3.1. Transient and averaged spatial distribution of the temperature and electric charge

Each simulation runs for 8 million time steps, which corresponds to the dimensionless time $t^*=t/(H^2/\nu )=8.4$ , where $\nu$ is the kinematic viscosity. The time-averaged quantities, such as temperature $\theta$ , electric charge $q$ and $\textit{Nu}$ , are averaged during the latter 4 million time steps, where the system has equilibrated. First, figures 2(a) and 2(b) display the instantaneous temperature, velocity and charge fields at the characteristic time $t^* =5$ for a low Rayleigh number $Ra=100$ and a high electric Rayleigh number $T=1000$ . It is well established that, for pure thermoconvection ( $T=0$ ), a steady state is attained at $Ra = 100$ . Our results show that a sufficiently strong electric field can induce turbulent flows at low $Ra$ . As depicted in figure 2(a), two vortices emerge, and the temperature field exhibits pronounced spatial fluctuations. Correspondingly, similar behaviours can also be observed in the electric charge field. The charge plumes along the sidewalls, originating from the bottom wall, emit downward dissipative structures into the charge void region. This behaviour arises because the impinging plumes generate a corner vortex, as shown in figure 2(a). This vortex deflects the sidewall plumes toward the central region, where they descend and diffuse into the charge void region.

Figure 2.

(a) Representative transient temperature and velocity field, and (b) electric charge field at $t^*=5$ , $Ra=100$ and $T=1000$ .

To illustrate the transient and turbulent characteristics of the fluid flow and heat transfer, the temporal evolution of the Reynolds number $Re_A=U_{\mathrm{max}}H/\nu$ , where the maximum velocity $U_{\mathrm{max}}=\mathrm{max}|\boldsymbol{u}|$ , is depicted in figure 3(a). It clearly shows that the flow field varies chaotically and exhibits turbulent behaviours at $Ra=100$ and $T=1000$ . In addition, the black line in figure 3(b) shows the spatially averaged Nusselt number along the top wall, defined as $\textit{Nu}_A = -({1}/{H})\int _0^H \partial _y \theta (x,H)\mathrm{d}x$ , which characterises the total heat transfer rate. This further confirms that, at low buoyancy effect ( $Ra = 100$ ), the flow can become turbulent and exhibit high-frequency oscillations at high electric Rayleigh number $T = 1000$ . The moving average of $\textit{Nu}_A$ over a window of 1000 data points is shown as the red line in figure 3(b). Its near-constant profile indicates that, despite substantial instantaneous fluctuations, the long-term average $\textit{Nu}_A$ remains approximately steady.

Figure 3.

(a) Time series of the averaged Reynolds number $Re_A$ . (b) Time series of the averaged Nusselt number $\textit{Nu}_A$ . The red line represents the moving average of $\textit{Nu}_A$ . The values of $Ra$ and $T$ are the same as those in figure 2.

Figures 4(a) and 4(b) show the contours of the averaged dimensionless temperature and electric charge for different $Ra$ and $T$ . The case of $T=0$ corresponds to pure thermal convection without charge injection. At low and moderate Rayleigh numbers ( $Ra = 10^2$ and $10^4$ ), two thermal plumes begin to emerge near the top-left and bottom-right corners of the cavity as $T$ increases. The thermal plume originating from the bottom-right corner rises upward as the electric field becomes increasingly dominant. As $Ra$ increases further, namely $Ra=10^6$ , the influence of $T$ on the plume structure becomes negligible, as evidenced by the similar temperature contours at different $T$ . Figure 4(b) presents the corresponding electric charge fields, revealing the formation of a charge void region near the cavity centre. With increasing $Ra$ and $T$ , this void region expands with more charges diffusing into it. This indicates that the buoyancy-driven circulation and thermal plumes advect and redistribute the injected charge. Furthermore, this redistribution directly alters the spatial pattern of Coulomb force and therefore modifies the resulting flow organisation. This phenomenon clearly demonstrates the coupled interaction between buoyancy and Coulomb forces.

Figure 4.

(a) Time-averaged dimensionless temperature at different $Ra$ and $T$ . (b) The corresponding averaged dimensionless electric charge field. The square box with diagonals represents no electric charge.

Moreover, figure 4(a) shows a noticeable reduction in the thickness of the thermal boundary layer. To quantitatively assess the effect of the electric field on thermal transport, time-averaged temperature profiles $\theta$ along the vertical mid-plane are analysed under varying electric Rayleigh numbers $T$ at $Ra=10^4$ . As shown in figure 5(a), the temperature profile under an electric field can be divided into two regions: the boundary layers, characterised by steep temperature gradients near the top and bottom walls ( $y \gt 0.9$ and $y \lt 0.1$ ), and the bulk region, where the temperature remains nearly uniform within the range $y \in (0.1, 0.9)$ . It can also be observed that the bulk temperature increases with increasing $T$ . Moreover, the temperature gradient near the bottom wall becomes steeper with increasing $T$ , as illustrated in the bottom inset of figure 5(a), where the enlarged near-boundary portions of the temperature profiles $\theta$ are plotted. The inset shows that the temperature $\theta$ varies linearly with $y$ near the bottom plate. The corresponding thermal boundary layer thicknesses $\delta _t$ are calculated and presented in figure 5(b), where the inset presents the definition of $\delta _t$ indicated by the red dashed lines (Zhou & Xia Reference Zhou and Xia2013). The results reveal that $\delta _t$ progressively decreases with increasing $T$ , reflecting an enhancement in heat transfer under stronger electric fields.

Figure 5.

(a) The time-averaged temperature profile along the vertical mid-plane (white dashed line in the contour inset) at different $T$ with $Ra=10^{4}$ . The inset at the bottom left is the zoomed-in temperature profile near the bottom centre. (b) The thermal boundary layer thickness $\delta _t$ at different $T$ . The inset shows a temperature profile, where the red dashed lines illustrate the definition of $\delta _t$ .

3.2. Heat transfer characterisation and regime transition

In this section, the averaged $\textit{Nu}$ is calculated over a wide range of $Ra$ ( $0$ – $10^6$ ) and $T$ (0–1000) as shown in figure 6(a). In the absence of an electric field, namely $T=0$ , $\textit{Nu}$ first remains constant with the increase of $Ra$ . We further note that this constant value exceeds unity. This does not, however, indicate convective enhancement or the presence of fluid motion in the limit $Ra \to 0$ . With sidewall heating, the conductive reference state is intrinsically two-dimensional and exhibits a finite mean vertical temperature gradient at the top wall. As a result, $\textit{Nu}_A\gt 1$ may arise even in the absence of flow. Then, a power-law scaling of $\textit{Nu} = 0.56Ra^{0.22}$ is observed at $Ra\geqslant 10^5$ . The modification of the scaling law compared with classical RBC ( $\textit{Nu} =0.22 Ra^{0.29}$ ) is attributed to the sidewall heating, a phenomenon also reported in previous studies (Hébert et al. Reference Hébert, Hufschmid, Scheel and Ahlers2010; Huang & Zhang Reference Huang and Zhang2023). In the presence of an electric field, figure 6(a) shows that $\textit{Nu}$ also exhibits two distinct regimes. At low $Ra$ , $\textit{Nu}$ remains approximately constant for each $T$ , indicating a Coulomb-dominated regime, where the buoyancy is too weak to further influence heat transfer compared with the Coulomb force. Once $Ra$ exceeds the critical threshold $Ra_T$ , $\textit{Nu}$ increases rapidly with $Ra$ , signifying a transition to a buoyancy-dominated regime where the Coulomb effect becomes secondary. The formation of these regimes is predominantly influenced by the competing effects between the characteristic buoyancy force $\boldsymbol{F}_{\!B}=\rho g \beta \Delta \theta$ and Coulomb force under unipolar charge injection $\boldsymbol{F}_{\!C}=\boldsymbol{E}q$ . To characterise this competition and quantify the transition between different regimes, a dimensionless parameter defined as the ratio of the buoyancy to Coulomb force is proposed

(3.1) \begin{equation} B_C=\frac {\boldsymbol{F}_{\!B}}{\boldsymbol{F}_{\!C}}=\frac {\rho g \beta \Delta \theta }{\boldsymbol{E}q}, \end{equation}

Figure 6.

(a) Averaged $\textit{Nu}$ at different $Ra$ and $T$ . The results with red circles denote the averaged $\textit{Nu}$ at $Ra=0$ . (b) Normalised $\textit{Nu}$ versus $B_C$ .

After rearrangement, the above equation can be reformulated in terms of other dimensionless parameters as follows:

(3.2) \begin{equation} B_C= \frac {Ra M^2}{\textit{Pr} C T^2}. \end{equation}

Figure 6(b) shows the normalised $\textit{Nu}_A$ as a function of the proposed dimensionless parameter $B_C$ . The data collapse onto a single master curve, demonstrating that $B_C$ effectively predicts the transition regime. Depending on the value of $B_C$ , two distinct regimes are identified, which are Coulomb- (at low $B_C$ ) and buoyancy-dominated regimes (at high $B_C$ ), with the transition at $B_C \approx 1$ (black-dashed line in figure 6 b). The physical origin of this transition is a nonlinear feedback between charge transport and flow organisation. When $B_C \ll 1$ , the flow is primarily organised by charge injection: the space-charge distribution concentrates near the injecting electrode and drives strong local shear and recirculation. In this regime, the electroconvection largely dictates the large-scale organisation and near-wall mixing. Consequently, the thermal boundary layers are reshaped predominantly by Coulomb-driven recirculation, and the global heat transfer is only weakly dependent on $Ra$ . As $Ra$ increases at fixed $T$ , buoyancy-driven circulation and the thermal plume dynamics become progressively more effective in controlling the bulk transport and the near-wall temperature gradients. Importantly, this effect is not a simple superposition on electroconvection: the buoyancy-driven large-scale circulation actively reorganises the charge-transport field by advecting and stretching space-charge structures and by modifying the regions of charge accumulation and depletion within the cavity. This redistribution alters the spatial distribution of the Coulomb force and weakens the ability of charge-driven recirculation alone to control the global heat transport. The transition near $B_C = 1$ marks a reorganisation of the coupled temperature–charge–momentum dynamics: kinetic energy is transferred from charge-driven recirculation to buoyancy-supported large-scale circulation. Thermal plume emission becomes the primary mechanism governing the near-wall temperature gradients, and the Nusselt number $\textit{Nu}$ recovers a clear dependence on $Ra$ . Thus, $B_C$ delineates the transition from an electroconvection-dominated regime, where charge-driven motion controls the flow and heat transfer, to a buoyancy-dominated regime, where the buoyancy effect governs the flow structure and the scaling of the global heat flux.

Moreover, in the Coulomb-dominated regime, $B_C$ exhibits a power-law dependence on the vertical coordinate, characterised by an exponent of $-0.2$ . By rearranging (3.2) with $B_C=1$ , the transitional Rayleigh number $Ra_T$ can be written as $Ra_T=\textit{Pr} C T^2/M^2$ . Given that $\textit{Pr}$ , $C$ and $M$ remain constant, it shows that $Ra_T$ scales quadratically with $T$ , i.e. $Ra_T \propto T^2$ .

3.3. Flow structure characterisation and transition

In addition to modifying heat transfer, the coupled effects of the side-heated wall and charge injection also alter the fluid flows. Figure 7(a) presents the time-averaged streamlines under various $Ra$ and $T$ , where four types of flow structures are revealed. At $Ra = 0$ and low $T$ , no flow motion is observed, as indicated by the square symbol with diagonals. When $Ra$ and $T$ are small, a single vortex develops near the centre of the cavity. At high $Ra$ , a large vortex accompanied by a smaller one near the bottom-right corner appears. However, two horizontally aligned vortices of comparable size are observed at high $T$ . To quantify the flow structures for different $Ra$ and $T$ , a Fourier mode decomposition is employed to evaluate the relative strengths of the dominant flow modes. Specifically, instantaneous velocity fields $(u,v)$ are projected onto the Fourier basis as follows (Chen et al. Reference Chen, Huang, Xia and Xi2019)

(3.3) \begin{equation} u(x,y,t) = \sum _{m,n}\hat {u}(m,n,t)[-2\mathrm{sin}(m\pi x)\mathrm{cos}(n\pi y)], \end{equation}

(3.4) \begin{equation} v(x,y,t) = \sum _{m,n}\hat {v}(m,n,t)[2\mathrm{cos}(m\pi x)\mathrm{sin}(n\pi y)], \end{equation}

where $m$ and $n$ are positive integers. A flow mode denoted by indices $(m,n)$ corresponds to a flow structure consisting of $m$ horizontally stacked rolls and $n$ vertically stacked rolls. Therefore, the large-scale circulation (one big vortex) is characterised by the $(1,1)$ mode, whereas other flow structures, such as corner rolls, are represented by modes with higher indices (Chen et al. Reference Chen, Huang, Xia and Xi2019). In this study, we only present the flow structures corresponding to three dominant modes, as indicated in figure 7(b) (i–iii).

Figure 7.

(a) Time-averaged streamlines at different $Ra$ and $T$ . (b) The $Ra$ – $T$ phase diagram showing the dominant time-averaged Fourier mode. Different symbols denote different flow modes. The colour bar represents the ratio of the kinetic energy of the dominant flow mode to the total kinetic energy. The black dashed line denotes the $Ra_T$ line. The hollow diamond at $Ra=0$ indicates the absence of induced flow, while the asterisk marks the critical electric Rayleigh number $T_c=240$ reported in the literature, corresponding to the onset of convection in the pure electroconvection case. Insets (i), (ii) and (iii) are the three typical snapshots dominated by three modes (1, 1), (1, 2) and (2, 1), respectively. The colour bar represents the magnitude of the velocity.

The amplitude of the modes is computed as $\hat {u}(m,n,t)=\langle -2u(x,y,t)\mathrm{sin}\,(m\pi x){} \mathrm{cos}\,(n\pi y)\rangle _A$ and $\hat {v}(m,n,t)=\langle 2v(x,y,t)\mathrm{cos}\,(m\pi x)\mathrm{sin}\,(n\pi y)\rangle _A$ , and the instantaneous kinetic energy contained in a mode $(m,n)$ is given by

(3.5) \begin{equation} E(m,n,t) = |\hat {u}(m,n,t)|^2 + |\hat {v}(m,n,t)|^2. \end{equation}

The time-averaged kinetic energy of each flow mode, $E(m,n)$ , is defined as $E(m,n)=\langle E(m,n,t)\rangle _t$ . To quantify the contribution of an individual flow mode $(m,n)$ , we further introduce the parameter $\eta$ , defined as the ratio of the time-averaged energy contained in the $(m,n)$ Fourier mode to the time-averaged total kinetic energy of the flow $\langle E_{\textit{total}} \rangle$

(3.6) \begin{equation} \eta = \frac {E(m,n)}{\langle E_{\textit{total}} \rangle } \end{equation}

where $\langle E_{\textit{total}} \rangle = \sum _{(m,n)}E(m,n)$ . By definition, $0\lt \eta \leq 1$ , with values of $\eta$ close to unity indicating that the flow is dominated by a single large-scale mode that contains the majority of the kinetic energy, whereas smaller values imply that the kinetic energy is distributed across multiple flow modes.

Based on the obtained $E(m,n)$ and $\eta$ , the dominant flow modes for different $Ra$ and $T$ can be identified. According to our calculation, three modes, namely (1,1), (1,2) and (2,1), are dominant. Accordingly, a $Ra$ – $T$ phase diagram is constructed in figure 7(b) to characterise the relative strengths of the dominant flow structures and the associated mode transitions. When $Ra \neq 0$ , the flow structure shifts from mode (1,1) to (1,2) with the increase of $Ra$ at $T \leqslant 100$ . Nevertheless, at higher $T$ , the transition pathway changes, with the flow evolving from mode (2,1) to (1,2) as $Ra$ increases. At large $T$ , the flow is primarily driven by the Coulomb force, which is aligned with the imposed vertical electric field. This produces vertically directed charge-driven jets in the vicinity of the injecting electrode. After impingement on the upper wall, these jets spread laterally and form horizontally extended recirculation cells, leading to the dominance of the (2,1) mode. When $Ra$ is increased at fixed large $T$ , the buoyancy force associated with sidewall heating becomes increasingly important. Thermal plumes emitted from the heated sidewall introduce an additional source of vertical momentum in the bulk that is spatially separated from the charge-injection region. This buoyancy-driven momentum input is not aligned with the Coulomb-controlled circulation and progressively destabilises the horizontally elongated (2,1) structure. As a result, kinetic energy is transferred from the Coulomb-dominated mode to vertically stacked circulation patterns, and the flow reorganises into a configuration characterised by the (1,2) mode. The observed transition from (2,1) to (1,2) therefore reflects a change in the dominant momentum-production mechanism, from Coulomb to buoyancy driven, in the bulk of the flow.

The colour bar of $\eta$ in figure 7(b) quantifies how strongly the time-averaged flow is dominated by a single Fourier mode. At low $T$ , $\eta$ is around 0.6 in the $(1,1)$ mode, indicating that most kinetic energy is concentrated in one coherent large-scale circulation. As $T$ increases into the Coulomb-dominated regime, $\eta$ generally decreases because the Coulomb force introduces additional recirculations, redistributing energy into other modes. At larger $Ra$ and near the regime transition boundary, $\eta$ is typically lower. The increase in $Ra$ intensifies the thermal plume activity and enhances competition among modes, redistributing kinetic energy across multiple large-scale structures rather than concentrating it in a single dominant mode. The further reduction of $\eta$ near the boundary is consistent with the coupled reorganisation: charge redistribution, together with thermal-plume-induced forcing, erodes single-mode dominance and strengthens competition between large-scale structures during the transition

In the phase diagram, the transitional Rayleigh number $Ra_T$ at different $T$ is plotted as the black dashed line. It clearly shows that this line is capable of reliably predicting the flow structure transition. However, it should be noted that the prediction is invalid when $T\lt 200$ , as the Coulomb force in this regime is too weak to induce fluid motion. This is consistent with the previous study on the pure electroconvection case ( $Ra=0$ ), where the critical electric Rayleigh number $T_c$ for the onset of convection was reported to be $T_c \approx 240$ (Wu et al. Reference Wu, Traoré, Vázquez and Pérez2013), as indicated by the asterisk in figure 7(b). This behaviour is also accurately captured in the present study: for $Ra=0$ , no flow is observed when $T\lt T_c$ , as indicated by the hollow diamonds in figure 7(b), whereas two horizontally aligned vortices develop once $T\gt T_c$ .