Effect of an Adiabatic Obstacle on the Symmetry of the Temperature, Flow, and Electric Charge Fields during Electrohydrodynamic Natural Convection

Abstract

This study explores the impact of an adiabatic obstacle on the symmetry of temperature, flow, and electric charge fields during electrohydrodynamic (EHD) natural convection. The configuration studied involves a square, differentially heated cavity with an adiabatic obstacle subjected to a destabilizing thermal gradient and a potential difference between horizontal walls. A numerical analysis was performed using the finite volume method combined with Patankar’s “blocked-off-regions” technique, employing an in-house FORTRAN code. The study covers a range of dimensionless electrical Rayleigh numbers (0 to 700) and thermal Rayleigh numbers (10² to 10⁵), with various obstacle positions. Key findings indicate that while the obstacle reduces heat transfer, this can be counterbalanced by electric field effects, achieving up to 165% local heat transfer improvement and 100% average enhancement. Depending on the obstacle’s position and size, convective transfer can increase by 27% or decrease by 21%. The study introduces five multiparametric mathematical correlations for rapid Nusselt number determination, applicable to numerous engineering scenarios. This work uniquely combines passive (adiabatic obstacle) and active (electric field) techniques to control heat transfer, providing new insights into the flow behaviour and charge distribution in electro-thermo-hydrodynamic systems.

1. Introduction

Electro-thermo-hydrodynamics (ETHD) is an interdisciplinary field that explores the mutual thermal and electric field effects. Its potential to serve various industrial applications in energy, electronics, mechanics, and chemistry has garnered significant attention from researchers. Studies have shown that applying an electric field to a dielectric liquid can result in extra instabilities and generate secondary flows in the fluid [1,2]. As a result, numerous numerical investigations on electro-thermo-hydrodynamic systems in square spaces have been published in recent years. Traoré et al. [3] have numerically studied the electro-convective problems occurring in a dielectric fluid exposed to unipolar electric injection. They succeeded for the first time to solve the coupled EHD equations using the finite volume technique. A specific non-oscillatory scheme was used to solve a density transport equation. The authors determined the non-linear and linear criteria stability in the cases of low and high injection. They demonstrated the appearance of a hysteresis loop related to both values of the stability parameter. Subsequently, numerical simulations were conducted by Wu et al. [4] to study the impact of sidewalls of a 2D cavity on the development of electro-convective instability. They discovered that an unpredicted change in the bifurcation behaviour arises for many cavity form factors. The authors showed that beyond the linear stability limit, a supercritical bifurcation emerges. This bifurcation occurs at a critical electric Rayleigh number. Also, the authors recorded a secondary subcritical bifurcation that appears at a different value of the electrical Rayleigh number, thus making a hysteresis loop. The study of the ETHD problem, in particular its potential to intensify the heat transfer in cavities filled with dielectric fluids, was also investigated. Castellanos et al. [5] analysed the flow behaviour of a dielectric fluid subjected to a unipolar injection. In this study, the electrical properties of the fluid such as dielectric constant and ion mobility were not assumed to be constant but varied with temperature. The authors were able to detect and define the regions where the convective instabilities were oscillatory by using the “Galerkin” method. In their study, Zhang et al. [6] investigated how electro-hydrodynamics forces impact the convection in a rectangular enclosure for a high Prandtl number fluid (116.6). Their findings showed that the electric field caused a significant mixing between hot and cold liquid. However, they also mentioned that the Nusselt number increased only under a strong electric field magnitude. Later, Truong et al. [7] studied the heat transfer characteristics of an ethylene glycol-based Fe₃O₄ nanofluid within a curved porous cavity subjected to an electric field. The impact of electrohydrodynamic (EHD) force, radiation, and nanoparticle shape on nanofluid behaviour was analysed. The results revealed that the Darcy number and applied voltage enhanced convection, while the presence of an electric field and radiation contributed to improved thermal performance. Dantchi et al. [8] numerically investigated the two-dimensional ETHC for several types of dielectric fluids. A parametric study dealing with the combined effects of buoyancy and electrical forces was performed. The authors found that only the electric forces control the flow. Indeed, when the electric field value is strong enough, the convective transfer (Nusselt number) becomes completely independent of the thermal gradient. Recently, Wang et al. [9] conducted a numerical investigation into the electro-thermo-hydrodynamic convection system induced by an electric field. Their findings reveal the temporal and spatial distributions of the flow field, temperature field, electric field, and charge density within the system. Influenced by electric force and a destabilizing buoyancy force, the system loses its symmetrical behaviour and displays electro-thermo-convective vortices, ultimately transitioning to a chaotic flow field with irregular temporal fluctuations. The research demonstrates that a strong electric driving force can double the heat transfer effects, as indicated by the Nusselt number (Nu).

Studies dealing with ETHD in the case of rectangular cavities containing an adiabatic obstacle are still rare, although the existence of the blocks causes interesting disturbances and instabilities that may change the flow structure and affect the heat transfer efficiency. In their study, Bhave et al. [10] investigated how the addition of an adiabatic obstacle in an enclosure can enhance the convection. The authors examined the effects of the block dimensions on the convective flow for different types of fluids. Their findings indicate that the appropriate dimension of the adiabatic block to maximize heat transfer depends on the type of fluid used and the applied thermal gradient. Furthermore, they demonstrated that adjusting the size of the obstacle can lead to a heat transfer improvement of approximately 10% compared to the non-obstacle scenario. Mahmoodi and Sebdani [11] numerically simulated the free convection in a differentially heated cavity having an incorporated adiabatic block. A comparative study between the case of an enclosure filled with pure fluid and the case where it is filled with a nanofluid (Cu–water) has been developed. Particular care has been given to the effect of the aspect ratio on the heat transfer. It has been seen that for low Ra values, the heat transfer rate is inversely proportional to the adiabatic block size. However, for high Rayleigh numbers, the obstacle dimension no longer has an important impact on the convective transfer.

It should also be noted that several recent studies [12,13,14] have demonstrated that employing an adiabatic obstacle can indeed be a highly effective technique for controlling heat transfer. In general, the length and positioning of the obstacle play pivotal roles in enhancing the heat transfer rate. Numerous industrial applications can be affected by such configurations, including but not limited to the following: energy efficiency and building design, electronics cooling, renewable energy systems like solar water heaters with baffles, cooling systems for nuclear reactor safety, materials processing, food processing, heat storage systems, and heat transfer enhancement.

Unlike the preceding investigations, as far as our current knowledge extends, the combination of a passive technique, encompassing the incorporation of an adiabatic obstacle, with an active technique involving the imposition of an electric field to control heat transfer, remains unexplored. The main focus of the study is the investigation of the impact of these techniques on the flow structure temperature field and electric charge density. The governing equations related to ETHD processes will be solved numerically. The expected results will help us to better understand the flow behaviour, charge density distribution, isotherm profiles, and heat transfer efficiency for different obstacle sizes and positions. Finally, using the linear regression method, five multiparametric mathematical correlations were established to quickly determine the average Nusselt number. These correlations offer a significant contribution to practical engineering by providing a simplified method for predicting heat transfer in complex scenarios.

2. Materials and Methods

A Newtonian incompressible dielectric liquid, enclosed in a square differentially heated cavity is considered in the present paper. Figure 1 shows the considered configuration. The horizontal upper wall (y = H) is the grounded electrode with an electrical potential V₀, while the lower electrode (lower horizontal wall, y = 0), with a higher electrical potential (V₁ > V₀), acts as an electrical emitter. The potential difference creates a unipolar injection from the emitter electrode, considered to be stable, homogeneous, and autonomous, so the charge density of the lower electrode is assumed to be uniform (q = 1). The cavity is therefore subjected to a destabilizing thermal gradient between the vertical walls and a destabilizing potential difference between the horizontal walls.

Inside this enclosure, an adiabatic obstacle of length and width L’ and H’, respectively, was inserted. Depending on the position and size of this obstacle, several configurations, summarised in

Table 1. Details of the different configurations.

Configuration	H’/H	L’/L	Obstacle Position
Cf1	0.4	0.3	Centred on the bottom wall
Cf2	0.3	0.4	Centred on the right-side wall
Cf3	0.4	0.3	Centred on the top wall
Cf4	0.34	0.34	Centred
Cf5	-	-	Without obstacle

, were analysed.

It is interesting to note that the horizontal walls of the studied configuration are both thermally adiabatic and electrically conductive. This seems rather unusual; however, some materials exhibit this property, such as certain polymers like polypyrrole or polythiophene, graphite, amorphous selenium, and certain metallic alloys like “Kanthal” (an alloy of iron and chromium). In fact, all these materials demonstrate a certain level of electrical conductivity while retaining thermal insulating properties.

Basic governing equations

The set of governing equations of the electro-thermo-convection flow includes the conservation equation of mass, momentum, energy, and the Poisson equation:

$$\overrightarrow{\nabla }·\overrightarrow{\mathrm{U}}=0$$

(1)

$$\mathsf{\rho }\left[\frac{\partial \overrightarrow{\mathrm{U}}}{\partial \mathrm{t}}+\left(\overrightarrow{\mathrm{U}}·\overrightarrow{\nabla }\right)\overrightarrow{\mathrm{U}}\right]=-\overrightarrow{\nabla }\mathrm{P}+\mathsf{\mu }{\nabla }^2\overrightarrow{\mathrm{U}}+{\mathsf{\rho }}_0\overrightarrow{\mathrm{g}}\mathsf{\beta }\left(\mathsf{\theta }-{\mathsf{\theta }}_0\right)+\mathrm{q}\overrightarrow{\mathrm{E}}$$

(2)

$$\mathsf{\rho }{\mathrm{C}}_{\mathrm{P}}\frac{\partial \mathsf{\theta }}{\partial \mathrm{t}}+\overrightarrow{\nabla }·\overrightarrow{\mathrm{U}}=\mathsf{\lambda }{\nabla }^2\mathsf{\theta }$$

(3)

$$\frac{\partial \mathrm{q}}{\partial \mathrm{t}}+\overrightarrow{\nabla }·\overrightarrow{\mathrm{j}}=0$$

(4)

$$\overrightarrow{\nabla }·\overrightarrow{\mathrm{E}}=\frac{\mathrm{q}}{\mathsf{\epsilon }}$$

(5)

$$\overrightarrow{\mathrm{E}}=-\overrightarrow{\nabla }\mathrm{V}$$

(6)

With

$$\overrightarrow{\mathrm{j}}=\mathrm{q}\overrightarrow{\mathrm{U}}+\mathrm{K}\mathrm{q}\overrightarrow{\mathrm{E}}$$

(7)

Based on the “stream function–vorticity” (ψ − ω) formulation and taking the below non-dimensional variables:

$$\mathrm{x}=\frac{{\mathrm{x}}^{\mathrm{*}}}{\mathrm{L}}$$	$$\mathrm{y}=\frac{{\mathrm{y}}^{\mathrm{*}}}{\mathrm{L}}$$	$${\mathrm{U}}_{\mathrm{x}}={\mathrm{U}}_{\mathrm{x}}^{\mathrm{*}}\frac{\mathrm{L}}{\mathrm{a}}$$	$${\mathrm{U}}_{\mathrm{y}}={\mathrm{U}}_{\mathrm{y}}^{\mathrm{*}}\frac{\mathrm{L}}{\mathrm{a}}$$	$$\mathsf{\Psi }=\frac{{\mathsf{\Psi }}^{\mathrm{*}}}{\mathrm{a}}$$	$$\mathsf{\omega }={\mathsf{\omega }}^{\mathrm{*}}\frac{{\mathrm{L}}^2}{\mathrm{a}}$$
$$\mathrm{t}={\mathrm{t}}^{\mathrm{*}}\frac{\mathrm{a}}{{\mathrm{L}}^2}$$	$$\mathsf{\theta }=\frac{\left({\mathsf{\theta }}^{\mathrm{*}}-{\mathsf{\theta }}_{\mathrm{C}}\right)}{\left({\mathsf{\theta }}_{\mathrm{H}}-{\mathsf{\theta }}_{\mathrm{C}}\right)}$$	$$\mathrm{q}=\frac{{\mathrm{q}}^{*}}{{\mathrm{q}}_0}$$	$$\mathrm{E}={\mathrm{E}}^{\mathrm{*}}\frac{\mathrm{L}}{\left({\mathrm{V}}_1-{\mathrm{V}}_0\right)}$$	$$\mathrm{V}=\frac{\left({\mathrm{V}}^{\mathrm{*}}-{\mathrm{V}}_0\right)}{\left({\mathrm{V}}_1-{\mathrm{V}}_0\right)}$$

The resulting non-dimensional governing equations are as follows:

$$\mathsf{\omega }=-\left(\frac{{\partial }^2\mathsf{\Psi }}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathsf{\Psi }}{\partial {\mathrm{y}}^2}\right)$$

(8)

$$\frac{\partial \mathsf{\omega }}{\partial \mathrm{t}}+{\mathrm{U}}_{\mathrm{x}}\frac{\partial \mathsf{\omega }}{\partial \mathrm{x}}+{\mathrm{U}}_{\mathrm{y}}\frac{\partial \mathsf{\omega }}{\partial \mathrm{y}}=\frac{{\partial }^2\mathsf{\omega }}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathsf{\omega }}{\partial {\mathrm{y}}^2}+\mathrm{R}\mathrm{a}\mathrm{P}\mathrm{r}\frac{\partial \mathsf{\theta }}{\partial \mathrm{x}}+\frac{\mathrm{C}{\mathrm{T}}^2}{{\mathrm{M}}^2}\left[\frac{\partial \left(\mathrm{q}{\mathrm{E}}_{\mathrm{y}}\right)}{\partial \mathrm{x}}-\frac{\partial \left(\mathrm{q}{\mathrm{E}}_{\mathrm{x}}\right)}{\partial \mathrm{y}}\right]$$

(9)

$$\frac{\partial \mathsf{\theta }}{\partial \mathrm{t}}+{\mathrm{U}}_{\mathrm{x}}\frac{\partial \mathsf{\theta }}{\partial \mathrm{x}}+{\mathrm{U}}_{\mathrm{y}}\frac{\partial \mathsf{\theta }}{\partial \mathrm{y}}=\frac{{\partial }^2\mathsf{\theta }}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathsf{\theta }}{\partial {\mathrm{y}}^2}$$

(10)

$$\frac{\partial \mathrm{q}}{\partial \mathrm{t}}+\frac{\partial }{\partial \mathrm{x}}\left[\mathrm{q}\left({\mathrm{U}}_{\mathrm{x}}+\frac{\mathrm{T}}{{\mathrm{M}}^2}{\mathrm{E}}_{\mathrm{x}}\right)\right]+\frac{\partial }{\partial \mathrm{y}}\left[\mathrm{q}\left({\mathrm{U}}_{\mathrm{y}}+\frac{\mathrm{T}}{{\mathrm{M}}^2}{\mathrm{E}}_{\mathrm{y}}\right)\right]=0$$

(11)

$$\left(\frac{{\partial }^2\mathrm{V}}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathrm{V}}{\partial {\mathrm{y}}^2}\right)=-\mathrm{C}·\mathrm{q}$$

(12)

$${\mathrm{E}}_{\mathrm{x}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{x}}$$

(13)

$${\mathrm{E}}_{\mathrm{y}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{y}}$$

(14)

With the dimensionless velocities given below:

$${\mathrm{U}}_{\mathrm{x}}=\frac{\partial \mathsf{\Psi }}{\partial \mathrm{y}} \mathrm{and} {\mathrm{U}}_{\mathrm{y}}=-\frac{\partial \mathsf{\Psi }}{\partial \mathrm{x}}$$

(15)

The dimensionless numbers appearing in Equations (8)–(14) are as follows:

Electric Rayleigh number: $\mathrm{T}=\frac{{\mathsf{\epsilon }}_0·\mathsf{\Delta }\mathrm{V}}{\mathsf{\rho }·\mathsf{\nu }·{\mathrm{K}}_0}$.

Thermal Rayleigh number: $\mathrm{R}\mathrm{a}=\frac{\mathsf{\rho }·\mathsf{\beta }·\mathsf{\Delta }\mathsf{\theta }·{\mathrm{L}}^3}{\mathsf{\nu }·\mathrm{a}}$.

Prandtl number: $\mathrm{P}\mathrm{r}=\frac{\mathsf{\nu }}{\mathsf{\alpha }}$.

Injection strength: $\mathrm{C}=\frac{{\mathrm{q}}_{\mathrm{i}}·{\mathrm{L}}^2}{{\mathsf{\epsilon }}_0·\mathsf{\Delta }\mathrm{V}}$.

Ion mobility $\mathrm{M}=\frac{1}{{\mathrm{K}}_0}{\left(\frac{{\mathsf{\epsilon }}_0}{\mathsf{\rho }}\right)}^{\frac{1}{2}}$

For the calculation of the local and mean Nusselt number, the following expressions are used:

$$\mathrm{N}{\mathrm{u}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}=\frac{\mathrm{d}\mathsf{\theta }}{\mathrm{d}\mathrm{x}}$$

(16)

$$\mathrm{N}{\mathrm{u}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}=\frac{1}{\mathrm{H}}\underset{0}{\overset{1}{\int }}\mathrm{N}{\mathrm{u}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}\mathrm{d}\mathrm{y}$$

(17)

Boundary conditions

The system of Equations (8)–(15) has been solved by considering the following boundary conditions:

At the hot wall (x = 0)

$\mathsf{\theta }=1$

$\frac{\partial \mathrm{V}}{\partial \mathrm{x}}=0$

$\frac{\partial \mathrm{q}}{\partial \mathrm{x}}=0$

$\mathsf{\Psi }=\frac{\partial \mathsf{\Psi }}{\partial \mathrm{x}}=0$

At the cold wall (x = L)

$\mathsf{\theta }=0$

$\frac{\partial \mathrm{V}}{\partial \mathrm{x}}=0$

$\frac{\partial \mathrm{q}}{\partial \mathrm{x}}=0$

$\mathsf{\Psi }=\frac{\partial \mathsf{\Psi }}{\partial \mathrm{x}}=0$

At the adiabatic emitter electrode (y = 0)

$\frac{\partial \mathsf{\theta }}{\partial \mathrm{y}}=0$

$\mathrm{V}=1$

$\mathrm{q}=1$

$\mathsf{\Psi }=\frac{\partial \mathsf{\Psi }}{\partial \mathrm{y}}=0$

At the adiabatic collector electrode (y = H)

$\frac{\partial \mathsf{\theta }}{\partial \mathrm{y}}=0$

$\mathrm{V}=0$

$\frac{\partial \mathrm{q}}{\partial \mathrm{y}}=0$

$\mathsf{\Psi }=\frac{\partial \mathsf{\Psi }}{\partial \mathrm{y}}=0$

At the obstacle

$\frac{\partial \mathsf{\theta }}{\partial \mathrm{n}}=0$

$\frac{\partial \mathrm{V}}{\partial \mathrm{n}}=0$

$\frac{\partial \mathrm{q}}{\partial \mathrm{n}}=0$

$\mathsf{\Psi }=\frac{\partial \mathsf{\Psi }}{\partial \mathrm{n}}=0$

Numerical procedure

The governing equations described above have been solved using the finite volume method (Patankar [15]) implemented with an in-house FORTRAN code. For the time scheme selection, the fully implicit Euler scheme of order one has been applied. To guarantee the stability of the computations, a small time step of 10⁻⁴ was taken [16]. After discretisation, all the algebraic equations obtained were solved by the Gauss–Seidel method with a successive relaxation algorithm (SOR) [17]. During each calculation time step, the following convergence tolerance is considered in this study:

$$\frac{\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathsf{\Psi }}^{\mathrm{k}+1}-{\mathsf{\Psi }}^{\mathrm{k}}\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathsf{\Psi }}^{\mathrm{k}+1}\right)}+\frac{\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathsf{\theta }}^{\mathrm{k}+1}-{\mathsf{\theta }}^{\mathrm{k}}\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathsf{\theta }}^{\mathrm{k}+1}\right)}+\frac{\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{q}}^{\mathrm{k}+1}-{\mathrm{q}}^{\mathrm{k}}\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{q}}^{\mathrm{k}+1}\right)}\le {10}^{-5}$$

(18)

Here, the superscript k refers to the kth iteration of the SOR procedure.

Given the stability constraints and to reduce numerical diffusion, a power law scheme is implemented to process the convective terms. For interested readers, details regarding the resolution method and the different schemes used can be found in our publication [18].

The iterative solver, during a single iteration loop denoted as ‘’k’’ in the equation-solving algorithm, operates as follows:

① All dependent variables are set to zero (ψ = ω = q = θ=U_x = U_y = E_x = E_y = V = 0);

② Solve the discrete form of the vorticity Equation (9);

③ Solve the discrete form of the Poisson equation for the stream function Equation (8), with the updated vorticity as left-hand side;

④ Calculate the velocity profile by Equation (15);

⑤ Solve the discrete form of the charge density transport Equation (11) and the heat Equation (10) using the newly calculated velocity profile obtained in Step 4;

⑥ Solve the discrete form of the Poisson Equation (12) for the electric potential with the updated charge density as right-hand side;

⑦ Calculate the electric field profile by Equations (13) and (14);

⑧ Steps ① to ④ are repeated until the stopping criteria of the SOR loop Equation (18) is satisfied.

Once the criteria outlined in Equation (18) are satisfied, all newly computed independent variables are adopted as the initial field, and we switch to the next time step.

Choice of the mesh and validation of the code

To study the mesh sensitivity, several tests were carried out. In

Table 2. Mesh effect.

	N × N
	31 × 31	61 × 61	101 × 101	201 × 201
Nu mean (hot wall)	2.19	2.96	3.04	3.06
Deviation (%)	(27.96)	(2.63)		(0.65)
Nu mean (Cold wall)	1.45	1.69	1.72	1.73
* Deviation (%)	(15.70)	(1.74)		(0.58)
Ψmax	13.35	15.29	15.61	15.63
Deviation (%)	(14.48)	((2.05)		(0.128)

* Deviation is relative to the 101 × 101.

, we have chosen to represent some simulation results for the configuration Cf.1, C = 10, T = 300, and Ra = 10⁴. The calculation mainly concerns the variation in the average Nusselt number on the cold and hot walls and the value of the maximum stream function, in relation to the mesh size. To obtain a balance between the time of runs and the independence of the results to the cell number, a 101 × 101 mesh size was adopted.

The results of our in-house written numerical code were compared to those of the study of Asan [19] (natural convection of air with a hot obstacle) to validate the thermal side. Then, in a second step, a comparison was performed with the results of Traoré et al. [20] (i.e., horizontal cavity filled with a dielectric fluid, subject to an electric filled) to validate the electric side.

Figure 2a,b show that the results are in excellent agreement with the references [19,20], with a maximum error that never exceeds 2%. It can be stated that the purely thermal case as well as the electrical-thermal case are validated.

3. Results and Discussion

3.1. Charge Density Distribution

In general, in electro-thermo-hydro-convection, in addition to the viscous forces trying to dampen and slow down the flow, there are two other destabilising forces, namely the electrical and thermal forces. If these two driving forces are sufficiently large, two different regimes can be observed. The thermal dominated regime where thermal buoyancy forces dominate and the electrical dominated regime where electrical forces control the flow.

Figure 3 shows the electrical Rayleigh number impact on the electric charge density contours. For T = 10, the electric field is weak enough to allow the thermal forces to drive the flow. These buoyancy thermal forces act in the upward direction close to the hot wall and in the downward one close to the cold wall. Under these conditions, thermal forces tend to drag electrical charges to the collector at the left side of the cavity and conversely stop their migration by blocking them on the emitter electrode at the right side. At T > 200, the electrical forces are stronger and start to overcome the thermal buoyancy forces mainly on the cold wall side. For T = 700, the electric field is so strong that the regime becomes electrically dominant and characterised by a symmetrical migration of electric charges close to the vertical walls of the cavity.

The opposite effect is observed in Figure 4, showing the contours of charge density for various thermal Rayleigh number values (with T set at 400). For values of Ra > 10⁵, the electric charges can no longer reach the collecting electrode on the right side because the regime is thermally dominated. For smaller thermal Rayleigh numbers, the electric charge distribution returns to a symmetrical aspect, where the migration of electric charges takes place mainly on both sides of the cavity.

It should be noted in Figure 3 and Figure 4, that the presence of the obstacle on the emitter electrode helps to create a kind of compartmentalisation (left–right) in the electrical charge distribution.

In Figure 5, several types of configurations have been presented. For the case without an obstacle Cf.5 and for low values of T, the electric charges migrate to the collecting electrode only at the left side. This is the result of the collaboration between the buoyancy forces and electrical forces. The addition of obstacles at various positions (Cf.1–Cf.4) has no significant effect at low electrical Rayleigh numbers. Indeed, as always, the buoyancy forces drive the electric charges upwards on the hot side of the cavity.

By increasing T to 200, in addition to the charge injection already observed at the left side, a new electric charge penetration starts to occur on the right side. Indeed, the electric charges diffuse and exceed the middle region on the side of the cold wall for all the configurations except for the case Cf.2 where, on the contrary, the existence of an obstacle attracts the electric charges on the bottom emitting wall.

The electric forces increase in intensity, which gives rise to a new topography of the electric charge distribution. In this case, the presence of two electric charge injection paths was noted and two electric charge plumes were imposed along the vertical walls. These plumes are symmetrical for the configurations Cf.1 and Cf.4, almost symmetrical for Cf.3 and Cf.5, and completely dissymmetrical for the case Cf.2. Indeed, in the situation Cf.2, the obstacle on the left wall blocks the movement of the electric charges and obliges them to bypass it, creating a quasi-chaotic behaviour.

3.2. Flow Structure

Figure 6 shows the effect of the electrical Rayleigh number on the flow structure for Ra = 10⁴ and C = 10. In the case of classical natural convection (T = 0), the flow is characterised by a single-cell regime with a vortex centre located just above the obstacle. By increasing the electrical Rayleigh number, the system evolves towards a bicellular regime with a larger left cell. This was expected since the left side of the cavity is where the electrical and buoyancy forces cooperate, while the right side is where the upward electrical forces compete with the downward thermal forces. For T = 400, the electrical forces are more significant compared to the thermal forces; the charge injection, described in the previous paragraph, creates a symmetrical behaviour characterised by two counter-rotating cells. All these findings are confirmed by Figure 7; indeed, for Ra =1000 and 5000, the thermal forces are quite weak and an electrically dominated regime is established. Two symmetrical counter-rotating cells are then observed. This symmetrical flow behaviour is completely lost when the thermal and electrical forces have similar strengths.

Figure 8 presents the maximum stream function according to the electrical Rayleigh number. The effect of the location and obstacle dimension has a major impact on the flow intensity. There is an increase in flow intensity reaching 244% for T = 300 between the Cf.4 configuration (centred obstacle) and the Cf.5 configuration (without the obstacle). Figure 9 can confirm all these findings. In all configurations, the presence of the obstacle slows down the flow, but the central position is the most disabling one compared to the case without an obstacle. Also, from Figure 9, it is possible to state that the number of cells (mono or bicellular) is not dictated by the size or the position of the obstacles, but only by the electrical forces (electrical Rayleigh number). Finally, for high electric Rayleigh numbers, the flow is multicellular, composed by two counter-rotating cells, except for Cf.2.

3.3. Temperature Distribution

Figure 10 illustrates the isothermal lines evolution for Ra = 10⁴, C = 10, and various electrical Rayleigh numbers. For T = 0, the case of classical natural convection, the temperature isocontours present a horizontal stratification which shows the dominance of a conductive regime (thermal boundary layer) at the corners of the active walls. Some isotherms distortions are recorded in the middle of the cavity pointing to the presence of the convective mode. For higher values of T (300 and 700), the coulomb force becomes strong enough to control the heat transfer. In this case, it is noticed that the cavity is divided into two zones: a hot zone, located on the left side, where thermal and electrical forces cooperate, and a second cold zone, where thermal and electrical forces compete. The same findings are seen in Figure 11 for a low Rayleigh number where the thermal forces make way for the electrical forces to impose this kind of compartmentalisation at Ra = 10² and 10³. For Ra = 10⁵, the thermal buoyancy forces have enough energy to drive up the hot fluid and down the cold one, this time creating a vertical stratification in the central region. The increase in T significantly affects the symmetry of the temperature field by splitting the cavity into distinct thermal zones: a warmer zone where the thermal and electric forces collaborate (left side) and a cooler zone on the opposite side where these forces are in opposition. This causes the appearance of two distinct thermal zones and the asymmetry of the temperature field.

Figure 12 shows the isotherm lines for the different studied configurations. The presence and position of the adiabatic obstacle have considerable effects on the symmetry of temperature distribution. In the configuration without an obstacle (Cf.5), at low T values there is a diagonal symmetry, which becomes altered for higher values. When the obstacle is introduced (Cf.1 to Cf.4), depending on its location it can either enhance or disrupt the symmetry by altering the convective flow and consequently the temperature gradient. In fact, for the cases Cf.1 and Cf.4, the presence of the obstacle boosts the symmetry of the temperature field, while the opposite occurs for the cases Cf.2 and Cf.3. For the reference case without obstacles, the increase in the electrical forces (T = 0, 200, and 400) makes the heat transfer switch, respectively, from a convective regime with a horizontal and then a vertical stratification and finally towards a compartmentalisation. A thermal boundary layer in the lower left and upper right corners has been created, and a thermal gradient inversion is also recorded for T = 200. The same findings are observed by incorporating an obstacle with the following exceptions:

• The vertical stratification disappears for the configurations Cf.1 and Cf.4.
• For T < 400, the convective regime disappears to a conductive regime for Cf.4.
• The thermal gradient inversion disappears for Cf.1, Cf.3, and Cf.4.

To assess the effect of the electrical forces and the size and position of the obstacle on the heat transfer, the local and average Nusselt number profiles on the hot cavity wall have been plotted.

Figure 13 and Figure 14 display the local Nusselt number at the hot wall for various values of the electrical Rayleigh number and various used obstacle configurations. Whatever the value of “T” or the used obstacle, the highest heat transfer rate occurs at the bottom of the wall, especially at Y = 0.15. The intensification induced by the electric forces compared to the reference case (T = 0) is extremely important; an improvement of more than 165%, at Y = 0.15, was detected between T = 0 and T = 800. The choice of the size and position of the obstacle (Cf.1–Cf.5) can either improve or reduce the convective transfer. In fact, compared to the reference configuration Cf.5 (without obstacles), the cases Cf.2 and Cf.1 improve the convective transfer by 10.5% while the cases Cf.3 and Cf.4 reduce the convective heat transfer up to 25.5% at Y = 0.15.

Figure 15 presents the variations in the mean Nusselt number on the hot wall (Ra = 10⁴ and C = 10) for various positions of the obstacle. As expected, regardless of the studied configuration, the heat transfer rate increases proportionally with the electric Rayleigh number. The Coulomb force has a more important impact on the intensification of the heat transfer for Cf.1 and Cf.2 (see

Table 3. Electric Rayleigh number effect on heat transfer intensification.

Configuration	Mean Nu
	T = 0	T = 700	% Increase
Cf.1	2.11	3.73	76.77
Cf.2	2.20	4.35	97.72
Cf.3	1.70	3.40	100.0
Cf.4	1.50	2.81	87.33
Cf.5	1.89	3.40	79.89

). Compared to the reference case without obstacle Cf.5, it can be confirmed that for 0 < T < 700 the convective heat transfer can be intensified from 7 to 27% (Cf.1 or Cf.2) or attenuated from 0 to 21% (Cf.3 or Cf.4) just by a simple modification of the obstacle dimension and location.

Following the execution of several hundred numerical simulations, an empirical correlation was established for each configuration through multiparameter regression analysis. This indicated that the suggested correlations are applicable for a range of thermal Rayleigh numbers from 10² to 10⁵, electrical Rayleigh numbers from 100 to 700, and injection levels varying between 1 to 10. Consequently, the modelled multiparameter correlations, which provide the approximated values of the Nusselt number, are expressed and presented in

Table 4. Multiparameter correlations.

Configuration	Empirical Correlation	R2
Cf.1	$\mathrm{N}\mathrm{u}=0.182\times {\mathrm{R}\mathrm{a}}^{0.120}\times {\mathrm{T}}^{0.269}\times {\mathrm{C}}^{0.066}$	0.954
Cf.2	$\mathrm{N}\mathrm{u}=0.178\times {\mathrm{R}\mathrm{a}}^{0.103}\times {\mathrm{T}}^{0.314}\times {\mathrm{C}}^{0.083}$	0.949
Cf.3	$\mathrm{N}\mathrm{u}=0.053\times {\mathrm{R}\mathrm{a}}^{0.184}\times {\mathrm{T}}^{0.364}\times {\mathrm{C}}^{0.036}$	0.943
Cf.4	$\mathrm{N}\mathrm{u}=0.099\times {\mathrm{R}\mathrm{a}}^{0.187}\times {\mathrm{T}}^{0.216}\times {\mathrm{C}}^{0.091}$	0.952
Cf.5	$\mathrm{N}\mathrm{u}=0.064\times {\mathrm{R}\mathrm{a}}^{0.228}\times {\mathrm{T}}^{0.260}\times {\mathrm{C}}^{0.074}$	0.955

.

4. Conclusions

Numerical simulations were conducted to investigate the impact of an electric field on natural convection in a square dielectric fluid layer fitted with an adiabatic obstacle.

To detect the impact of different variables on the flow behaviour, a parametric study based on varying the electrical Rayleigh number, the thermal Rayleigh number, and the size and the position of the obstacle was performed. Special attention was given to the evaluation of the heat transfer rate, and mathematical correlations were deduced for the average Nu by combining all the control variables.

The most interesting results of this study are the following:

• Electrical force can either promote or disturb the symmetry of the flow and temperature fields, depending on its strength relative to the thermal forces.
• Low T values boost the thermal-driven symmetry, disrupting the natural convective symmetry, due to the electrical force dominance.
• The position of the adiabatic obstacle has an important effect on the symmetry of the flow, temperature, and electric charge distributions.
• Symmetrical flow structures are generally observed for balanced electrical and buoyancy forces, while asymmetries appear when one force dominates.
• For a low T value, the flow is always described by a single-cell regime. By adding high electrical forces (T > 400), the system evolves towards a multi-cell regime characterised by two counter-rotating cells.
• The applied electric field intensifies the heat transfer rate; depending on the used configuration, a local and average improvement of about 165% and 100% can be achieved, respectively.
• Compared to the reference case without an obstacle, the choice of the dimension and position of the obstacle can either improve the convective transfer by 27% or reduce it by about 21%.
• Five multiparameter mathematical correlations to determine the average Nusselt number were established using the linear regression method. These correlations, which have a very high coefficient of determination, can be useful in certain practical engineering scenarios.