1. Introduction
Rayleigh–Bénard convection (RBC) has been studied extensively in the past few decades (Tilgner, Belmonte & Libchaber Reference Tilgner, Belmonte and Libchaber1993; Belmonte, Tilgner & Libchaber Reference Belmonte, Tilgner and Libchaber1994; Niemela et al. Reference Niemela, Skrbek, Sreenivasan and Donnelly2000; Funfschilling et al. Reference Funfschilling, Brown, Nikolaenko and Ahlers2005; Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009; Lohse & Xia Reference Lohse and Xia2010) for its vastly broad applications in geophysics (Zhong & Zhang Reference Zhong and Zhang2005; Meakin & Jamtveit Reference Meakin and Jamtveit2010; Huang et al. Reference Huang, Zhong, Zhang and Mertz2018; Wang & Zhang Reference Wang and Zhang2023; Huang Reference Huang2024), solar physics (Nordlund Reference Nordlund1985; Stein & Nordlund Reference Stein and Nordlund1989; Dudok de Wit et al. Reference Dudok de Wit2020), atmospheric science (Emanuel Reference Emanuel1994; Salmon Reference Salmon1998) and ocean dynamics (Jorgensen & LeMone Reference Jorgensen and LeMone1989; Vallis Reference Vallis2006). In RBC, a domain of fluid is heated from below and cooled from above, so the near-boundary fluid is either warmed up or cooled off, leading to thermal convection. As the warm and cold fluids move between the top and bottom plates, they effectively create an upward heat flux. At a fixed bottom-to-top temperature difference 
$\Delta T$, this convective heat transfer rate 
$Q_{conv}$ is much greater than that which would pass through a quiescent fluid body, 
$Q_{cond}$, which is the so-called conductive heat transfer rate. Defined as the ratio between the convective and conductive heat transfer rates, the Nusselt number 
$Nu{} = Q_{conv}/ Q_{cond}$ measures the heat-passing capability in RBC, which depends on imposed control parameters such as the Rayleigh number 
$Ra{} = \alpha g L^3 \Delta T / (\kappa \nu )$ and the Prandtl number 
$Pr {} = \nu /\kappa$. Here, 
$g, L, \alpha, \kappa, \nu$ are the acceleration due to gravity, fluid depth, thermal expansion coefficient, thermal diffusivity and kinematic viscosity, respectively.
Once a fluid and its containing geometry are given, the heat flux is uniquely determined by the imposed temperature difference 
$\Delta T$. In particular, the famous scaling relationship 
$Nu{}\propto Ra^{\beta }$ has been observed in the range from 
$Ra{}=10^6$ to 
$Ra{} = 10^{14}$ (Niemela et al. Reference Niemela, Skrbek, Sreenivasan and Donnelly2000; Funfschilling et al. Reference Funfschilling, Brown, Nikolaenko and Ahlers2005), despite local deviations. Grossmann and Lohse (Grossmann & Lohse Reference Grossmann and Lohse2000; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013) developed a theory that incorporates the heat transfer contributed by both the thermal boundary layers and the bulk, revealing a 
$Nu{}$–
$Ra{}$ relationship that has been found to be consistent with experiments in a large range of parameters.
At a given Rayleigh number, a controllable 
$Nu{}$ is often desired; anomalies that deviate from the usual scaling laws found in nature also demand explanations (Barry Reference Barry1992). In the past, there have been many attempts to modify the 
$Nu{}$–
$Ra{}$ dependency. For example, surface roughness was added to the boundary, which modifies the boundary layer structure and enhances the Nusselt number (Du & Tong Reference Du and Tong1998; Jiang et al. Reference Jiang, Zhu, Mathai, Verzicco, Lohse and Sun2018). Adding rotation to the entire fluid changes the bulk flow structure (Stevens et al. Reference Stevens, Zhong, Clercx, Ahlers and Lohse2009; Zhong et al. Reference Zhong, Stevens, Clercx, Verzicco, Lohse and Ahlers2009; Zhong & Ahlers Reference Zhong and Ahlers2010) and increases the Nusselt number at moderate rotation rate. Stronger confinement on the convective fluid increases the flow structure coherence (Xia & Lui Reference Xia and Lui1997; Chong & Xia Reference Chong and Xia2016; Huang & Xia Reference Huang and Xia2016), and leads to a higher Nusselt number. Bao et al. (Reference Bao, Chen, Liu, She, Zhang and Zhou2015) reported a strong 
$Nu{}$ enhancement when vertical partition walls are inserted in the bulk fluid. There, the convective fluid self-organizes and circulates around these partitions, intruding into the thermal boundary layers and leading to increased heat flux.
Inspired by the fact that large-scale circulation can be induced by an imposed horizontal temperature gradient (Belmonte, Tilgner & Libchaber Reference Belmonte, Tilgner and Libchaber1995), we have demonstrated that the addition of horizontal flux can greatly enhance the large-scale circulation and heat transfer in RBC. In that work (Huang & Zhang Reference Huang and Zhang2022), a pair of heating–cooling vertical walls is added so a net horizontal flux can flow through the RBC cell. While such a study is numerically feasible, it is difficult to add and remove the same amount of heat experimentally. We thus focus only on the side-heating effect here, and demonstrate experimentally how an added horizontal flux can change the dynamical and thermal properties of RBC.
In this study, a classical RBC system in a cubical domain is perturbed by a heat flux injected from one of its four vertical sides, as shown in figure 1(a). During each experiment, the top cooling temperature 
$T_t$ is fixed, while 
$Ra{}$ and 
$Nu{}$ are measured as functions of the side and bottom heating powers 
$Q_s$ and 
$Q_b$. To the best of our knowledge, this unconventional configuration has not been studied in the past, despite many experiments having explored configurations such as sidewall heating–cooling without bottom heating (Belmonte et al. Reference Belmonte, Tilgner and Libchaber1995), horizontal convection (Miller Reference Miller1968; Wang & Huang Reference Wang and Huang2005; Hughes & Griffiths Reference Hughes and Griffiths2008), and different sidewall boundary conditions (Stevens, Lohse & Verzicco Reference Stevens, Lohse and Verzicco2014).
Figure 1. A cubical thermal convection cell with sidewall heating is filled with water as the working fluid. (a) Schematic showing the convection cell and the side heater. Thermistors (red dots) are distributed to measure local temperature at various locations. (b) Shadowgraph visualizing the density difference and thus the flow structures, with bottom heating power 
$Q_b = 100$ W, and side-heating power 
$Q_s = 40$ W. Heat injected from the bottom and the side exits the fluid through the top cooling plate, while the clockwise large-scale circulation is enhanced and dictated by the side heating.
In what follows, we will introduce the experimental set-up in § 2, and show the main results regarding the flow patterns and heat transfer properties of the side-heated RBC in § 3. And finally, in § 4, we will summarize our results and outline future research plans.
2. Experimental set-up
As shown in figure 1(a), our experiments are carried out in a cubical cell of side length 
$L = 20$ cm. The top and bottom plates are made of anodized aluminium that ensures high thermal conductivity and uniform temperature distribution within each plate. In the top plate, coolant water is circulated at a constant temperature 
$40\,^{\circ }$C that is regulated by a water circulator. An electrical film heater is embedded in the bottom plate, providing bottom heating to the fluid. To ensure temperature uniformity, both the top and bottom plates are 2.54 cm thick and made of aluminium. Another film heater covered by an aluminium sheet, 2 mm in thickness and in contact with the fluid, is attached to one side of the vertical walls. The covering aluminium sheet is intended to improve temperature uniformity, whose top and bottom edges are cut away by 5 mm in order to prevent a ‘short circuit’ of heat flowing from/to the boundary layers. During all experiments excluding the flow visualization, insulating materials cover the RBC cell to minimize heat leak. Local temperature is measured by thermistors distributed in the convection cell. As shown in figure 1(a), two thermistors measuring the bottom temperature are embedded 0.5 mm below the bottom surface. One thermistor is located at the centre of the side heater to monitor the side temperature, and two thermistors are mounted on a support that can traverse vertically to measure the temperature profile of the fluid.
We perform all temperature and flow measurements at each 
$(Q_b,Q_s)$, while varying them independently in the range 0–120 W. While the heat mainly transfers through the fluid, there are three possible heat leaks that may affect our measurements: (1) the conduction beneath the bottom plate; (2) the vertical conduction of heat through the sidewalls; (3) the horizontal heat leak from the sidewalls to the air.
To minimize the conductive heat leak, the bottom plate is placed on top of layers of insulating foam, beneath which a compensating heater is installed so that the temperature gradient in the foam can be monitored and minimized by a PID algorithm. The sidewalls are made of 
$5$ mm thick acrylic, whose thermal conductivity is approximately 
$0.2\ {\rm W}\ {\rm m}^{-1}\ {\rm K}^{-1}$. Considering that the typical temperature difference between top and bottom is about 
$10\,^\circ$C, the vertical conducting power through the sidewalls is estimated as 
$0.04$ W, much smaller than the convective power carried by the fluid.
The main contribution to heat leakage is in fact the heat transfer between the sidewalls and the surrounding air. Typically, the convection cell operates at a bulk temperature 
$40\,^\circ$C, and the surrounding air is at the room temperature 
$25\,^\circ$C. The convection cell, when operating, is covered by many layers of aluminium-coated bubble wrap whose total thickness is of the order of 
$10$ cm, and whose conductivity is approximately 
$0.02\ {\rm W}\ {\rm m}^{-1}\ {\rm K}^{-1}$. This allows us to estimate the total side leaking power as 
$0.5$ W, which is much smaller than the minimum bottom heating power (
$10$ W) used in the study. Thus the maximum error that we may have in measuring 
$Nu$ is 5 %, and this error decays inversely with 
$Q_b$.
Whenever a control parameter is changed, the system runs for 4 hours to reach a dynamical equilibrium, and each measurement takes another 4 hours to collect data. With degassed water as the working fluid, the system works at a Rayleigh number in the range 
$3\times 10^8$ to 
$5\times 10^9$, and Prandtl number in the range 
$3.4$ to 
$4.1$. At this 
$Ra{}$, the convection is turbulent and the large-scale circulation develops spontaneously.
As shown in figure 1(b), a shadowgraph of the convecting fluid reveals local density differences and thus provides a visualization of the convective flows. A convex lens with focal length 20 cm converts light from a point source to near-parallel light, which passes through the convection cell and casts a shadow on a translucent screen. Bulk flow speed 
$U$ can be inferred by tracing the thermal plumes at a given region, which we choose to be 1 cm from the centre of the heated sidewall. This flow speed 
$U$ represents the strength of the large-scale circulation. Typically, trajectories of 40 thermal plumes are timed and metered during each experiment, so both the mean flow speed and its standard deviation can be calculated.
3. Results
3.1. Enhanced large-scale circulation and the modified 
$Nu{}$–$Ra{}$ relationship
We expect the addition of side heating to enhance the large-scale circulation, and perhaps also to increase the vertical heat transfer rate. As shown in figure 1(b), the fluid near the sidewall is heated and becomes lighter, thus forming an upwelling jet that feeds a clockwise large-scale circulation. Therefore, the thermal energy provided by sidewall heating is partly converted into the fluid kinetic energy that enhances the large-scale circulation.
With a constant bottom heating power 
$Q_b = 100$ W and side-heating power in the range 
$Q_s = 10\unicode{x2013}100$ W, the flow speed can be non-dimensionalized as the Reynolds number 
$Re{} = UL/\nu$ shown in figure 2(a). In figure 2(a), the sidewall power is also non-dimensionalized as the power Grashof number 
$Gr{}_q = \alpha gL^2 Q_s/(\lambda \nu ^2)$, which is a measure of the relative strength between buoyancy and viscous effects (Schlichting & Gersten Reference Schlichting and Gersten2003). Here, 
$\lambda$ is the fluid thermal conductivity. In comparison with the Grashof number 
$Gr{} = \alpha g L^3 (T_s - T_\infty )/\nu ^2$, the power Grashof number is defined directly through the input heating power 
$Q_s$ instead of the sidewall temperature difference 
$T_s - T_\infty$. Far field temperature 
$T_\infty$ in our case can be taken as the bulk temperature.
Figure 2. Flow speed and heat transfer measured by Reynolds and Nusselt numbers. (a) Plume speed increases with side-heating power. With bottom heating fixed at 100 W, the Reynolds number 
$Re{} =UL/\nu$ measured at 1 cm from the sidewall centre shows a monotonic dependence on the power Grashof number 
$Gr_q$, consistent with the power-law scaling 
$Re{}\propto Gr{}_q^{1/5}$ from the boundary layer theory (Schlichting & Gersten Reference Schlichting and Gersten2003). The inset shows a logarithmic-scale plot of Reynolds number versus power Grashof number. (b) Nusselt number 
$Nu{}$ measured at various values of 
$Ra{}$ and side-heating power 
$Q_s$. The bottom heating power is 
$Q_b = 10,20,\ldots, 120$ W. Without side-heating, 
$Nu{}$ shows a classical power-law dependence on 
$Ra{}$. Fixing the bottom power, 
$Nu{}$ is observed to decrease with increasing side power, while 
$Nu{}\propto Ra{}^{-1}$. Coloured solid curves are the 
$Nu{}$–
$Ra{}$ curve at each fixed 
$Q_s$ predicted by a theory introduced in § 3.2.
The monotonic increase of 
$Re{}$ with 
$Gr{}_q$ shows that the bulk flow speed is indeed accelerated by the side-heating power. From the boundary layer theory (Schlichting & Gersten Reference Schlichting and Gersten2003), the buoyancy-driven flow speed near a vertical wall scales with the Grashof number 
$Gr{}$ as 
$U \propto Gr{}^{1/4}$. Using the relationship between the two Grashof numbers (Schlichting & Gersten Reference Schlichting and Gersten2003), namely 
$Gr{} \propto Gr{}_q^{4/5}$, we have the scaling law 
$Re{}\propto Gr{}^{1/4} \propto Gr{}_q^{1/5}$. This is found to be consistent with our experimental data, as shown in figure 2(a). As 
$Q_s$ reduces to 0, flow velocity drops to the conventional RBC value, which corresponds to 
$Re{}\sim 1000$ in our range of 
$Ra{}$ (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). Without side heating, the direction of large-scale circulation becomes arbitrary but stays along one of the vertical diagonal planes of the convection cell.
The Nusselt number in the classical cubical RBC is 
$Nu{} = Q_b / (\lambda \,\Delta T\,L)$, and figure 2(b) shows the Nusselt number measured at various values of 
$Q_s$ and 
$Q_b$. With sidewall heating added, the bulk circulation is enhanced as expected, but the bottom temperature 
$T_b$ and the resulting 
$\Delta T = T_b - T_t$ are also found to increase. By definition, 
$Ra{} \propto \Delta T$ and 
$Nu{} \propto 1 / \Delta T$, so an increased 
$\Delta T$ due to side heating causes 
$Ra{}$ to increase and 
$Nu{}$ to decrease. Moreover, with 
$Q_b$ fixed at a constant, the product 
$Nu{}\,Ra{}$ is independent of 
$\Delta T$. Thus the dotted lines in figure 2(b) have a common slope 
$-1$ in the log-log scale.
As a reference, the 
$Nu{}$–
$Ra{}$ measurement without sidewall heating, 
$Nu{}|_{Q_s = 0} = Nu{}_0$ in figure 2(b), agrees well with previous experiments (Funfschilling et al. Reference Funfschilling, Brown, Nikolaenko and Ahlers2005; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009) and follows a power-law scaling 
$Nu{} = 0.12\, Ra{}^{0.304}$ that is consistent with the Grossmann–Lohse theory. When 
$Q_s \neq 0$, the observed data deviate from the power-law relation – a simple scaling relationship is absent, particularly at large 
$Q_s$.
3.2. Temperature measurement explaining the $Nu{}$–$Ra{}$ scaling
To investigate the modified 
$Nu$–
$Ra{}$ relationship due to sidewall heating, the vertical temperature distribution is examined along the central vertical axis of the RBC cell shown in figure 1(a). Figure 3(a) shows several temperature profiles measured at the same bottom power 
$Q_b = 100$ W but different side power 
$Q_s$, where we have defined the heating power ratio as 
$S = Q_s/Q_b$. On each curve, temperature changes linearly within thermal boundary layers, through which heat transfers by conduction. In the bulk, strong turbulent mixing makes the bulk temperature 
$T_c$ nearly uniform (Belmonte et al. Reference Belmonte, Tilgner and Libchaber1994; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009), which, together with the bottom temperature 
$T_b$, increases with 
$S$.
Figure 3. Vertical temperature profiles measured at 
$Q_b = 100$ W and various values of 
$Q_s$, and along the centre vertical line as shown in figure 1(a). (a) Temperature at different heights. The bulk and bottom temperatures are found to increase with the heating power ratio 
$S = Q_s/Q_b$. Top, bottom and side temperatures (
$T_t, T_b$ and 
$T_s$) are marked with ticks and triangles. (b) Zoom-in profiles of the bottom thermal boundary layer: 
$T_b$ increases with the bulk temperature, while the temperature gradient stays nearly constant. (c) Zoom-in profiles of the top thermal boundary layer. With top temperature fixed, increased bulk temperature leads to a higher temperature gradient within the boundary layer. (d) Thermal boundary layer (BL) thicknesses 
$\delta$ of the top and bottom boundary layers. (e) With 
$Q_b$ fixed, the increase of bulk temperature 
$\Delta T_c = T_c|_S - T_c|_{S=0}$ has a 
$4/5$ power-law scaling relationship to the side-heating power 
$Q_s$. (f) Nusselt number as a function of power ratio 
$S$, i.e. 
$Nu{}_0/Nu{} = 1+ 0.25 S^{4/5}$. Plots (a–e) share the same colour scheme; (f) uses the colour scheme and definition of 
$Nu{}$ and 
$Nu_0$ as in figure 2(b).
Zooming in near the bottom, figure 3(b) shows that the bottom temperature 
$T_b$ increases with 
$Q_s$, while the temperature gradient 
$\partial T / \partial y$ stays approximately constant within the boundary layer. This is due to the fact that 
$\partial T / \partial y = -Q_b / (\lambda L^2)$ and the bottom power 
$Q_b$ is fixed at a constant. The bottom thermal boundary layer thickness 
$\delta _b = 1.4 \pm 0.1$ mm (or 
$\delta _b / L = (0.70 \pm 0.05 )\%$) is roughly constant for varying 
$S$, as shown in figure 3(d). This, together with a constant 
$\partial T/\partial y \sim \Delta T_b / \delta _b$, indicates that the temperature difference between the bottom and the bulk 
$\Delta T_b = T_b - T_c$ is not changing with 
$S$ either, as shown in figure 3(b). Figure 3(c) shows the zoom-in temperature profiles near the top plate. The addition of side heating 
$Q_s$, which has to leave the system through the top plate, leads to a higher temperature gradient 
$\partial T / \partial y = -(Q_b+Q_s)/(\lambda L^2)$ in the top thermal boundary layer. In our experiment, the top thermal boundary layer thickness is 
$\delta _t = 1.1 \pm 0.1$ mm (or 
$\delta _t / L = (0.55 \pm 0.05)\%$), as shown in figure 3(d).
Interestingly, the top thermal boundary layer is thinner than the bottom in figure 3(d), even when 
$S=0$. This is the opposite of the classic RBC, where the non-Boussinesq effect of working fluid makes the bulk warmer and the bottom thermal boundary layer thinner (Zhang, Childress & Libchaber Reference Zhang, Childress and Libchaber1997). We have confirmed that this anomaly is a consequence of adding the aluminium side heater, which both conducts heat and modifies the flow structure even when the heating is off. For 
$S \neq 0$, the boundary layer thickness also varies in space, so the temperature distribution shown in figure 3 only reflects a local profile covered by the range of moving thermistors. To better understand the temperature and flow distributions in the side-heated RBC, we are currently working on a numerical study that may provide further insight.
As the only variable in 
$Nu = Q_b / (\lambda \,\Delta T\, L)$ is 
$\Delta T$ when holding 
$Q_b$ constant, understanding how 
$\Delta T$ depends on 
$S$ can directly explain how 
$Nu{}$ depends on 
$S$. From the data shown in figure 3(b), the bottom to bulk temperature difference stays unchanged while the bulk temperature increases with 
$S$. Therefore, the increase of bulk temperature 
$\Delta T_c (S)= T_c|_S - T_c|_{S=0}$ directly contributes to an increased bottom–top temperature difference 
$\Delta T$. As shown in figure 3(e), the bulk temperature change has a scaling 
$\Delta T_c \propto S^{4/5}$, which is a direct consequence of the scaling 
$Gr{} \propto Gr{}_q^{4/5}$ discussed earlier, in § 3.1.
We therefore estimate 
$\Delta T = (1 + \gamma S^{4/5})\, \Delta T |_{S=0}$, where 
$\gamma$ is a positive constant. Substituting this into the definition of the Nusselt number, we have 
$Nu{}_0/Nu{}(S) = \Delta T/\Delta T|_{S=0} = 1 + \gamma S^{4/5}$. Plotting 
$Nu{}_0/Nu{} - 1$ against 
$S^{4/5}$, all the data points in figure 2(b) land on a straight line, as shown in figure 3(f), whose slope is 
$\gamma \approx 0.25$. We can also apply this relation to 
$S = Q_s/Q_b$ with fixed 
$Q_s$ and varying 
$Q_b$, which is shown as the 
$Ra{}\unicode{x2013}Nu{}$ curve for each 
$Q_s$ in figure 2(b).
The above analysis provides a relationship between 
$Nu{}$ and 
$S$, thus suggesting that the heat transfer in RBC can be controlled by simply changing 
$S$. However, an obvious question remains: why does a stronger large-scale circulation shown in figure 2(a) lead to a reduced 
$Nu{}$ for side-heated RBC? In the next subsection, we will measure the true heat flux by redefining the Nusselt number, and show how large-scale circulation enhances the heat transfer of side-heated RBC.
3.3. Redefined Nusselt number and its enhancement
In this subsection, we redefine the Nusselt number by counting all heat inputs to the RBC system as the total convective heat 
$Q^*_{conv}$, and compare it with the overall conductive heat 
$Q^*_{cond}$ associated with the wall temperature distribution. The resulting analysis will show that the newly defined Nusselt number 
$Nu{}^* = Q^*_{conv} / Q^*_{cond}$ indeed increases with 
$S$.
As we have argued above, the sum of the side and bottom heat has to flow out through the top plate. In terms of power, the total convective power through the cell is 
$Q^*_{conv} = Q_t = Q_b + Q_s$, where 
$Q_t$, 
$Q_b$ and 
$Q_s$ represent the magnitudes of power passing through the top, bottom and side, respectively. The side heating also leads to a side temperature 
$T_s > T_c> T_t$, and it contributes to the total conductive heat flowing through the top plate. Hence 
$Q_{cond}^*$ has to be determined by solving a steady-state heat equation with 
$T_b$, 
$T_t$ and 
$T_s$ satisfying Dirichlet boundary conditions for the bottom, top, and one of the sidewalls, and 
$\partial _n T = 0$ for the other three sidewalls. To prevent short-circuiting, the heating sidewall is separated into three regions: one central region with constant temperature 
$T_s$, and two stripes on its top and bottom with adiabatic boundary conditions, whose size matches the experimental configuration 5 : 190 : 5 mm. The steady-state heat equation 
$\partial _x^2 T + \partial _y^2 T = 0$, together with these mixed boundary conditions, can be solved numerically. Here, we use a finite elements package COMSOL Multiphysics with MATLAB Livelink, and compute the conductive heat in two dimensions, as we assume no significant structural change in the third direction. The overall conductive heat transfer rate, computed along the top boundary, is thus determined as 
$Q^*_{cond} = - \lambda L \int _0^{L} ({\partial T}/{\partial y}) (x, L) \,{{\rm d}\kern0.7pt x}$.
As shown in figure 4(a), the redefined 
$Nu{}^*=Q^*_{conv}/Q^*_{cond}$ for each 
$Ra{}$ and 
$S>0$ lies above the unperturbed 
$Ra{}$–
$Nu{}_0$ curve (dashed line in figure 4a), indicating that side heating indeed results in a higher heat transfer rate. At large 
$Ra{}$ (small 
$S$), the contribution from the bottom heating 
$Q_b$ dominates the overall heat transfer, so the classical 
$Nu{}_0$–
$Ra{}$ relationship becomes the asymptotic limit. At small 
$Ra{}$ (large 
$S$), however, the redefined 
$Nu{}^*$ deviates from the conventional power-law scaling. The degree of this enhancement, measured as 
$Nu{}^*/Nu{}_0$ in figure 4(b), increases monotonically with 
$S$ and reaches enhancement 
$66\,\%$ at 
$S=9$ – the highest 
$S$ reached in our experiments. We attribute this 
$Nu{}^*$ boost to the strengthening of large-scale circulation due to side heating, which enhances the bulk mixing and perhaps thins the boundary layers (figure 3d).
Figure 4. Redefined Nusselt number 
$Nu{}^*$ as the ratio of total convective heat to total conductive heat. (a) With redefinition, data shown in figure 2(b) are now above the 
$Ra{}$–
$Nu_0$ curve when 
$Q_s>0$, showing an enhancement of heat transfer. (b) The ratio 
$Nu{}^*/Nu_0$ increases monotonically with the heating power ratio 
$S = Q_s/Q_b$, reaching the highest enhancement 
$66\,\%$ at 
$S = 9$ in the experiment. Plots (a,b) share the same colour scheme as in figure 2(b).
4. Discussion
Through laboratory experiments, we have investigated the effects of adding side heating to the classical RBC system. In particular, we have found an empirical law 
$Nu{}(S) = Nu{}_0 (1 + \gamma S^{4/5})^{-1}$, which deviates from the classical 
$Nu{}$–
$Ra{}$ scaling. We further show that a redefinition of 
$Nu{}$ is required to capture the unconventional boundary conditions in our study, and the redefined 
$Nu{}^*$ increases monotonically with the side-heating power, thus allowing for a direct control to the RBC.
Such controllability was also demonstrated in Huang & Zhang (Reference Huang and Zhang2022), where adjusting the horizontal heat flux leads to a response of 
$Nu{}$ similar to figure 4(a). In that work, 
$Nu{}$ also increases monotonically with 
$S$, with the highest relative enhancement achieved near the critical Rayleigh number. In the limit 
$S\to 0$, the horizontal flux becomes negligible compared to the vertical one, so 
$Nu{}\to Nu {}_0$ and the RBC converges to the classic configuration asymptotically. On the other hand, taking 
$S\gg 1$ would bring the system closer to the configuration of Belmonte et al. (Reference Belmonte, Tilgner and Libchaber1995), which is an RBC that is turned 
$90^\circ$ to the side.
Thus both the present experiment and the previous numerical simulation (Huang & Zhang Reference Huang and Zhang2022) provide a simple means of controlling the RBC – adjust the horizontal heat flux, and the vertical flux responds accordingly. In electronics, current is a direct analogy to the heat flux in thermal dynamics. Therefore, our side-heated RBC functions much like an electronic NPN transistor, where the current flowing between the collector and the emitter can be controlled by adding a current to the base (Scherz & Monk Reference Scherz and Monk2013).
But unlike the electronic transistors, the response of our ‘thermal transistor’ is slow, usually taking hours to reach dynamical equilibrium after adjusting the side-heating power. In Huang & Zhang (Reference Huang and Zhang2022), we have experimented with a time-dependent perturbation and identified a relaxation time for the system to reach dynamical equilibrium. How does the side-heated RBC respond to a change of side-heating power? Our observation is that bulk quantities such as 
$Re{}$ and 
$Nu{}$ relax to their equilibrium values exponentially, with a relaxation time scale of hours.
With water as the working fluid, the 
$Ra{}$ range for the present study is limited. We plan to conduct numerical studies and also work with other fluids to extend this range of 
$Ra{}$. At very high 
$Ra{}$, the RBC will transition into its ultimate regime (Lohse & Shishkina Reference Lohse and Shishkina2023), and a noticeable perturbation from the side heating would require a significantly higher power in that case. However, at very low 
$Ra{}$, the fluid may stay in place if 
$Ra{}$ is below its critical value – side heating could thereby bring motion to an otherwise motionless fluid. A more detailed perturbation study is currently underway, and we hope to better understand the dynamical interplay between the boundary conditions and the bulk fluid through such fluid–structure interaction investigations.
Acknowledgements
We thank S. Childress, M. Shelley, J.-Q. Zhong and L. Ristroph for helpful discussions.
Funding
We acknowledge funding provided by the National Natural Science Foundation of China (92252204, 12272237) and the NYU Shanghai Major Grants Seed Fund.
Declaration of interests
The authors report no conflict of interest.
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