Numerical analysis of local thermal non-equilibrium free convection in a porous enclosure with a wavy cold side wall

Abstract

In this piece of communication, numerical heat transfer and fluid flow in a Darcy porous enclosure are investigated. The right wavy wall is maintained at a constant low temperature, with the top and bottom preserved thermally insulated, while the left vertical wall is uniformly heated. The governing equations for non-dimensional systems are solved using the Galerkin finite element method; a local thermal non-equilibrium appears between the base fluid and porous matrix. The effects of interface heat transfer parameter $(0\le Nhs\le {10}^3)$ along with porosity of porous media $(0.1\le ϵ\le 0.9)$ , Rayleigh-Darcy number $(10\le Ra\le {10}^3)$ , and parameters involving waviness of cavities, such as the number of undulations in a unit length $(1\le N\le 5)$ and amplitude of the wave $(0.05\le a\le 0.25)$ are investigated. The results show that increment waviness augments heat transfer enhancement for fluid and solid phases. The local thermal equilibrium cases for various parameters are pointed out at a 5% significance level of the average temperature difference between the two phases.

Introduction

Numerous numerical and experimental studies have been carried out over many years in order to better understand the numerical examinations of fluid heat transport via convection. These studies are essential since many applications of convective heat transfer are reported by researchers, among them Baytas and Pop, ¹ Sheremet et al., ² Nield and Bejan ³ reported the use of porous cavities in geothermal systems, solar power collectors, dehydrating technologies, production of thermal isolators, pollutant dispersion in aquifers, the control of pollutant spread in groundwater, the insulation of buildings, solar power collectors, multishield structures used in the insulation of nuclear, design of nuclear reactors, compact heat exchangers, the utilization of geothermal energy, and so on. The research papers by Vafai and Sozen, ⁴ Kuznetsov and Vafai ⁵ show the importance of non-equilibrium at a high Reynolds number and higher porosity.

Numerous articles have been published about the investigation of free convection heat transfer in a porous cavity with local thermal equilibrium (LTE) and local thermal non-equilibrium (LTNE). Among them, for height-to-gap ratios of 1.46, 1.0, and 0.545 and a radius ratio of 5.338, experimental findings on free convection in a vertical annulus filled with a saturated porous media are described by Prasad et al. ⁶ The inner and outside walls of these experiments are kept at consistent temperatures. If Darcy's law does not apply, they came to the conclusion that there is a distinct link between the Nusselt and Rayleigh numbers. Kumar ⁷ presented the FEA of buoyancy-driven fluid flow and heat transfer in a porous enclosure with heat flux at the wavy wall. According to the author, the cavity's flow and convection processes depend on the Rayleigh-Darcy number, and geometrical factors like wave number, wave phase, and wave amplitude in the cavity's vertical dimension. Alves et al. ⁸ analyzed the stability of free convection in porous cavities through integral transforms. Reported that for a low-dimensional system (Lorenz equations), the qualitative dependency of the Rayleigh number at the start of chaos on the transient behavior and aspect ratio is provided, and its convergence behavior for higher expansion orders is examined. Hossain and Rees ⁹ have investigated the free convection flow of an incompressible, viscous fluid in a rectangular, porous chamber that is heated from the bottom wall and has chilly sides and reported that the presence of sidewall dominates the resulting flow field, particularly for the range of Grashof numbers taken into consideration here. The sidewall boundary conditions have substantially less of an impact on the inner cells of the cavity as the aspect ratio rises. Saeid and Mohamad ¹⁰ investigated how the heated sidewall temperature's wave number and amplitude affect the cavity's natural convection. The average Nusselt number is shown to alter sinusoidally as the wave number increases. According to Al-Amiri et al., ¹¹ the numerical analysis of regular conjugate free convection in a fluid-saturated porous cavity revealed that for small Rayleigh numbers, energy is largely transported through conduction heat transfer in the wall and porous medium. In a computational study of natural convection heat transfer in a partially opened cavity filled with porous media, Oztop et al. ¹² found that the local Nusselt number from a partially heated vertical wall increases as the Grashof number rises because buoyancy-driven flows are strengthened. In a numerical simulation of natural convection in partly porous, layered cavities that had been differentially heated, Chamkha and Ismael ¹³ found that the convective heat transfer might be improved even at low permeability porous mediums. Khanafer et al. ¹⁴ studied the presence of a thin porous fin in free convection heat transfer in an enclosure heated non-uniformly and analyzed different angles of the attached fin. Cheong et al.^15,16 studied free convection heat transfer inside a wavy porous cavity with a cold sidewall and reported the effect of a different heating mode and the effect of various aspect ratios of a wavy porous cavity. Also, natural convection inside wavy porous cavities having different aspect ratios and partial heat flux is studied by Rao and Barman ¹⁷ and Barman and Rao. ¹⁸

The numerical study of free convection taking place in a cavity that is square and porous in nature, considering LTNE was started by Baytas and Pop ¹ using the finite volume method. They found that the LTNE model significantly changes the behavior of the flow characteristics, particularly those of the local heat transfer coefficients, for a square porous cavity with the temperature difference between sidewalls and other walls thermally insulated. Also, Baytas ¹⁹ numerically studied the free convection in a square cavity, which is filled with non-Darcy porous media; in this study, the heat generation inside the cavity is from porous matrix only and reported that the temperature of the fluid phase is higher than the temperature of the solid phase in the upper portion of the cavity for small values of modified conductivity ratio. Misirlioglu et al. ²⁰ implemented the finite element method to analyze the free convection in a porous cavity having undulated wall in the LTNE model and reported that at $Ra={10}^3$ and aspect ratio less than three with surface waviness might lead to a negative Nusselt number, which signifies heat transfer from hot wall to the cold wall is not possible in such configuration. Rees and Pop ²¹ discussed LTNE natural and forced convection inside cavities and boundary layers. A non-implicit condition applicable to local thermal equilibrium for heat conduction in porous media is analyzed by Vadasz, ²² and it is reported the importance of local thermal equilibrium describing the relationship among mean temperature variation and energy flow over the fluid–solid interface. Misirlioglu et al. ²³ studied the free convection occurring in a cavity that is undulated in nature and is filled with porous media which generates heat and they presented the results in the form of average Nusselt numbers, isotherms and streamlines. Using a computational LTNE model, Mansour et al. ²⁴ investigated natural convection in a wavy porous cavity under the effect of heat radiation and found that raising the modified conductivity ratio causes a drop in the overall average Nusselt number. Sheremet et al. ² considered Tiwari and Das nanofluid model to investigate LTNE-free convection in an enclosure and demonstrated that an increase in Nhs causes the convective flow to intensify and the thickness of the boundary layer to significantly rise because of the more intense micro-structural heat transfer between two phases. Also, Alsabery et al. ²⁵ investigated the non-Darcian effect on natural convection in a wavy cavity; their investigation is primarily focused on local thermal non-equilibrium conditions. Raizah et al. ²⁶ focus on the impact of a magnetic field on the free convection of a hybrid nanofluid in a wavy porous enclosure under an LTNE condition. The properties of heat transmission and nanofluid motions inside an undulating cavity are efficiently adjusted by the length and position of the partial heat. Also, Alsabery et al.^27,28 considered LTNE conditions for Forchheimer–Brinkman-extended Darcy porous layer. In their works, they studied the effect of the waviness of walls from different sides.

Based on the literature review discussed above and to the best of our knowledge, the proposed LTNE natural convection occurring in a cavity that is porous in nature and has a cold wavy wall will be discussed for the first time, and this will be a good initiative for new research fields.

Mathematical equations

A square cavity (height and width is D), placed in a horizontal plane, has an adiabatic top and bottom wall, and is heated uniformly from a vertical sidewall keeping another side at low temperature. Considering the cavity bottom right corner as the origin, the horizontal ( $x$ -direction) and vertical ( $y$ -direction) lines are considered as axes of references, and the domain of computation in dimensional coordinates is shown in Figure 1(a). The cavity is filled with fluid-structured porous media, and the fluid and porous matrix are in local thermal non-equilibrium. To analyze the heat transfer and fluid flow inside the cavity, governing equations based on Darcy-Boussinesq approximation are expressed in the two-dimensional coordinate system as:

Figure 1.

The domain of computation in dimensional coordinate and meshing: (a) Domain of computation and (b) Triangular mesh distribution.

Continuity equation:

$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0}}$$

(1)

Momentum equation removing the pressure term:

$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}-{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }x}=-{\displaystyle \frac{Kg{(\rho \beta )}_f}{{\mu }_f}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}(T_f-T_c)}}}}$$

(2)

Energy equation for fluid phase:

$$\begin{array}{rl}{\displaystyle \frac{1}{ϵ}\left(u{\displaystyle \frac{\mathrm{\partial }{T}_f}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }{T}_f}{\mathrm{\partial }y}}}\right)}& ={\displaystyle \frac{k_f}{{(\rho C_p)}_f}({\displaystyle \frac{{\mathrm{\partial }}^2{T}_f}{\mathrm{\partial }{x}^2}+{\displaystyle \frac{{\mathrm{\partial }}^2{T}_f}{\mathrm{\partial }{y}^2})}}}\\ & +{\displaystyle \frac{h_{fs}(T_s-T_f)}{ϵ{(\rho C_p)}_f}}\end{array}$$

(3)

Energy equation for solid phase:

$${\displaystyle \frac{k_s}{{(\rho C_p)}_s}\left({\displaystyle \frac{{\mathrm{\partial }}^2{T}_s}{\mathrm{\partial }{x}^2}+{\displaystyle \frac{{\mathrm{\partial }}^2{T}_s}{\mathrm{\partial }{y}^2}}}\right)={\displaystyle \frac{h_{fs}}{(1-ϵ){(\rho C_p)}_s}(T_s-T_f)}}$$

(4)

Subjected to the boundary conditions:

$$\begin{array}{rl}& \mathrm{On}\mathrm{bottom},{\mathrm{\Gamma }}_1(0\le x\le D,y=0):\\ & u=v=0,{\displaystyle \frac{\mathrm{\partial }{T}_f}{\mathrm{\partial }y}={\displaystyle \frac{\mathrm{\partial }{T}_s}{\mathrm{\partial }y}=0}}\\ & \mathrm{On}\mathrm{right},{\mathrm{\Gamma }}_2(x=D-aD(1-\cos (2\pi NDy))),0\le y\le D):\\ & u=v=0,T_f=T_s=T_h\\ & \mathrm{On}\mathrm{top},{\mathrm{\Gamma }}_3(0\le x\le D,y=D):\\ & u=v=0,{\displaystyle \frac{\mathrm{\partial }{T}_f}{\mathrm{\partial }y}={\displaystyle \frac{\mathrm{\partial }{T}_s}{\mathrm{\partial }y}=0}}\\ & \mathrm{On}\mathrm{left},{\mathrm{\Gamma }}_4(x=0,\le y\le D):\\ & u=v=0,T_f=T_s=T_c\end{array}$$

(5)

Above mentioned governing equation and boundary conditions are subject to the following assumptions:

• The porous media inside the cavity is homogeneous, and isotopic, and Darcy's law is valid.

• The fluid and solid porous matrix are in local thermal non-equilibrium.

• The fluid is incompressible, Newtonian, and experiences negligible viscous dissipation.

• The fluid flow is laminar, steady, and two-dimensional.

• The Boussinesq approximation is considered, and the only body force, gravity $(g)$ , acts negatively to the $y$ -axis.

$(u,v)$ is the dimensional Darcy velocity, K is the porous media permeability, $ϵ$ is the porosity of porous medium, T is the dimensional temperature, k is the thermal conductivity, $\mu$ is the dynamic viscosity of the fluid, the heat capacitance, and buoyancy coefficient are denoted by $\rho \beta$ and $\rho C_p$ , respectively. The solid phase and fluid phase are distinguished by f and s as subscripts, respectively.

The governing equations (1) to (4) with boundary conditions (5) are transformed into non-dimensional form, using the following parameters, $(X,Y)=\frac{(x,y)}{D}$ , $(U,V)=\frac{(u,v)D}{{\alpha }_f}$ , ${\mathrm{\Theta }}_f=\frac{T_f-T_c}{T_h-T_c}$ , ${\mathrm{\Theta }}_s=\frac{T_s-T_c}{T_s-T_c}$ , $Nhs=\frac{h_{fs}{D}^2}{k_f}$ is the fluid-solid interface heat transfer parameter, where $h_{fs}$ is the fluid-solid heat transfer coefficient, and $Ra=\frac{Kg{\beta }_f(T_h-T_c)D}{{\alpha }_f{\nu }_f}$ is Rayleigh-Darcy number. Also, a non-dimensional stream function $(\mathrm{\Psi })$ is introduced, such that $U=\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }Y}$ , $V=-\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }X}$ . $\gamma$ is modified thermal conductivity ratio expressed as $\gamma =\frac{k_f}{[k_s(1-ϵ)]}$ , the porous media is taken as aluminum foam and base fluid is water, so, substituting $k_f$ and $k_s$ values we have, $\gamma =\frac{0.613}{[205.0(1-ϵ)]}$ . The non-dimensional forms of governing equations are:

$${\displaystyle \frac{{\mathrm{\partial }}^2\mathrm{\Psi }}{\mathrm{\partial }{X}^2}+{\displaystyle \frac{{\mathrm{\partial }}^2\mathrm{\Psi }}{\mathrm{\partial }{Y}^2}=-Ra{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }X}}}}$$

(6)

$$\begin{array}{rl}& {\displaystyle \frac{{\mathrm{\partial }}^2{\mathrm{\Theta }}_f}{\mathrm{\partial }{X}^2}+{\displaystyle \frac{{\mathrm{\partial }}^2{\mathrm{\Theta }}_f}{\mathrm{\partial }{Y}^2}={\displaystyle \frac{1}{ϵ}\left(U{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }X}+V{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }Y}}}\right)}}}\\ & -{\displaystyle \frac{1}{ϵ}Nhs.({\mathrm{\Theta }}_s-{\mathrm{\Theta }}_f)}\end{array}$$

(7)

$${\displaystyle \frac{{\mathrm{\partial }}^2{\mathrm{\Theta }}_s}{\mathrm{\partial }{X}^2}+{\displaystyle \frac{{\mathrm{\partial }}^2{\mathrm{\Theta }}_s}{\mathrm{\partial }{Y}^2}=Nhs.\gamma .({\mathrm{\Theta }}_s-{\mathrm{\Theta }}_f)}}$$

(8)

Subjected to the boundary conditions:

$$\begin{array}{rl}& \mathrm{On}\mathrm{bottom},{\mathrm{\Gamma }}_1(0\le X\le 1,Y=0):\\ & \mathrm{\Psi }=0,{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }Y}={\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_s}{\mathrm{\partial }Y}=0}}\\ & \mathrm{On}\mathrm{right},{\mathrm{\Gamma }}_2(x=1+a(\cos (2\pi Ny)-1)),\le Y\le 1):\\ & \mathrm{\Psi }=0,{\mathrm{\Theta }}_f={\mathrm{\Theta }}_s=0\\ & \mathrm{On}\mathrm{top},{\mathrm{\Gamma }}_3(0\le X\le 1,Y=1):\\ & \mathrm{\Psi }=0,{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }Y}={\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_s}{\mathrm{\partial }Y}=0}}\\ & \mathrm{On}\mathrm{left},{\mathrm{\Gamma }}_4(X=0,\le Y\le 1):\\ & \mathrm{\Psi }=0,{\mathrm{\Theta }}_f={\mathrm{\Theta }}_s=1\end{array}$$

(9)

The non-dimensional parameters Nusselt number $(Nu)$ and average Nusselt number $(\overline{Nu})$ are calculated by:

$$\begin{array}{ccccc}& Nu{(Y)}_{f,s}& & =-{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_{f,s}}{\mathrm{\partial }X}{|}_{X=0}}& \\ & {\overline{Nu}}_{f,s}& & ={\int }_0^{1}Nu{(Y)}_{f,s}dY& \end{array}$$

(10)

Also, the temperature difference between the fluid phase and solid phase $(\mathrm{\Delta }\mathrm{\Theta })$ and the average temperature deference $(\overline{\mathrm{\Delta }\mathrm{\Theta }})$ are calculated by:

$$\begin{array}{rl}\mathrm{\Delta }\mathrm{\Theta }& ={\mathrm{\Theta }}_f-{\mathrm{\Theta }}_s\\ \\ \overline{\mathrm{\Delta }\mathrm{\Theta }}& ={\displaystyle \frac{1}{\mathrm{area}\mathrm{of}\mathrm{\Omega }}\underset{\mathrm{\Omega }}{\int \int }|\mathrm{\Delta }\mathrm{\Theta }|d\mathrm{\Omega }}\end{array}$$

(11)\end

Following Tayebi et al., ²⁹ LTNE and LTE cases are classified at a 5% value of $\overline{\mathrm{\Delta }\mathrm{\Theta }}$ .

Numerical method

The coupled non-dimensional governing equations (6) to (8) approximated results are computed iteratively using the finite element method based on the Galerkin formulation. The domain of computation $(\mathrm{\Omega })$ bounded by ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2,{\mathrm{\Gamma }}_3,\mathrm{and}{\mathrm{\Gamma }}_4$ is subdivided into triangular elements, grid distribution is shown in Figure 1(b) and bi-quadratic triangular shape functions ${\varphi }_0,{\varphi }_1,\mathrm{and}{\varphi }_2$ are considered for unknowns $\mathrm{\Psi },{\mathrm{\Theta }}_s,$ and ${\mathrm{\Theta }}_f$ , respectively. The governing equations (6) to (8) along with boundary conditions (9) are solved in coupled form, imposing natural boundary conditions from (9), weak form of the problem is obtained as:

$${\int \int }_{\mathrm{\Omega }}\left({\displaystyle \frac{\mathrm{\partial }{\varphi }_0}{\mathrm{\partial }X}{\displaystyle \frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }X}+{\displaystyle \frac{\mathrm{\partial }{\varphi }_0}{\mathrm{\partial }Y}{\displaystyle \frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }Y}-Ra\cdot {\varphi }_0{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }X}}}}}}\right)\mathrm{\partial }\mathrm{\Omega }=0$$

(12)

$$\begin{array}{r}{\int \int }_{\mathrm{\Omega }}({\displaystyle \frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }X}{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }X}+{\displaystyle \frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }Y}{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }Y}+{\displaystyle \frac{{\varphi }_1}{ϵ}\left(U{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }X}+V{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_f}{\mathrm{\partial }Y}}}\right)}}}}}\\ -{\displaystyle \frac{{\varphi }_1}{ϵ}Nhs\cdot ({\mathrm{\Theta }}_s-{\mathrm{\Theta }}_f)})\mathrm{\partial }\mathrm{\Omega }=0& \end{array}$$

(13)

$$\begin{array}{r}{\int \int }_{\mathrm{\Omega }}({\displaystyle \frac{\mathrm{\partial }{\varphi }_2}{\mathrm{\partial }X}{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_s}{\mathrm{\partial }X}+{\displaystyle \frac{\mathrm{\partial }{\varphi }_2}{\mathrm{\partial }Y}{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Theta }}_s}{\mathrm{\partial }Y}}}}}\\ +Nhs\cdot \gamma \cdot {\varphi }_2\cdot ({\mathrm{\Theta }}_s-{\mathrm{\Theta }}_f))\mathrm{\partial }\mathrm{\Omega }=0\end{array}$$

(14)

along with essential boundary conditions, $\mathrm{\Psi }=0$ on ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2,{\mathrm{\Gamma }}_3,{\mathrm{\Gamma }}_4$ and ${\mathrm{\Theta }}_f={\mathrm{\Theta }}_s=0$ on ${\mathrm{\Gamma }}_2$ , ${\mathrm{\Theta }}_f={\mathrm{\Theta }}_s=1$ on ${\mathrm{\Gamma }}_4$ . The non-dimensional momentum equation (6) and two energy equations (7), (8) are solved with the help of FreeFem++, and the variational formulations (12) to (14) are programmed according to the FreeFem++ syntax, in this case, local as well as global stiffness matrices and force vectors are generated by the FreeFem++ and solved by LU decomposition. Since the matrix formulations and solution process are performed by FreeFem++, we skip the matrix representations of the problem, details about the solver can be found at. ³⁰ The coupled equations are solved iteratively taking the convergence criteria as,

$$\sqrt{{\int }_{\mathrm{\Omega }}({({\mathrm{\Psi }}^{i+1}-{\mathrm{\Psi }}^i)}^2+{({\mathrm{\Theta }}_f^{i+1}-{\mathrm{\Theta }}_f^i)}^2+{({\mathrm{\Theta }}_s^{i+1}-{\mathrm{\Theta }}_s^i)}^2)}<{10}^{-6}$$

where $i=0,1,2,\dots$ number of iterations. We achieved the threshold value of the $L_2$ norm of error precision ${10}^{-6}$ which is quite good and optimum beyond that the changes are almost negligible for this case.

With the help of open-source multi-physics solver FreeFem++, the CFD code is developed, and it is tested and validated by comparing $\overline{Nu}$ with previously published research papers representing LTNE natural convection in a square porous cavity. An analogy of ${\overline{Nu}}_f$ and ${\overline{Nu}}_s$ at different $\gamma$ values is shown in Table 1, also comparisons at different Nhs are shown in Figure 2. Both comparisons give adequate agreement with the results which have been published. Hence, the code may be considered appropriate for investigating LTNE natural convection in a porous cavity.

Figure 2.

Comparison of $\overline{N{u}_f}$ (left) and $\overline{N{u}_s}$ (right) with Sheremet et al. ² for pure fluid at $Ra={10}^3$ , $ϵ=0.6$ .

Table 1.

Comparisons of $\overline{Nu}$ at $ϵ=0.9$ and $Ra={10}^3$ .

		Obtained	Bayats and Pop 1	Sheremet et al. 2
$\gamma =1.0$	${\overline{Nu}}_f$	11.08	11.01	10.97
	${\overline{Nu}}_s$	7.32	7.248	7.27
$\gamma =10.0$	${\overline{Nu}}_f$	14.08	13.96	13.96
	${\overline{Nu}}_s$	12.16	12.19	11.99

Further, the code is validated for a cavity having a wavy wall, from literature streamlines and isotherms are compared for natural convection in a wavy cavity having thermal equilibrium between fluid and solid matrix, Figure 3 shows a satisfactory comparison between obtained results and published results.

Figure 3.

Comparisons of streamlines and isotherms: (a) Isotherms and (b) Streamlines (in each sub-figure, left: presented by Cheong et al., ¹⁵ right: present study).

This shows the finite element code developed in FreeFem++ may be suitable for the study of convection in a wavy porous cavity.

The mesh is created using the Delaunay triangulation technique, and the number of elements depends on the number of points measured along the borders. The effect of mesh is observed between wide number of degrees of freedom (DOF) and calculated $\overline{N{u}_f}$ and $\overline{N{u}_s}$ values for $Ra={10}^3,ϵ=0.6,Nhs=100.0,a=0.15,$ and $N=5$ are displayed in Table 2. Also, the grid convergence index (GCI) and Richardson extrapolation are calculated with respect to flow strength $|\mathrm{\Psi }{|}_m$ considering $r\sim 1.3$ (Roache ³¹ ), the ratio between a number of elements, and the results are presented in Supplemental Table 3. Observing the relative changes and the computational time taken to converse, a mesh having DOF 120322 at $a=0.15$ and $N=5$ is considered for further investigation. The code which has been developed may be considered appropriate for this investigation on the basis of the comparisons presented above and the independence of grids.

Table 2.

Effect of mesh on $\overline{Nu}$ at $Ra={10}^3$ at $a=0.15$ and $N=3$ .

i	DOF	$\overline{N{u}_f}$	$100\times \frac{|{\overline{N{u}_f}}^{(i)}-{\overline{N{u}_f}}^{(3)}|}{|{\overline{N{u}_f}}^{(i)}|}$	$\overline{N{u}_s}$	$100\times \frac{|{\overline{N{u}_s}}^{(i)}-{\overline{N{u}_s}}^{(3)}|}{|{\overline{N{u}_s}}^{(i)}|}$
0	30,829	14.5328	1.3067%	1.3944	0.0072%
1	58,306	14.3226	0.1417%	1.3944	0.0072%
2	96,754	14.2862	0.3969%	1.3945	0.0%
3	120,322	14.3429	–	1.3945	–
4	175,338	14.3763	0.2323%	1.3945	0.0%

Results and discussion

The obtained results are analyzed in terms of streamlines, isotherms for solid and fluid phases, the dimensionless temperature difference between two phases and heat transfer coefficient (Nusselt number). Figures 4 to 6 represent the contour plots for $\mathrm{\Psi }$ , ${\mathrm{\Theta }}_f$ , ${\mathrm{\Theta }}_s$ , and $\mathrm{\Delta }\mathrm{\Theta }$ at different Ra and Nhs value. From the obtained figures at $ϵ=0.5$ , $N=3$ , $a=0.15$ , a circulation zone (in the clockwise direction) for $\mathrm{\Psi }$ is observed covering most of the free spaces of the cavity; and the isotherms for ${\mathrm{\Theta }}_f$ and ${\mathrm{\Theta }}_s$ are spread between top and bottom adiabatic wall, and for a fixed Ra the effect of Nhs are strong on ${\mathrm{\Theta }}_f$ than ${\mathrm{\Theta }}_s$ . Also, it is observed that incomplete contour lines having positive and negative values for $\mathrm{\Delta }\mathrm{\Theta }$ are present. In the case of $\mathrm{\Delta }\mathrm{\Theta }>0$ , the contour lines in the upper section of the cavity indicate fluid phase is having more temperature than the solid phase $({\mathrm{\Theta }}_f>{\mathrm{\Theta }}_s)$ and in the lower half of the cavity the solid phase have more temperature than the fluid phase $({\mathrm{\Theta }}_s>{\mathrm{\Theta }}_f)$ inside the wavy porous cavity.

Figure 4.

Streamlines and isotherms at $Ra=10$ , $ϵ=0.5$ , $N=3$ , and $a=0.15$ .

Figure 5.

Streamlines and isotherms at $Ra={10}^2$ , $ϵ=0.5$ , $N=3$ , and $a=0.15$ .

Figure 6.

Streamlines and isotherms at $Ra={10}^3$ , $ϵ=0.5$ , $N=3$ , and $a=0.15$ .

Figure 4, streamlines and isotherms at $Ra=10$ , shows that the effect of heat transfer parameter $(Nhs)$ has negligible effect for $Nhs>0$ at $ϵ=0.5$ , the flow strength $(|\mathrm{\Psi }{|}_m)$ have no change as Nhs varies from 100 to 1000 and it is $0.7$ , a minor increment from value at $Nhs=0(|\mathrm{\Psi }{|}_m=0.67)$ . At the same time, there are minor changes in isolines for both ${\mathrm{\Theta }}_f$ and ${\mathrm{\Theta }}_s$ in case $Nhs\ne 0$ . The flow field has a similar trend at an increased Ra but observed significant flow strength changes. The $\mathrm{\Delta }\mathrm{\Theta }$ contours are symmetric with respect to $Y=0.5$ for $Nhs>0$ , but due at $Nhs=0$ the symmetric line is left side up and right side down.

From Figures 5 and 6 for $Ra={10}^2$ and $Ra={10}^3$ , respectively, it is observed that at a fixed Ra, the formation of boundary layers and thermal boundary layer are obvious at low Nhs and the $|\mathrm{\Psi }{|}_m$ is increased as Nhs increases, indicating higher Nhs leads to more fluid circulation at the center of the cavity than close to the boundary. The streamlines are wavy near the undulated wall, but an increase in Nhs decreases the wavy nature of the contour close to the wavy boundary.

There is a significant change in $\mathrm{\Delta }\mathrm{\Theta }$ with respect to Nhs at $Ra={10}^2$ , the negative and positive contour lines are partitioned diagonally at $Nhs=0$ and gradually the line of separation become horizontal as Nhs reaches 1000. This indicates near the left bottom corner of the cavity solid phase has more temperature than a fluid phase, and close to the top and right corner, the fluid phase exhibits more temperature than the solid phase but the increment in Nhs boosts heat transfer between the two phases. At $Ra={10}^3$ , since fluid velocity increases, as a result, the fluid phase carries more thermal energy in the top right corner of the cavity, so the positive contour lines are close to the top right corner, and the negative lines are at the bottom left corner of the cavity for Nhs between 0 and 1000.

At $ϵ=0.5$ , $N=3,a=0.15$ from Figures 4 to 6 it is observed that at a fixed Ra, there is no major change in isotherms for the fluid phase, the lines are parallel to the left vertical wall and have wavy nature close to the wavy wall, although the manifestation of thermal boundary layers is obvious at low Nhs for $Ra>10$ .

For the fluid phase $(\overline{N{u}_f})$ and solid phase $(\overline{N{u}_s})$ , the average Nusselt numbers are calculated and presented in Figures 7 and 8, Supplemental Figure 9 for cavities having $N=1,3,5,$ respectively, and the presented results shows that in a wavy porous cavity for a fixed Ra and $ϵ$ , the $\overline{N{u}_f}$ is increasing order and $\overline{N{u}_s}$ is in decreasing order as Nhs increases from 0 to 1000. Also, the effect of Nhs on $\overline{N{u}_f}$ and $\overline{N{u}_s}$ are negligible at $Ra=10$ and the effect increases as Ra increases to ${10}^3$ . The porosity $(ϵ)$ of Darcy porous media controls the heat transfer in solid and fluid phase at high Ra, and for a fixed $ϵ$ , heat transfer parameter has the role to increase heat transfer by the solid phase restricting the heat transfer by fluid phase, also from Figures 7 and 8, Supplemental Figure 9, the $\overline{N{u}_f}$ , and $\overline{N{u}_s}$ are parallel to Nhs axis at $Nhs\ge 1000$ . Using curve feting method the flowing correlations are drawn for $Nhs=100$ and $a=0.15,N=3$ ,

$$\begin{array}{rl}N{u}_f& =1.0+0.007737{ϵ}^{-0.5083}R{a}^{1.041}\\ N{u}_s& =1.0+0.327{ϵ}^{0.03665}R{a}^{0.02796}\end{array}$$

Figure 7.

Nhs versus $\overline{Nu}$ plots at $N=1$ : (a) Fluid phase and (b) Solid phase (linestyle: $-.-$ lines for $Ra={10}^3$ , $--$ lines for $Ra={10}^2$ , and $-$ lines for $Ra=10$ ).

Figure 8.

Nhs versus $\overline{Nu}$ plots at $N=3$ : (a) Fluid phase and (b) Solid phase (linestyle: $-.-$ lines for $Ra={10}^3$ , $--$ lines for $Ra={10}^2$ , and $-$ lines for $Ra=10$ ).

The effect of wall undulation on heat transfer can be explained in terms of change in N and a. From Figures 7 and 8, Supplemental Figure 9, the wall waviness has a direct impact on the heat transfer coefficient. The increment in N and a helps the heat transfer for both cases. The enhancement of heat transfer for the solid phase is more than the fluid phase due to wall waviness.

The maximum absolute difference between the two phases $(|\mathrm{\Delta }\mathrm{\Theta }{|}_m)$ are presented in Supplemental Figure 10, from the results it is observed that at $N=3$ and $a=0.15$ the $|\mathrm{\Delta }\mathrm{\Theta }{|}_m$ values are in decreasing order with the increment in Nhs for a fixed Ra or $ϵ$ . The increment in Ra from 10 to ${10}^3$ shows augmentation of $|\mathrm{\Delta }\mathrm{\Theta }{|}_m$ for a fixed Nhs, this is due to the increment of the convective heat transfer of fluid with Ra. Also, the $|\mathrm{\Delta }\mathrm{\Theta }{|}_m$ is decreasing with respect to $ϵ$ for a fixed Nhs.

Considering 5% significance level of $\overline{\mathrm{\Delta }\mathrm{\Theta }}$ LTNE and LTE cases are presented in Supplemental Figure 11 at $N=3$ and various parameters. The cases falling below the 5% value of $\overline{\mathrm{\Delta }\mathrm{\Theta }}$ are considered as LTE cases and cases falling above are LTNE. At $Ra={10}^3$ the convection inside the wavy cavity are LTNE and in the case of $Ra=10$ and ${10}^2$ the cases are LTNE at low Nhs. From the presented figure, it’s observed that LTNE cases for $Ra=10$ are close to $Nhs=0$ and for $Ra={10}^2$ LTNE cases are observed for a maximum value of Nhs close to 300. The wall waviness has a minimal effect on $\overline{\mathrm{\Delta }\mathrm{\Theta }}$ , and the wall amplitude $(a)$ reduces $\overline{\mathrm{\Delta }\mathrm{\Theta }}$ for a fixed number of undulations $(N)$ , $ϵ$ , Nhs, and Ra. The change of $\overline{\mathrm{\Delta }\mathrm{\Theta }}$ is prominent in the case of low Ra; the change of a for a fixed N increases LTE cases.

Comparing the present results with previously published LTNE convection by Baytas and Pop, ¹ an improvement in Nu due to undulated cold wall is observed. From Supplemental Figure 12, presented graphs show that the convection for both the phases is augmented by the wavy cold wall.

Conclusions

The finite element method is used to perform this investigation of LTNE-free convection in a wavy porous enclosure numerically, and the findings are grid independent. The discussion leads to the following astounding conclusions:

The influence of Nhs on flow strength is minor at a low Reynolds number $(Ra=10)$ , and the influence is very high as Ra increases $(Ra={10}^3)$ . At $N=3$ , $a=0.15$ , and $ϵ=0.5$ , for a fixed Ra, say, at $Ra={10}^3$ the $|\mathrm{\Psi }{|}_{max}$ increases 283% as Nhs changes from 0 to 1000.

The wavy cavity helps heat transfer enhancement for the solid phase and fluid phase. The monotonicity of $\overline{N{u}_f}$ and $\overline{N{u}_s}$ is present in the wavy cavity.

The dimensionless temperature difference between the two cases is at most for low Nhs and higher Ra.

Considering a 5% significance level the LTNE cases are observed of $Ra={10}^3$ and at $Ra<{10}^3$ the LTNE cases are observed only for low Nhs.

To solve the governing equation, a dense mesh is considered; mesh adaptation would be an efficient way to investigate wavy wall problems.

A limitation of the study is that it considers a square cavity that has been placed horizontally, and the effect of aspect ratio and angle of inclination will be an important extension to this study.

Supplemental Material

sj-pdf-1-pie-10.1177_09544089231154363 - Supplemental material for Numerical analysis of local thermal non-equilibrium free convection in a porous enclosure with a wavy cold side wall

Supplemental material, sj-pdf-1-pie-10.1177_09544089231154363 for Numerical analysis of local thermal non-equilibrium free convection in a porous enclosure with a wavy cold side wall by Prabir Barman and Pentyala Srinivasa Rao in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering