3.1. Validations

The literature data for the verification of the present profiles include those from Versteegh & Nieuwstadt (Reference Versteegh and Nieuwstadt1999), Ng et al. (Reference Ng, Ooi, Lohse and Chung2017) and Howland et al. (Reference Howland, Ng, Verzicco and Lohse2022), where the data set by Versteegh & Nieuwstadt (Reference Versteegh and Nieuwstadt1999) has been used widely as the benchmark in many studies (Hölling & Herwig Reference Hölling and Herwig2005; Pallares et al. Reference Pallares, Vernet, Ferre and Grau2010; Ng et al. Reference Ng, Chung and Ooi2013), and Ng et al. (Reference Ng, Ooi, Lohse and Chung2017) and Howland et al. (Reference Howland, Ng, Verzicco and Lohse2022) present the latest numerical data in channel VC. In figure 2, we compare the present profiles with the data collected from the literature. It is seen clearly that all the data show a linear relationship for $u_+$ ($\kern 0.06em y_+$) and $T_+$ ($\kern 0.06em y_+$) near the hot wall (figures 2a,b), consistent with the predictions in (2.8a,b). Meanwhile, with $(\mathcal {K} y_+)^{1/3}-\mathcal {K}_0^{1/3}$ and $(\mathcal {K} y_+)^{-1/3}-\mathcal {K}_0^{-1/3}$ being the horizontal axes, these data collapse into straight lines close to the channel centre (figures 2c,d), as predicted by the power profiles (2.14a,b). From figures 2(c,d), one could further obtain the unknown coefficients $c_1$ and $c_2$ in (2.14a,b) via linear fittings as

(3.1a,b)\begin{equation} c_1 \approx - 10.0 \quad {\rm and} \quad c_2 \approx - 3.8. \end{equation}

To illustrate the universality of the fitted $c_1$ and $c_2$, more data from Howland et al. (Reference Howland, Ng, Verzicco and Lohse2022) are employed for further comparisons (figure 3). It is worth emphasising that the data presented in figure 3 are produced at rather different ${Ra}$ from those in figure 2. Additionally, as a direct indicator of the turbulence level, the corresponding values of the Reynolds number ($Re$) are also indicated in figure 3, which are estimated from the scaling correlation developed by Howland et al. (Reference Howland, Ng, Verzicco and Lohse2022):

(3.2)\begin{equation} {Re}=\frac{u_* H}{\nu} \sim {Ra}^{0.362}\,{Pr}^{ - 0.446}. \end{equation}

It is clear in figure 3 that the applied range of the cubic velocity profile (2.9a) is expanded greatly compared with the linear one (2.8a), as expected. Moreover, the power mean profiles (2.14a,b), computed with the fitted values of $c_1$ and $c_2$ (3.1a,b), agree well with the literature data in the vicinity of the channel centre. At the same time, figure 3 also indicates that away from the channel centre, the applied range for the power mean profiles and the fitted coefficients diminishes gradually with decreasing Reynolds number, eventually becoming insignificant at very low $Re$. This behaviour is entirely expected and does not suggest any issues with the present profiles or fitted coefficients, since the power profiles arise from turbulent mixing, and naturally their applied range will diminish as the turbulence level (the value of ${Re}$) decreases. Similar behaviour has also been observed in wall shear flows, where the applied range for the well-known log velocity profile and the von Kármán constant (typically approximately 0.4) also diminishes with decreasing ${Re}$, eventually vanishing at very low ${Re}$, such as those indicative of a laminar state (Kundu, Cohen & Dowling Reference Kundu, Cohen and Dowling2012, p. 586). Nevertheless, it is widely recognised that the log velocity profile and its von Kármán constant are universal in wall shear flows. Likewise, the present power profiles and the corresponding fitted coefficients can also be deemed as universal in channel VC, as they describe effectively all available literature data, except for cases at very low ${Re}\leq 109.1$, as shown in figure 3 and in the next subsection.

Figure 2.

Comparisons between the present profiles (red lines) and the literature data (discrete symbols) (Versteegh & Nieuwstadt Reference Versteegh and Nieuwstadt1999; Ng et al. Reference Ng, Ooi, Lohse and Chung2017; Howland et al. Reference Howland, Ng, Verzicco and Lohse2022). (a,b) Mean velocity and temperature profiles near the wall (2.8a,b). (c,d) Mean velocity and temperature profiles near the channel centre (2.14a,b).

Figure 3.

Three-sublayer structure of the mean fields in channel VC. The orange and blue backgrounds denote the viscosity-dominated and the turbulence-dominated sublayers respectively, where the prediction relative error of the present mean profiles is less than $10\,\%$. The grey background represents the intermediate sublayer that connects smoothly the two aforementioned sublayers. Only the results in the left half of the channel are illustrated, and those in the right half can be obtained according to the antisymmetry of the mean fields. All the discrete data points (black squares) presented in this figure are from Howland et al. (Reference Howland, Ng, Verzicco and Lohse2022). Also note that the horizontal axis is in log scale.

In order to further quantify the effect of ${Re}$, we introduce a relative error ($E$) defined as

(3.3a,b)\begin{equation} {E_u}=\frac{|u_{{pre}}-u_{{rep}}|}{u_{{rep}}}\quad {\rm and}\quad {E_T}=\frac{|T_{{pre}}-T_{{rep}}|}{T_{{rep}}}, \end{equation}

where the subscripts ‘$pre$’ and ‘$rep$’ refer to the predicted mean profiles and the corresponding reported data in the literature, respectively. With this definition, we restrict the viscosity-dominated velocity and temperature sublayers to regions $E_u \leq 10\,\%$ and $E_T \leq 10\,\%$, when $u_{{pre}}$ and $T_{{pre}}$ are calculated from the viscous-form profiles (2.9a,b). The procedure for identifying the turbulence-dominated velocity and temperature sublayers is analogous, with the only difference being that $u_{{pre}}$ and $T_{{pre}}$ are calculated from the turbulent-form profiles (2.14a,b). In this way, the applied range of the viscous-form profiles and that of the turbulent-form profiles can be represented by the widths ($W$) of viscosity-dominated and turbulence-dominated sublayers, respectively. These widths are further normalised by the channel width ($H$), and reflect the space occupancy (${SO}$) of each sublayer:

(3.4a,b)\begin{equation} {SO_u}=\frac{2 W_u}{H}\quad {\rm and}\quad {SO_T}=\frac{2 W_T}{H}, \end{equation}

where the subscripts $u$ and $T$ denote the sublayers of the velocity BL and those of the temperature BL, respectively. The prefactor 2 is due to the fact that each sublayer appears not only in the left half of the channel, but also in the right half. Obviously, the space occupancy of each sublayer is a dimensionless quantity that measures the applied range of the profile within that sublayer.

The space occupancy of each sublayer shown in figure 3 is calculated, and presented as a function of the Reynolds number in figure 4. Clearly, it provides a confirmation that the space occupancy of the turbulence-dominated sublayer, representing the applied range of the power profiles, increases with increasing $Re$. Thus the narrow applied range of the power profiles observed at low $Re \sim 43.9$ (figures 3i,l) is an expected consequence.

Figure 4.

Space occupancy for the three sublayers of (a) mean velocity and (b) mean temperature. The data here are from figures 3 and 5. It is important to note that the space occupancy of each sublayer also represents the dimensionless applied range for the profile within that sublayer.

In figure 4, other results are suggested as well. First, as ${Re}$ increases to a sufficiently high value, the space occupancy of the turbulence-dominated sublayer will be approaching unity, meaning that this sublayer will eventually occupy the entire channel. In other words, channel VC will be entirely turbulent, being consistent with the expectation for the ultimate regime. Second, as the turbulence-dominated sublayer extends, the intermediate sublayer will be squeezed gradually until vanishing, indicating that this sublayer is an impermanent structure without fixed boundaries. Finally, although the space occupancies of the power profiles are visually limited close to the channel centre (e.g. figure 3h), they are much larger than they look since the horizontal axis is in log scale. In fact, following the criterion $E \leq 10\,\%$, it is found that when ${Re}>122$, the predicted region of the velocity power profile extends beyond $30\,\%$ of the channel, and that of the temperature power profile extends beyond $80\,\%$ of the channel, as illustrated in figure 4. This demonstrates the validity of the present mean profiles.

In addition, it is noticed that the temperature power profiles present a much wider applied range than the velocity power profiles in figure 4. This is actually a natural outcome, and can be understood in the following way. Although the power profiles for velocity and temperature both arise from turbulent mixing, the scalars transferred via turbulent mixing are different. Specifically, the velocity power profile results from the mixing of kinetic energy, thus its applied range depends on the turbulent viscosity $\nu _t$, which measures the mixing intensity of kinetic energy. By contrast, the temperature power profile originates from the mixing of heat, thus its applied range depends on the turbulent thermal diffusivity $\alpha _t$, which measures the mixing intensity of heat. Consequently, the ratio of applied range of the velocity power profile to the temperature power profile reads

(3.5)\begin{equation} \frac{\text{applied range of velocity power profile}}{\text{applied range of temperature power profile}}\propto \frac{\nu_t}{\alpha_t}= {Pr}_t, \end{equation}

where ${Pr}_t$ is the turbulent Prandtl number and has been investigated thoroughly by Jischa & Rieke (Reference Jischa and Rieke1979), who concluded that ${Pr}_t<1$ for ${Pr}\gtrsim 1$. Later, Hölling & Herwig (Reference Hölling and Herwig2005) found ${Pr}_t \approx 0.9$ numerically at ${Pr} =0.709$. Thus in the parameter space considered in this study ($100 \geq {Pr} \geq 0.709$), ${Pr}_t$ should be less than 1, indicating that the applied range of the velocity power profile should be lower than that of the temperature power profile (3.5). This explains why it is much more difficult for the space occupancy of the velocity turbulence-dominated sublayer to approach unity as ${Re}$ increases, compared with the temperature turbulence-dominated sublayer (figures 4a,b). Finally, it should be pointed out that different applied ranges for the velocity and temperature power profiles are not in contradiction with the derivations, where the mean velocity and temperature profiles are derived simultaneously at the channel centre ($\kern 0.06em y_+ \to \infty$) only (see (2.7a,b)). This requires that the upper bounds of these applied ranges must be overlapping at the channel centre, but does not require the lower bounds to be the same, indicating that the two applied ranges can be different.

3.2. Comparisons with the reported profiles in the classical regime

In the previous studies, considerable efforts have been devoted to determining the mean profiles, and a consensus has been reached in the near-wall region, where the mean profiles follow linear laws (2.8a,b). As stated before, the cubic velocity profile (2.9a) will reduce to the linear one (2.8a) at $y_+ \to 0$. However, the reported mean profiles in the region at a distance from the wall are rather debatable, which in the pioneer work of George & Capp (Reference George and Capp1979) reads

(3.6a)$$\begin{gather} \frac{\bar{u}}{U_l}=\phi_1\left(\frac{y}{\eta} \right)^{1/3}+A({Pr}), \end{gather}$$

(3.6b)$$\begin{gather}\frac{T_h-\bar{T}}{T_l}= - \phi_2\left(\frac{y}{\eta} \right)^{ - 1/3}-B({Pr}), \end{gather}$$

where $\phi _1$ and $\phi _2$ are dimensionless coefficients, and $A({Pr})$ and $B({Pr})$ are unknown functions of the Prandtl number. By incorporating $T_l=|F_0|^{3/4}(g \beta \alpha )^{-1/4}$, $U_l=( g\beta \,|F_0|\,\alpha )^{1/4}$ and $\eta =[ \alpha ^3/(g\beta \,|F_0|)]^{1/4}$, these profiles are transformed into

(3.7a)$$\begin{gather} u_+ = \phi_1\left(\mathcal{K}y_+ \right)^{1/3}+A({Pr})\,(\alpha g \beta T_*/u_*^3)^{1/4}, \end{gather}$$

(3.7b)$$\begin{gather}T_+ = -\phi_2\left(\mathcal{K}y_+ \right)^{ - 1/3}-B({Pr})\,(\alpha g \beta T_*/u_*^3)^{ - 1/4}. \end{gather}$$

These results were developed originally for plate VC, and they were applied to channel VC by subsequent studies (Versteegh & Nieuwstadt Reference Versteegh and Nieuwstadt1999; Ng et al. Reference Ng, Chung and Ooi2013). In these studies, the mean temperature profile has been well confirmed, and the dimensionless coefficients were fitted as $\phi _2=4.2$ and $B=-5.0$ when ${Pr}=0.709$ (Versteegh & Nieuwstadt Reference Versteegh and Nieuwstadt1999; Ng et al. Reference Ng, Chung and Ooi2013), while the mean velocity profile was found to be problematic, and no fitted values for $\phi _1$ and $A({Pr})$ were documented. As a result, Versteegh & Nieuwstadt (Reference Versteegh and Nieuwstadt1999) proposed a modified linear velocity profile

(3.8)\begin{equation} u_+ = y_+ - \frac{U_l}{u_*}\,\frac{9.7(y/\eta)^2}{2.8+y/\eta}, \end{equation}

and Shiri & George (Reference Shiri and George2008) proposed a log velocity profile

(3.9)\begin{equation} u_+ = \mathcal{P}\ln(y_+)+\mathcal{Q}({Pr}). \end{equation}

However, this log profile was not compared with any numerical or experimental data, and the fitted coefficients of $\mathcal {P}$ and $\mathcal {Q}({Pr})$ were not provided either. Then Hölling & Herwig (Reference Hölling and Herwig2005) proposed another set of mean profiles:

(3.10a)\begin{align} u_{{\times}}&=\frac{0.427\,{Pr}}{0.9}\,y_{{\times}} \left[ 0.427(\ln y_{{\times}}-2)+1.93-T_{{\times} 0}\right] \nonumber\\ &\quad +\left[0.49 \left(\frac{\partial u_{{\times}}}{\partial y_{{\times}}}\right)_w -2.27\right]\ln y_{{\times}}+1.28\left(\frac{\partial u_{{\times}}}{\partial y_{{\times}}}\right)_w+1.28, \end{align}

(3.10b)\begin{align} &T_{{\times}}=0.427\ln y_{{\times}}+1.93, \end{align}

where the dimensionless variables are

(3.11) \begin{equation} \left\{\begin{array}{l} y_{{\times}}=y_+({Pr}^3 \mathcal{K})^{1/4},\\ T_{{\times}}=T_+\left(\dfrac{\mathcal{K}}{Pr}\right)^{1/4},\\ T_{{\times} 0}=0.5\,{\rm \Delta} T \left(\dfrac{g\beta \alpha}{u_*^3T_*^3}\right)^{1/4},\\ u_{{\times}}=u_+ \left(\dfrac{{Pr}^5}{\mathcal{K}}\right)^{1/4},\\ \left(\dfrac{\partial u_{{\times}}}{\partial y_{{\times}}}\right)_w= \left(\dfrac{\partial u_{+}}{\partial y_{+}}\right)_w \left(\dfrac{Pr}{\mathcal{K}}\right)^{1/2}. \end{array}\right. \end{equation}

However, a subsequent study by Kiš & Herwig (Reference Kiš and Herwig2012) pointed out that the constant coefficient 0.427 in (3.10b) was problematic and should be replaced by a probability density function. As a result, the log profile was modified as (Kiš & Herwig Reference Kiš and Herwig2012)

(3.12)\begin{equation} T_{{\times}}=0.5+\ln(2y_{{\times}})+\mathcal{G}(y_\times)-\mathcal{G}(0.5), \end{equation}

with

(3.13)\begin{equation} \mathcal{G}(y_\times)=\frac{0.8}{0.877}\,\frac{\sqrt{\rm \pi}}{2}\,{\rm erf}[0.877(\ln y_{{\times}}-0.64)]. \end{equation}

These reported profiles are compared with the present profiles in figure 5. It is evident that the present mean profiles are the only ones that are able to interpret data from different literatures, thus demonstrating much better versatility compared with the reported ones. The improved versatility originates from the fact that the characteristic scales for velocity, temperature and length in the present profiles come naturally from the normalisation of the governing equations (2.4a,b), thus they possess a more sound mathematical basis than the previous characteristic scales obtained from various heuristic understandings of channel VC.

Figure 5.

Mean profiles from the present study and those reported in the previous studies (George & Capp Reference George and Capp1979; Versteegh & Nieuwstadt Reference Versteegh and Nieuwstadt1999; Hölling & Herwig Reference Hölling and Herwig2005; Kiš & Herwig Reference Kiš and Herwig2012) in channel VC. The discrete data are from (a,d) Ng et al. (Reference Ng, Ooi, Lohse and Chung2017) and (b,c,e, f) Howland et al. (Reference Howland, Ng, Verzicco and Lohse2022). The orange and blue backgrounds denote the viscosity-dominated and turbulence-dominated sublayers, respectively, where the prediction relative error of the present mean profiles is less than $10\,\%$. The grey background represents the intermediate sublayer that connects smoothly the two aforementioned sublayers. Only the results in the left half of the channel are illustrated, and those in the right half can be obtained according to the antisymmetry of the mean fields. Also note that the horizontal axis is in log scale.

Further, figures 5(a–c) show that the mean velocity profiles can be described fully by the cubic law (prior to the maximum velocity) and the present $1/3$ power law (subsequent to the maximum velocity), except the intermediate sublayer enclosing the velocity peak. Obviously, this sublayer cannot be described by the log law. In addition, although a log law (3.9) is proposed for the mean velocity by Shiri & George (Reference Shiri and George2008), this was not verified. Moreover, recent numerical studies by Ng et al. (Reference Ng, Ooi, Lohse and Chung2017) and Howland et al. (Reference Howland, Ng, Verzicco and Lohse2022) have also failed to observe the log law behaviour in the classical regime. Thus it is reasonable to conclude that in the classical regime of turbulence, the mean velocity profile is unlikely to be logarithmic in channel VC.

Similarly, figures 5(d–f) show that the mean temperature profiles can be described fully by the linear law (prior to the maximum velocity) and the present $-1/3$ power law (subsequent to the maximum velocity), except the intermediate sublayer. This sublayer will finally vanish at sufficiently high ${Re}$ (figure 4b), and obviously cannot be described by the log law. In addition, although the log law (3.10b) was proposed for the mean temperature by Hölling & Herwig (Reference Hölling and Herwig2005), it can be applied only to certain parameters (figure 5d). Moreover, even at the certain parameter (figure 5d), the log law (3.10b) exhibits only a narrower agreement range, compared with the present $-1/3$ power profile. Thus it is reasonable to conclude that in the classical regime of turbulence, the mean temperature profile in channel VC is unlikely to be logarithmic as well.

In summary, in the classical regime of channel VC, the mean velocity and temperature profiles should not conform to log laws, in the region before, around or after the velocity peak. As for the ultimate regime in channel VC, there are still no literature data available to confirm the form of mean profiles, since the corresponding Rayleigh number needs to be extremely high and exceeds the critical value ${Ra}_c=4.27\times 10^{11}\times {Pr}^{1.89}$ (Howland et al. Reference Howland, Ng, Verzicco and Lohse2022). This correlation provides ${Ra}_c=2.2\times 10^{11}$ at ${Pr}=0.709$, aligning with the expectation of Ng et al. (Reference Ng, Ooi, Lohse and Chung2017), in which it is suggested that the ultimate regime in channel VC should occur at ${Ra} \gtrsim 10^{11}$ for the same Prandtl number.

3.3. Mean profiles in the ultimate regime

In the ultimate regime of turbulence, the mean profiles are assumed to follow log laws in channel VC, by analogy with RBC systems (Ng et al. Reference Ng, Ooi, Lohse and Chung2017; Howland et al. Reference Howland, Ng, Verzicco and Lohse2022). Herein, we find that this assumption could encounter some challenges, and that the form of mean profiles in the ultimate regime may still follow power laws. Below, we present discussions regarding the form of mean profiles in the ultimate regime. It is important to emphasise that the discussions in this subsection are based merely on theoretical arguments, and there are not yet reported data or direct evidence to eventually judge the form of the mean profiles in the ultimate regime. Consequently, these envisioned power-law or log-law forms in the ultimate regime badly need further numerical or experimental confirmation in the future.

On the one hand, in fully developed turbulent channel VC, the turbulence statistics of the velocity and temperature would not vary along streamwise and spanwise directions, but depend on the normal coordinate only. Consequently, the equations of the mean flow will remain identical in both classical and ultimate regimes, resulting in the same laws for the mean profiles. In other words, once channel VC is in a fully developed turbulent state, the form of the mean profiles will not be altered by increasing ${Re}$. This implies that the linear, cubic and power laws derived in the classical regime may be still valid in the ultimate regime, and that increasing $Re$, i.e. the transition of classical–ultimate regimes, will not bring a new log profile. In fact, the ${Re}$-independent profile form has been confirmed in turbulent RBC systems with infinite horizontal extent (abbreviated as ‘channel RBC’ hereafter) in previous studies (Grossmann & Lohse Reference Grossmann and Lohse2012; Ahlers et al. Reference Ahlers, Bodenschatz and He2014). It is found that in channel RBC, the mean temperature profile always follows a linear law close to the wall and a log law at a distance from the wall, and the transition of classical–ultimate regimes does not introduce any new profile (Grossmann & Lohse Reference Grossmann and Lohse2012; Ahlers et al. Reference Ahlers, Bodenschatz and He2014). In addition, in turbulent wall shear flows, increasing ${Re}$ continuously will not bring any new form of profile either, and the well-known linear law near the wall and the log law away from the wall always hold.

On the other hand, in the ultimate regime, the velocity and temperature BLs are fully turbulent. This indicates that in this regime, both the viscosity-dominated and intermediate sublayers will be completely squeezed out by the turbulence-dominated sublayer. Thus if the log law is observed in the ultimate regime of channel VC, then it could appear only in the turbulence-dominated sublayer. In channel RBC, the log-law profile was indeed proposed in the turbulence-dominated sublayer (Grossmann & Lohse Reference Grossmann and Lohse2012; Ahlers et al. Reference Ahlers, Bodenschatz and He2014). Since the turbulence-dominated sublayer is the only possible region to host the log-law profile, channel VC and channel RBC must then share the same flow equations in this sublayer, to ensure the validity of the profile analogy. In this sublayer, the equations of the mean flow in channel VC are given in (2.7a,b). In channel RBC, the equations governing the mean flow are derived as follows.

Consider a channel RBC system, with the horizontal (streamwise) and vertical (wall-normal) directions denoted by $x$ and $y$, respectively, and the fluid pressure denoted by $p$. After the system reaches the fully developed turbulent state, the mean velocity and temperature vary only along the vertical direction, thus the corresponding governing equations reduce to

(3.14a)$$\begin{gather} 0=\nu\,\frac{\partial^2 \bar u}{\partial y^2}-\frac{1}{\rho}\,\frac{\partial \bar p}{\partial x}-\frac{\partial \overline{u^\prime v^\prime}}{\partial y}, \end{gather}$$

(3.14b)$$\begin{gather}0= - \frac{1}{\rho}\,\frac{\partial \bar p}{\partial y}+g\beta(\bar T-T_{{ref}}), \end{gather}$$

(3.14c)$$\begin{gather}0=\alpha\,\frac{\partial^2 \bar T}{\partial y^2}-\frac{\partial \overline{v^\prime T^\prime}}{\partial y}. \end{gather}$$

Differentiating (3.14b) with respect to $x$, the buoyancy term $g\beta (\bar T-T_{{ref}})$ disappears because the mean temperature field $\bar {T}$ depends only on $y$. Consequently, one gets

(3.15)\begin{equation} \frac{\partial^2 \bar{p}}{\partial x\,\partial y}=0, \end{equation}

which indicates that the streamwise pressure gradient $\partial \bar {p}/\partial x$ stays the same along the normal direction, thus it can be estimated at the outside of the velocity BL. Beyond the velocity BL, the viscous effect becomes less important, and the corresponding streamwise pressure gradient can be modelled by Bernoulli's equation (Shishkina, Horn & Wagner Reference Shishkina, Horn and Wagner2013) as

(3.16)\begin{equation} \frac{\partial \bar{p}}{\partial x}= - \frac{\partial \rho \bar{u}^2(y)/2}{\partial x}=0. \end{equation}

Then the equations of the mean flow (3.14a–c) can be simplified further into

(3.17a)$$\begin{gather} 0=\nu\,\frac{\partial^2 \bar u}{\partial y^2}-\frac{\partial \overline{u^\prime v^\prime}}{\partial y}, \end{gather}$$

(3.17b)$$\begin{gather}0=\alpha\,\frac{\partial^2 \bar T}{\partial y^2}-\frac{\partial \overline{v^\prime T^\prime}}{\partial y}. \end{gather}$$

Following the same procedure as the derivations (2.1)–(2.4) for channel VC, the dimensionless integration equations resulting from (3.17a,b) read

(3.18a)$$\begin{gather} \frac{\partial u_+}{\partial y_+}+k_1 y_+^2\,\frac{\partial u_+}{\partial y_+}\,\frac{\partial u_+}{\partial y_+}=1, \end{gather}$$

(3.18b)$$\begin{gather}\frac{\partial T_+}{\partial y_+}+k_2\,{Pr}\,y_+^2\,\frac{\partial u_+}{\partial y_+}\,\frac{\partial T_+}{\partial y_+}={Pr}. \end{gather}$$

These equations in the turbulence-dominated sublayer ($\kern 0.06em y_+ \to \infty$) reduce to

(3.19a)$$\begin{gather} k_1 y_+^2\,\frac{\partial u_+}{\partial y_+}\,\frac{\partial u_+}{\partial y_+}=1, \end{gather}$$

(3.19b)$$\begin{gather}k_2 y_+^2\,\frac{\partial u_+}{\partial y_+}\,\frac{\partial T_+}{\partial y_+}=1. \end{gather}$$

Obviously, in the turbulence-dominated sublayer, the equations of the mean flow in channel VC (2.7a,b) are different from those in channel RBC (3.19a,b). Thus it is unlikely that the log profiles, which appear in channel RBC, will also manifest in channel VC. By comparison, the difference between the power profiles in channel VC and the log profiles in channel RBC arises from the buoyancy effect. In channel VC, the buoyancy is parallel to the mean flow, thus the buoyancy term should be included in the streamwise momentum equation (2.7a). This then results in the mean gradients $\partial u_+/ \partial y_+ \sim y_+ ^{-2/3}$ and $\partial T_+ /\partial y_+ \sim y_+ ^{-4/3}$ (2.13a,b), further leading to power profiles. By contrast, in channel RBC, the buoyancy is perpendicular to the mean flow, thus the buoyancy term is absent from the streamwise momentum equation (3.19a). Accordingly, the mean gradients become $\partial u_+/ \partial y_+ \sim y_+ ^{-1}$ and $\partial T_+/ \partial y_+ \sim y_+ ^{-1}$, and eventually result in the log profiles, recovering the results derived from a single mean temperature equation (Grossmann & Lohse Reference Grossmann and Lohse2012; Ahlers et al. Reference Ahlers, Bodenschatz and He2014). In summary, in the turbulence-dominated sublayer where the log profile could be possible, although the mean temperature equations are the same in channel VC and channel RBC, (2.7b) and (3.19b), the streamwise momentum equations are quite different due to the buoyancy effect, (2.7a) and (3.19a). As a result, the log profiles in channel RBC may not be extended directly to channel VC in the ultimate regime by analogy.