On the boundary-layer asymmetry in two-dimensional annular Rayleigh–Bénard convection subject to radial gravity

Abstract

Radial unstable stratification is a potential source of turbulence in the cold regions of accretion disks. To investigate this thermal effect, here we focus on two-dimensional Rayleigh–Bénard convection in an annulus subject to radially dependent gravitational acceleration $g \propto 1/r$. Next to the Rayleigh number $Ra$ and Prandtl number $Pr$, the radius ratio $\eta$, defined as the ratio of inner and outer cylinder radii, is a crucial parameter governing the flow dynamics. Using direct numerical simulations for $Pr=1$ and $Ra$ in the range from $10^7$ to $10^{10}$, we explore how variations in $\eta$ influence the asymmetry in the flow field, particularly in the boundary layers. Our results show that in the studied parameter range, the flow is dominated by convective rolls and that the thermal boundary-layer (TBL) thickness ratio between the inner and outer boundaries varies as $\eta ^{1/2}$. This scaling is attributed to the equality of velocity scales in the inner ($u_i$) and outer ($u_o$) regions. We further derive that the temperature drops in the inner and outer TBLs scale as $1/(1+\eta ^{1/2})$ and $\eta ^{1/2}/(1+\eta ^{1/2})$, respectively. The scalings and the temperature drops are in perfect agreement with the numerical data.

1. Introduction

Thermal convection is a pivotal process in astrophysical and geophysical systems, critically influencing energy transport in a variety of environments. It drives the convective zones within stars (Hotta & Kusano 2021; Vasil, Julien & Featherstone 2021), shapes the interiors of planets (Samuel et al. 2021), governs atmospheric dynamics in gas giants (Christensen 2001; Christensen & Wicht 2008) and plays a significant role in the behaviour of protoplanetary disks (Klahr 2006; Hirose et al. 2014). Recent research has highlighted convection driven by radial stratification, with gravity pointing to the centre, as a potential driver of turbulence within the cold regions in protoplanetary disks, especially within the 1–10 AU range (Teed & Latter 2021). The cold regions inside protoplanetary disks, often modelled as cylindrical systems with central heating sources, present a unique challenge for understanding convection dynamics. To address this, our study employs two-dimensional (2-D) direct numerical simulations (DNS) within an annular geometry to investigate convection driven by radially dependent gravity, $g \propto 1/r$. While planar thermal convection, particularly Rayleigh–Bénard convection (RBC) between horizontal plates, has been extensively studied (Ahlers, Grossmann & Lohse 2009; Lohse & Xia 2010; Chillà & Schumacher 2012; Shishkina 2021; Lohse & Shishkina 2024), convection in annular configurations remains comparatively underexplored. Because of the relatively cheap computational cost, the 2-D simulations can provide valuable insights into annular RBC across a broad parameter space, advancing our understanding of the complex dynamics at play.

A key focus of this study is the asymmetry in thermal boundary-layer (TBL) thickness that arises in annular systems due to curvature effects. This asymmetry can significantly impact flow dynamics and heat transfer, making it essential for accurately characterizing convection processes. Our research examines these asymmetries using data from DNS, considering critical control parameters such as the Rayleigh number ($Ra$) and the Prandtl number ($Pr$). The radius ratio $\eta \equiv r_i/r_o$, where $r_i$ and $r_o$ are the inner and outer radii of the domain, respectively, serves as the primary geometric variable influencing the convection behaviour.

In comparison, previous studies have focused mainly on the boundary-layer (BL) asymmetry within the spherical shell convection (see Sharpe & Peltier 1978; Schubert & Zebib 1980; Zebib et al. 1983; Tilgner 1996). For instance, Jarvis (1993) attributed this asymmetry to differences in BL Rayleigh numbers ($Ra_{\lambda }$) for marginally stable BLs, following the criteria established by Malkus (1954). Extending this work, Vangelov & Jarvis (1994) explored 2-D axisymmetric convection and similarly observed TBL asymmetries, which they explained satisfactorily through the same marginal stability criteria. However, later work by Deschamps, Tackley & Nakagawa (2010) in spherical convection simulations challenged these findings. Their study revealed that the ratio of inner to outer $Ra_{\lambda }$ scales as $\eta ^2$, diverging from the previous assumption of equality. Notably, these earlier studies focused on simulating Earth's mantle conditions, assuming infinite $Pr$ and a constant gravitational acceleration $g$. Recently, Gastine, Wicht & Aurnou (2015) conducted simulations of thermal convection in spherical geometries, with radius ratios $\eta$ ranging from 0.2 to 0.95, $Ra$ spanning from $10^3$ to $10^9$, and $Pr$ fixed at 1. They explored various gravity profiles; including $g \propto r, 1, 1/r^2, 1/r^5$. Their results revealed that none of the previously established scaling arguments could adequately explain the asymmetry observed in their data. Instead, they proposed a new scaling argument based on the equality of interplume areas in the inner and outer shells, which explained the asymmetry. The only study that specifically addresses BL asymmetry in cylindrical geometry was conducted by Wang et al. (2022). In this work, they performed numerical simulations where a cold inner shell and a hot outer shell rotated coaxially, with centrifugal force serving as the buoyancy source. Their research explored a range of $\eta$ from 0.3 to 0.9 and $Ra$ from $10^6$ to $10^8$, using $Pr$ of 4.3. To quantify the TBL asymmetry observed in their system, they employed the scaling methodology developed by Wu & Libchaber (1991) for non-Oberbeck–Boussinesq (NOB) RBC, providing valuable insights into the complexities of TBL behaviour in such configurations.

The present study aims to quantify the asymmetry of TBL within a 2-D annular RBC system driven by radial gravity which is directed toward the centre and inversely proportional to the distance from the centre, as opposed to the centrifugal-force-driven convection explored by Wang et al. (2022). While previous studies have proposed various scaling laws for TBL asymmetry in different geometries, these scalings do not adequately describe the behaviour observed in our system. Therefore, the main motivation of this work is to derive new scaling laws tailored to this specific configuration, providing a more comprehensive understanding of the mechanisms driving TBL asymmetry in radially stratified convection systems.

2. Numerical model

The configuration considered in this study consists of a heated inner shell and a cooler outer shell, with radially inward gravity providing the buoyancy (see figure 1). This differs from set-ups that use the outward centrifugal force as the buoyancy source (see Jiang et al. 2020,  2022; Wang et al. 2022,  2023; Zhong, Wang & Sun 2023; Zhong, Li & Sun 2024; Yao et al. 2025) where the outer shell is heated and the inner shell is cooled. Since the considered gravitational acceleration equals ${\boldsymbol {g}}\equiv -g\boldsymbol {e_r}$, where $g\equiv r_o/r$ and $\boldsymbol{e}_{\boldsymbol{r}}$ is the unit vector in the radial direction, the dimensionless equations governing the system are

(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 +\sqrt{\frac{Pr}{Ra}}\nabla^2\boldsymbol{u}+gT\boldsymbol{e}_{\boldsymbol{r}}, \end{gather}

(2.3)\begin{gather}\frac{\partial T}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla} T=\sqrt{\frac{1}{PrRa}}\nabla^2 T, \end{gather}

where $\boldsymbol {u}$, $p$, $T$ and $t$ denote the velocity, pressure, temperature and time, respectively. The control parameters include the Rayleigh number $Ra = \alpha g_o \Delta L^3/(\nu \kappa )$, the Prandtl number $Pr = \nu /\kappa$, and the radius ratio $\eta = r_i/r_o$. The non-dimensionalizing parameters are the characteristic length $L = r_o - r_i$, temperature $\varDelta = T_i - T_o$, and free-fall velocity $U = \sqrt {\alpha g_o \Delta L}$. Here, $\alpha$ denotes the isobaric expansion coefficient, $g_o$ is the gravitational constant at the outer shell, $(\nu)$ is the kinematic viscosity and $(\kappa)$ is the thermal diffusivity while $r_i$ and $r_o$ are the inner and outer radii. We used a second-order finite-difference code, as referenced in Verzicco & Orlandi (1996), Van Der Poel et al. (2015) and Zhu et al. (2018b). All simulations enforce no-slip and isothermal boundary conditions.

Figure 1. Sketch of the geometry used in the DNS with a hot inner shell at the inner radius $r_i$ and a cold outer shell at the outer radius $r_o$.

3. Flow field

In our simulations, $Ra$ and $\eta$ span from $10^7$ to $10^{10}$ and $0.2$ to $0.8$, respectively. The averaged temperature fields, shown in figure 2, reveal that the midpoint temperature deviates from the average boundary temperatures, with this deviation becoming more pronounced at lower $\eta$. Additionally, the inner TBL width is smaller than the outer TBL width, with this asymmetry increasing as $\eta$ decreases. Notably, for $Ra = 10^7$ at $\eta = 0.2$, a slight temperature gradient is observed in the bulk region of the flow.

Figure 2. Azimuthally ($\phi$) and temporally $(t)$ averaged temperature fields for (a) $Ra=10^7$ and (b) $Ra=10^{10}$, and varying $\eta$. In (a), data from Zhu et al. (2018a) are added for comparison with planar RBC for the aspect ratio $\varGamma =2$.

Due to the 2-D nature of our simulations, the flow is dominated by convective rolls, as depicted in figure 3. These rolls are distorted by the flow geometry, becoming narrower near the inner shell and wider near the outer shell. At lower $\eta$, due to the increasing asymmetry in the flow field, the cold plume becomes fragmented. This is due to the shearing of the cold plume by the convective roll. Mass conservation requires any radially outward mass flux to be balanced by radial inflow. Since the plume widths scale with TBL widths (Ching et al. 2004), the radial flow velocity in the inner plume exceeds that in the outer plume. The convective rolls, driven by the hot inner plume, exhibit higher velocities compared with the outer plume. At lower $\eta$, this shear effect fragments the outer plume. This also causes cold fluid from the outer BL to be mixed into the bulk flow. Although higher $Ra$ levels lead to increased mixing and temperature homogenization, at low $\eta$ and low $Ra$ (e.g. $\eta = 0.2$ and $Ra = 10^7$), the temperature gradient persists in the bulk region. As $Ra$ increases, both inner and outer plumes break up, forming vortices that eventually merge into the convective rolls.

Figure 3. Instantaneous temperature fields for different $\eta$ at $Ra= 10^8$: (a) $\eta =0.2$, (b) $\eta =0.4$, (c) $\eta =0.6$, (d) $\eta =0.8$.

4. Boundary layers

For an annular geometry, the mismatch between the circumferences of the inner and outer shells leads to unequal heat fluxes through them. In our simulations, we have detected the BLs using the slope method, see e.g. Shishkina et al. (2010). Assuming purely conductive heat transfer through the TBLs, the following relation holds true in 2-D annular RBC:

(4.1)\begin{equation} \eta\frac{\Delta T_i}{\lambda_i}=\frac{\Delta T_o}{\lambda_o}, \end{equation}

where $\lambda _i(\lambda _o)$ is the TBL width near the inner (outer) wall, and $\Delta T_i(\Delta T_o)$ is the temperature drop within the corresponding TBLs. Moreover, based on our simulations (see figure 2) we conclude that, similar to planar RBC, the temperature drop across the shells occurs only in the BLs if the flow is turbulent ($Ra$ is large). Hence, the following relation also holds true:

(4.2)\begin{equation} \Delta T_i+\Delta T_o=1. \end{equation}

By the definition of $Nu$, evaluated at the walls, we have $Nu=r({\partial T}/{\partial r})\ln \eta$. From this, we obtain

(4.3)\begin{gather} \frac{\Delta T_i}{\lambda_i}\approx\frac{1-\eta}{\eta}\frac{Nu}{\ln{\eta}}, \end{gather}

(4.4)\begin{gather}\frac{\Delta T_o}{\lambda_o}\approx\left( 1-\eta \right)\frac{Nu}{\ln{\eta}}. \end{gather}

Equations (4.3) and (4.4) combine to give (4.1). In order to quantify the variables $\Delta T_i$, $\Delta T_o$, $\lambda _i$ and $\lambda _o$, (4.1), (4.2), (4.3) and (4.4) are not sufficient. Additional relations between the parameters are required.

4.1. Wu & Libchaber assumption

An asymmetric temperature field is also observed in planar NOB RBC (Wu & Libchaber 1991; Zhang, Childress & Libchaber 1997; Bodenschatz, Pesch & Ahlers 2000; Ahlers et al. 2006; Weiss, Emran & Shishkina 2024). Several authors have previously attempted to quantify the asymmetry of the temperature field due to the effects of NOB conditions on planar RBC systems (see Ahlers et al. 2006; Horn, Shishkina & Wagner 2013; Horn & Shishkina 2014; Weiss et al. 2018; Yik, Valori & Weiss 2020). Possibly the simplest model was proposed in the seminal work of Wu & Libchaber (1991). Based on their experimental results, they inferred that in the central region, temperature fluctuation scales of the colder region are equal to that of the hotter region and obtained the following equality:

(4.5)\begin{equation} \theta_i \equiv \frac{\nu\kappa}{\alpha g_i{\lambda_i}^3}=\frac{\nu\kappa}{\alpha g_o{\lambda_o}^3}\equiv \theta_o. \end{equation}

Here, $(g_i)$ and $(g_o)$ are the gravitational accelerations at the inner and the outer TBLs, respectively. The proposed model could predict both the midpoint temperature and the heat transfer rate in an RBC cell with NOB effects. Wang et al. (2022) used this model to obtain relations between different BL parameters for RBC in a rotating cylindrical geometry. For the present flow configuration, it results in the following expressions:

(4.6a)\begin{gather} \lambda_i = \frac{\eta\ln{\eta}}{(1+\eta^{2/3})(1-\eta)Nu}, \quad \Delta T_i = \frac{1}{1+\eta^{2/3}}, \end{gather}

(4.6b)\begin{gather}\lambda_o = \frac{\eta^{2/3}\ln{\eta}}{(1+\eta^{2/3})(1-\eta)Nu}, \quad \Delta T_o = \frac{\eta^{2/3}}{1+\eta^{2/3}}. \end{gather}

Figure 4(c) shows the comparison of the analytical expression with the results from the present simulations. A significant deviation can be seen towards the lower $\eta$. Gastine et al. (2015) demonstrated that in spherical geometries, this model deviated from observed results under various gravity profiles, including uniform gravity. In the original study by Wu & Libchaber (1991), the model's validity was confirmed for $Pr \sim 1$, and Zhang et al. (1997) extended this to $Pr$ values ranging from 300 to 7000. Thus, it appears that the $Pr$ should not significantly impact the model's applicability. We hypothesize that the geometry of the container and in particular its curvature has a stronger impact. It influences the plume thickness along its path through the bulk, leading to changes in the temperature fluctuation scales and contributing to the observed deviations from the above model.

Figure 4. Comparison of the various scaling criteria with the present data. (a) Ratio of the BL $Ra$ based on the argument of Jarvis (1993). (b) Ratio of average plume spacings in the inner and outer shells (Gastine et al. 2015). (c) Fluctuating temperature scales as defined in Wu & Libchaber (1991). (d) Ratio of the velocity scales of the inner and outer BLs.

4.2. Gastine et al. assumption

Based on the disparity between the sizes of the plumes ejected from the inner and outer shells, Gastine et al. (2015) proposed that the average plume spacing in the inner and outer shells are equal. Their hypothesis was based on the observance that the plumes ejected from the outer BL are thicker than those ejected from the inner BL. To prove the hypothesis, they obtained probability distributions of the interplume area in both the inner and outer shells and found them to be overlapping. Furthermore, as the interplume area is proportional to the interplume spacing, they derived an expression for the interplume spacing in the inner (outer) shell as $l_{i}=\sqrt {\alpha g_{i} \Delta T_{i}{\lambda _{i}}^5/(\nu \kappa )}$ ($l_{o}=\sqrt {\alpha g_{o} \Delta T_{o}{\lambda _{o}}^5/(\nu \kappa )}$) which when equated leads to the following:

(4.7)\begin{equation} \sqrt{\frac{\alpha g_{i} \Delta T_{i}{\lambda_{i}}^5}{\nu \kappa}} =\sqrt{\frac{\alpha g_{o} \Delta T_{o}{\lambda_{o}}^5}{\nu \kappa}}. \end{equation}

Using (4.7) along with (4.1) and (4.2) we obtain that $\lambda _i/\lambda _o=\eta ^{1/3}$. This leads to the same set of equations as (4.6a)–(4.6b). We have already seen that these equations do not concur with the results of present simulations. We conjecture that this might be due to the dominance of convective rolls in our set-up. Moreover, this model is based on geometrical constraints and does not depend on the gravity profile; it was efficient across various gravity profiles in Gastine et al. (2015). However, it has only been applied to the case of $Pr=1$. Hence, it is unknown how it depends on $Pr$.

4.3. Jarvis assumption

Based on the argument of marginal BL stability (Malkus 1954), Jarvis (1993) equate the BL Rayleigh numbers ($Ra_i=\alpha g_{i} \Delta T_{i}{\lambda _{i}}^3/(\nu \kappa )$ and $Ra_o=\alpha g_{o} \Delta T_{o}{\lambda _{o}}^3/(\nu \kappa )$) and obtain

(4.8)\begin{equation} \frac{\alpha g_{i} \Delta T_{i}{\lambda_{i}}^3}{\nu \kappa} =\frac{\alpha g_{o} \Delta T_{o}{\lambda_{o}}^3}{\nu \kappa}. \end{equation}

Equations (4.1), (4.2) and (4.8) lead to the following equations:

(4.9a)\begin{gather} \lambda_i=\frac{\eta\ln{\eta}}{(1+\eta^{1/2})(1-\eta)Nu}, \quad \Delta T_i=\frac{1}{1+\eta^{1/2}}, \end{gather}

(4.9b)\begin{gather}\lambda_o=\frac{\eta^{1/2}\ln{\eta}}{(1+\eta^{1/2})(1-\eta)Nu}, \quad \Delta T_o=\frac{\eta^{1/2}}{1+\eta^{1/2}}. \end{gather}

Figure 5(a) shows that the ratio of the TBLs follow $\lambda _i/\lambda _o \sim \eta ^{1/2}$, indeed. Validation of the TBL thickness is further supported in figure 6(b). While the assumption $Ra_i = Ra_o$ can explain these observations, we find no solid physical justification for this assumption (see figure 4a). This model was originally validated for mantle convection with an infinite Prandtl number. However, since it is based on the marginal stability of the BLs, it is possible that the deviations between the model and our data are due to the lack of turbulence in the studied 2-D flows. Indeed, Gastine et al. (2015) found that this model did not match their simulation results for any of the gravity profiles they simulated. As a result, we refrain from adopting this argument and instead proceed with an alternative approach.

Figure 5. The TBL ratio compensated by (a) $\eta ^{1/2}$ and (b) $\eta ^{1/3}$. Temperature drop in the inner (c) and outer (d) TBL compared with the analytical results in (4.6a) and (4.6b) as well as (4.9a) and (4.9b).

Figure 6. (a) A schematic of the velocity and length scales chosen to obtain scaling arguments. (b) Comparison of the normalized TBL width with the analytical result, (4.9a) and (4.9b).

4.4. Proposed model

The flow is dominated by convective rolls which interact with the BLs and plumes. We assume that the viscous term of the temperature equation scales with the advective term at the BL width $u_{\phi }(({1}/{r})({\partial T}/{\partial {\phi }}))\sim \kappa (({1}/{r})({\partial }/{\partial r})(r({\partial T}/{\partial r})))$. Here, $(r)$ and $(\phi)$ are the radial and azimuthal coordinate, respectively. This assumption is derived from the balance of the leading terms in the temperature equation (2.3). At the edge of the TBL, the magnitude of the azimuthal velocity component $u_{\phi }$ is significantly larger than the radial velocity component $u_r$. As a result, $u_{\phi }(({1}/{r})({\partial T}/{\partial \phi })) \gg u_r({\partial T}/{\partial r})$. Similarly, for the diffusive term, the radial gradient of $T$ dominates the azimuthal gradient. Therefore, $\kappa (({1}/{r})({\partial }/{\partial r})(r({\partial T}/{\partial r}))) \gg \kappa (({1}/{r^2})({\partial ^2 T}/{\partial \phi ^2}))$. Hence we define an advective length scale for the inner (outer) shells as ${\mathcal {L}_{i}}\sim r_i\,{\rm d} {\phi }$ (${\mathcal {L}_{o}}\sim r_o \,{\rm d} {\phi }$) and take the BL thickness for the inner (outer) shell to be the vertical length scale. We define $u_{i}$ ($u_{o}$) to be the velocity scale for the convective roll near the inner (outer) shell; we then obtain the following scaling relations:

(4.10a,b)\begin{equation} \frac{u_{i}\Delta T_{i}}{{\mathcal{L}_{i}}}\sim\frac{\kappa \Delta T_{i}}{\lambda_{i}^2} \quad\textrm{and}\quad\frac{u_{o}\Delta T_{o}}{{\mathcal{L}_{o}}}\sim\frac{\kappa \Delta T_{o}}{\lambda_{o}^2}. \end{equation}

Therefore,

(4.11)\begin{equation} \frac{u_i}{u_o}\sim\frac{{\mathcal{L}_{i}} \lambda^2_o}{{\mathcal{L}_{o}} \lambda^2_i}, \end{equation}

where ${{\mathcal {L}_{i}}}/{{\mathcal {L}_{o}}}=\eta$, see figure 6. We further propose that the velocity scales of the convective roll near the inner and outer shells are equal, i.e. $u_i=u_o$ (see figure 4d). The ratio $u_i/u_o$ shown in figure 4(d) is that of the peaks of the root-mean-squared velocity near the inner and outer shell. The equality $u_i=u_o$ can be justified as the 2-D nature of the flow ensures the azimuthal mass flux near the outer shell equals that towards the inner shell, and as the convective roll is symmetrically placed along the radial direction the velocity scales should be equal. Figure 4(d) shows that at lower $\eta$, there is a higher scatter of the ratio of $u_i$ and $u_o$ and of the ratio of the TBL thickness (figure 5a). This might be attributed to the breaking of the cold plume at lower radius ratios as discussed in § 3. Thus from (4.11), $u_i=u_o$ and ${\mathcal {L}_{i}}/{\mathcal {L}_{o}}=\eta$ we conclude that $\lambda _i/\lambda _o{\sim }\eta ^{1/2}$, which is supported by our DNS results, see figure 5(a). This physics-based model can also lead to the derivations of (4.9a) and (4.9b).

5. Conclusion

In this study, we derived scaling laws for the TBL width ratio in RBC within a 2-D annulus, focusing on radius ratios ranging from 0.2 to 0.8 and Rayleigh numbers ($Ra$) from $10^7$ to $10^{10}$. The system features a gravity profile $g \propto 1/r$ with a fixed $Pr=1$. We observed an asymmetry in the TBL widths between the inner and outer shells, consistent with previous studies. However, existing scaling laws failed to quantify this asymmetry. To address this, we proposed a new scaling argument based on the assumptions of the isothermal bulk and constant velocity along the path of all large-scale convection rolls that develop in the annulus RBC. This new scaling successfully predicted the temperature drops and TBL widths across the parameter range studied.