Transient conjugate free convection within flexible enclosure having discrete heat source

Abstract

Transient conjugate free convection in flexible enclosure is examined numerically. Isoflux heat sources are mounted in a left wall of finite thermal conductivity while a right wall is assumed to be hyper-flexible. The finite element method (FEM) is adopted to solve the governing partial differential equations, an arbitrary Lagrangian–Eulerian (ALE) approach inherent in the unstructured mesh. The governing parameters under consideration are: the number of the heat source, $1\le n\le 3$ , the thickness of the heat source, $0\le a\le 0.25$ , and the Rayleigh number, ${10}^5\le \mathrm{Ra}\le {10}^8$ . It is determined that the development of conjugate convection heat transfer experiences through an initial phase, a transition phase, and a steady state phase. Each phase interval is shifted by adjusting the thickness of the heat source. Higher number and thicker of heat source cases had an ignorant effect on the shape of the flexible wall, but they tend to suppress the heat transfer rate. Increasing the block amplitude by 25% for $n=1,2,3$ results in reductions of the values of $\overline{\mathrm{Nu}}$ to 24%, 20%, and 28% respectively.

Introduction

Free convection has been selected as the heat transfer mechanism for electronic cooling. This is due to low-cost operation, the quietest, independent from electromagnetic effect and the most important this is a very high reliability. The electronic circuits or components may be mounted to one vertical wall of an electronic casing or an enclosure. The component heating is discrete and can be highly non-uniform heat source. An optimal arrangement of the discrete heaters has been addressed in the literatures. Ho and Chang ¹ study of free convection in an enclosure with discrete heater. Heindel et al. ² found that the discrete heater no longer have their thermal identity. Sezai and Mohamad ³ put a discrete heater on the bottom of the enclosure. da Silva et al. ⁴ concluded that the optimal distribution is not in equidistant arrangement and as the heating intensity increases the heat sources must be installed near the tip of a boundary layer with zero spacings. Three discrete heaters was attached on the vertical wall by Bae and Hyun. ⁵ Chen and Chen ⁶ obtained the two different thermal boundary conditions appear following heat sources arrangement on the vertical and horizontal walls. Müftüoğlu and Bilgen ⁷ investigated when the discrete heater were varied from one to three various size. They concluded that the heat transfer and flow rates are increasing function of heating intensity, heater size, and number of the heaters. Sankar and Do ⁸ reported that the thermal performance is always higher at the bottom heat source. Lopez et al. ⁹ indicated that the flow evolves into three categories: the first when vertical levels below the heat source, an essentially low temperature, stagnant pool is formed, and the heat flow through the outer pool is zero. Nardini and Paroncini ¹⁰ and Nardini et al.^11,12 sequentially investigated free convection with multiple discrete sources using experimental and theoretical approach. Different configuration of hot wall locations have been identified by Mahapatra et al. ¹³ to enhance the heat transfer, thermal mixing, and thermal uniformity distribution. Elsherbiny and Ismail ¹⁴ considered the tilted enclosure having two localized heat sources. Naffouti et al. ¹⁵ concluded that the increasing heating intensity and the distance between the heaters increase the heat transfer rate. They also showed that the most influential factors on flow and heat transfer rate were heater height and aspect ratio. Nayaki et al. ¹⁶ set the array isoflux heaters. El-Moutaouakil et al. ¹⁷ studied that the heated cavity from six identical isothermal elements put on the walls and observed that the case uniform arrangement 2 × 2 × 2 leads to the maximum thermal performance. Parveen and Mahapatra ¹⁸ considered nanofluid heat transfer in a wavy enclosure with a central heating element.

Most studies of discrete heaters in enclosures have focused on rigid surfaces. However, recent advances have introduced a novel material that provides a flexible surface with customized elasticity, extreme thinness, and unique thermal properties. For example, in applications such as inelastic water distribution tasks under expected solar radiation for drying processes. In such cases, a very thin membrane case shields sensitive electronic components. This membrane exhibits high elasticity and is influenced by fluid circulation. The flexible wall undergoes periodic displacements or oscillations within a deformable domain subject to stress boundary conditions as the fluid exerts force on it. In their study, Engel and Griebel ¹⁹ pointed out that the effect of solid deformations on the fluid is determined by the location and velocities of the dynamic boundary. They show that the velocities and displacement contours of the interface resulting from the forces exerted by the motion of the fluid on the elastic structure. Notably, there were significant differences in the flow field structures between flexible and rigid wall models in aortic artery experiments, as reported by Khanafer et al. ²⁰ In a numerical study, Al-Amiri and Khanafer ²¹ investigated combined convection in a lidded cavity with an elastic bottom surface. Khanafer ²² compared convective flow and Nusselt numbers between rigid and elastic right walls and showed that heating intensity with surface elasticity significantly affects flexible surface shapes and consequently enhances heat transfer. Subsequent research by Khanafer ²³ involved a comparison between flexible and corrugated geometries for heated bottom surfaces. Raisi and Arvin ²⁴ investigated heat transfer enhancement with an elastic top wall, where higher Rayleigh numbers lead to larger deformations of the flexible wall. Selimefendigil and Öztop ²⁵ studied a locally heated triangular cavity filled with nanofluid, with a partially flexible surface and internal heat generation. Nanofluid-magnetic field interaction in a lidded cavity with flexible sidewalls was studied by Selimefendigil et al., ²⁶ who later extended their research to the flexible top wall.

Further studies were conducted to analyze the impact of flexible bottom walls incorporating conductive internal L-shaped obstacles by Selimefendigil et al. ²⁷ and inclined L-enclosures by Selimefendigil and Oztop ²⁸ . These studies demonstrated an 11% enhancement in heat transfer. Ghalambaz et al. ²⁹ studied an enclosure separated by a flexible thin wall and controlled by uneven temperature heating. Mehryan et al. ³⁰ found that the elastic plate experiences less deformation and stress when located in the center, while the heat output and wall stress increase with heating intensity. Shahabadi et al. ³¹ noted that the thermal performance along the hot wall does not follow the value of Young’s modulus. Haghani et al. ³² considered transient heat transfer from fluid to solid in a cavity with elastic baffles on the sidewalls. Ghalambaz et al. ³³ studied non-Newtonian working fluids in an enclosure containing a thin flexible heating plate. Al-Amir et al. ³⁴ made a comparison between vertical and horizontal flexible walls in an open enclosure filled with nanofluid, highlighting the high sensitivity of the bottom wall to the flexible surface. Yaseen et al. ³⁵ evaluated several layouts of flexible step barriers. They noticed a step of vertical flexible wall to increase the Nusselt number/pressure drop percentage. Ismael et al. ³⁶ discovered that the flexible sheet can increase the pressure drop by up to 200% in some situations, causing the thermal performance requirement to fall below unity. The elastic baffles, which have a smaller modulus of elasticity, were found by Salih et al. ³⁷ to increase the Nusselt number only marginally. Yaseen et al. ³⁸ verified the applicability of a flexible wall and a certain displacement of the upper wall. Alshuraiaan ³⁹ found that calculating the average Nusselt number was highly dependent on the heating intensity and wall flexibility. Recently, Alhashash and Saleh ⁴⁰ discovered that the evolution of fluid flow passes through distinct stages, including an initial phase, a transition phase, and a steady state phase. Each of these phases experiences a shift in its duration based on changes in the Darcy or Rayleigh numbers.

In summary, previous research has emphasized the relationships between convective heat transfer and wall flexibility. The studies reviewed above have primarily examined the effect of heating intensity on the shape flexibility of various working fluids in different configurations. However, limited information is available regarding discrete block heaters within an elastic enclosure. An electronic enclosure with elastic walls or membranes offers the advantage of protecting sensitive electronic circuitry compared to rigid walls. The vertical enclosure wall may contain electronic circuitry or components, and these discrete, finite thickness components can serve as highly non-uniform heat sources. This study aims to contribute to the existing knowledge by numerically investigating the unsteady conjugate free convection problem within an elastic enclosure. The objective is to explore the scenario of an elastic wall with discrete heaters optimally arranged.

Mathematical formulation

A schematic diagram and rectangular coordinate system for this model is shown in Figure 1. The model represents an unsteady conjugate natural convection and heat transfer involving a Newtonian and incompressible fluid enclosed in an enclosure of size $h$ with a flexible right wall. The flexible wall exhibits the characteristics of a hyperelastic material, responding in a non-linear manner to the forces imposed by the fluid, demonstrating its ability to deform and react dynamically in the presence of buoyancy and gravitational forces acting within the fluid. Several discrete heat sources $(q_h)$ are attached to the left wall, while the flexible wall is kept at an isothermal cold temperature $(T_c)$ . The horizontal walls and the gap wall are thermally insulated. The fluid within the flow field is assumed to have constant thermophysical properties except for density variations, which introduce a body force term into the momentum equation. To relate density changes to temperature changes, the Boussinesq approximation is used to relate fluid properties, thereby coupling the temperature field to the flow field. All boundaries are impermeable, and radiation, viscous dissipation, and Joule heating considerations are ignored.

Figure 1.

Schematic model of the enclosure having two discrete isoflux block.

Given these assumptions, the dimensional governing equations can be expressed in the Arbitrary Lagrangian-Eulerian formulation as follows ⁴¹ :

$$\nabla ·{\mathbf{u}}^{*}=0,$$

(1)

$$\begin{array}{c}\frac{\partial {\mathbf{u}}^{*}}{\partial t}+({\mathbf{u}}^{*}-{\mathbf{w}}^{*})·{\nabla }^{*}{\mathbf{u}}^{*}=-\frac{1}{{\rho }_f}{\nabla }^{*}{P}^{*}+{\nu }_f{\nabla }^{*2}{\mathbf{u}}^{*}\hfill \\ +{\beta }_f\mathbf{g}(T_f-T_c),\hfill \end{array}$$

(2)

$$\frac{\partial T_f}{\partial t}+({\mathbf{u}}^{*}-{\mathbf{w}}^{*})·{\nabla }^{*}{T}_f={\alpha }_f{\nabla }^{*2}{T}_f,$$

(3)

$$\frac{\partial T_b}{\partial t}={\alpha }_b{\nabla }^{*2}{T}_b$$

(4)

with the subscript of $f$ represents the fluid, $b$ represents the block of the discrete heat source. ${\mathbf{u}}^{*}$ refers to the fluid velocities, ${\mathbf{w}}^{*}=(u_s^{*},v_s^{*})$ denotes the velocities of the mesh as it moves through space. In this context, $P^{*}$ and $T$ correspond to the pressure and temperature fields, respectively. The quantities ${\alpha }_f$ , ${\rho }_f$ , ${\beta }_f$ , and ${\nu }_f$ represent the thermal diffusivity, density, thermal expansion constant, and kinematic viscosity of the fluid, respectively. The quantities ${\alpha }_b$ represent the thermal diffusivity of the block. The expression $\mathbf{g}$ represents the gravitational field acceleration.

Flexible thin wall deformation is calculated using ⁴² :

$${\rho }_s\frac{d^2{\mathbf{d}}_s^{*}}{d{t}^2}-{\nabla }^{*}{\sigma }^{*}={\mathbf{F}}_{\nu }^{*},$$

(5)

In this context, ${\sigma }^{*}$ , ${\mathbf{d}}_s^{*}$ , and ${\mathbf{F}}_{\nu }^{*}$ represent the stress tensor, the motion of the flexible structure, and the applied volumetric forces. The motion of the wall structure was modeled using a hyperelastic model, and the stress tensor ${\sigma }^{*}$ was evaluated using the neo-Hookean model:

$${\sigma }^{*}=J^{-1}\mathrm{FS}{F}^T.$$

(6)

where

$$F=(I+{\nabla }^{*}{\mathbf{d}}_s^{*}),J=\det (F)\mathrm{and}S=\frac{\partial W_s}{\partial \epsilon },$$

(7)

In the equation earlier, the symbol $(T)$ is the transpose operator. The strain $(\epsilon )$ and its associated energy density $(W_s)$ can be introduced as follows:

$$\begin{array}{c}{W}_s=\frac{1}{2}{\mu }_l(J-I_l-3)-{\mu }_l\ln (J)\hfill \\ +0.5\lambda (\ln (J){)}^2,\hfill \end{array}$$

(8)

$$\epsilon =12(\nabla {\mathbf{d}}_s^{*}+\nabla {\mathbf{d}}_s^{*T}+{\nabla }^{*}{\mathbf{d}}_s^{*T}\nabla {\mathbf{d}}_s^{*}),$$

(9)

In this context, the first and second constants of Lame can be derived using the expressions ${\mu }_l=E_{\tau }/(2(1+\nu ))$ and $\lambda =E_{\nu }/[(1+\nu )(1-2\nu )]$ , where $E_{\tau }$ and $E_{\nu }$ are the tangential and normal Young’s moduli, respectively. The main invariant of the Cauchy Green displacement tensor is denoted by $I_l$ . The ratio of Poisson and Young’s modulus are represented by $\nu$ and $E$ , respectively. Furthermore, boundary conditions are imposed to ensure continuity of displacement and force at the interface between the wall and the fluid:

$$\frac{\partial {\mathbf{d}}_s^{*}}{\partial t}={\mathbf{u}}^{*},{\sigma }^{*}·\mathbf{n}=-P^{*}+{\mu }_f{\nabla }^{*}{\mathbf{u}}^{*}.$$

(10)

The continuity of heat flux and temperatures was used at the thin wall/fluid junction:

$$k_b\frac{\partial T_b}{\partial \mathbf{n}}=k_f\frac{\partial T_f}{\partial \mathbf{n}}\mathrm{and},$$

(11)

in which, $\mathbf{n}$ indicates the wall-normal vector.

Now, the next set of non-dimensional quantities is invoked:

$$\begin{array}{c}a=\frac{A}{h},X=\frac{x}{h},Y=\frac{y}{h},\mathbf{u}=\frac{{\mathbf{u}}^{*}h}{{\alpha }_f},\mathbf{w}=\frac{{\mathbf{w}}^{*}h}{{\alpha }_f},\hfill \\ \multicolumn{1}{c}{\nabla =\frac{{\nabla }^{*}}{h},{\nabla }^2=\frac{{\nabla }^{*2}}{h^2},\mathrm{\Theta }=\frac{T_f-T_c}{\mathrm{\Delta }T},{\mathbf{d}}_s=\frac{{\mathbf{d}}_s^{*}}{h},}\\ \multicolumn{1}{c}{\sigma =\frac{{\sigma }^{*}}{E^{*}},t_p=\frac{t_p^{*}}{h^2},\tau =\frac{t{\alpha }_f}{h^2},P=\frac{h^2}{{\rho }_f{\alpha }_f^2}{P}^{*},}\\ \multicolumn{1}{c}{{\mathbf{F}}_{\nu }^{*}=\frac{({\rho }_f-{\rho }_s)\mathrm{hg}}{E_{\nu }},E=\frac{E_{\nu }{L}^2}{{\rho }_f{\alpha }_f^2},{\rho }_r=\frac{{\rho }_f}{{\rho }_s},}\\ \multicolumn{1}{c}{\mathrm{Ra}=\frac{\mathbf{g}{\beta }_f{\rho }_f\mathrm{\Delta }T{h}^3}{{\mu }_f{\alpha }_f},\mathrm{\Delta }T=\frac{\mathrm{qh}}{k_f}}\end{array}$$

(12)

The dimensionless expression of the governing equations was achieved as:

$$\frac{1}{{\rho }_r}\frac{d^2{\mathbf{d}}_s}{d{\tau }^2}-E\nabla \sigma =E{\mathbf{F}}_{\nu },$$

(13)

$$\nabla ·\mathbf{u}=0,$$

(14)

$$\frac{\partial \mathbf{u}}{\partial \tau }+(\mathbf{u}-\mathbf{w})·\nabla \mathbf{u}=-\nabla P+\mathrm{Pr}{\nabla }^2\mathbf{u}+\mathrm{Pr}\mathrm{Ra}{\mathrm{\Theta }}_f,$$

(15)

$$\frac{\partial {\mathrm{\Theta }}_f}{\partial \tau }+(\mathbf{u}-\mathbf{w})·\nabla {\mathrm{\Theta }}_f={\nabla }^2{\mathrm{\Theta }}_f,$$

(16)

$$\frac{\partial {\mathrm{\Theta }}_b}{\partial \tau }={\nabla }^2{\mathrm{\Theta }}_b,$$

(17)

In this context, the thermophysical ratios for thermal diffusivity and density, denoted as ${\alpha }_r$ and ${\rho }_r$ respectively, are introduced, where ${\alpha }_r$ is defined as ${\alpha }_f/{\alpha }_b$ and ${\rho }_r$ is defined as ${\rho }_f/{\rho }_b$ . Additionally, the Prandtl number, denoted as $\mathrm{Pr}$ , is defined as the ratio ${\nu }_f/{\alpha }_f$ . It is worth noting that in this scenario, the influence of buoyancy forces on the structure has been ignored, leading to ${\mathbf{F}}_{\nu }=0$ .

The nondimensional boundary conditions of equations (13)–(16) are:

On the isoflux block:

$$\frac{\partial {\mathrm{\Theta }}_b}{\partial X}=-1,\frac{1}{7}\le Y\le \frac{3}{7},\frac{4}{7}\le Y\le \frac{6}{7},$$

(18)

On the block-fluid interface:

$$u=v=0,\frac{\partial {\mathrm{\Theta }}_f}{\partial n}=k_r\frac{\partial {\mathrm{\Theta }}_b}{\partial n},$$

(19)

On the flexible wall:

$$u=v=0,{\mathrm{\Theta }}_f=0,$$

(20)

On the top and bottom horizontal walls:

$$u=v=0,\frac{\partial {\mathrm{\Theta }}_f}{\partial Y}=0,$$

(21)

On the left adiabatic wall:

$$\begin{array}{c}u=v=0,\frac{\partial {\mathrm{\Theta }}_f}{\partial X}=0,0\le Y\le \frac{1}{7},\hfill \\ \frac{3}{7}\le Y\le \frac{4}{7},\frac{6}{7}\le Y\le 1.\hfill \end{array}$$

(22)

where $k_r=k_f/k_b$ is the thermal conductivity ratio. The above boundary condition is case for $n=2$ . The boundary conditions on the elastic surface are derived as:

$$\frac{\partial {\mathbf{d}}_s}{\partial \tau }=\mathbf{u}\mathrm{and}E\sigma ·n=-P+\mathrm{Pr}\nabla \mathbf{u}.$$

(23)

The Nusselt number $(\mathrm{Nu})$ serves as a key parameter in the field of heat transfer, providing valuable information on the rate of heat transfer into the enclosure. The local heat transfer rate at the isoflux surface $(\frac{1}{7}\le Y\le \frac{3}{7},\frac{4}{7}\le Y\le \frac{6}{7})$ is:

$$Y_h=\frac{q_h}{T_b(Y)-T_c}$$

(24)

where $Y_h$ refers to local heat transfer coefficient. The $\mathrm{Nu}$ with constant heat flux are defined as:

$$\mathrm{Nu}=\frac{Y_{h}h}{k_b}=\frac{1}{{\mathrm{\Theta }}_b(Y)}$$

(25)

By integrating the local heat transfer values, the mean Nusselt number $(\overline{\mathrm{Nu}})$ , representing the average heat transmission across the entire enclosure, is obtained:

$$\overline{\mathrm{Nu}}={\int }_{\frac{1}{7}}^{\frac{3}{7}}\mathrm{Nu}\mathrm{d}Y+{\int }_{\frac{4}{7}}^{\frac{6}{7}}\mathrm{Nu}\mathrm{d}Y$$

(26)

The above case is for $n=2$ . Case for $n=3$ is stated as:

$$\overline{\mathrm{Nu}}={\int }_{\frac{1}{10}}^{\frac{3}{10}}\mathrm{Nu}\mathrm{d}Y+{\int }_{\frac{4}{10}}^{\frac{6}{10}}\mathrm{Nu}\mathrm{d}Y+{\int }_{\frac{7}{10}}^{\frac{9}{10}}\mathrm{Nu}\mathrm{d}Y$$

(27)

Previous case for $n=1$ could be treated in similar manner.

Computational methodology

The non-dimensional governing equations in equations (13)–(16) subjected to the initial and boundary conditions in equations (18)–(23) are determined numerically by finite element method (FEM) via Comsol. The computational region is discretized toward triangular components that performed in various times as drawn in Figure 2. Features of the finite element method procedure in comsol are explicitly explained by Alhashash and Saleh. ⁴⁰ In addition, for each of the current variables, the convergence to the solution continues when the relative error satisfies the following condition in each time step:

$$\left|\frac{{\mathrm{\Gamma }}^{i+1}-{\mathrm{\Gamma }}^i}{{\mathrm{\Gamma }}^{i+1}}\right|\le {10}^{-6},$$

Figure 2.

Grid-points distribution for a mesh size, Extra-Fine with 6369 total elements at different time for $n=2$ , $E={10}^{12}$ , $a=0.15$ , and $\mathrm{Ra}={10}^8$ : (a) $\tau$  = 0.0003, (b) $\tau$  = 0.007, (c) $\tau$  = 0.02, and (d) $\tau$  = 1.5.

In this study, the crucial selection of an appropriate time step for coupling fluids and structures is automated using a backward differentiation formula (BDF). BDF methods, being implicit numerical integration schemes, employ backward differences for derivative terms, expressing them in relation to previous time steps. The conditionally stable nature of BDF methods imposes restrictions on time step size, requiring a balance between accuracy and computational efficiency. The chosen time step follows a geometric sequence with a ratio of ${10}^{0.01}$ – starting with small steps and exponentially increasing for computational efficiency while maintaining stability.

To assure the robustness of the computational solution and the independence of the mesh size within the computational domain, a series of simulations with different mesh sizes was conducted to evaluate three key parameters, the minimum strength of the flow circulation $({\mathrm{\Psi }}_{\min })$ , the average Nusselt number $(\overline{\mathrm{Nu}})$ , and CPU time. These simulations were performed for a specific case with the parameters $n=2$ , $E={10}^{12}$ , $a=0.15$ , and $\mathrm{Ra}={10}^8$ . The results, as shown in the Table 1, show negligible discrepancies between the “Extra-Fine” mesh and higher mesh resolutions. The choice of a simulation approach using an exceptionally refined mesh resulted in superior accuracy for the model. However, due to computational efficiency and CPU time constraints, it was decided to use the “Extra-Fine” mesh size for all of the computations detailed in this paper. Balancing accuracy and computational resources, the chosen the “Extra-Fine” mesh size provided a practical compromise for achieving accurate results within a reasonable time frame throughout the study.

Table 1.

Grid sensitivity for ${\mathrm{\Psi }}_{\min }$ and $\overline{\mathrm{Nu}}$ at various grid sizes for $n=2$ , $E={10}^{12}$ , $a=0.15$ , and $\mathrm{Ra}={10}^8$ .

Predefined mesh size	Number of elements	$\overline{\mathrm{Nu}}$	${\mathrm{\Psi }}_{\min }$	CPU time
Coarser	3394	9.7504	−40.9133	12
Coarse	3892	9.8119	−40.7188	17
Normal	4799	9.8611	−40.3044	24
Finer	5339	9.8777	−39.9322	35
Extra-Fine	6379	9.8978	−39.7911	43
Extremely-Fine	7847	9.9608	−39.8966	50

To validate, the streamlines and isotherms were compared against that of Öğüt ⁴³ for the absence of block in the case $a=0.0$ at $\mathrm{Ra}={10}^6$ , $n=1$ , and rigid cold wall (see Figure 3). The result demonstrates the credibility of our solution in representing the thermal boundary layer as well as the configuration and intensity of the circulation in terms of isotherms and streamlines. Additional validation conduct by comparing the result of the existing work and the result of Mehryan et al. ⁴² for the FSI problem and free convection inside a square enclosure that evenly divided with flexible membrane for $\mathrm{Ra}={10}^7$ and $E={10}^{14}$ as demonstrated in Figure 4. It is observed that the present solution is in good agreement with published works.

Figure 3.

Comparing the streamlines and isotherms obtained from existing literature (a) with the currently calculated streamlines and isotherms (b) excluding the block at $a=0.0$ at $\mathrm{Ra}={10}^6$ , $n=1$ , and rigid cold wall.

Figure 4.

Streamlines (a) and isotherms (b) with the left side showing results from the study by Mehryan et al. ⁴² and the right side showing results from our current investigation under the conditions of $\mathrm{Ra}={10}^7$ , $E={10}^{14}$ , and $\mathrm{Pr}=6.2$ .

Results and discussion

The study in this work is carried out toward the following specializing on the linked dimensionless compositions: the number of the heat source, $1\le n\le 3$ , the thickness of the heat source, $0\le a\le 0.25$ , the Rayleigh number, ${10}^5\le \mathrm{Ra}\le {10}^8$ . The modulus Young is set at $E={10}^{12}$ , the conductivity ratio is set to $k_r=10$ , $\mathrm{Pr}=6.2$ . The investigation analyzes characteristics of streamlines, isotherms, local temperature, and the average Nusselt number, $\overline{\mathrm{Nu}}$ .

Figures 5 and 6 present the time evaluation of streamlines and isotherms for the case of $n=2$ , $E={10}^{12}$ , $a=0.15$ , and $\mathrm{Ra}={10}^8$ . At the initial time, that is, $\tau =0.0001$ , the flexible cold surface is straight, and there is a minor free convection circulation in the enclosure. The distance between the streamlines is quite large. Figure 6 also displays that the entire enclosure is at the steady cold temperature. At the same time, a single isotherm appears inside the block. Later, the conduction heat transfer evolves to the convection heat transfer. Figure 6(b) shows commencing the deflections of the flexible surface at the upper right corner. Figure 6(c) depicts a plume of heated fluid moving toward the top wall. This due to the enclosure was at the cold temperature previously then the isoflux wall heats the fluid, reducing the density and lead to the fluid moves upward by the buoyancy. The hot fluid leaves the isoflux wall, the interface and moves along the upper region to the upper elastic wall. At these points, the flow changes its direction to the vertical movement toward the bottom plate. Thus, the cold wall will experience a force in a direction opposite to the convective flow. Then the flexible wall is concave in the upper zone, while the lower zone is convex. At $\tau =0.01$ , a strong free convection flow appeared in the enclosure. The convective flow almost reaches the steady state. The hot fluid moves up and follows the upper wall until it reaches the cold flexible wall. The fluid next to the cold wall moves down and loses its energy. Figure 5(g) shows a large deflection of the flexible wall at the lower zone, which is a convex shape. The streamlines of Figure 5(g) show that the cold fluid moves along the elastic wall and leave the cold wall before reaching the bottom plate. At the time the fluid leaves the cold surface, it generates a minimum pressure zone at the base, creating a weak force for the elastic wall to continue moving. Finally, in the steady state, due to the conservation of the mass, the shape of the elastic wall remains constant. The single eye of the vortex are formed in the concave zone. The boomerang shapes are formed inside the gap as shown in Figure 5(h) and (i).

Figure 5.

Time dependent of the streamlines at $n=2$ , $E={10}^{12}$ , $a=0.15$ , and $\mathrm{Ra}={10}^8$ : (a) $\tau$  = 0.0001, (b) $\tau$  = 0.0006, (c) $\tau$  = 0.002, (d) $\tau$  = 0.004, (e) $\tau$  = 0.01, (f) $\tau$  = 0.04 (g) $\tau$  = 0.08, (h) $\tau$  = 0.1, and (i) $\tau$  = 1.0.

Figure 6.

Time dependent of the isotherms at $n=2$ , $E={10}^{12}$ , $a=0.15$ , and $\mathrm{Ra}={10}^8$ : (a) $\tau$  = 0.0001, (b) $\tau$  = 0.0006, (c) $\tau$  = 0.002, (d) $\tau$  = 0.004, (e) $\tau$  = 0.01, (f) $\tau$  = 0.04 (g) $\tau$  = 0.08, (h) $\tau$  = 0.1, and (i) $\tau$  = 1.0.

Figure 7 displays the influence number of the heat source on the streamlines and temperature fields in the enclosures at the steady-state conditions for $E={10}^{12}$ , $a=0.15$ , and $\mathrm{Ra}={10}^8$ . As can be seen, the variation of the heater number mainly affects the cell circulation. The characteristics of the streamlines at the top of the enclosure are almost similar but denser streamlines are formed at the bottom as the heater number increases. This is due to the hot fluids reaching the cold, flexible wall then falling and being pushed by the wall bending and circulating along the lower part of the enclosure. Thus, the change of the flexible wall and the heater number are considerable in these areas. Figure 7 also reveals that the heater number has small effect on deformation shape of the elastic wall. This is because the inertial forces are dominant at the lower part of the enclosure. Hence, the heater number cannot modify the shape of the elastic wall, notably. The fluid development follows the shape of the hot block for the considered heater number, but the eye of the vortex is longer when the heater number takes higher. So, the impact of the flow development can be recognized in the isothermal profile, especially in the upper region.

Figure 7.

Variety of the streamlines (left) with isotherms (right) by number of the heat source for $E={10}^{12}$ , $a=0.15$ , and $\mathrm{Ra}={10}^8$ : (a) n = 1, (b) n = 2, and (c) n = 3.

Figure 8 shows the impact of the heater thickness on the streamlines and isotherms contours for $E={10}^{12}$ , $n=2$ , and $\mathrm{Ra}={10}^8$ . This interesting illustration shows that changes in the thickness of the heat source have a very minimal impact on the global patterns of streamlines-isotherms inside the enclosure, with the exclusion of the number and position of the eddy core. The isoflux heats the wall and transfers the heat to the fluid and the block, resulting in a flow of warm fluid forming across the gap and the interface. Focusing on the isotherms at the adiabatic wall and the interface indicates that a thermal boundary layer exists at these interfaces regardless of the block size. This layer gets sparser by increasing the block size. Indeed, when the block is thick, the thermal boundary layer covers the entire interface and dampens the hydraulic influence on the interface shape. Impact of block thickness is minimal on the hydraulic behavior of the enclosure when the thermal boundary layer is kept persistent covering the interface. Moreover, the conjugate convection flow is the driving buoyancy. When the fluid is trapped in the gap, the fluid absorbs heat, and the gravitational force increases. Thus, the block thickness provides a minimal impact on the overall behavior of the enclosure. However, a thicker block could reduce of the heat transfer as the distance between the interface and the cold wall increases.

Figure 8.

Variety of the streamlines (left) with isotherms (right) by heater thickness for $E={10}^{12}$ , $n=2$ , and $\mathrm{Ra}={10}^8$ : (a) a = 0.05, (b) a = 0.15, and (c) a = 0.25.

Figure 9 shows the effect of Rayleigh number on the convective flows at the steady-state conditions for $E={10}^{12}$ , $n=2$ , and $a=0.15$ . When the Rayleigh number becomes low, $\mathrm{Ra}={10}^5$ , the flow will circulate poorly, and thus the momentum disposition between the flexible wall and the fluid flow will be weak. Therefore, as seen in Figure 9(a) and (b), the deflections of the cold wall from the rigid line are insignificant. The streamlines show that the convective flow does not enter the gap when the Rayleigh number is low, and the fewer isotherms are trapped in the gap. By increasing the Rayleigh number, the convective flow enters the gap and the isotherms are well distributed. Later, the advantages of using finite thickness of the heat source and elastic wall are pronounced at the high heating intensity. The isotherms present boomerang shape inside the gap.

Figure 9.

Variety of the streamlines (left) with isotherms (right) by Rayleigh number for $E={10}^{12}$ , $n=2$ , and $a=0.15$ : (a) Ra = 10⁵, (b) Ra = 10⁷, and (c) Ra = 10⁸.

The impact of the number and thickness of the heat source on the left wall temperature are displayed in Figure 10 for $\mathrm{Ra}={10}^8$ and $E={10}^{12}$ . Note that significant convective heat transfer occurs at the top of the left wall regardless of the number and thickness of the heat sources. This is where the flow of hot fluid arrives at the wall. The global trend of the temperature increases with increasing $Y$ position. However, the heat source geometry of the isoflux wall changes the left wall temperature. As can be seen, the temperature increases near the geometric peaks of the heater. This is where the heater is exposed to a fresh and cold moving fluid. Between the peaks, the fluid is generally trapped in the gap and reaches a hot temperature. Hence, there is not much of flow circulation in the trapped fluid inside the gap to contribute to the thermal variation. Interestingly, the minimum temperature is close to zero where the corner $0,0$ are surrounded by a uniform hot fluid. Moreover, the maximum peaks of the temperature in the case of $n=3$ and $a=0.15$ are incomparable with that of $n=1,2$ . As the number of heat sources increases, the total length of the hot interface increases, so the thicker the heater, the larger the heater volume. A thin heater results in a local temperature minimum at the gap. However, a small heater leads to a larger overall free space, which could increase the convective flow.

Figure 10.

Variation of the left wall temperature at $a=0.15$ (a) and $n=2$ (b) for $\mathrm{Ra}={10}^8$ and $E={10}^{12}$ .

Figure 11 examines the effect of the Rayleigh number on the average Nusselt number for two samples of $n=1$ (a) and $n=3$ (b) at $\mathrm{Ra}={10}^8$ and $E={10}^{12}$ . This figure shows that the number and thickness of the heat source are crucial in the very beginning. Initially, the thickness of the heat source induces a moderate effect on the Nusselt number, but later it induces a small effect on the average Nusselt number in the transition phase. The transitional phase interval could be shifted by adjusting the heat source thickness. In the steady phase, the variation of the number and thickness of the heat source could persuade small effect on the average Nusselt number. This conclusion agrees with the streamline and isothermal contours described in previous figures where the shortest distance between the interface and the cold wall occurs in the middle. This is where the maximum local Nusselt number takes place. By lowering the heater thickness, the convection flows get stronger and the heater number has an insignificant effect on the steady convection.

Figure 11.

Variation of average $\mathrm{Nu}$ at the isoflux wall for different $a$ when heater number, $n=1$ (a) and heater number, $n=3$ (b) at $\mathrm{Ra}={10}^8$ and $E={10}^{12}$ .

The impact of the arrangement of discrete heater on the average Nusselt number is plotted in Figure 12. In the presence of three heaters, a significant decrease in the mean Nusselt number is observed as the heater thickness increases, possibly due to the increased volume of the heat source. This pronounced decrease may be related to the increased thermal inertia resulting from the thicker heaters. For the $n=2$ scenario, a modest decrease in Nusselt number is associated with an increase in heater thickness. At the same time, an increase in block thickness generally results in a decrease in the average Nusselt number. However, this decrease is offset by the accumulation of hot fluid at the interface, which contributes to an increase in overall heat transfer. In the case of a single heater, increasing the number of heat sources worsens the degradation of the average Nusselt number. Moreover, increasing the block amplitude by 25% for $n=1,2,3$ results in corresponding reductions in the values of $\overline{\mathrm{Nu}}$ to 24%, 20%, and 28%, respectively. This highlights the complexity of the interaction between the various parameters and their visible effect on the heat transfer characteristics.

Figure 12.

Variation of $\overline{\mathrm{Nu}}$ with $a$ for different $n$ at $\mathrm{Ra}={10}^8$ $E={10}^{12}$ .

Conclusions

The present study scrutinized the fluid structure interaction on conjugate free convection in an enclosure with a flexible right wall and isoflux heaters mounted on the left wall. The partial differential equations of the governing equations are solved numerically by means of the FEM via Comsol. The graphical visualization of computational results displays the convective flow and the heat transfer rate. The arrangement of the discrete heat source and the Rayleigh number were interconnected to the design of streamlines, thermal pattern, and heat transfer forms. The optimal discrete heat source arrangement must follow our results:

1. The development of convection heat transfer goes through an initial phase, a transition phase, and a steady state phase. Each phase interval is shifted by adjusting the Rayleigh number and arrangement of discrete heat source.

2. The steady phase of circulation flow formation and thermal pattern are based on the Rayleigh number and arrangement of discrete heater. Multiple discrete heat sources $(n=3)$ produce the steady maximum temperature at the top hot corner of the enclosure.

3. Thickness and number of heat sources had indistinguishable impact on the appearance of the flexibility shape.

4. Thicker and higher number of heat source generate higher temperature near the isoflux block, but create a sparser thermal boundary layer.

5. Increasing the block amplitude by 25% for $n=1,2,3$ results in reductions of the values of $\overline{\mathrm{Nu}}$ to 24%, 20%, and 28% respectively.