3.2.1. Scaling for $R_F=0.1$

In this section we fix $R_F=0.1$, meaning that $90\,\%$ of the power transferred through the lower boundary goes out through the side, and we seek scaling laws for the mean temperature systematically varying $Ra_{\phi }$ and $\varGamma$. For each simulation, the statistically stationary state is reached, which typically takes $5$ diffusive time scales, and we compute the mean temperature of the system noted $\langle T\rangle$ using the space–time average operator defined in (2.7).

The mean temperature evolution with $Ra_{\phi }$ and $\varGamma$ is reported in figure 3. When the value of $Ra_{\phi }$ increases, the mean non-dimensional temperature of the system decreases as expected (see figure 3a). This is because a higher Rayleigh–Roberts number results in more efficient heat transfers within the system, leading to an average temperature getting closer to the side temperature that is zero in our dimensionless units (2.1). Note however that the dimensional temperature obviously increases when we increase the heat flux. When $Ra_{\phi }$ exceeds $10^5$, a power-law behaviour emerges with an exponent that appears to be unaffected by $\varGamma$. Upon estimating the power exponent (best fit using the least square method), it has been found that $\langle T\rangle \sim Ra_{\phi }^{-0.20\pm 0.03}$. To determine the mean exponent, we compute the average of the exponents associated with $\varGamma =4,5,8,16$ considering $Ra_{\phi } \geqslant 10^5$ data only, while the variability is quantified by the largest difference between these exponents. This scaling law is derived by considering DNS and filtered simulation data. The exclusion of the filtered data from the analysis only leads to a slight alteration in the scaling law, resulting in $\langle T\rangle \sim Ra_{\phi }^{-0.21\pm 0.03}$.

Figure 3.

Log-log plot of the mean temperature evolution with Rayleigh number for different aspect ratios in (a) and with the aspect ratio for different Rayleigh numbers in (b). The width of the thick light lines is equal to $3$ times the standard deviation of the mean temperature time series at the statistically stationary state. The dash-dotted lines show the best power-law fit for each $\varGamma$ in (a) and $Ra_{\phi }$ in (b). Empty/full symbols respectively indicate filtered/DNS simulations. Here $Pr=0.1$ and $R_F=0.1$.

Additionally, when $\varGamma$ increases at constant $Ra_{\phi }$, the mean temperature of the system increases. Indeed, when $\varGamma$ increases for a fixed $Ra_{\phi }$, the ratio between the heating bottom surface and the cooling lateral surface also increases, leading to a larger global energy input into the system. Figure 3(b) shows that when $Ra_{\phi }$ is greater than $10^5$, a power-law behaviour in $\varGamma$ also emerges. Considering both DNS and filtered simulation data, and employing a consistent methodology for exponent estimation as applied to the $Ra_{\phi }$ dependence, we obtain $\langle T\rangle \sim \varGamma ^{0.83 \pm 0.11}$ whereas we find that $\langle T\rangle \sim \varGamma ^{0.89 \pm 0.08}$ without taking filtered data into account. Once again these results indicate a minor influence of the filtered simulations on the overall scaling behaviour.

The two scalings can be combined leading to the following final power law:

(3.1)\begin{equation} \left\langle T\right\rangle \sim Ra_{\phi}^{{-}1/5}\varGamma^{4/5} . \end{equation}

Here the particular exponent values will be theoretically justified below (see § 3.2.2) and fall well into the fitted range of exponents found from our simulations.

Figure 4 shows the mean temperature, compensated by scaling (3.1) as a function of $Ra_{\phi }$. All rescaled data converge to the same constant close to unity. For $\varGamma =4$, the scaling law seems to apply when $Ra_{\phi }>10^5$, whilst it is necessary to wait until $Ra_{\phi }=10^8$ when $\varGamma =16$. In the following section, we use dimensional analysis to gain insight into the physics underlying this scaling.

Figure 4.

Compensated mean temperature following (3.1) as a function of $Ra_{\phi }$. Different symbols/colours correspond to different aspect ratios. The coloured areas indicate $3$ times standard deviations of the mean temperature time series at the statistically stationary state multiplied by $Ra_{\phi }^{1/5}\varGamma ^{-4/5}$. Empty/full symbols respectively indicate filtered/DNS simulations. Here $Pr=0.1$ and $R_F=0.1$.

3.2.2. Theoretical analysis for $R_F=0.1$

We focus on the sidewall considering the dimensionless steady axisymmetrical governing equations using cylindrical coordinates

(3.2)$$\begin{gather} u\frac{\partial u}{\partial r}+ w\frac{\partial u}{\partial z}={-}Pr\frac{\partial P}{\partial r} +Pr\nabla^2 u , \end{gather}$$

(3.3)$$\begin{gather}u\frac{\partial w}{\partial r} + w\frac{\partial w}{\partial z}={-}Pr\frac{\partial P}{\partial z} + Ra_{\phi}PrT+ Pr\nabla^2 w , \end{gather}$$

(3.4)$$\begin{gather}\frac{1}{r}\frac{\partial ru}{\partial r}+\frac{\partial w}{\partial z}=0 , \end{gather}$$

(3.5)$$\begin{gather}u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} = \nabla^2 T . \end{gather}$$

Due to the low Prandtl regime, the viscous boundary layer is nested within the thermal one. A diagram of this configuration is shown in figure 5. The behaviour of the vertical ($w_s$) and radial ($u_s$) velocities at the edge of the viscous boundary layer is derived from the conservation of mass (3.4) and energy (3.5). We assume that the typical scale of variation in the radial direction is the dimensionless thickness of the side boundary layer ($\partial /\partial _r \sim 1/\delta$). This thickness may either represent the thermal boundary layer ($\delta _{th}$) or the viscous boundary layer ($\delta _v$), depending on whether the radial variation under consideration pertains to temperature or velocity. Moreover, we assume that the scale of variation in height is the dimensionless height equal to $1$ ($\partial /\partial _z \sim 1$). Then mass conservation (3.4) leads to

(3.6)\begin{equation} \frac{u_s}{\delta_v}\sim w_s. \end{equation}

Figure 5.

Sketch of the side boundary layers when $Pr<1$.

On the left-hand side of (3.5), both advection terms, $u\partial _r T$ and $w\partial _z T$, scale as $w_s T$, due to mass conservation (3.6). Because radial variations are much larger than height variations near the lateral boundaries, we assume that the dominant term in the Laplacian operator scales with the dimensionless thickness of the side boundary layer squared ($\nabla ^2 \sim 1/\delta ^2$). Therefore, (3.5) leads to

(3.7)\begin{equation} w_s\sim 1/\delta_{th}^2. \end{equation}

Let us now approximate the temperature variation across the thermal boundary layer ($\delta T_s$) by the average temperature of the system $\langle T\rangle$ (recall that the dimensionless temperature on the sidewall is zero). Within the thermal boundary layer, we expect a force balance between the inertia term and the buoyancy term, so that the vertical momentum balance (3.3) reduces to

(3.8)\begin{equation} u \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z} \simeq RaPrT. \end{equation}

On the left-hand side of (3.8), each advection term, $u\partial _r w$ and $w\partial _z w$, scales like $w_s^2$, due to mass conservation (3.6). Then (3.8) reduces to $w_s^2 \sim Ra_{\phi }Pr\langle T\rangle$ or, equivalently, using (3.7),

(3.9)\begin{equation} \delta_{th}\sim\left(Ra_{\phi}Pr\left\langle T\right\rangle\right)^{{-}1/4} . \end{equation}

Finally, to link the heat flux applied on the bottom surface to the thermal characteristics of the side, a global flux balance is required. Integrating (3.5) over the volume at steady state leads to

(3.10)\begin{equation} {\rm \pi}\varGamma^2\left(1-R_F\right)=2{\rm \pi}\varGamma\frac{\left\langle T\right\rangle}{\delta_{th}} , \end{equation}

where the left-hand side corresponds to the power mismatch between the lower and upper boundaries, which is balanced by the conducting flux across the thermal boundary layer on the right-hand side. Thus, the averaged temperature $\langle T\rangle$ is proportional to the aspect ratio and to the thickness of the thermal boundary layer

(3.11)\begin{equation} \left\langle T\right\rangle\sim\delta_{th}(1-R_F)\varGamma . \end{equation}

Combining (3.9) and (3.11), the mean temperature can therefore be expressed in terms of the control parameters through the relationship

(3.12)\begin{equation} \left\langle T\right\rangle\sim Ra_{\phi}^{{-}1/5}\varGamma^{4/5}\left(1-R_F\right)^{4/5} Pr^{{-}1/5} . \end{equation}

These simple dimensional arguments allow us to recover the scaling (3.1) obtained via numerical simulations. In addition, this reveals the dependencies on $R_F$ and $Pr$, which we did not observe since both these parameters have been fixed for now. Note that assuming that the average temperature is representative of the temperature difference across the radial thermal boundary layer is presumably only valid when the Rayleigh–Roberts number is large enough to mix efficiently the bulk of the convective system. In order to compare our results with existing literature, it is standard to use the Nusselt number (the ratio of convective to diffusive heat flux). However, this measurement is only meaningful when the heat transfer can be unambiguously defined, that is, when the average isothermal surfaces are parallel. Or in other words, when temperature varies only along one dimension in the system. In our system there is no specific direction for heat transfer except in the asymptotic regime of high or low flux ratio. In these scenarios, it is conceivable to determine the Nusselt value, which indicates the main heat flux direction (vertical for $R_F \rightarrow 1$ and horizontal for $R_F=0$).

In the low flux ratio regime, the heat flux mainly goes in the horizontal direction; therefore, we define the Nusselt number by the relationship

(3.13)\begin{equation} \phi_{in} \equiv \frac{k}{R}\left(\left\langle\theta\right\rangle-\theta_0\right)Nu . \end{equation}

With the choice made for the non-dimensional temperature, the Nusselt number reads as the inverse of the mean temperature, hence

(3.14)\begin{equation} Nu\sim Ra_{\phi}^{1/5}\varGamma^{1/5}\left(1-R_F\right)^{{-}4/5} Pr^{1/5}. \end{equation}

Although we find a $1/5$ exponent for the Rayleigh dependency, reminiscent of the horizontal convection scaling of the Rossby (Reference Rossby1965) regime, it is important to note that the force balance in the boundary layer is different. In Rossby (Reference Rossby1965) buoyancy and viscosity at the bottom both play a role, while in our case, the balance is between inertia and buoyancy on the vertical side boundary. The equivalent Rayleigh exponent based on a temperature scale (instead of a flux scale as done here) is $1/4$ and corresponds to the scaling of vertical convection identified by Batchelor (Reference Batchelor1954). Furthermore, in a two-dimensional rectangular system with identical thermal boundary conditions to the present case (except at the top boundary where $R_F$ was set to $0$), Ganzarolli & Milanez (Reference Ganzarolli and Milanez1995) identified a Nusselt scaling law. We obtain identical exponents for the Rayleigh–Roberts number and the aspect ratio as those found by Ganzarolli & Milanez (Reference Ganzarolli and Milanez1995). Both exhibited $1/5$ exponents. It is worth mentioning that the validity of the Rayleigh scaling was recently shown to be limited to laminar boundary layers by Shishkina (Reference Shishkina2016), so that a different scaling is expected at even large Rayleigh–Roberts numbers (we considered $Ra_{\phi }\leqslant 10^9$). Previous works (George & Capp Reference George and Capp1979) also pointed out this possibility. Finally, our results show the same $1/5$ Prandtl exponent dependence predicted by Shishkina (Reference Shishkina2016) and recently emphasized byZwirner et al. (Reference Zwirner, Emran, Schindler, Singh, Eckert, Vogt and Shishkina2022). However, we recall here that we have not checked this Prandtl scaling with our numerical simulations, which were all performed at $Pr=0.1$.

To validate the underlying physical approximations required to derive our scaling law, we now examine the secondary variables, specifically the thickness of the thermal boundary layer and the vertical velocity. Thanks to the relations (3.7) and (3.11), the scaling laws for those quantities can be written as

(3.15)\begin{equation} \left.\begin{gathered} \delta_{th}\sim Ra_{\phi}^{{-}1/5}\varGamma^{{-}1/5}\left(1-R_F\right)^{{-}1/5} Pr^{{-}1/5} ,\\ w_s\sim Ra_{\phi}^{2/5}\varGamma^{2/5}\left(1-R_F\right)^{2/5} Pr^{2/5} . \end{gathered}\right\} \end{equation}

The thickness of the thermal boundary layer is estimated by determining the radius at which $90\,\%$ of the maximum temperature near the edge is reached, while the vertical velocity is based on seeking the minimum vertical velocity near the edge. More precisely, we first compute the vertical velocity at mid-height of the domain to avoid the influence of the top and bottom boundaries, and we average in time and in the azimuthal direction. We then search for the first peak of negative vertical velocity, starting from the boundary and moving toward the centre of the domain. We conducted these measurements for aspect ratios of $4\leqslant \varGamma \leqslant 16$ and for $10^5\leqslant Ra_{\phi }\leqslant 10^8$. Results shown in figure 6 are in excellent agreement with the scaling laws (3.15), hence, further validating our approach.

Figure 6.

Log-log plot of (a) the size of the thermal boundary layer and (b) the absolute value of the vertical peak velocity near the edge (defined as the first minimum of the negative vertical velocity at mid-height, starting from the boundary and moving toward the centre of the domain) versus the determined scaling laws (3.15) for different $Ra_{\phi }$ and aspect ratios. The dash-dotted lines correspond to the scaling laws and for all the simulations. Empty/full symbols respectively indicate filtered/DNS simulations. The input parameters are $Pr=0.1$ and $R_F=0.1$.

3.3.1. Scaling for $R_F=0.9$

We now fix $R_F=0.9$, meaning that $90\,\%$ of the heating power goes out through the top surface. Similarly to the precedent section, we seek power laws systematically varying $Ra_{\phi }$ and $\varGamma$. However, in this second regime closer to the classical Rayleigh–Bénard configuration, heat transfers are mainly along the vertical direction. We therefore focus on the mean temperature difference between the top and bottom surfaces, denoted $\Delta T_v$ and computed as

(3.16)\begin{equation} \Delta T_v=\left\langle T(t,r,\varphi,z=0)-T(t,r,\varphi,z=1)\right\rangle_{r\varphi} , \end{equation}

where the average corresponds to a temporal and surfacic average along radius and azimuthal angle. As we will see below, this averaged quantity is more relevant than the mean temperature of the system that was more representative of the radial temperature differences between the bulk and the side boundary when $R_F=0.1$. The evolution of the mean temperature difference with $Ra_{\phi }$ and $\varGamma$ is plotted in figure 7. We observe that, as $Ra_{\phi }$ increases, $\Delta T_v$ decreases, suggesting that the system becomes more homogeneous vertically. The decrease in the top–bottom temperature difference appears to be independent of the aspect ratio, which suggests a local mechanism. Note also that this vertical temperature difference is the same at different locations, as will be further discussed below in § 3.5. When $Ra_{\phi }$ is larger than $10^5$, a power-law scaling emerges with an exponent $Ra_{\phi }^{-0.20\pm 0.02}$ independently of $\varGamma$. As we will show below, the closest relevant scaling is given by

(3.17)\begin{equation} \Delta T_v\sim Ra_{\phi}^{{-}1/5} . \end{equation}

Figure 7.

Log-log plot of the bottom–top temperature difference for different $\varGamma$ values. The error bars are computed by taking 3 times the standard deviation of the bottom–top temperature difference time series at the statistically stationary state. The dash-dotted line corresponds to the best-fit scaling law $Ra_{\phi }^{-1/5}$, the blue diamond ($\Diamond$) plot shows the horizontally homogeneous Chapman & Proctor (Reference Chapman and Proctor1980) configuration ($R_F=1$ in a doubly periodic Cartesian box) and the continuous black line corresponds to the Grossmann & Lohse (Reference Grossmann and Lohse2000) $I_l$ regime written in terms of flux Rayleigh number $Ra_{\phi }$. Empty/full symbols respectively indicate filtered/DNS simulations. The input parameters are $Pr=0.1$ and $R_F=0.9$.

3.3.2. Theoretical analysis for $R_F=0.9$

In this configuration close to Rayleigh–Bénard convection, one might initially assume that the system is controlled by the heat flux across a thin thermal boundary layer (Malkus Reference Malkus1954). Let us define the Rayleigh number based on the temperature difference $Ra_{\Delta T_v}=(\beta g \Delta T_v H^3)/(\nu \kappa )$, which is an output parameter in our case since the temperature difference is not known a priori. Note that in the asymptotic limit where $R_F$ tends to one, $Ra_{\Delta T_v}$ and $Ra_{\phi }$ are getting proportional to each other with the Nusselt number (as defined in Rayleigh–Bénard convection) being the proportionality coefficient (Cioni, Ciliberto & Sommeria Reference Cioni, Ciliberto and Sommeria1997). The classical scaling $\Delta T_v\sim Ra_{\Delta T_v}^{-1/3}$ would equivalently give $Ra_\phi ^{-1/4}$, which is not compatible with our result (3.17). Our regime is closer to the regime $I_l$ predicted by Grossmann & Lohse (Reference Grossmann and Lohse2000) in which an energetic approach is used to estimate the viscous and thermal dissipation rates for the $Pr<1$ case. In the $I_l$ regime, dissipation rates are dominated by their boundary layer contributions, therefore, the $I_l$ regime is obtained at relatively low $Ra_{\Delta T_v}$, i.e. when the turbulence is sufficiently underdeveloped, so that the dissipation is mostly concentrated within the boundary layers.

Considering a homogeneous bulk (reached at sufficiently high Rayleigh–Roberts numbers) and thanks to the $I_l$ regime of Grossmann & Lohse (Reference Grossmann and Lohse2000), the Nusselt scaling $Nu\sim \delta _b^{-1}\sim Ra_{\Delta T_v}^{1/4}$, where $\delta _b$ is the thickness of the bottom thermal boundary layer, leads to $\Delta T_v \sim Ra_{\phi }^{-1/5}$ consistent with (3.17). It is noteworthy that analogy with the $I_l$ regime only makes sense if one assumes an equivalence between the upper and lower thermal boundary layers, which is not necessarily the case here since the velocity boundary conditions are mixed (free slip at the top, no slip at the bottom). In addition, all the results of the study of Grossmann & Lohse (Reference Grossmann and Lohse2000) are obtained with an imposed temperature difference contrary to the imposed heat flux here.

When $R_F$ tends to $1$, our configuration is close to the one studied by Chapman & Proctor (Reference Chapman and Proctor1980), where a constant flux is imposed at the bottom surface and goes out by the top surface ($R_F=1$) in a horizontally infinite domain ($\varGamma = \infty$). This limiting case can be explored with a Cartesian box of size $2\times 2\times 1$ periodic in both horizontal directions, with the same imposed flux at the top and bottom and with no-slip and free-slip conditions, respectively, at the bottom and top. Figure 7 shows $\Delta T_v$ measured after the statistically stationary state is obtained. We also plot the temperature difference $\Delta T_v$ predicted by the $I_l$ regime scaling, keeping the prefactor determined for rigid boundaries and imposed temperatures (Grossmann & Lohse Reference Grossmann and Lohse2000). We find good agreement between the local Cartesian set-up and the $I_l$ regime, with the same Rayleigh exponent being found. A slight difference in the prefactor is nevertheless noted. Modifying the thermal and velocity boundary conditions did not have a significant impact, with only a minor change in the prefactor. Our $R_F=0.9$ case is also found to be in good agreement with the $I_l$ regime, but with a small difference. Here $10\,\%$ of the flux exits through the side, leading to a disruption in the vertical temperature gradient and, therefore, a difference in the temperature difference from bottom to top caused by the baroclinic flow.

Another way to verify the relevance of the $I_l$ regime is to check the bottom thermal boundary layer behaviour. The scaling law predicts that $\delta _{b} \sim Ra_{\phi }^{-1/5}$. A measure of the bottom thermal boundary layers has been done based on the temperature variance computed as

(3.18)\begin{equation} \sigma_T=\left\langle\left(T(t,r=\varGamma/2,\varphi,z)-\left\langle T(t,r=\varGamma/2,\varphi,z)\right\rangle_{\varphi}\right)^2\right\rangle_{\varphi}^{1/2} . \end{equation}

We focus here on the temperature at $r=\varGamma /2$ to avoid the side and centre areas that are more significantly affected by the overall baroclinic circulation driven at the sidewall. In figure 8(a), $\sigma _T$ as a function of the altitude $z$ is plotted for different $Ra_{\phi }$ and for $\varGamma =16$. Near the top and bottom boundaries, the temperature variance is rapidly varying, which indicates the existence of boundary layers. In the bulk $0.2< z<0.8$, the system is more homogeneous especially when $Ra_{\phi }$ is high. Vertical dotted lines indicate the thickness of the bottom boundary layer $\delta _b$. It is estimated as the position $z$ for which the temperature variance has decreased by $90\,\%$ compared with its value at the boundary. We used the temperature variance rather than the vertical temperature profile because the recirculation flow (induced by the cold side) involves a heat transport mechanism similar to the horizontal convection (Mullarney, Griffiths & Hughes Reference Mullarney, Griffiths and Hughes2004) altering the thermal boundary layer structure. In figure 8(b) the corresponding boundary layer thickness is plotted as a function of $Ra_{\phi }$ for several $\varGamma$ at $R_F=0.9$. It is independent of $\varGamma$ and scales like $\delta _b \sim Ra_{\phi }^{-0.20\pm 0.05}$ (determined by the least square method). The $I_l$ regime law (Grossmann & Lohse Reference Grossmann and Lohse2000) is also plotted and is consistent with our measurements at $R_F=0.9$ in terms of exponent, with again an offset on the prefactor presumably due to residual effects of the large-scale circulation driven at the sidewall.

Figure 8.

(a) Plot of the temperature variance with the altitude $z$ for different $Ra_{\phi }$ at $\varGamma =16$, $Pr=0.1$ and $R_F=0.9$. The irregular profile for $Ra_{\phi }=10^8$ is due to a lack of statistical samples for this very costly computation. The vertical dotted lines indicate the boundary layer altitudes based on the $90\,\%$ decrease of the temperature variance near the bottom. (b) Log-log plot of the bottom boundary layer with $Ra_{\phi }$ for different $\varGamma$. The continuous black line represents the Grossmann & Lohse (Reference Grossmann and Lohse2000) scaling corresponding to the $I_l$ regime. Empty/full symbols respectively indicate filtered/DNS simulations. The input parameters are $Pr=0.1$ and $R_F=0.9$.

3.5.1. Radial temperature gradient

First, let us compute the slope of the radial temperature profile $G_T$ as $|\langle {\mathrm {d}\!\langle T\rangle _{\varphi z}}/{\mathrm {d}r}\rangle _{r} |$ and the radial velocity modulus computed as $\sqrt {\langle \langle u\rangle _{\varphi z}^2\rangle _r}$ restricted to $r\leqslant 5/8\varGamma$ and $r\leqslant 0.95\varGamma$, respectively, for $R_F=0.1$ and $0.9$, $10^5< Ra_{\phi }<10^8$ and considering $\varGamma =4,8$ and $16$. All data are plotted in figure 12. Both the mean temperature gradient and the radial velocity do not significantly depend on the flux ratio, nor on the aspect ratio, and they seem to be anti-correlated with each other. Indeed, they both follow a power-law behaviour with $Ra_{\phi }$, with the same approximate exponent $1/3$ but positive for the radial velocity and negative for the temperature gradient. This suggests that, to understand the physical origin of the thermal gradient, a closer look at the flow structure is necessary.

Figure 12.

Log-log plot of the radial temperature gradient in (a) and of the RMS radial velocity in (b) as a function of $Ra_{\phi }$, with various values of $\varGamma$ shown with symbols and of $R_F$ shown with colours. The dash-dotted lines indicate the best-fit power law. The coloured areas are computed by taking the standard deviation of the radial temperature gradient and the square of the radial velocity in the bulk for (a,b), respectively. Empty/full symbols respectively indicate filtered/DNS simulations. The input parameter is $Pr=0.1$.

To do so, we consider the representative case $Ra_{\phi }=10^7$ and $\varGamma =8$. We compute in a $r,z$ plane and, for both regimes, $R_F=0.1$ and $R_F=0.9$, the norm of the velocity field denoted ${U_n}$, shown in the top panels of figure 13, as well as the velocity fluctuations field denoted ${u_n}^\prime$, shown at the bottom. Those fields are computed as follows:

(3.21a,b)\begin{align} {U_n}(r,z)=\big[\langle u\rangle_{\varphi}^2+\langle w\rangle_{\varphi}^2\big]^{1/2},\quad {u_n}^\prime(r,z)= \big[\langle(u-\langle u\rangle_{\varphi})^2\rangle_{\varphi}+ \langle(w-\langle w\rangle_{\varphi})^2\rangle_{\varphi}\big]^{1/2} . \end{align}

In both regimes a global circulation surrounds the whole domain, with intense mean velocities close to the boundaries. In the bulk, fluctuations dominate the flow, at least for $0< r<5$ at $R_F=0.1$ and up to the side boundary layer for $R_F=0.9$. These regions correspond to the domain with a significant radial gradient of the mean temperature, shown in figure 11. In order to understand the emergence of the radial temperature gradient, let us look at the heat transport equation. We average it in time, in azimuth, but also over a particular thickness $h$. This thickness corresponds to the altitude at which the vertical profile of the radial velocity $\langle u\rangle _{\varphi r}$ changes sign. We denote this averaging operation as $\langle \bullet \rangle _{\varphi h}$. To clarify, we do not average over the whole depth because the terms related to radial advection would vanish owing to continuity.

Figure 13.

Map in the $r,z$ plane of the mean (a,b) and fluctuating (c,d) velocity norm for (a,c) $R_F = 0.1$ and (b,d) $R_F = 0.9$. The dashed lines indicate the $r_p$ value. The input parameters are $Pr = 0.1$, $\varGamma = 8$ and $Ra_{\phi }=10^7$.

Our averaging procedure allows us to distinguish between the mean flow carrying hot fluid from the centre to the edge of the domain in the upper region $z>h$ and cold fluid advected from the edge to the centre in the bottom region $z< h$.

By decomposing the temperature and velocity fields between mean and fluctuating components, the heat equation can be written as

(3.22)\begin{equation} \frac{\partial T'}{\partial t} +(\boldsymbol{\mathcal{U}}+\boldsymbol{u'})\boldsymbol{\cdot}\boldsymbol{\nabla}(\mathcal{T}+T')= {\nabla}^2 (\mathcal{T}+T') , \end{equation}

where $\boldsymbol {\mathcal {U}}=({U},{V},{W})^\textrm {T}=\langle \boldsymbol {u}\rangle _{\varphi h}$ and $\mathcal {T}=\langle T\rangle _{\varphi h}$ respectively represent the velocity and temperature mean component and $\boldsymbol {u'}=(u',v',w')^\textrm {T}$, $T'$ the fluctuating ones.

Thus, the heat equation reads

(3.23)\begin{equation} {U}\frac{\mathrm{d}\mathcal{T}}{\mathrm{d}r}+\left\langle u'\frac{\partial T'}{\partial r}\right\rangle_{\varphi h} +\left\langle({W}+w')\frac{\partial T'}{\partial z}\right\rangle_{\varphi h}=\frac{1}{h} + \frac{1}{r}\frac{\mathrm{d}}{\mathrm{d} r}\left(r\frac{\mathrm{d}\mathcal{T}}{\mathrm{d} r}\right)+ \frac{1}{2{\rm \pi} h}\int_{0}^{2{\rm \pi}}\frac{\partial T'}{\partial z}_{|z=h} \,\mathrm{d\varphi} . \end{equation}

Neglecting the mean vertical transport (third term on the left-hand side), diffusive terms (second and third terms on the right-hand side) and the transport induced by fluctuations (second and fourth terms on the left-hand side), (3.23) reduces to

(3.24)\begin{equation} \frac{\mathrm{d}\mathcal{T}}{\mathrm{d}r} \simeq \frac{1}{\mathrm{U}h} . \end{equation}

The radial temperature gradient appears to be inversely proportional to the radial velocity. Note that our approach here was guided and validated by empirical observations, and we did not perform any formal ordering between the various terms of (3.23). In figure 14 we plot the radial temperature profiles and the radial velocity modulus profiles for both regimes $R_F=0.1$ and $R_F=0.9$, with $\varGamma =16$ and $Ra_{\phi }$ between $10^5$ and $10^8$. We also plot the theoretical radial temperature gradient estimated by (3.24). To do so, ${U}$ was estimated as the mean value of the radial velocity modulus ($\langle |\langle u\rangle _{\varphi }|\rangle _z$) when $2< r<10$ and $2< r<12$, respectively, for $R_F=0.1$ and $R_F=0.9$, i.e. when the velocity is relatively constant. It can be seen in figure 14 that when $Ra_{\phi }$ increases, the values of the radial velocity modulus profile increases while the radial temperature gradient decreases. Furthermore, we can see that our estimate (3.24) matches very well to the radial temperature profiles observed in simulations.

Figure 14.

Radial profile of the mean temperature (a,b) and of the radial velocity modulus (c,d) for different $Ra_{\phi }$ and for (a,c) $R_F=0.1$ and (b,d) $R_F=0.9$. The other input parameters are $Pr=0.1$ and $\varGamma =16$. The dash-dotted lines correspond to the theoretical slope (3.24) for the top row and to the mean value of the radial velocity modulus for the bottom row.

This model derives from the assumption that the mean flow is responsible for the radial temperature gradient. One could have argued that on the contrary, the radial temperature gradient creates the flow by a baroclinic torque: the temperature gradient would then be proportional (and not inversely proportional as we observe) to $U$. Thus, as will be further detailed in the next sections, we conclude that the side cold temperature generates the mean flow that then builds up the radial temperature gradient. Incidentally, the radial gradient scaling with $Ra_{\phi }$ shown in figure 12, $G_T\sim Ra_{\phi }^{-1/3}$, implies that it decreases faster with $Ra_{\phi }$ than the mean temperature of the system ($\langle T\rangle \sim Ra_{\phi }^{-1/5}$, see (3.1)). This hints that the radial thermal gradient is a secondary aspect of the thermal structure. At first order, the plateau scaling is the dominant factor, which explains why the scaling of the low flux ratio regime works well at high $Ra_{\phi }$ values for all $R_F$, at least in the vicinity of the sidewall.

3.5.2. Temperature plateau value

The physical reason behind the uniform radial temperature profile near the sidewall, particularly visible at low $R_F$, is not trivial. Indeed, figure 15 shows that the temperature map averaged over azimuth and time, but not over depth, is very heterogeneous along the vertical direction. A cold zone is localized at the bottom close to the edge, in close connection with the localized, strong velocity zone seen in figure 13(a,c). Indeed, close to the sidewall, there is a strong downward flow that then turns into a cold jet with a characteristic radial extent, seemingly corresponding to the plateau area on the temperature profile. The larger $Ra_{\phi }$, the more extended this area. Actually, as we shall see, when $Ra_{\phi }$ increases, the cold jet that drives the global recirculation gets more inertia and propagates more into the domain. Averaging over depth, the $T_p$ plateau value is well predicted by (3.11) resulting from the analysis of the heat transfer through the side boundary layer made in the first regime. We now focus on understanding the cold jet dynamics in order to estimate the $r_p$ parameter.

Figure 15.

Map in the $r,z$ plane of the temperature field averaged in time and azimuthal direction. The input parameters are $Ra_{\phi }=10^7$, $\varGamma =8$, $R_F=0.1$ and $Pr=0.1$.

3.5.3. Cold jet penetration length

We assume that the cold jet is a turbulent, self-similar structure, where a given amount of momentum initially injected at the side is propagating in the radial direction and heated from below by the lower plate. Following the seminal work of Morton, Taylor & Turner (Reference Morton, Taylor and Turner1956), it is well known that the thickness of jets and thermal plumes increases linearly along their propagation direction due to the turbulent entrainment of the ambient fluid (see also, e.g. List Reference List1982; Turner Reference Turner1986). Our structure can be seen as a mixture between a jet and a plume, and so it is natural to fit its thickness by a linear growth as

(3.25)\begin{equation} b = \alpha (\varGamma-r) + r_0, \end{equation}

where $r_0$ is the radius close to the sidewall and $\alpha$ a constant known as the entrainment coefficient. In figure 16 the thickness of the cold jet is plotted as a function of the radius for the case where $Ra_{\phi }=10^8$, $\varGamma =8$, and for $R_F=0$ and $0.7$. We estimate the cold jet thickness at a given $r$ by seeking the depth where the ratio between the radial velocity (averaged in time and azimuthal direction) and the maximal value of this velocity located in the cold jet is equal to $0.5$. We see the linear behaviour corresponding to the cold jet spreading starting near the edge and stopping where $r\approx 5/8 \varGamma$ for $R_F=0$ and $r\approx 0.8\varGamma$ for $R_F=0.7$. The experimental entrainment coefficient ranges between $0.091$ and $0.121$. These values are consistent in terms of order of magnitude with values reported in the jets/plumes literature for other configurations (e.g. Carazzo, Kaminski & Tait Reference Carazzo, Kaminski and Tait2006; Fischer et al. Reference Fischer, List, Koh, Imberger and Brooks2007; Van Reeuwijk & Craske Reference Van Reeuwijk and Craske2015). Note that it does not make sense to be more quantitative here, since each configuration gives a different value for $\alpha$.

Figure 16.

Plot of the cold plume thickness $b$ as a function of the radius. The blue points are the data, while the continuous line in orange is the best fit. The insets show the radial velocity normalized by the maximal velocity inside the jet, as a function of the height normalized by the jet thickness. Several radii indicated by the colour bar are shown going from near the edge (red) to the bulk (blue). Here $Ra_{\phi }=10^8$, $\varGamma =8$, $Pr=0.1$ and $R_F=0$ and $0.7$, respectively, for (a,b).

Furthermore, in the encapsulated graphs of figure 16, we plot the normalized radial velocity with the height normalized by the thickness of the jet. The radial velocity is normalized by the maximum values observed within the jet denoted $v_{max}$, which are determined using the expression

(3.26)\begin{equation} v_{max}(r)=\mathrm{max}\left[|\left\langle u(t,r,\varphi,0\leqslant z\leqslant b) \right\rangle_{\varphi}|\right]. \end{equation}

This analysis is made for various radii ranging from the edge to the bulk. For both flux ratios in figure 16, all velocity profiles collapse on a unique profile that indicates a self-similar structure. In addition, when considering a fixed value of $Ra_{\phi }$ and $\varGamma$, a lower flux ratio results in a larger radial extent of the cold jet (not shown). The propagation of a jet on a heated plate is a well-studied topic (Schneider & Wasel Reference Schneider and Wasel1985; Steinrück Reference Steinrück1995; Higuera Reference Higuera1997; Fernandez-Feria, del Pino & Fernández-Gutiérrez Reference Fernandez-Feria, del Pino and Fernández-Gutiérrez2014; Fernandez-Feria & Castillo-Carrasco Reference Fernandez-Feria and Castillo-Carrasco2016), often examined in the context of a jet with uniform inlet velocity on a uniformly heated (imposed temperature) plate. The extension of the jet is determined by the competition between inertia and buoyancy forces. The velocity of the jet decreases due to entrainment of surrounding fluid, reducing its inertia, while lateral heating generates a vertical buoyancy force. The transition point is typically defined when the inertia is balanced by the buoyancy forces, often using a local Froude number to quantify this transition (Daniels & Gargaro Reference Daniels and Gargaro1993; Fernandez-Feria & Castillo-Carrasco Reference Fernandez-Feria and Castillo-Carrasco2016),

(3.27)\begin{equation} \mathcal{F}r(r)=\left (\frac{v_{max}^2(r)}{\beta g \Delta T_{jet}(r)b^3(r)}\right)^{1/2} . \end{equation}

Here, $\Delta T_{jet}$ represents the temperature difference between the bottom and the outside of the cold plume $(\langle T\rangle _{\varphi }(r,z=0) -\langle T\rangle _{\varphi }(r,z=b) )$ at a given $r$. In figure 17 the local Froude number is plotted as a function of the radius for several $R_F$ going from $0$ to $0.9$. Two areas can be distinguished, near the edge where the cold jet inertia dominates ($\mathcal {F}r>1$) and in the bulk where buoyancy finally dominates ($\mathcal {F}r<1$). In order to reach the equilibrium between buoyancy and inertia ($\mathcal {F}r\approx 1$), the cold jet travels a greater distance as the flux ratio is lower. When the Froude number falls below a certain threshold (function of $Pr$, $\varGamma$ and $Ra_{\phi }$), the cold jet is unable to be maintained, resulting in the radially inward motions being hindered by the adverse pressure gradient caused by buoyancy (Daniels & Gargaro Reference Daniels and Gargaro1993; Higuera Reference Higuera1997). In figure 18 we plot the radial temperature profile ($\langle T\rangle _{\varphi z}(r)$) rescaled by the plateau temperature ($T_p$) as a function of the radius rescaled by the radius where $\mathcal {F}r=1$, for various $R_F$. We observe that as $r/r(\mathcal {F}r=1)$ approaches 1, the rescaled temperature profiles converge towards the plateau temperature. This convergence is particularly clear when the flux ratio is low. As $R_F$ increases, the cold jet region becomes less meaningful, and in the asymptotic case where $R_F=1$, it effectively disappears since all the heat flux is evacuated at the top. The transition radius in (3.20) is thus well determined by the parameter $r_p = r(\mathcal {F}r=1)$. We might notice on figure 18 that accounting for this threshold is satisfying at first order only: while the critical Froude number is close to 1, its exact threshold value might also slightly depend on the aspect ratio and the Rayleigh number, which is left to future works.

Figure 17.

Local Froude number as a function of radius for $Ra_{\phi }=10^8$, $\varGamma =8$, $Pr=0.1$ and $R_F$ going from $0$ to $0.9$. The dotted horizontal line indicates $\mathcal {F}r=1$ and the encapsulated plot represents a zoom on the area where $4.5< r<7$.

Figure 18.

Radial temperature profile rescaled by the plateau temperature as a function of the radius rescaled by the radius where $\mathcal {F}r=1$, for $R_F$ going from $0$ to $0.9$. The input parameters are $Ra_{\phi }=10^8$, $\varGamma =8$ and $Pr=0.1$.