2.1. Governing equations

We consider RBC with Oberbeck–Boussinesq approximation in spherical shells with inner radius $r_{i}$ and outer radius $r_{o}$. The temperatures are fixed as $T_i$ and $T_o$ at the inner and the outer boundary, respectively. The mechanical boundary conditions are no-slip at both boundaries. The dimensionless equations were adopted by using the shell gap $d=r_{o}-r_{i}$, the viscous dissipation time scale $d^2/\nu$, the temperature difference between the inner and outer boundaries $\Delta T=T_i-T_o$, the momentum diffusive velocity $\nu /d$, and the characteristic pressure $\rho \nu ^2 /d^2$ as the reference scales. Gravity is non-dimensionalized using its reference value at the outer boundary $g_o=g(r_o)$. The dimensionless governing equations for the problem read

(2.1)$$\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} =0, \end{gather}$$

(2.2)$$\begin{gather}\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} ={-}\boldsymbol{\nabla} p + \frac{Ra}{Pr}gT\boldsymbol{e_r}+\nabla^2 \boldsymbol{u}, \end{gather}$$

(2.3)$$\begin{gather}\frac{\partial T}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} T = \frac{1}{Pr} \nabla^2 T. \end{gather}$$

Here, the symbols $\boldsymbol {u}$, $p$ and $T$ represent velocity, pressure and temperature, respectively. Additionally, $\boldsymbol {e_r}$ represents the unit vector in the radial direction. In our study, we focus on spherical shell convection under the influence of a centrally condensed mass with the gravity profile, $g(r) = (r_{o}/r)^2$ (Chandrasekhar Reference Chandrasekhar1981). By adopting this gravity profile, exact relations connecting dissipation rates with driving forces can be established (see § 2.3), enabling us to conduct scaling analysis effectively.

The dimensionless equations (2.1)–(2.3) are controlled by three input parameters – Rayleigh number, Prandtl number and the radius ratio, which are defined as

(2.4a–c)\begin{equation} Ra=\frac{\alpha g_o \Delta T d^3}{\nu \kappa}, \quad Pr=\frac{\nu}{\kappa}, \quad \eta=\frac{r_i}{r_o}, \end{equation}

respectively. Here, $\alpha$ is the thermal expansion coefficient, and $\nu$ and $\kappa$ are the viscous and thermal diffusivities, respectively.

2.2. Response parameters

In RB, there are two key response parameters in the system. One of them is the Nusselt number, $Nu$, which represents the dimensionless heat flux. Within the Oberbeck–Boussinesq approximation, one obtains

(2.5) \begin{equation} Nu = \frac{\overline{\langle u_rT \rangle_s}-\dfrac{1}{Pr}\dfrac{{\rm d} \vartheta}{{\rm d} r}}{-\dfrac{1}{Pr} \dfrac{{\rm d} T_c}{{\rm d} r}} ={-}\eta \frac{{\rm d} \vartheta}{{\rm d} r} \Bigg|_{r=r_i} ={-}\frac{1}{\eta} \frac{{\rm d} \vartheta}{{\rm d} r} \Bigg|_{r=r_o},\end{equation}

where $\overline {({\cdot })}$ represents the time average, and $\langle {\cdot } \rangle _s$ and $\langle {\cdot } \rangle$ represent the spatial average over the horizontal surface and the whole volume of the spherical shell, respectively. The time and horizontally averaged radial temperature profile is $\vartheta =\overline {\langle T \rangle _s}$, and $T_c$ is the conductive temperature profile which, for spherical shells with fixed thermal boundary condition, reads

(2.6)\begin{equation} T_c(r) = \frac{\eta}{(1-\eta)^2} \frac{1}{r} - \frac{\eta}{1-\eta}. \end{equation}

Another key response parameter is the Reynolds number,

(2.7)\begin{equation} Re = \overline{\sqrt{\langle u^2 \rangle}} = \overline{\sqrt{\langle u_r^2+u_{\theta}^2+u_{\phi}^2 \rangle}},\end{equation}

which represents the dimensionless velocity. The radial profile of the time and horizontally averaged horizontal velocity is

(2.8)\begin{equation} Re_h(r)=\overline{\sqrt{\langle u_{\theta}^2+u_{\phi}^2 \rangle}_s}. \end{equation}

2.3. Exact dissipation relations in spherical shells

There are two exact relations for the time- and volume-averaged kinetic energy dissipation rate $\epsilon _u$ and the thermal energy dissipation rate $\epsilon _{\vartheta }$. In spherical shells with a centrally condensed mass gravity profile $g(r) = (r_{o}/r)^2$, the exact relations can be written as (Gastine et al. Reference Gastine, Wicht and Aurnou2015)

(2.9)$$\begin{gather} \epsilon_u = \frac{3}{1+\eta+\eta^2} \frac{Ra}{Pr^2} (Nu-1), \end{gather}$$

(2.10)$$\begin{gather}\epsilon_{\vartheta} = \frac{3\eta}{1+\eta+\eta^2} Nu. \end{gather}$$

In our dimensionless quantities, $\epsilon _u$ and $\epsilon _{\vartheta }$ are calculated as follows:

(2.11a,b) \begin{equation} \epsilon_u = \overline{\biggl\langle \left(\frac{\partial u_i}{\partial x_j} \right)^2 \biggr\rangle}, \quad \epsilon_{\vartheta} = \overline{\langle (\boldsymbol{\nabla} T )^2 \rangle}.\end{equation}

Using the relations (2.9), (2.10) and (2.11a,b), we can express the Nusselt numbers based on the kinetic and thermal energy dissipation rates as

(2.12) \begin{equation} \left. \begin{array}{@{}c@{}} \displaystyle Nu_{\epsilon_u} = \dfrac{1+\eta+\eta^2}{3} \dfrac{Pr^2}{Ra} \epsilon_u +1,\\ \displaystyle Nu_{\epsilon_{\vartheta}} = \dfrac{1+\eta+\eta^2}{3\eta} \epsilon_{\vartheta}. \end{array}\right\}\end{equation}

The relative error between the estimates of the Nusselt number obtained from (2.5) and (2.12) is used as a measure of the resolution in our simulations. For more details, please refer to Appendix A.

2.4. Numerical settings and simulation parameters

In this work, we have used the magnetohydrodynamic code MagIC (Wicht Reference Wicht2002; Christensen & Wicht Reference Christensen and Wicht2007; Lago et al. Reference Lago, Gastine, Dannert, Rampp and Wicht2021) to solve (2.1)–(2.3). MagIC is a pseudo-spectral code in which all the unknown variables are expanded into complete sets of functions in radial and horizontal directions. Chebyshev polynomials are applied in the radial direction while spherical harmonic functions are used in the azimuthal and latitudinal directions. Since the velocity field is solenoidal under the Oberbeck–Boussinesq approximation, MagIC decomposes it into poloidal and toroidal components,

(2.13)\begin{equation} \boldsymbol{u} = \boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times W \boldsymbol{e_r}) + \boldsymbol{\nabla} \times Z \boldsymbol{e_r}. \end{equation}

The velocity field $\boldsymbol {u}$, which has three components, can thus be replaced by two scalar fields which are the poloidal potential $W$ and the toroidal potential $Z$. The equations are time-stepped by advancing the nonlinear terms using an explicit second-order Adams–Bashforth scheme, while the linear terms are time-advanced using an implicit Crank–Nicolson algorithm. At each time step, the linear terms are calculated in the spectral space while the nonlinear terms are calculated in the physical space.

We conducted six sets of simulations with $\eta = 0.2$, $0.3$, $0.4$, $0.5$, $0.6$, $0.8$ and $Pr=1$. For $\eta =0.2, 0.3$ and $0.4$ simulations, $Ra$ is varied in the range $3 \times 10^3 \leq Ra \leq 5 \times 10^8$; for the $\eta =0.5$ and $0.8$ cases, we have $3 \times 10^{5} \leq Ra \leq 5 \times 10^8$. The simulations for the $\eta =0.6$ case are the same as those of Gastine et al. (Reference Gastine, Wicht and Aurnou2015). For more details about the simulation parameters, grid resolution and the balance of the turbulent kinetic energy budget, please refer to Appendix A.

4.1. Asymmetry of temperature and velocity profiles

RBC in spherical shell geometry is significantly different from its planar counterpart, chiefly due to the curvature of the bounding surfaces. The presence of curvature in spherical RBC leads to asymmetry of the radial profiles of temperature and velocity fluctuations. Owing to the thermal shortcut (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009), the temperature drop mostly happens inside the thermal BLs. As a result, the properties of BLs are essential, especially in the quantification of the scaling relations of the global response parameters. In this work, we use the slope method (Shishkina et al. Reference Shishkina, Stevens, Grossmann and Lohse2010) to define the kinetic as well as the thermal BL thicknesses for consistency with the BL analyses of Prandtl (Reference Prandtl1905) and Blasius (Reference Blasius1907). The definitions of the boundary layer thicknesses are the same as those of Gastine et al. (Reference Gastine, Wicht and Aurnou2015).

Time and horizontally averaged radial temperature profiles $\vartheta (r)$ exhibit a strong asymmetry in spherical shell convection. Figure 12 shows $\vartheta (r)$ and $Re(r)$ at different radius ratios. At small radius ratios, the differences in curvature and gravity between the inner and outer boundaries are significantly larger. This results in a stronger asymmetry in both the temperature as well as the velocity profiles. For example, as illustrated in figure 12(a), the inner BL temperature drop is approximately $57\,\%$ of the total temperature drop at $\eta =0.8, Ra=7 \times 10^{7}$. However, at $\eta =0.2, Ra=7 \times 10^{7}$, the inner BL temperature drop constitutes approximately $90\,\%$ of the total temperature drop, implying a stronger asymmetry in the temperature field. For the velocity field, represented by $Re(r)$, the profile is almost symmetrical when $\eta =0.8$. As $\eta$ becomes smaller, the difference between $Re(r)$ in the vicinity of the inner and the outer boundaries becomes larger. At $\eta =0.2$, the peak value of $Re(r)$ near the inner boundary is approximately twice that of the value near the outer boundary, as shown in figure 12(b). The asymmetry of the velocity field also illustrates that the flow is more turbulent in the inner half-shell than in the outer half-shell.

Figure 12. Radial profiles of time and horizontally averaged (a) temperature $\vartheta (r)$ and (b) Reynolds number $Re(r)$, at different $\eta$. All the simulations presented here are at $Ra=7 \times 10^7$ and $Pr=1$.

For the temperature field, the asymmetry is quantified by the temperature drop $\Delta \vartheta _{i}$ ($\Delta \vartheta _{o}$) and the thermal BL thickness $\lambda _{\vartheta }^{i}$ ($\lambda _{\vartheta }^{o}$) at the inner (outer) BL. We assume that the temperature drop occurs only inside the BLs, which leads to

(4.1)\begin{equation} \Delta \vartheta_{i} + \Delta \vartheta_{o} = 1,\end{equation}

so that we have

(4.2a,b)\begin{equation} \Delta \vartheta_{i} = 1 - \vartheta_{m}, \quad \Delta \vartheta_{o} = \vartheta_{m}, \end{equation}

where $\vartheta _{m}=\vartheta ((r_i+r_o)/2)$ is the time and horizontally averaged temperature at the mid-depth. Gastine et al. (Reference Gastine, Wicht and Aurnou2015) gives an analytical expression for $\Delta \vartheta _{i}, \Delta \vartheta _{o}$ and $\lambda _{\vartheta }^{i}/\lambda _{\vartheta }^{o}$. For the present paper to be self-contained, an outline of the scaling arguments used by Gastine et al. (Reference Gastine, Wicht and Aurnou2015) is presented below. From (2.5), we obtain

(4.3)\begin{equation} \frac{\Delta \vartheta_{i}}{\lambda_{\vartheta}^{i}} = \frac{1}{\eta^2} \frac{\Delta \vartheta_{o}}{\lambda_{\vartheta}^{o}}.\end{equation}

Gastine et al. (Reference Gastine, Wicht and Aurnou2015) proposed that the average plume density in the inner and outer BLs should be the same, which yields

(4.4)\begin{equation} \rho_{p}^{i} = \rho_{p}^{o} \rightarrow \frac{\alpha g_{i} \Delta \vartheta_{i} {\lambda_{\vartheta}^{i}}^{5}}{\nu \kappa} = \frac{\alpha g_{o} \Delta \vartheta_{o} {\lambda_{\vartheta}^{o}}^{5}}{\nu \kappa},\end{equation}

where $\rho _{p}^{i}$ ($\rho _{p}^{o}$) is the average plume density in the inner (outer) boundary, $g_{i}(g_{o})$ being the corresponding values of gravitational acceleration. Combining (4.1)–(4.3) and (4.4), Gastine et al. (Reference Gastine, Wicht and Aurnou2015) finally derive

(4.5a–c)\begin{equation} \Delta \vartheta_{i} = \frac{1}{1+\eta^{5/3}\chi_{g}^{1/6}}, \quad \Delta \vartheta_{o} = \vartheta_{m} = \frac{\eta^{5/3} \chi_{g}^{1/6}}{1+\eta^{5/3} \chi_{g}^{1/6}}, \quad \frac{\lambda_{\vartheta}^{i}}{\lambda_{\vartheta}^{o}} = \frac{\eta^{1/3}}{\chi_{g}^{1/6}}, \end{equation}

where $\chi _{g}=g(r_i)/g(r_o)$ denotes the ratio of the gravitational acceleration between the inner and the outer boundary. For the $g(r)$ profile considered in the present work, $\chi _{g}=1/\eta ^{2}$. Hence, (4.5a–c) can be written as

(4.6a–c)\begin{equation} \Delta \vartheta_{i} = \frac{1}{1+\eta^{4/3}}, \quad \Delta \vartheta_{o} = \vartheta_{m} = \frac{\eta^{4/3}}{1+\eta^{4/3}}, \quad \frac{\lambda_{\vartheta}^{i}}{\lambda_{\vartheta}^{o}} = \eta^{2/3}. \end{equation}

However, there are some other models for $\Delta \vartheta _{i}, \Delta \vartheta _{o}$ and $\lambda _{\vartheta }^{i}/\lambda _{\vartheta }$ (e.g. Wu & Libchaber Reference Wu and Libchaber1991; Zhang, Childress & Libchaber Reference Zhang, Childress and Libchaber1997; Wang et al. Reference Wang, Jiang, Liu, Zhu and Sun2022). Gastine et al. (Reference Gastine, Wicht and Aurnou2015) shows that the model specified in (4.5a–c) is better than the others when $Pr=1$. Thus, we employ (4.5a–c) and (4.6a–c) in our work. Defining the asymmetry factor $\chi$ by employing the temperature drop at the inner BL,

(4.7)\begin{equation} \chi = \frac{2 \Delta \vartheta_{i}}{\Delta \vartheta} = \frac{2}{1+\eta^{4/3}},\end{equation}

where $\Delta \vartheta$ is the total temperature drop across the spherical shell. With $Pr=1$, the ratio of the inner and the outer kinetic BL thickness should be the same as that for thermal BL thicknesses (Shishkina et al. Reference Shishkina, Stevens, Grossmann and Lohse2010), such that

(4.8)\begin{equation} \frac{\lambda_{u}^{i}}{\lambda_{u}^{o}} = \frac{\lambda_{\vartheta}^{i}}{\lambda_{\vartheta}^{o}} = \eta^{2/3}. \end{equation}

In figure 13, (4.7) and (4.8) are successfully validated against our DNS data.

Figure 13. (a) Asymmetry factor $\chi$ and (b) ratio of the inner and outer BL thicknesses, with respect to $\eta$. Symbols represent DNS data. Dashed curves represent analytical predictions given by (4.7) and (4.8).

From (4.6a–c) and the $Nu$ definition in (2.5), we can directly obtain the following relations:

(4.9a,b)\begin{equation} \lambda_{\vartheta}^{i} = \frac{\eta}{1+\eta^{4/3}} \frac{1}{Nu}, \quad \lambda_{\vartheta}^{o} = \frac{\eta^{1/3}}{1+\eta^{4/3}} \frac{1}{Nu}. \end{equation}

Thus, we can define a normalized thermal boundary layer thickness

(4.10)\begin{equation} \tilde{\lambda}_{\vartheta} = \frac{1+\eta^{4/3}}{\eta} \lambda_{\vartheta}^{i} = \frac{1+\eta^{4/3}}{\eta^{1/3}} \lambda_{\vartheta}^{o} = \frac{1}{Nu}.\end{equation}

Figure 14 shows a validation of (4.10) and it shows that the relation perfectly aligns with our DNS data.

Figure 14. Normalized thermal boundary layer thickness $\tilde {\lambda }_{\vartheta }$ as a function of $Nu$ at different $\eta$. Symbols represent DNS data, while the dashed black line denotes $\tilde {\lambda }_{\vartheta } = 1/Nu$.

4.2. Radius ratio dependence of Nusselt number

In spherical RBC, the geometrical asymmetry present in the system is expected to have an effect on the scaling law of the dimensionless heat transport. Figure 15(a) shows $Nu$ as a function of $Ra$ at different $\eta$. In the considered $Ra$ range, the smaller $\eta$ cases have a smaller value of $Nu$ at a fixed $Ra$. As $\eta \rightarrow 1$, $Nu$ scaling will asymptotically approach the planar case with an infinite aspect ratio and the differences in the exponent become smaller at larger $\eta$; for $\eta \geq 0.4$, the differences in $Nu$ at different $\eta$ are vanishingly small and almost indistinguishable in the plot. Moreover, $Ra$ dependence of $Nu$ also varies with $\eta$. For example, the data fit for $Nu$ at $\eta =0.2$ is $Nu |_{\eta =0.2}=0.101 Ra^{0.308}$, while as for the $\eta =0.8$ case, it reads $Nu |_{\eta =0.8}=0.156 Ra^{0.291}$. Taking into consideration the effect of $\eta$ on $Nu$ as well as the $\eta$-dependence of the scaling exponent with respect to $Ra$, a power law of the form $Nu=f(\eta ) Ra^{\gamma (\eta,Ra)}$ is deemed appropriate (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009).

Figure 15. (a) Nusselt number, $Nu$, and (b) the compensated Nusselt number, $Nu (1+\eta ^{4/3})/(\eta ^{1/2})$, as functions of $Ra$ at different $\eta$. Symbols represent DNS data and the dashed black line corresponds to the equation given by (4.14).

Now we come to the scaling for $Nu$ with radius ratio dependence. It is well known that the width of the BL is related to $Nu$. Equation (2.5) can be written as

(4.11)\begin{equation} Nu ={-}\eta \frac{{\rm d} \vartheta}{{\rm d} r} |_{r=r_i} \approx \eta \frac{\Delta \vartheta_{i}}{\lambda_{\vartheta}^{i}}.\end{equation}

Equation (4.6a–c) gives us the $\eta$-dependence of the temperature drop in the inner BL (Gastine et al. Reference Gastine, Wicht and Aurnou2015), while the radius ratio dependence of $\lambda _{\vartheta }^{i}$ remains to be found.

Within the BL, the viscous term balances the advection term in (2.2), which leads to

(4.12)\begin{equation} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} \sim \nu \nabla^2 \boldsymbol{u} \Rightarrow \frac{U}{L} \sim \nu \frac{1}{\delta^2},\end{equation}

where $U$ is the velocity of the large-scale circulation, $L$ is the associated length scale and $\delta$ represents the length scale relevant inside the BL. Inside the inner BL, if we assume $L=L_{i} \sim (r_{i}/r_{o}) d$ and $\delta = \tilde {\lambda }_{u}^{i} \approx \tilde {\lambda }_{\vartheta }^{i}$, we have

(4.13)\begin{equation} \tilde{\lambda}_{\vartheta}^{i} \sim \left( \frac{\nu d}{U} \right)^{1/2} \eta^{1/2} \Rightarrow \lambda_{\vartheta}^{i} = \frac{\tilde{\lambda}_{\vartheta}^{i}}{{\rm d}} \sim Re^{1/2} \eta^{1/2},\end{equation}

where $\lambda _{u}^{i}, \lambda _{\vartheta }^{i}$ are the normalized inner viscous and thermal BL thicknesses, respectively; whereas $\tilde {\lambda }_{u}^{i}, \tilde {\lambda }_{\vartheta }^{i}$ represent their dimensional counterparts. The shell thickness $d=r_o-r_i$ is used to non-dimensionalize $\tilde {\lambda }_{u}^{i}$ and $\tilde {\lambda }_{\vartheta }^{i}$. Here, $Re=Ud/\nu$ is the global Reynolds number as defined in (2.7). In our simulations, it is observed that $Re$ has only a weak dependence on $\eta$, as shown in figure 16. Hence, it is reasonable to regard $Re$ as being independent of the radius ratio. We will investigate the radius ratio dependence of $Re$ more precisely by applying the GL theory (Grossmann & Lohse Reference Grossmann and Lohse2000) in subsequent sections.

Figure 16. Reynolds number $Re$ as a function of the Rayleigh number $Ra$ at different $\eta$. Symbols represent DNS data, while the dashed black line denotes $Re=0.34Ra^{0.48}$.

Figure 17 shows that the DNS data collapse well with $\lambda _{\vartheta }^{i} \eta ^{-1/2} = 3.3 Ra^{-0.295}$, which validates the result that $\lambda _{\vartheta }^{i}$ is proportional to $\eta ^{1/2}$. Finally, using the $\eta$-dependence of $\Delta \vartheta _{i}$ and $\lambda _{\vartheta }^{i}$ from (4.6a–c) and (4.13), respectively, in (4.11), we get the following scaling relation for the Nusselt number:

(4.14)\begin{equation} Nu \approx \eta \frac{\Delta \vartheta_{i}}{\lambda_{\vartheta}^{i}} \sim \frac{\eta^{1/2}}{1+\eta^{4/3}} Ra^{\gamma}.\end{equation}

Here, $\gamma \approx 0.295$ if we fit a single scaling exponent in the whole data range. A similar exponent has also been reported by Brown et al. (Reference Brown, Nikolaenko, Funfschilling and Ahlers2005) and Funfschilling et al. (Reference Funfschilling, Brown, Nikolaenko and Ahlers2005) in the cylindrical geometries. Figure 15 also shows that (4.14) exhibits a good agreement with our DNS data.

Figure 17. (a) Inner thermal BL thickness $\lambda _{\vartheta }^{i}$ and (b) compensated inner BL thickness $\lambda _{\vartheta }^{i} \eta ^{-1/2}$, as functions of $Ra$ at different $\eta$. Symbols represent DNS data, while the dashed black line represents the fit $\lambda _{\vartheta }^{i} \eta ^{-1/2} = 3.3 Ra^{-0.295}$, which follows from the scaling argument (4.13).

4.3. Dissipation analysis

It must be stressed that in (4.14), although $\gamma \approx 0.295$ agrees well with the DNS data if fitting with one single scaling exponent in the whole $Ra$ range, the local effective heat transfer scaling exponent varies with $Ra$ (Grossmann & Lohse Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001). To investigate $\eta$ and $Ra$ dependence of $Nu$ and $Re$ in more detail, we apply the GL theory to our dataset. There are basically two key assumptions in GL theory (Grossmann & Lohse Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). One is that there exists a large-scale circulation (LSC) with only one typical velocity scale $U$. Another is that the kinetic BLs are scaling-wise characterized by a single effective thickness $\lambda _{u}$ regardless of the positions along the boundaries. For the first assumption, we observe that there exist large-scale structures in our simulations, as demonstrated in § 3. These large-scale structures can be regarded as indicative of the LSC. Moreover, we use the time- and volume-averaged velocity (2.7) as the typical velocity scale $U$, and use the $Re$ based on this velocity scale to derive the scaling relations for $\epsilon _{u}^{bulk}(Re), \epsilon _{u}^{BL}(Re), \epsilon _{\vartheta }^{bulk}(Re)$ and $\epsilon _{\vartheta }^{BL}(Re)$. These scaling relations are validated by our DNS data, as will be shown in this section. For the assumption of kinetic BLs, Gastine et al. (Reference Gastine, Wicht and Aurnou2015) have already shown that the radial profile of the horizontal velocity looks very similar to the Prandtl–Blasius solution. We also validated that in our simulations. Hence, it would be reasonable to state that in the parameter space explored in our simulations, these two key assumptions of the GL theory are also valid in spherical shell RBC.

The basic idea of GL theory is to separate the whole flow into two parts, i.e. the laminar part and the turbulent part. The laminar part resides in the vicinity of the boundaries while the turbulent part concentrates in the bulk region. Following this idea, the kinetic and thermal energy dissipation rates can be separated into two contributions, which are the BL contribution and the bulk contribution. Therefore, the two dissipation rates can be expressed as

(4.15a,b)\begin{equation} \epsilon_u = \epsilon_{u}^{BL} + \epsilon_{u}^{bulk}, \quad \epsilon_{\vartheta} = \epsilon_{\vartheta}^{BL} + \epsilon_{\vartheta}^{bulk}.\end{equation}

Here, we use the subscripts $BL$ and $bulk$ for the BL and bulk contributions, respectively. The bulk and BL contributions are defined as follows:

(4.16a,b) $$\begin{gather} \epsilon_{u}^{bulk} = \frac{4 {\rm \pi}}{V} \int_{r_i+\lambda_{u}^{i}}^{r_o-\lambda_{u}^{o}} \overline{\biggl\langle \left(\frac{\partial u_i}{\partial x_j} \right)^2 \biggr\rangle}_{s} r^{2}\,\mathrm{d}r, \quad \epsilon_{\vartheta}^{bulk} = \frac{4 {\rm \pi}}{V} \int_{r_i+\lambda_{\vartheta}^{i}}^{r_o-\lambda_{\vartheta}^{o}} \overline{\langle (\Delta T )^2 \rangle}_{s} r^{2}\,\mathrm{d}r, \end{gather}$$

(4.17) $$\begin{gather} \left. \begin{array}{@{}l@{}l@{}} \displaystyle \epsilon_{u}^{BL} = \dfrac{4 {\rm \pi}}{V} \int_{r_i}^{r_i+\lambda_{u}^{i}} \overline{\biggl\langle \left(\dfrac{\partial u_i}{\partial x_j} \right)^2 \biggr\rangle}_{s} r^{2}\,\mathrm{d}r + \dfrac{4 {\rm \pi}}{V} \int_{r_o-\lambda_{u}^{o}}^{r_o} \overline{\biggl\langle \left(\dfrac{\partial u_i}{\partial x_j} \right)^2 \biggr\rangle}_{s} r^{2}\,\mathrm{d}r, \\ \displaystyle \epsilon_{\vartheta}^{BL} = \dfrac{4 {\rm \pi}}{V} \int_{r_i}^{r_i+\lambda_{\vartheta}^{i}} \overline{\langle (\Delta T )^2 \rangle}_{s} r^{2}\,\mathrm{d}r + \dfrac{4 {\rm \pi}}{V} \int_{r_o-\lambda_{\vartheta}^{o}}^{r_o} \overline{\langle (\Delta T )^2 \rangle}_{s} r^{2}\,\mathrm{d}r, \end{array}\right\} \end{gather}$$

where $V=(4/3) {\rm \pi}(r_{o}^3-r_{i}^3)$ is the volume of the spherical shell. Assuming the flow in the bulk region is highly turbulent, Grossmann & Lohse (Reference Grossmann and Lohse2000) proposed the following scalings for $\epsilon _{u}^{bulk}$ and $\epsilon _{\vartheta }^{bulk}$ at a fixed $Pr$:

(4.18a,b) \begin{equation} \epsilon_{u}^{bulk} \sim Re^{3}, \quad \epsilon_{\vartheta}^{bulk} \sim Re. \end{equation}

Figures 18 and 19 show the bulk dissipation rates as a function of $Re$ at different $\eta$. The data fits for $Ra \geq 10^{5}$ yield $\epsilon _{u}^{bulk}$ scales between $\epsilon _{u}^{bulk} \sim Re^{2.72}$ (at $\eta =0.2$) and $\epsilon _{u}^{bulk} \sim Re^{2.80}$ (at $\eta =0.8$), and $\epsilon _{\vartheta }^{bulk}$ scales between $\epsilon _{\vartheta }^{bulk} \sim Re^{0.66}$ (at $\eta =0.2$) and $\epsilon _{\vartheta }^{bulk} \sim Re^{0.71}$ (at $\eta =0.6$). For both dissipation rates, our data show considerable deviations from the predictions of GL theory. These deviations can mainly occur because of two potential reasons: (1) the bulk regions are still not turbulent enough for the considered Reynolds number range, $80 < Re < 7000$; and (2) the bulk region defined in this paper is the region outside the BLs, which, as a result of plume ejection from the inner and outer boundaries, may contain plumes that are mostly laminar and can be considered as detached BLs (Grossmann & Lohse Reference Grossmann and Lohse2004). The presence of these plumes in the bulk region will decrease the scaling exponents of the bulk contributions in the two dissipation rates. Similar deviations have also been reported, for example, by Calzavarini et al. (Reference Calzavarini, Lohse, Toschi and Tripiccione2005).

Figure 18. (a) Bulk contribution of kinetic energy dissipation rate $\epsilon _{u}^{bulk}$ and (b) the compensated $\epsilon _{u}^{bulk} Re^{-3}$, as functions of $Re$. Symbols represent the DNS data and dashed lines correspond to the fitting relations.

Figure 19. (a) Bulk contribution of the thermal energy dissipation rate $\epsilon _{\vartheta }^{bulk}$ and (b) the compensated $\epsilon _{\vartheta }^{bulk} Re^{-1}$, as functions of $Re$. Symbols represent the DNS data and the dashed lines correspond to the fitting relations.

Based on the Prandtl–Blasius BL theory and the fact that the BL thickness is significantly smaller than the gap length, Grossmann & Lohse (Reference Grossmann and Lohse2000) derived the following scaling relations for $\epsilon _{u}^{BL}$ and $\epsilon _{\vartheta }^{BL}$ at $Pr=1$,

(4.19a,b) \begin{equation} \epsilon_{u}^{BL} \sim Re^{5/2}, \quad \epsilon_{\vartheta}^{BL} \sim Re^{1/2}. \end{equation}

BL contributions for kinetic and thermal energy dissipation rates are shown in figures 20 and 21, respectively. As shown in figure 20, the best fit to DNS data with $Ra \geq 10^{5}$ yields $\epsilon _{u}^{BL} \sim Re^{2.54}$ (at $\eta =0.8$) and $\epsilon _{u}^{BL} \sim Re^{2.62}$ (at $\eta =0.2$). These fits are very close to the theoretical scaling of $\epsilon _{u}^{BL} \sim Re^{5/2}$. However, our data suggest the local scaling of $\epsilon _{u}^{BL}$ becomes larger with increasing $Ra$. It is even more clear in the compensated plot, as shown in figure 20(b). The local exponents of $\epsilon _{u}^{BL}(Re)$ initially decrease with $Re$ and then increase with $Re$, with the transitional $Re$ being approximately $500$. This observation reveals that a single power law is not sufficient for describing the $\epsilon _{u}^{BL}(Re)$ relation and there exists a secondary $Re$ dependence on $\epsilon _{u}^{BL}$. For $\epsilon _{\vartheta }^{BL}$ scaling, the best fit to DNS data with $Ra \geq 10^{5}$ shows $\epsilon _{\vartheta }^{BL}$ scales between $\epsilon _{\vartheta }^{BL} \sim Re^{0.57}$ (at $\eta =0.4$) and $\epsilon _{\vartheta }^{BL} \sim Re^{0.61}$ (at $\eta =0.2$), which are close to the expected exponents. It is observed that $\epsilon _{\vartheta }^{BL}$ scaling exhibits a similar behaviour to $\epsilon _{u}^{BL}$; that the local exponent becomes larger at a higher $Re$. Although this is less obvious for $\epsilon _{\vartheta }^{BL}$, we can still observe a gradual steepening of the slope when $Re$ increases, as shown in figure 21(b). This increase in the local exponent with respect to $Re$ points to the deviations from the simple power law predicted by GL theory for the BL dissipation rates. Similar deviations have also been reported by Gastine et al. (Reference Gastine, Wicht and Aurnou2015), as a result of the difficulty to perfectly separate the BL from the bulk.

Figure 20. (a) BL contribution of the kinetic energy dissipation rate $\epsilon _{u}^{BL}$ as a function of $Re$. Since the data fits at different $\eta$ are indistinguishable from each other, only the fit for the $\eta =0.8$ case is displayed on the plot. (b) Compensated $\epsilon _{u}^{BL} Re^{-5/2}$ as a function of $Re$. Symbols represent the DNS data and the dashed lines correspond to the fitting relations.

Figure 21. (a) BL contribution of the thermal energy dissipation rate $\epsilon _{\vartheta }^{BL}$ and (b) the compensated $\epsilon _{\vartheta }^{BL} Re^{-1/2}$, as functions of $Re$. Symbols represent the DNS data and the dashed lines correspond to the fitting relations.

Despite potential deviations of the local assumptions from the current DNS data, GL theory remains remarkably effective for our system across various radius ratios, as explained in detail below. Following GL theory, the two dissipation rates can be represented by the sum of bulk and BL contributions,

(4.20)\begin{equation} \left. \begin{array}{@{}c@{}} \hat{\epsilon}_{u} = \alpha_{1}(\eta) Re^{5/2} + \alpha_{2}(\eta) Re^{3}, \\ \hat{\epsilon}_{\vartheta} = \beta_{1}(\eta) Re^{1/2} + \beta_{2}(\eta) Re, \end{array}\right\}\end{equation}

where the bulk and BL contributions for viscous and thermal energy dissipation rates from (4.18a,b) and (4.19a,b) are used. Here, we denote the curve fits by the tilded variables and the simulation data by the plain variables. The coefficients $\alpha _{1}, \alpha _{2}, \beta _{1}$ and $\beta _{2}$ depend only on geometry (Grossmann & Lohse Reference Grossmann and Lohse2000), which means they only depend on $\eta$ in a spherical shell configuration. To fit (4.20) to our data, we proceed as

(4.21) \begin{equation} \left. \begin{array}{@{}c@{}} \displaystyle \log_{10} \epsilon_{u} = \log_{10}\left[\alpha_{1}(\eta) \exp \left(\dfrac{5}{2} \dfrac{\log_{10}Re}{\log_{10}e}\right) + \alpha_{2}(\eta) \exp\left(3 \dfrac{\log_{10}Re}{\log_{10}e} \right)\right], \\ \displaystyle \log_{10} \epsilon_{\vartheta} = \log_{10} \left[\beta_{1}(\eta) \exp\left(\dfrac{1}{2} \dfrac{\log_{10}Re}{\log_{10}e} \right) + \beta_{2}(\eta) \exp\left(\dfrac{\log_{10}Re}{\log_{10}e} \right) \right]. \end{array}\right\}\end{equation}

For each radius ratio, the coefficients $\alpha _1, \alpha _2$, $\beta _1$ and $\beta _2$ are determined by applying the least squares method to $\log _{10}(\epsilon _{u})$–$\log _{10}(Re)$ and $\log _{10}(\epsilon _{\vartheta })$–$\log _{10}(Re)$ data obtained from our DNS database. The least squares estimate of these coefficients at different $\eta$ are listed in table 2.

Table 2. Least squares estimate of coefficients $\alpha _1, \alpha _2$, $\beta _1$ and $\beta _2$ in (4.20).

In figures 22 and 23, the curve fits represented by (4.21) are shown. Figure 22(a) shows $\epsilon _{u}$ as a function of $Re$. Figure 22(b) shows $\epsilon _{u}$ normalized by the curve fit $\hat {\epsilon _{u}}$. The curve fits show good agreement with the DNS data in both the log-log and the semi-log plots. Figure 23(a) shows the thermal energy dissipation rate $\epsilon _{\vartheta }$ and the corresponding fits $\hat {\epsilon _{\vartheta }}$ at all $\eta$, while figure 23(b) shows the $\epsilon _{\vartheta }$ data normalized by $\hat {\epsilon _{\vartheta }}$. The curve fits in this case also agree well with the DNS data.

Figure 22. (a) Volume-averaged kinetic energy dissipation rate $\epsilon _{u}$ as a function of $Re$. Symbols represent DNS data at different $\eta$. Dashed light and dark red lines represent the predictions for $\eta = 0.2$ and $\eta = 0.8$, respectively, given by (4.20). (b) Ratio between volume-averaged kinetic energy dissipation rates obtained from the DNS data ($\epsilon _{u}$) and the predictions ($\hat {\epsilon _{u}}$) given by (4.20), as a function of $Re$.

Figure 23. (a) Volume-averaged thermal energy dissipation rate $\epsilon _{\vartheta }$ as a function of $Re$. Symbols represent DNS data, while the dashed lines with the corresponding colours denote predictions given by (4.20). (b) Ratio between volume-averaged thermal energy dissipation rates obtained from the DNS data ($\epsilon _{\vartheta }$) and the predictions ($\hat {\epsilon _{\vartheta }}$) given by (4.20), as a function of $Re$.

Using (2.3) in (4.20), we can write

(4.22) \begin{equation} \left. \begin{array}{@{}c@{}} \displaystyle \dfrac{3}{1+\eta+\eta^2} \dfrac{Ra}{Pr^2} (Nu-1) = \alpha_{1}(\eta) Re^{5/2} + \alpha_{2}(\eta) Re^{3}, \\ \displaystyle \dfrac{3 \eta}{1+\eta+\eta^2} Nu = \beta_{1}(\eta) Re^{1/2} + \beta_{2}(\eta) Re, \end{array}\right\}\end{equation}

with the coefficients $\alpha _1, \alpha _2, \beta _1$ and $\beta _2$ at each $\eta$ estimated as introduced above. At a fixed $Pr$ and $\eta$, from (4.22), we can numerically estimate $Nu$ and $Re$ as functions of $Ra$. Furthermore, to quantify the trend of the exponents, we can also define the local effective exponents as

(4.23a,b)\begin{equation} \alpha_{eff} = \frac{\partial \ln Nu }{\partial \ln Ra}; \quad \beta_{eff} = \frac{\partial \ln Re }{\partial \ln Ra}. \end{equation}

The radius ratio dependence of $Nu \sim f(Ra)$ is shown in figure 24. In figure 24(a), a good agreement between the DNS data and the GL theory predictions is observed. It can be seen from figure 24(a) that in the considered $Ra$ range, $Nu$ is small for the smaller radius ratios. The data corresponding to $\eta =0.2$ and $0.3$ clearly show this trend. For the rest of $\eta$, the values of $Nu$ are very close to each other. A look at the GL theory predictions reveals that $Nu$ for $\eta =0.3$ exceeds those for other $\eta$ in the range $7 \times 10^{8} < Ra < 3 \times 10^{9}$, and it reaches the largest value at $Ra=10^{10}$. We can also see that the GL theory predictions for $\eta =0.2$ will exceed all the other radius ratio cases (except $\eta =0.3$) in the range $4 \times 10^9 < Ra < 9 \times 10^{9}$. This trend can be explained clearly from the behaviour of the local effective exponents of $Nu(Ra)$. As shown in figure 24(b), at a fixed $Ra$, the smaller radius ratio ($\eta = 0.2, 0.3$) cases have a larger local effective exponent. A larger local effective exponent indicates a faster increase in $Nu$ with respect to $Ra$. Based on the magnitude of the local effective exponents, $Nu$ at $\eta =0.2$ is expected to finally exceed the $\eta =0.3$ curve beyond a certain $Ra$. However, the local effective exponents for the $\eta = 0.5, 0.6$ and $0.8$ cases are indistinguishable, which is consistent with the observation that the response of the system asymptotes to the planar case as $\eta \rightarrow 1$.

Figure 24. (a) $NuRa^{-1/3}$ as a function of $Ra$. Symbols represent DNS data; solid lines with corresponding colours are numerical results calculated from (4.22). (b) Local exponents for $Nu(Ra)$ using GL theory predictions given by (4.23a,b). The data are plotted at all radius ratios.

In figure 24(a), the data show that for all $\eta$, $Nu$ approaches $Nu \sim Ra^{1/3}$ at $Ra \approx 10^{9}$. By using the GL theory, we can predict the transitional $Ra$, where $Nu(Ra)$ attains the local scaling $Nu \sim Ra^{1/3}$. In figure 24(b), by looking at the local effective exponents predicted by the GL theory, we find that the $\eta =0.2$ curve reaches the $1/3$ local exponent at approximately $Ra = 1.4 \times 10^{8}$, while the $\eta =0.3$ and $0.4$ curves reach this local exponent at approximately $Ra = 4.1 \times 10^{8}$ and $8.6 \times 10^{8}$, respectively. For $\eta =0.5, 0.6$ and $0.8$, the transitional $Ra$ are so close that they are indistinguishable in the plot. The transitional $Ra$ for $\eta =0.5, 0.6$ and $0.8$ systems is roughly at $Ra = 10^{9}$. The behaviour of the transitional $Ra$ indicates that the transition to an enhanced heat transport regime would occur at a much lower $Ra$ in spherical shells than in Cartesian or cylindrical cells. For example, experiments by Roche et al. (Reference Roche, Gauthier, Kaiser and Salort2010) in cylindrical cells show an enhanced scaling of $Nu \sim Ra^{0.33}$ for $Ra = 5 \times 10^{11}$, which is much higher than the transitional $Ra$ observed in our system, although the thermal boundary conditions in their experiments are different from our work. In essence, GL theory predicts that the smaller radius ratio system may reach the $Nu \sim Ra^{1/3}$ scaling at a relatively lower $Ra$. This prediction may also imply that the smaller radius ratio system may reach the ultimate regime at a lower $Ra$. Furthermore, figure 24(b) shows that the differences in the local effective exponent at different $\eta$ are smaller at larger $\eta$. This is again in line with our expectation that the system will asymptotically approach the planar case with an infinite aspect ratio as $\eta \rightarrow 1$.

The radius ratio dependence of the compensated $Re \sim f(Ra)$ is shown in figure 25. The predictions of GL theory are in a good agreement with the DNS data. It is interesting to note that the behaviour of the $Re(Ra)$ relation is quite different for the smaller radius ratio ($\eta =0.2, 0.3$ and $\eta =0.4$) cases compared with the larger radius ratio ($\eta =0.5, 0.6$ and $\eta =0.8$) cases. The $Re(Ra)$ curves for the $\eta =0.5, 0.6$ and $0.8$ cases follow the same trend and are almost parallel to each other. However, the $Re(Ra)$ curves for $\eta =0.2, 0.3$ and $0.4$ show different trends and are observed to have some cross-overs with the $Re(Ra)$ curves at larger $\eta$. For example, at $Ra = 3 \times 10^{5}$, the $\eta =0.2$ case has the lowest $Re$, which increases with increasing $Ra$, faster than the rest of the cases. As mentioned above, some cross-overs with the rest of the $Re$ curves are also observed. The predictions from the GL theory in figure 25(a) indicate that at $Ra \approx 10^{10}$, $Re$ for the $\eta =0.2$ case will increase to be the largest. Different trends for different radius ratio cases can be explained by looking at the local effective exponents for $Re$ (see (4.23a,b)) in figure 25(b). At a fixed $Ra$, the local effective exponent is the largest at $\eta =0.2$, followed by the $\eta = 0.3$ and $0.4$ cases. For the $\eta \geq 0.5$ cases, the local effective exponents are indistinguishable from each other in figure 25(b). Similar to the larger $\alpha _{eff}$ for $Nu$, a larger local effective exponent $\beta _{eff}$ for $Re$ indicates a faster increase in $Re$ with respect to $Ra$, which is consistent with the steep rise of $Re$ for the $\eta =0.2$ cases as shown in figure 25(a). Furthermore, GL theory predictions of $\beta _{eff}$ indicate that the smaller radius ratio system may reach the ultimate scaling for $Re(Ra)$ at a relatively lower $Ra$, similar to what we observe for $Nu(Ra)$. Moreover, figure 25(b) shows that the differences in $\beta _{eff}$ at different $\eta$ are smaller for larger $\eta$, in line with the fact that the system approaches the planar case as $\eta \rightarrow 1$.

Figure 25. (a) $ReRa^{-1/2}$ as a function of $Ra$. Symbols represent DNS data; solid lines with corresponding colours are the GL theory predictions given by (4.22). (b) Local exponents for $Re(Ra)$ using GL theory predictions given by (4.23a,b). The data are plotted at all radius ratios.

In the preceding analysis, it is natural to question how the scaling behaviour of $Nu$ and $Re$ relates to the degree of supercriticality $Ra/Ra_c$ instead of $Ra$, where $Ra_{c}$ is the critical $Ra$ for the onset of convection. Plotting the scaling relations $Nu\sim Nu(Ra/Ra_c)$ and $Re\sim Re(Ra/Ra_c)$, the corresponding local exponents for different $\eta$ revealed that the use of $Ra/Ra_c$ as an independent variable does not change any of our conclusions and adds nothing new or interesting to the discussion.

6.1. Asymmetry of thermal energy dissipation rate

Figure 28 shows the relative contributions of the bulk and BL regions for the thermal energy dissipation rate as a function of $Ra$, at different $\eta$. The total BL contribution is always larger than the bulk contribution for the considered $Ra$ range. With increasing $Ra$, the bulk contribution will also increase. In figure 28, it is seen that for the thermal energy dissipation rate, the inner BL contribution is always larger than the outer BL contribution. The bulk and the total BL contributions as functions of both $Ra$ and $\eta$ are shown in figure 29. The relative contributions of the bulk and the sum of the two BLs are almost independent of $\eta$.

Figure 28. Contributions of the bulk and BL regions to the thermal energy dissipation rate as functions of $Ra$. Curves are plotted for all $\eta$. Solid circles represent the bulk contribution, while the solid squares represent the total BL contributions. Open downward triangles and open upward triangles denote the inner and the outer BL contributions, respectively. (a) $\eta =0.2$, (b) $\eta =0.3$, (c) $\eta =0.4$, (d) $\eta =0.5$, (e) $\eta =0.6$ and (f) $\eta =0.8$.

Figure 29. Relative contributions of the bulk and the BLs (inner $+$ outer) to the thermal energy dissipation rate as a function of $Ra$ at different $\eta$.

In figure 23, for $\eta \geq 0.3$, we observe cross-over points between the bulk and the outer BL contribution curves. From the GL theory, it is known that the energy dissipation rates will gradually transition from BL-dominant to bulk-dominant, as $Ra$ increases. At small $Ra$, the flow is quasi-laminar with a thicker BL and a less turbulent bulk, implying a larger fraction of the BL dissipation and a relatively smaller fraction of the bulk dissipation. As $Ra$ increases, the BL shrinks in size and the bulk becomes more turbulent, leading to an increase in the fraction of the bulk dissipation. Since the outer BL contribution is smaller than the inner BL contribution for the thermal energy dissipation rate, the bulk contribution will first exceed the outer BL contribution, followed by the inner BL contribution, and will finally exceed the total BL contribution at an even larger $Ra$. This explains the cross-overs observed in the thermal dissipation rate data. As a result of the limited $Ra$ range considered in this simulation campaign, the cross-over point between the inner BL dissipation and the bulk dissipation has not been observed yet. However, in figure 28, we can clearly observe a trend that the bulk contribution would exceed the inner BL contribution at an even higher $Ra$.

In figure 28, it is also observed that $Ra$ at the cross-over point is larger for higher $\eta$ cases. For example, for $\eta =0.3$, the bulk contribution exceeds the outer BL contribution at approximately $Ra=3 \times 10^{3}$, while for $\eta =0.8$, the cross-over point is at $Ra > 5 \times 10^{8}$. Since both $\epsilon _{\vartheta }^{BL}$ and $\epsilon _{\vartheta }^{bulk}$ stay almost the same with increasing $\eta$, and $\epsilon _{\vartheta }^{BL,outer}$ increases, the cross-over point shifts towards the higher $Ra$ end of the plot.

To quantify the asymmetry of the thermal energy dissipation rate in the inner and the outer BLs, we estimate

(6.4)\begin{equation} \epsilon_{\vartheta}^{BL,inner} = \frac{4 {\rm \pi}}{V} \int_{r_i}^{r_i+\lambda_{\vartheta}^{i}} \overline{\langle (\Delta T )^2 \rangle}_{s} r^{2}\,\mathrm{d}r \approx \biggl(\frac{\Delta \vartheta_{i}}{\lambda_{\vartheta}^{i}}\biggr)^{2} \frac{4{\rm \pi} r_{i}^{2}}{V} \lambda_{\vartheta}^{i}\end{equation}

and

(6.5)\begin{equation} \epsilon_{\vartheta}^{BL,outer} = \frac{4 {\rm \pi}}{V} \int_{r_o-\lambda_{\vartheta}^{o}}^{r_o} \overline{\langle (\Delta T )^2 \rangle}_{s} r^{2}\,\mathrm{d}r \approx \left(\frac{\Delta \vartheta_{o}}{\lambda_{\vartheta}^{o}} \right)^{2} \frac{4{\rm \pi} r_{o}^{2}}{V} \lambda_{\vartheta}^{o}.\end{equation}

Combining the above equations with (4.6a–c) leads to

(6.6)\begin{equation} \frac{\epsilon_{\vartheta}^{BL,inner}}{\epsilon_{\vartheta}^{BL,outer}} \sim \left(\frac{\Delta \vartheta_{i}}{\Delta \vartheta_{o}} \right)^{2} \frac{\lambda_{\vartheta}^{o}}{\lambda_{\vartheta}^{i}} \eta^{2} = \eta^{{-}4/3}.\end{equation}

In figure 30, it can be seen that (6.6) agrees well with the DNS data.

Figure 30. Ratio of the inner and outer BL contributions of thermal energy dissipation rate as a function of $\eta$. The dashed line denotes $\epsilon _{\vartheta }^{BL,inner}/\epsilon _{\vartheta }^{BL,outer} = 1$. Open symbols represent the DNS data.

6.2. Asymmetry of kinetic energy dissipation rate

Relative contributions of the kinetic energy dissipation rate $\epsilon _{u}^{bulk}$, $\epsilon _{u}^{BL}$, $\epsilon _{u}^{BL,inner}$ and $\epsilon _{u}^{BL,outer}$ are shown in figure 31. The kinetic energy dissipation rate is bulk dominant since the bulk contribution is always larger than the total BL contribution. For the $\eta \geq 0.3$ cases, $\epsilon _{u}^{bulk}$ keeps increasing with increasing $Ra$. However, at $\eta = 0.2$, $\epsilon _{u}^{bulk}$ will first increase and then decrease at a higher $Ra$. Furthermore, figure 32 shows that there is no $\eta$ dependence on the fractions of the bulk and the total BL contributions for kinetic energy dissipation rate.

Figure 31. Bulk and BL (inner $+$ outer) contributions of the kinetic energy dissipation rate as functions of $Ra$ for different radius ratios. Solid circles represent the bulk contribution and the solid squares represent the total BL contributions. Open downward and upward triangles denote inner and outer BL contributions, respectively. Dashed lines with corresponding colours are added to improve the readability of the plot. (a) $\eta =0.2$, (b) $\eta =0.3$, (c) $\eta =0.4$, (d) $\eta =0.2$, (e) $\eta =0.6$ and (f) $\eta =0.8$.

Figure 32. Relative contributions of the bulk and the BLs (inner $+$ outer) to the kinetic energy dissipation rate as the function of $Ra$ and different $\eta$.

In contrast to the thermal energy dissipation rate, the outer BL contribution is always larger than the inner BL contribution for the kinetic energy dissipation rate. Following a similar procedure as in § 6.1, the inner and the outer BL kinetic energy dissipation rates are estimated as

(6.7) \begin{equation} \epsilon_{u}^{BL,inner} = \frac{4 {\rm \pi}}{V} \int_{r_i}^{r_i+\lambda_{u}^{i}} \overline{\biggl\langle \left(\frac{\partial u_i}{\partial x_j} \right)^2 \biggr\rangle}_{s} r^{2}\,\mathrm{d}r \approx \left(\frac{Re_{i}}{\lambda_{u}^{i}} \right)^{2} \frac{4{\rm \pi} r_{i}^{2}}{V} \lambda_{u}^{i}\end{equation}

and

(6.8) \begin{equation} \epsilon_{u}^{BL,outer} = \frac{4 {\rm \pi}}{V} \int_{r_o-\lambda_{u}^{o}}^{r_o} \overline{\biggl\langle \left(\frac{\partial u_i}{\partial x_j} \right)^2 \biggr\rangle}_{s} r^{2}\,\mathrm{d}r \approx \left(\frac{Re_{o}}{\lambda_{u}^{o}} \right)^{2} \frac{4{\rm \pi} r_{o}^{2}}{V} \lambda_{u}^{o},\end{equation}

respectively. Using (5.6) in combination with (6.7) and (6.8), we get

(6.9) \begin{align} \frac{\epsilon_{u}^{BL,inner}}{\epsilon_{u}^{BL,outer}} \sim \left(\frac{Re_{i}}{Re_{o}} \right)^{2} \frac{\lambda_{u}^{o}}{\lambda_{u}^{i}} \eta^{2} =\frac{\eta^{4/3}\left(\dfrac{2}{1+\eta^{{4}/{3}}}\right)^{{1}/{2}}(\eta^2 +4\eta+7)}{8(\eta^2+\eta+1)-\left(\dfrac{2}{1+\eta^{{4}/{3}}}\right)^{{1}/{2}} (7\eta^2+4\eta+1)} < 1.\end{align}

Equation (6.9) ensures $\epsilon _{u}^{BL,outer} > \epsilon _{u}^{BL,inner}$.

Figure 33 shows $\epsilon _{u}^{BL,inner} / \epsilon _{u}^{BL,outer}$ as a function of $\eta$. The theoretical estimation of (6.9) predicts the trend reasonably well. However, there are significant deviations from the data, particularly at smaller $\eta$. The smaller the radius ratio, the larger the scatter in the $\epsilon _{u}^{BL,inner} / \epsilon _{u}^{BL,outer}$ data with respect to $Ra$. The deviations come from the $Ra$ dependence of $\epsilon _{u}^{BL,inner} / \epsilon _{u}^{BL,outer}$ especially in the small $\eta$ cases, since both $Re_i/Re_o$ and $\lambda _{u}^{i}/\lambda _{u}^{o}$ have $Ra$ dependence as illustrated in figures 13 and 27. Further investigations are still needed for demonstrating the $Ra$-dependence of $\epsilon _{u}^{BL,inner} / \epsilon _{u}^{BL,outer}$.

Figure 33. Ratio of the inner and outer BL contributions of kinetic energy dissipation rate as a function of $\eta$. Open symbols represent the DNS data.

It must be stressed here that owing to the dominance of $\epsilon _{\vartheta }^{BL} \textrm { and } \epsilon _{u}^{bulk}$, our simulations lie in regime II of the GL theory, in which the boundary layer and the bulk contributions dominate the thermal dissipation rate and the kinetic energy dissipation rate, respectively (Grossmann & Lohse Reference Grossmann and Lohse2000).