_{u}}{h} = \frac{a}{\sqrt{Re_0}}. \end{equation}

Note that the prefactor $a$ is obtained by match with a record of experimental results (Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013).

To obtain the Rayleigh dependence of the Nusselt and Reynolds numbers, $\overline {\epsilon _u}$ and $\overline {\epsilon _T}$ need to be estimated both in the bulk flow and in the BLs

(2.15)\begin{gather} \langle{\epsilon_u}\rangle_{Bu} \sim \frac{U_0^2}{h/U_0} \left(1-\frac{\delta_u}{h} \right)\approx \frac{\nu^3}{h^4} Re_0^3, \end{gather}

(2.16)\begin{gather}\langle{\epsilon_T}\rangle_{Bu} \sim \frac{((\rm \Delta) T)^2}{h/U_0^{edge}} \left(1-\frac{\delta_T}{h} \right) \approx \kappa \left ( \frac{(\rm \Delta) T}{h} \right)^2 Re_0 Pr\,f \left( \frac{2 a Nu_0}{\sqrt{Re_0}} \right), \end{gather}

(2.17)\begin{gather}\langle{\epsilon_u}\rangle_{BL} \sim \nu \left(\frac{U_0}{\delta_u} \right )^2 \frac{\delta_u}{h} = \frac{\nu^3}{h^4} \frac{Re_0^{5/2}}{2a}, \end{gather}

(2.18)\begin{gather}\langle{\epsilon_T}\rangle_{BL} \sim \kappa \left( \frac{(\rm \Delta) T}{\delta_T}\right )^2 \frac{\delta_T}{h} = 2 \kappa \left (\frac{(\rm \Delta) T}{h}\right)^2 Nu_0. \end{gather}

In (2.16), the relevant velocity at the edge between thermal BL and the thermal bulk can be less than $U_0$, depending on the ratio: $\delta _u / \delta _T = 2 a Nu_0 / \sqrt {Re_0}$. Grossmann & Lohse (Reference Grossmann and Lohse2001) introduced a function $0 \leq f \leq 1$ saying that the relevant velocity at the edge then becomes $U_0^{edge} = U_0 \,f(\delta _u / \delta _T)$, with $f \to 1$ when $\delta _u / \delta _T \to 0$ and $f \to 0$ for $\delta _u \gg \delta _T$. They gave $f(x) = (1 + x^n )^{-1/n}$, with $n\!=\!4$ as an example of function $f$.

Grossmann & Lohse (Reference Grossmann and Lohse2001) have also extended those estimations of the viscous and dissipation rates for very large Prandtl numbers for which (2.14) cannot stay valid. Indeed, when $Pr$ is high enough, $\delta _u$ must saturate to a maximum value $\delta _u(Re_c)$ lower than the height of the cell. The critical Reynolds number $Re_c$ was estimated from experimental data to 0.28 by Grossmann & Lohse (Reference Grossmann and Lohse2001) and 0.35 by Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013). However, for the sake of simplicity, only the case of $Re_0 \gg Re_c$ is considered here.

From decomposition of the two global dissipation rates (2.10) and (2.11), four regimes of convection can be defined depending on whether the bulk or the BL contributions dominate the global dissipations. Besides, each of these four regimes is in principle divided into two subregimes, depending on whether the thermal BL or the kinetic BL is larger. The two $\langle \overline {\epsilon _u}\rangle _{Bu}$ bulk-dominated regimes (referred to as II and IV) are first presented because most of the experimental and numerical results fall under one of these two regimes (see figure 8 from Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013)).

2.3.1. Regimes II and IV, $\langle {\epsilon _u} \rangle \sim \langle {\epsilon _u}\rangle _{Bu}$

For regimes II and IV, the kinetic energy dissipation rate is dominated by its bulk contribution. Combining (2.8) and (2.15), and assuming $Nu_0 \gg 1$, we obtain (2.5) again. Regime IV is obtained for high $Ra$ numbers for which thermal dissipation rate is dominated by its bulk contribution. Combining (2.9) and (2.16) yields

(2.19)\begin{equation} Nu_0 \sim Re_0 Pr\,f \left( \frac{2 a Nu_0}{\sqrt{Re_0}} \right)\quad \text{(Regime IV)}. \end{equation}

For lower $Ra$ numbers, the thermal dissipation rate is dominated by its BL contribution. However, combining (2.9) and (2.18) yields a trivial equation for $Nu_0$. To obtain a scaling relation between $Nu_0$ and $Re_0$, Grossmann & Lohse (Reference Grossmann and Lohse2000) proposed to consider, in each thermal BL, the order of magnitude of the different terms of energy equation

(2.20)\begin{equation} u_x \partial_x T + u_z \partial_z = \kappa \partial_{zz} T. \end{equation}

Both terms on the left-hand side are of order $U_0^{edge} (\rm \Delta) T /h$ whereas $\kappa \partial _{zz} T \sim \kappa (\rm \Delta) T / \delta _T^2$. Hence, using (2.1), one gets

(2.21)\begin{equation} Nu_0 \sim \sqrt{Re_0 Pr\,f \left( \frac{2 a Nu_0}{\sqrt{Re_0}} \right)} \quad \text{(Regime II)}. \end{equation}

Combining (2.5) and (2.19) or else (2.5) and (2.21), we obtain

(2.22)\begin{equation} Nu_0^{\theta_i} \sim (Nu_0 Ra Pr)^{1/3} f \left [ \frac{2a (Nu_0 Ra Pr)^{1/3} }{ (Ra /Nu_0)^{1/2} } \right ],\quad \text{with} \ \theta_{II}=2 \ \text{and} \ \theta_{IV}=1. \end{equation}

For Prandtl numbers small or large enough, $f(x) \approx 1$ ($\delta _T \gg \delta _u$) or $f(x) \approx 1/x$ ($\delta _u \gg \delta _T$), and (2.22) can be simplified as follows:

(2.23a,b)\begin{equation} {Nu_0 \sim} \begin{cases}(Ra Pr)^{1/(3\theta_i-1)},\quad \ \text{for} \ \delta_T \gg \delta_u, \\ Ra^{1/(2\theta_i+1)},\qquad \quad\text{for} \ \delta_u \gg \delta_T. \end{cases} \end{equation}

We can note that the sub-regime $\textrm {IV}_u$ ($\delta _u \gg \delta _T$) gives the same scaling as predicted by Malkus (Reference Malkus1954), i.e. $Nu_0 \sim Pr^0 Ra^{1/3}$.

2.3.2. Regimes I and III, $\langle {\epsilon _u} \rangle \sim \langle {\epsilon _u}\rangle _{BL}$

For these two regimes, (2.5) needs to be replaced by

(2.24)\begin{equation} Re_0^{5/2} \sim \frac{Ra Nu_0}{Pr^2}. \end{equation}

Equation (2.24) is obtained by combining (2.8) and (2.17). As for regime IV, the thermal dissipation rate is dominated by its bulk contribution in regime III and (2.19) is valid. On the contrary, in regime I, we use (2.21) instead of (2.19), as for regime II. The $Ra$- and $Pr$-dependent Nusselt number is then given by

(2.25)\begin{equation} Nu_0^{\theta_i} \sim (Nu_0 Ra \sqrt{Pr})^{2/5} f \left \{ \frac{2a (Nu_0 Ra \sqrt{Pr})^{2/5} } { [Ra^3 / (Nu_0^2 Pr)]^{1/5} } \right \},\quad \text{with} \ \theta_{I}=2 \ \text{and} \ \theta_{III}=1.\end{equation}

For Prandtl numbers small or large enough, (2.25) becomes

(2.26a,b)\begin{equation}{Nu_0 \sim} \begin{cases} (Ra \sqrt{Pr})^{2/(5\theta_i-2)},\qquad \quad\text{for} \ \delta_T \gg \delta_u, \\ Ra^{3/(5\theta_i+2)} Pr^{{-}1/(5\theta_i+2)},\quad \ \text{for} \ \delta_u \gg \delta_T. \end{cases} \end{equation}

2.3.3. Grossmann & Lohse (Reference Grossmann and Lohse2001) theory for the whole parameter $(Ra,Pr)$ plane

The four previous regimes can only be observed experimentally and numerically for extreme values of $Ra$ and $Pr$ numbers. For instance regime IV corresponds to very high $Ra$ numbers but in this case ultimate convection could appear while regime II is valid only for very small $Ra$ numbers for which convection is not really turbulent. Grossmann & Lohse (Reference Grossmann and Lohse2001) proposed to describe convection at any $Ra$ and $Pr$ numbers as a mixture of these four regimes. By replacing the expressions of $\langle {\epsilon _u}\rangle _{Bu}$ (2.15) and $\langle {\epsilon _u}\rangle _{BL}$ (2.17) in the balance equation for the viscous dissipation rate (2.10), they obtained this first generalised equation:

(2.27)\begin{equation} \frac{Ra Nu_0}{Pr^2} = c_1 \frac{Re_0^{5/2}}{2a} + c_2 Re_0^3. \end{equation}

Using (2.19) and (2.21), the second generalized equation can be written as

(2.28)\begin{equation} Nu_0 = c_3 \sqrt{Re_0 Pr\,f \left( \frac{2 a Nu_0}{\sqrt{Re_0}} \right)} + c_4 {Re_0 Pr\,f \left( \frac{2 a Nu_0}{\sqrt{Re_0}} \right)}. \end{equation}

Equations (2.27) and (2.28) give the dependency in $Ra$ and $Pr$ of both $Re_0$ and $Nu_0$ numbers, assuming the five coefficients ($a$,$c_1$–$c_4$) are known. Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013) determined these coefficients from previous experimental measurements in the literature.

3.1. Extension of the classical regime given by Malkus

In the classical regime by Malkus, (2.2) yields

(3.4)\begin{equation} \frac{\delta_{T}}{h} = \left ( \frac{2Ra^*}{Ra} \right )^{1/3} = \frac{1}{2 Nu_0}. \end{equation}

Using (3.4), (3.3) becomes

(3.5)\begin{equation} \frac{Nu}{Nu_0} = \frac{1}{ 1- 2 \tilde{l} Nu_0 \left[1 - \exp \left(- \dfrac{1}{2 \tilde{l} Nu_0 } \right)\right ] }. \end{equation}

In (3.4) and (3.5), $Nu_0$ is the Nusselt number for a standard RB experiment in the classical regime but it also represents the limit of $Nu$ when $\tilde {l} = l/h \to 0$. Even if $Nu$ depends on both parameters $\tilde {l}$ and $Ra$, (3.5) shows that the Nusselt ratio $Nu / Nu_0$ is a function of a single variable that is the product of $\tilde {l}$ and $Nu_0$. This is the main result of the present theory and is tested against experimental results in § 4.

The limits of (3.5) when $\tilde {l} \to 0$ and $\tilde {l} Nu_0 \gg 1$ are given in table 1. It can be noted that, when the product of $\tilde {l}$ and $Nu_0$ increases from 0 to $\infty$, the $Ra$-dependent Nusselt number ($Nu$) increases from a power law of one third to one of two thirds, i.e. with an exponent greater than 1/2 which characterizes the ultimate regime for a standard RB experiment (2.7).

Table 1.

Limits when $\tilde {l} Nu_0 \ll 1$ and $\tilde {l} Nu_0 \gg 1$ of the Nusselt number for a radiatively heated convection experiment. For the ultimate regime by Kraichnan, $\alpha = \ln ( Nu / Nu_0 ) / (3 \ln Re_0 )$ and $C_0^{{\mathcal {U}}} = {Pr } / { ( Pe^* \ln Re_0 )^3 }$. For regimes IV and III by GL, we assume that $\tilde {l} \ll 1$ to have ${\mathcal {C}} \to 0$.

3.2. Extension of Kraichnan's ultimate regime

Unlike the classical regime for which the thickness of the boundary layers depends only on $Ra$ whatever the type of experiment considered (see (3.4)), (2.4) shows that, in the ultimate regime, $\delta _{T}$ depends on the velocity fluctuations in the bulk ($v^*$) and therefore on the thermal power injected into the bulk flow. Assuming as before that $v^* \sim U / \ln Re$ (Kraichnan Reference Kraichnan1962), (2.4) becomes for a modified RB experiment

(3.6)\begin{equation} \frac{\delta_{T}}{h} = \frac{Pe^*}{Pr} \frac{\ln (Re)}{ Re } = \frac{(\delta_{T})_0}{h} \frac{Re_0}{ \ln (Re_0) } \frac{\ln (Re)}{ Re }. \end{equation}

For a standard RB experiment, $(\delta _{T})_0$ is given by (2.1) and thus (3.6) becomes

(3.7)\begin{equation} \frac{\delta_{T}}{h} = \frac{1}{2 Nu_0} \frac{Re_0}{ Re } \left [ 1 + \frac{ \ln (Re / Re_0) }{\ln (Re_0) } \right]. \end{equation}

As assumed previously for standard RB experiments, the Richardson number in the bulk flow is taken of order 1 i.e. $Ri = g \alpha (\overline {w' T'}) h / U^3 = g \alpha \kappa Q h / (\lambda U^3 ) \sim 1$ yielding $Re^3 \sim {Ra Nu / Pr^{2}}$, similarly to (2.5). Therefore, at constant Rayleigh number, the ratio of the Reynolds numbers for standard and modified RB experiments is proportional to the one-third power law of the ratio of the Nusselt numbers

(3.8)\begin{equation} \frac{Re}{Re_0} = \left ( \frac{Nu}{Nu_0} \right )^{1/3}. \end{equation}

Furthermore, (3.8) is valid both for ultimate and classical regimes of convection. Using (3.7) and (3.8), (3.3) can be written as

(3.9)\begin{equation} {\mathcal{N}}^2 = \frac{1}{ 1 + \alpha - 2\tilde{l} Nu_0 {\mathcal{N}} \left [ 1 - \exp \left ( - \dfrac{1+\alpha}{2 \tilde{l} Nu_0 {\mathcal{N}}} \right ) \right ] } , \end{equation}

where ${\mathcal {N}} = (Nu / Nu_0 )^{1/3}$ and $\alpha = \ln {\mathcal {N}} / \ln Re_0$.

In the ultimate regime and similarly to the classical regime case, the ratio $Nu / Nu_0$ is a function of the product $\tilde {l} \times Nu_0$. However, $\alpha$ also depends on the Rayleigh number through the Reynolds number $Re_0$. When $\tilde {l} \to 0$, $\alpha \approx 0$ since on the one hand ${\mathcal {N}} \to 1$ and on the other Reynolds numbers $Re_0$ must be large enough to reach the ultimate regime. The limit of (3.9) when $\tilde {l} \to 0$ is then given in table 1. For large values of $\tilde {l}$, (3.9) can be solved numerically for each chosen couple ($Ra$,$\tilde {l}$) to obtain ${\mathcal {N}}$ and then $Nu$. At high $Nu_0$ or else at very high Rayleigh numbers, $Nu$ scales asymptotically as $Ra^2$ i.e. with an exponent 2 well above 1/2 (see table 1).

3.3. Extension of the GL theory

The balances of the turbulent kinetic energy and of the thermal variance give the following two exact relations (Shraiman & Siggia Reference Shraiman and Siggia1990; Grossmann & Lohse Reference Grossmann and Lohse2000):

(3.10)\begin{gather} \langle\epsilon_u\rangle = \frac{g \alpha}{h} \left [ \int_{0}^h \frac{\bar{\Phi}(z)}{\rho c_p} \,\mathrm{d}z - \frac{\lambda (\rm \Delta) T}{\rho c_p} \right ], \end{gather}

(3.11)\begin{gather}\langle \epsilon_T \rangle = \frac{1}{h} \int_{0}^h \bar{T}(z) \frac{q_v(z)}{\rho c_p} \,\mathrm{d}z + \frac{T_h \bar{\Phi}(0) - T_c \bar{\Phi}(h)}{\rho c_p h } . \end{gather}

Actually, for a standard RB experiment, the convective flow is driven by the thermal boundary conditions ($(\rm \Delta) T$ or $\bar {\Phi }(z=0)$) whereas for the modified RB experiment presented in figure 1, the volumetric power source controls the intensity of the convective flow. Besides, for the second case, the lower and upper plates are assumed to be perfectly insulated conducting to $\bar {\Phi }(0) = \bar {\Phi }(h) = 0$. In steady state, energy conservation yields the following relation between the heat flux and volumetric power source:

(3.12)\begin{equation} \frac{\bar{\Phi}(z)}{Q} = \begin{cases} 1-\exp \left(-\dfrac{z}{l} \right) \quad \text{for} \ z \leq h/2 , \\ 1-\exp \left( \dfrac{h-z}{l} \right) \quad \text{for} \ h/2 \leq z \leq h.\end{cases} \end{equation}

Using (3.12) and (1.2), and assuming $Nu \gg 1$, (3.10) becomes

(3.13)\begin{equation} \langle\epsilon_u\rangle =\frac{\nu^3}{h^4} \frac{Nu Ra}{Pr^{2}} (1 - {\mathcal{C}} ). \end{equation}

The corrective term ${\mathcal {C}} = 2\tilde {l} [1-\exp (-{1}/{2\tilde {l}} ) ]$ only depends on $\tilde {l}$ and varies as $2\tilde {l}$ when $\tilde {l} \to 0$. Thus, the expression giving the dissipation rate of kinetic energy averaged over the whole cell (3.13) is very similar to that obtained for a standard RB experiment (2.8).

As for the thermal dissipation rate, its average over the cell is related to the profile of the mean temperature (3.11). As the GL theory is based on Prandtl–Blasius–Pohlhausen laminar boundary layers (Grossmann & Lohse Reference Grossmann and Lohse2000), the mean temperature can be written as

(3.14)\begin{equation} 2 \frac{\bar{T}(z) - T_b}{(\rm \Delta) T} =\begin{cases} 1 - \Theta_P \left(\dfrac{z}{\delta_T} \right) \quad \text{for} \ z \leq h/2 , \\ \Theta_P \left(\dfrac{h- z}{\delta_T} \right) - 1 \quad \text{for} \ h/2 \leq z \leq h , \end{cases} \end{equation}

with $\Theta _P$ the Pohlhausen temperature profile which is assumed to be independent of the Prandtl number. In particular, $\Theta _P(0)=0$ and $\Theta _P(\eta )\to 1$ when $\eta \gg 1$. Using (3.14) and (1.1), (3.11) then becomes

(3.15)\begin{equation} \langle \epsilon_T \rangle = \kappa \left ( \frac{(\rm \Delta) T}{h} \right )^2 Nu \frac{\delta_T}{l} \int_{0}^{h/(2\delta_T)} [1-\Theta_P(\eta)] \exp \left ( -\frac{\delta_T}{l} \eta \right ) \mathrm{d} \eta . \end{equation}

Equation (3.15) shows that $\langle \epsilon _T \rangle$ depends both on $\tilde {l}=l/h$ and $\delta _T/h$. The hypothesis adopted in § 3.1 for extending the classical regime of Malkus is again adopted here (3.4); $\delta _T/h$ is assumed to be only controlled by the Rayleigh number so that $\delta _T/h = 1/ [2 Nu_0(Ra)]$, where $Nu_0$ is the Nusselt number for a standard RB experiment. Equation (3.15) becomes

(3.16)\begin{equation} \langle \epsilon_T \rangle = \kappa \left ( \frac{(\rm \Delta) T}{h} \right )^2 Nu \, {\mathcal{G}}(2 Nu_0 \tilde{l}), \end{equation}

with

(3.17)\begin{equation} {\mathcal{G}}(y) = \frac{1}{y} \int_{0}^{\infty} [1-\Theta_P(\eta)] \exp \left ( -\frac{\eta}{ y} \right ) \mathrm{d} \eta . \end{equation}

Then, the central idea of the GL theory is to split the dissipation rates into two contributions (see (2.10)–(2.13a,b)). Generalization of (2.15)–(2.17) gives

(3.18)\begin{gather} \langle{\epsilon_u}\rangle_{Bu} \sim \frac{U^2}{h/U} \left(1-\frac{\delta_u}{h} \right) \approx \frac{\nu^3}{h^4} Re^3, \end{gather}

(3.19)\begin{gather}\langle{\epsilon_T}\rangle_{Bu} \sim \frac{((\rm \Delta) T)^2}{h/U^{edge}} \left(1-\frac{\delta_T}{h} \right) \approx \kappa \left ( \frac{(\rm \Delta) T}{h} \right )^2 Re Pr\,f \left( \frac{2 a Nu_0}{\sqrt{Re}} \right), \end{gather}

(3.20)\begin{gather}\langle{\epsilon_u}\rangle_{BL} \sim \nu \left(\frac{U}{\delta_u} \right )^2 \frac{\delta_u}{h} = \frac{\nu^3}{h^4} \frac{Re^{5/2}}{2a}. \end{gather}

To obtain (3.19), the relevant velocity at the edge between the thermal BL and bulk is assumed to be expressed as $U^{edge} = U f(\delta _u / \delta _T)$, with the same function $f$ used for standard RB convection. Besides, we have

(3.21)\begin{equation} \frac{\delta_u}{\delta_T} = \frac{2 Nu_0 }{\sqrt{Re}}. \end{equation}

Combining (3.13) and (3.18), and using (2.5), we obtain the following first equation valid for both regimes II and IV:

(3.22)\begin{equation} \frac{Nu}{Nu_0} (1 - {\mathcal{C}} ) = \left ( \frac{Re}{Re_0} \right)^{3}\quad \text{(regimes II and IV)}. \end{equation}

Forgetting the corrective term ${\mathcal {C}}$ which must be small since $\tilde {l} \ll 1$, (3.22) is the same equation as the one obtained both for extending the classical and ultimate regimes of Malkus and Kraichnan (see (3.8)). On the contrary, for regimes I and III, (3.22) needs to be replaced by

(3.23)\begin{equation} \frac{Nu}{Nu_0} (1 - {\mathcal{C}} ) = \left ( \frac{Re}{Re_0} \right)^{5/2} \quad (\text{regimes I and III}). \end{equation}

The results of this theory is first presented for regime IV because most of the experimental and numerical results fall into this regime. In addition, unlike regime II, the extension of regime IV to internally heated convection does not require the introduction of any adjustable parameters.

3.3.1. Regime IV, $\langle {\epsilon _u} \rangle \sim \langle {\epsilon _u}\rangle _{Bu}$ and $\langle {\epsilon _T} \rangle \sim \langle {\epsilon _T}\rangle _{Bu}$

For regime IV, the thermal dissipation rate is dominated by its bulk contribution. Combining (3.16) and (3.19), and using (2.19), we obtain

(3.24)\begin{equation} \frac{Nu}{Nu_0} {\mathcal{G}} (2 Nu_0 \tilde{l}) = \frac{Re}{Re_0} \frac{f ( {2 a Nu_0}/ {\sqrt{Re}}) }{f ( {2 a Nu_0}/ {\sqrt{Re_0}}) } . \end{equation}

The system of equations (3.22) and (3.24) gives the dependency of both $Nu/Nu_0$ and $Re/Re_0$ as a function of the 3 control parameters: $Ra$, $Pr$ and $\tilde {l}=l/h$. For Prandtl numbers small or large enough, (3.24) can be simplified as follows:

(3.25a,b)\begin{equation}{\frac{Nu}{Nu_0} = } \begin{cases}(1 - {\mathcal{C}} )^{-{1}/{2}} \, [{\mathcal{G}}(2 Nu_0 \tilde{l}) ]^{-{3}/{2}} ,\quad \text{for} \ \delta_T \gg \delta_u\quad (\text{regime IV}_l) \\ (1 - {\mathcal{C}} ) \, [{\mathcal{G}} (2 Nu_0 \tilde{l}) ]^{{-}2} ,\qquad\quad \text{for} \ \delta_u \gg \delta_T\quad (\text{regime IV}_u). \end{cases} \end{equation}

Besides, the limits of (3.25a) and (3.25b) when $\tilde {l} Nu_0$ tends to 0 or $\infty$ can be obtained saying that ${\mathcal {G}}(y) \overset {y \to 0}{\approx } 1 - \Theta _P^{\prime }(0) y$ or ${\mathcal {G}}(y) \overset {y \to \infty }{\approx } \delta _{\Theta }^d / y$, where $\delta _{\Theta }^d = \int _0^{\infty } [1-\theta _P(\eta )]\, \textrm {d} \eta$. A summary of the corresponding results is given in table 1.

3.3.2. Regime II, $\langle {\epsilon _u} \rangle \sim \langle {\epsilon _u}\rangle _{Bu}$ and $\langle {\epsilon _T} \rangle \sim \langle {\epsilon _T}\rangle _{BL}$

Following the idea of Grossmann & Lohse (Reference Grossmann and Lohse2000), we consider the order of magnitude of the different terms of energy equation i.e. $u_x \partial _x T + u_z \partial _z T = \kappa \partial _{zz} T + {q_v}/{(\rho c_p)}$. It yields

(3.26)\begin{equation} \frac{U^{edge} (\rm \Delta) T}{h} \sim \frac{\kappa (\rm \Delta) T}{\delta_T^2} + A \frac{q_v(\delta_T)}{\rho c_p}. \end{equation}

Using (1.1), (1.2), (2.1), (2.21) and $U^{edge} = U f(\delta _u / \delta _T)$, (3.26) becomes

(3.27)\begin{equation} 1 + \tilde{A} \frac{Nu}{Nu_0} {\mathcal{H}} (2 Nu_0 \tilde{l}) = \frac{Re}{Re_0} \frac{f({2a Nu_0} / {\sqrt{Re}})}{ f({2a Nu_0} / {\sqrt{Re_0}} ) }, \end{equation}

with ${\mathcal {H}} (y) = ({1}/{y}) \exp ( -{1}/{y} )$, $0\leq {\mathcal {H}} (y) \leq \exp (-1) \approx 0.37$ and $\tilde {A}$ a numerical constant of the order of one to be determined experimentally.

Equations (3.22) and (3.27) give the dependency of both $Nu/Nu_0$ and $Re/Re_0$ as a function of $Ra$, $Pr$ and $\tilde {l}=l/h$. Contrary to the regime IV, $Re/Re_0$ and $Nu/Nu_0$ both tend towards 1 when $Nu_0 \tilde {l} \gg 1$ for regime II because ${\mathcal {H}}(y) \approx 1/y$ when $y\gg 1$. For the two limits $Nu_0 \tilde {l} \ll 1$ and $Nu_0 \tilde {l} \gg 1$, we obtain

(3.28)\begin{equation} \frac{Nu}{Nu_0} (1 - {\mathcal{C}} ) = 1+ \frac{\tilde{A} \, \theta_i }{ 1 - {\mathcal{C}} } {\mathcal{H}} (2 Nu_0 \tilde{l}), \end{equation}

with $\theta _i=3$ for $\delta _T \gg \delta _u$ (regime $\textrm {II}_l$) and $\theta _i=2$ for $\delta _u \gg \delta _T$ (regime $\textrm {II}_u$).

For any value of $Nu_0 \tilde {l}$, we show in appendix A that $Nu/Nu_0$ can be given with a very good approximation by

(3.29)\begin{equation} \frac{Nu}{Nu_0} (1 - {\mathcal{C}}) = \left [ {\mathcal{S}}_{\beta_0} \left(\frac{ \tilde{A} \, {\mathcal{H}} }{1 - {\mathcal{C}}} \right ) \right ]^3, \end{equation}

with ${\mathcal {S}}_{\beta }(x)$ the real and positive solution of the equation: $1+x {\mathcal {S}}_{\beta }^3 = {\mathcal {S}}_{\beta }^{1+ \beta /2}$, and

(3.30)\begin{equation} \beta_0 = \left[\frac{2a Nu_0}{\sqrt{Re_0}} f\left( \frac{2a Nu_0}{\sqrt{Re_0}} \right) \right]^{{-}n}. \end{equation}

3.3.4. Regime III, $\langle {\epsilon _u} \rangle \sim \langle {\epsilon _u}\rangle _{BL}$ and $\langle {\epsilon _T} \rangle \sim \langle {\epsilon _T}\rangle _{Bu}$

For regime III, (3.23) and (3.24) give $Nu/Nu_0$ and $Re/Re_0$ as a function of $Ra$, $Pr$ and $\tilde {l}=l/h$. For Prandtl numbers large enough, we obtain

(3.31)\begin{equation} \frac{Nu}{Nu_0} = (1 - {\mathcal{C}}_{u} )^{3/2} [{\mathcal{G}} (2 Nu_0 \tilde{l}) ]^{{-}5/2}\quad \text{for} \ \delta_u \gg \delta_T\quad (\text{regime III}_u). \end{equation}

The limits of (3.31) when $\tilde {l} \times Nu_0$ tends to 0 or $\infty$ are given in table 1.

3.3.5. Theory in the whole parameter $(Ra,Pr,\tilde {l})$ plane

Following the idea of Grossmann & Lohse (Reference Grossmann and Lohse2001) (see paragraph 2.3.3), at given $Ra$, $Pr$ and $\tilde {l}$, radiatively driven convection can be described as a mixture of these 4 regimes. By replacing the expressions of $\langle {\epsilon _u}\rangle _{Bu}$ (3.18) and $\langle {\epsilon _u}\rangle _{BL}$ (3.20) in the balance equation for the viscous dissipation rate (3.13), the first generalized equation can be written as

(3.32)\begin{equation} \frac{Ra Nu}{Pr^2} (1 - {\mathcal{C}} ) = c_1 \frac{Re^{5/2}}{2a} + c_2 Re^3. \end{equation}

Using (3.24) and (3.29), the second generalised equation becomes

(3.33)\begin{equation} Nu = \frac{{c}_3}{ 1 - {\mathcal{C}} } \left [ {\mathcal{S}}_{\beta} \left(\frac{ \tilde{A} \, {\mathcal{H}} }{1 - {\mathcal{C}}} \right ) \right ]^3 {\sqrt{Re_0 Pr\,f \left( \frac{2 a Nu_0}{\sqrt{Re_0}} \right ) }} + \frac{{c}_4}{{\mathcal{G}}} {Re Pr\,f \left( \frac{2 a Nu_0}{\sqrt{Re}} \right)}.\end{equation}

By combining, on the one hand, (2.27) and (3.32), and, on the other hand, (2.28) and (3.33), we obtain the two equations which give $Nu$ and $Re$ numbers as a function of the three parameters $Ra$, $Pr$ and $\tilde {l}$

(3.34)\begin{gather} \frac{Nu}{Nu_0} = \frac{1}{1 -{\mathcal{C}}} \frac{{c}_1 \dfrac{Re^{5/2}}{2a} + {c}_2 Re^3}{{c}_1 \dfrac{Re_0^{5/2}}{2a} + {c}_2 Re_0^3}, \end{gather}

(3.35)\begin{gather}\frac{Nu}{Nu_0} = \frac{ \dfrac{{c}_3}{ 1 - {\mathcal{C}} } \left [ {\mathcal{S}}_{\beta} \left(\dfrac{ \tilde{A} \, {\mathcal{H}} }{1 - {\mathcal{C}}} \right ) \right ]^3 {\sqrt{Re_0 Pr\,f \left( \dfrac{2 a Nu_0}{\sqrt{Re_0}} \right ) }} +\dfrac{{c}_4}{{\mathcal{G}}} {Re Pr\,f \left( \dfrac{2 a Nu_0}{\sqrt{Re}} \right)}} {c_3 \sqrt{Re_0 Pr\,f \left( \dfrac{2 a Nu_0}{\sqrt{Re_0}} \right)} + c_4 Re_0 Pr\,f \left( \dfrac{2 a Nu_0}{\sqrt{Re_0}} \right)}. \end{gather}

Figures 2($a$) and 2($b$) show the variations of the ratio $Nu/Nu_0$ against $Ra$ and $Pr$ for fixed values of $\tilde {l}$, while $Nu/Nu_0$ is plotted against $\tilde {l}$ in figure 2($c$) for $Pr=1$ and for fixed values of $Ra$ between $10^6$ and $10^{14}$. As observed previously for the extensions of the Malkus and Kraichnan theories, the use of the variable $Nu_0 \times \tilde {l}$ allows us to gather the various curves drawn in figure 2($c$) (see figure 2$d$). As underlined by Grossmann & Lohse (Reference Grossmann and Lohse2001) for RB convection, pure regime $\textrm {IV}_u$ is only reached for very high $Ra$ and $Pr$ numbers (blue upper solid line in figure 2$d$) while for moderate values of $Ra$ and $Pr$ numbers, radiatively heated convection is described by a mixing of the 4 regimes I–IV.

Figure 2.

Results for the extension of GL theory for radiatively heated convection using (3.34) and (3.35) with $\tilde {A}=0.3$. The coefficient $a$ and prefactors $c_1$–$c_4$ are given by Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013). ($a$) $Nu/Nu_0$ versus $Ra$ for $Pr=0.01$ (red dotted lines) and $Pr=1$ (blue dashed lines), and for $\tilde {l} = 10^{-3}$ (lowest line), $\tilde {l} = 10^{-2}$ (middle line) and $\tilde {l} = 0.1$ (highest line). ($b$) $Nu/Nu_0$ versus $Pr$ for $Ra = 10^8$ (red dotted lines) and $Ra = 10^{10}$ (blue dashed lines), and for $\tilde {l} = 0.01$ (lowest line), $\tilde {l} = 0.02$ (middle line) and $\tilde {l} = 0.05$ (highest line). ($c$) $Nu/Nu_0$ versus $\tilde {l}$ for $Pr=1$ and for $Ra = 10^6$ (lowest line), $10^8$, $10^{10}$, $10^{12}$ and $10^{14}$ (highest line). ($d$) Same results as ($c$) but using $\tilde {l} \times Nu_0$ as $x$-coordinate. Also shown: regime $\textrm {IV}_u$ ($Nu/Nu_0 = {\mathcal {G}}^{-2}$) (blue solid line), regime $\textrm {IV}_l$ ($Nu/Nu_0 = {\mathcal {G}}^{-3/2}$) (blue dashed line), regime $\textrm {II}_u$ (red solid line) and regime $\textrm {II}_l$ (red dashed line).