Double Diffusive Natural Convection in a Square Cavity Filled with a Porous Media and a Power Law Fluid Separated by a Wavy Interface

Abstract

This study deals with the influence of a wavy interface separating two layers filled with power law fluid and porous media, respectively. The governing equations are solved using the Finite Element Method (FEM) and the numerical model is validated by comparing with experimental findings. The parameters governing the studied configuration are varied as: Rayleigh number (10³ ≤ Ra ≤ 10⁶), power law index (0.6 ≤ n ≤ 1.4), Darcy number (10⁻² ≤ Da ≤ 10⁻⁶), buoyancy ratio (0.1 ≤ N ≤ 10) and Lewis number (1 ≤ Le ≤ 10). It is inferred that the temperature gradient increases by augmenting the Rayleigh number, as the flow is observed from the vertical to horizontal direction in both layers. Constant enhancement in the heat and mass transfer is also observed by enriching the buoyancy effect. Moreover, the average Nusselt and Sherwood numbers decline by increasing the width of the porous layer.

1. Introduction

The double diffusion is a phenomenon that occurs in several engineering and industrial applications such as pulp paper, oil drilling, heat removal, heat storage, and food processing [1,2,3,4,5,6]. There are relatively few works that have been conducted on non-Newtonian fluid, instead of Newtonian fluid. Al-Amir et al. [7] reported the impact of the Prandtl number on the natural convection in cavity-containing non-Newtonian and nanofluid porous mediums, which are separated by the sinusoidal interface. It has been realized that the average Nusselt number rises by enhancing the Darcy and Prandlt number, and reduces by enriching the power law index. Alsabery et al. [8] examined the trapezoidal-shaped cavity including two layers (porous and non-Newtonian). The results confirmed that the flow rises remarkably when using silver nanofluid, and the effectiveness of heat transfer is perceived by varying the angle of inclination. The impact of wavy interfaces on the natural convection in non-Darcy porous cavities is studied by Nguyen et al. [9], using the incompressible smoothed particle hydrodynamics (ISPH) method. The results indicated that the average Nusselt number decreases by increasing the amplitude and undulation number of the interface separating the layers. Alsabery et al. [10] performed a numerical study to investigate the effect of the inclination on natural convection in a cavity filled with a porous media and a non-Newtonian fluid. The findings showed that convection is more intense for lower values of power law index. Power law fluids have attracted the researchers due to their importance in several engi-neering applications [11]. More related studies can be found in References [12,13,14,15,16]. The natural convection in porous layers has been studied Al-Srayyih et al. [17] using Galerkin Finite Element Method (GFEM). It was found that the enhancement of the heat transfer occurs using nanofluids. Barnoon et al. [18] investigated the coupled radiation-convection in a cavity filled with a non-Newtonian fluid and equipped with internal obstacles. The authors concluded that the tilt angle has an important effect of the values of Nusselt number. Jabber et al. [19] investigated the 2D natural convection in an enclosure filled with a porous medium saturated with a anofluid and equipped by wavy interfaces. The authors mentioned that power law index causes a reduction of the heat transfer rate. The effect of a power law fluid on natural convection in a cavity having wavy wall has been investigated by Chen et al. [20]. It was shown that the rate of heat transfer of pseudoplastic fluid is better compared to Newtonian fluid. Kefayati et al. [21] applied LBM to realize the behavior of non-Newtonian fluid under natural convection; a uniform magnetic field is also applied in this study. The results indicated that the heat transfer increases with the power law index and decreases with the Hartman number. Saleh et al. [22] studied a differentially heated cavity equipped with rotating obstacles. The results showed that the values of Nusselt number remain constant at L/D > 0.77. Other research papers related to the effect of separating interfaces in cavities are reported [23,24,25,26,27,28]. Turan et al. [29] studied the natural convection of a non-Newtonian fluid. They investigated the influence of the Prandtl number, and proposed new correlations for the Nusselt number for Newtonian and non-Newtonian fluids.

Based on the above-described literature review, and to the best of the authors’ knowledge, there is currently no published work on wavy interfaces in square cavities filled with two layers (non-Newtonian and porous (Newtonian)). The applications of the current study are in engineering and industry. We can further study this by increasing the layers. The effect of various controlled parameters, such as the Rayleigh number, power law index, Darcy number, buoyancy ratio, and Lewis number, is examined in detail.

2. Problem Formulation

The problem configuration is schematically illustrated in (Figure 1). It consists of 2D double-diffusive convection in a cavity containing two fluid layers (non-Newtonian and porous Newtonian fluid), which are separated by a sinusoidal wall. L and H represent the length and height of the cavity, respectively. ${\theta }_h$ and $C_h$ are the high temperature and concentration at the left wall, respectively. Similarly, ${\theta }_c$ and $C_c$ are the low temperature and concentration at the right wall, respectively. The remaining horizontal walls are considered to be adiabatic. The flow is considered to be laminar, steady, and incompressible, with the application of the Darcy–Brinkman–Forchheimer model.

The wavy interface is derived from the following equation:

$$X=Hp+A Sin\left(\frac{2\pi K}{L} Y+\phi \right)$$

(1)

where Hp and A represent the width of the porous layer and the amplitude, respectively. $\phi$ denotes the phase shift, which is taken as $\frac{\pi }{2}$.

3. The Governing Equations

Under the above-considered assumptions, the dimensionless systems of the equations for the fluid region and porous layer are written as follows [30,31,32,33,34,35,36]:

For the non-Newtonian fluid layer:

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$

(2)

$$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{Pr}{\sqrt{Ra}}\left[2\frac{\partial }{\partial X}\left(\frac{{\mu }_a}{M}\frac{\partial U}{\partial X}\right)+\frac{\partial }{\partial Y}\left(\frac{{\mu }_a}{M}\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)\right)\right]$$

(3)

$$\begin{gathered} U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+\frac{Pr}{\sqrt{Ra}}\left[2\frac{\partial }{\partial Y}\left(\frac{{\mu }_a}{M}\frac{\partial V}{\partial Y}\right)+\frac{\partial }{\partial X}\left(\frac{{\mu }_a}{M}\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)\right)\right] \\ +Pr\left(\theta +NC\right) \end{gathered}$$

(4)

$$U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{1}{\sqrt{Ra}}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)$$

(5)

$$U\frac{\partial C}{\partial X}+V\frac{\partial C}{\partial Y}=\frac{1}{Le \sqrt{Ra}}\left(\frac{{\partial }^{2}C}{\partial X^2}+\frac{{\partial }^{2}C}{\partial Y^2}\right)$$

(6)

$${\mu }_a=M{\left\{2\left[{\left(\frac{\partial U}{\partial X}\right)}^2+{\left(\frac{\partial V}{\partial Y}\right)}^2\right]+{\left(\frac{\partial V}{\partial X}+\frac{\partial U}{\partial Y}\right)}^2\right\}}^{\frac{n-1}{2}}$$

(7)

where N is the buoyancy ratio and n is the power law index.

The following also applies:

$$Pr=\frac{{\mu }_a}{\rho \alpha }$$

(8)

$$Ra=\frac{\rho \beta g_y{L}^3\left(T_H-T_C\right)}{{\mu }_a\alpha }$$

(9)

$$Le=\frac{a_e}{D}$$

(10)

For the porous fluid layer:

$$\frac{\partial \overline{U}}{\partial X}+\frac{\partial \overline{V}}{\partial Y}=0$$

(11)

$$\overline{U}\frac{\partial \overline{U}}{\partial X}+\overline{V}\frac{\partial \overline{U}}{\partial Y}=-\frac{\partial \overline{P}}{\partial X}+\frac{P{r}_m}{\sqrt{R{a}_m}}\left(\frac{{\partial }^2\overline{U}}{\partial X^2}+\frac{{\partial }^2\overline{U}}{\partial Y^2}\right)-\frac{{Pr}_m \overline{U}}{D{a}_m \sqrt{R{a}_m}}-\frac{\overline{U} \left|\overline{U}\right|}{\sqrt{D{a}_m}} \frac{1.75}{\sqrt{150}}$$

(12)

$$\begin{gathered} \overline{U}\frac{\partial \overline{V}}{\partial X}+\overline{V}\frac{\partial \overline{V}}{\partial Y}=-\frac{\partial \overline{P}}{\partial Y}+\frac{P{r}_m}{\sqrt{R{a}_m}}\left(\frac{{\partial }^2\overline{V}}{\partial X^2}+\frac{{\partial }^2\overline{V}}{\partial Y^2}\right)+P{r}_m\left(\overline{\theta }+N\overline{C}\right)-\frac{{Pr}_m \overline{V}}{D{a}_m \sqrt{R{a}_m}} \\ -\frac{\overline{V} \left|\overline{U}\right|}{\sqrt{D{a}_m}} \frac{1.75}{\sqrt{150}} \end{gathered}$$

(13)

$$\overline{U}\frac{\partial \overline{\theta }}{\partial X}+\overline{V}\frac{\partial \overline{\theta }}{\partial Y}=\frac{1}{\sqrt{R{a}_m}}\left(\frac{{\partial }^2\overline{\theta }}{\partial X^2}+\frac{{\partial }^2\overline{\theta }}{\partial Y^2}\right)$$

(14)

$$\overline{U}\frac{\partial \overline{C}}{\partial X}+\overline{V}\frac{\partial \overline{C}}{\partial Y}=\frac{1}{Le \sqrt{R{a}_m}}\left(\frac{{\partial }^2\overline{C}}{\partial X^2}+\frac{{\partial }^2\overline{C}}{\partial Y^2}\right)$$

(15)

where the following parameters are entered in the above problem:

$$Da=\frac{K}{L^2}$$

(16)

The relationship among the actual and modified Prandtl, Rayleigh and Darcy numbers can be written as follows: $P{r}_m=Pr\in , R{a}_m=Ra\in$ and $D{a}_m=\frac{Da}{\in }$.

The boundary conditions of the proposed problem in the corresponding regions are as follows:

• $\overline{U}$ = 0, $\overline{\theta }$ = 1, $\overline{V}$ = 0, $\overline{C}$ = 1 (left wall)
• U = 0, θ = 0, V = 0, C = 0 (right wall)
• $\frac{\partial \overline{\theta }}{\partial Y}=\frac{\partial \overline{C}}{\partial Y}=0$, $\overline{U}$ = $\overline{V}$ = 0 and $\frac{\partial \theta }{\partial Y}=\frac{\partial C}{\partial Y}=0$, U = V = 0 (top/bottom walls)

The numbers on the vertical hot wall are given as follows:

$$Nu={\left(-\frac{\partial \overline{\theta }}{\partial X}\right)}_{X=0}:\mathrm{Nusselt}\mathrm{number}(\mathrm{local})$$

(17)

$$Sh={\left(-\frac{\partial \overline{C}}{\partial X}\right)}_{X=0}:\mathrm{Sherwood}\mathrm{number}(\mathrm{local})$$

(18)

$$N{u}_{avg}={{\displaystyle \int }}_0^{1}Nu dY:\mathrm{Nusselt}\mathrm{number}(\mathrm{average})$$

(19)

$$S{h}_{avg}={{\displaystyle \int }}_0^{1}Sh dY :\mathrm{Sherwood}\mathrm{number}(\mathrm{average})$$

(20)

4. Solution Methodology

The mathematical models presented in Equations (2)–(7) and (11)–(15) is solved using the higher order Galerkin finite element method. As a first step, a weak formulation is developed by choosing a suitable test space. Afterwards, a hybrid mesh, consisting of both triangular and quadrilateral elements, is generated to cover the computational domain. A finite element method involving the cubic polynomials (P₃) is implemented to compute the velocity, temperature, and concentration fields, while the pressure is approximated by the quadratic (P₂) finite element space of functions. The stability and robustness of this higher-order pair of FEMs has been tested in [37]. The system of discretized equations is simplified using the adaptive Newton’s method. For further details regarding the solver, the reader is referred to [38].

4.1. Grid Convergence

As shown in

Table 1. Convergence of grid analysis.

Grid	NEL	DOFs	$\mathit{N}{\mathit{u}}_{\mathit{a}\mathit{v}\mathit{g}}$	$\mathit{S}{\mathit{h}}_{\mathit{a}\mathit{v}\mathit{g}}$
G1	230	3322	3.428355	5.767679
G2	352	4974	3.418098	5.727896
G3	540	7319	3.420159	5.715686
G4	1006	13007	3.422258	5.703224
G5	1520	19040	3.424014	5.701316
G6	2470	29757	3.424601	5.700223
G7	6544	76933	3.424613	5.698674
G8	16790	192148	3.424915	5.698709

, eight number of elements are compared to check the mesh independency at Gr = 10⁵, Da = 10⁻³, ϵ = 0.75 and Hp = 0.1. From this comparison it is clear that the deviations of Nu_av and Sh_av between the grids G7 and G8 are very small. Thus, for time economy and results accuracy the grid G7 was retained for all the performed simulations. The grid for the proposed computational model is displayed in Figure 2.

4.2. Code Validation

The current results are checked by comparing them to those of Gibanov et al. [39], by evaluating the average Nusselt number under the same conditions. As is shown in

Table 2. Validation of the code of average Nu with [39]. Adapted with permission from ref. [39]. Copyright 2017 Springer Nature.

Present Study	[39]	Error
12.57879	12.5335	0.36%
Present Model		Gibanov et al. [39].

, the error did not exceed 0.36%, which represents good agreement among the two results. Moreover, the numerical code is checked by comparing the isotherms and streamlines with those found by Gibanov et al. [39]. The results of the current numerical model are in concordance with the results of Gibanov et al. [39] (Figure 3).

In addition to the above validation, to strengthen the reliability of the implemented FEM, we also confirmed the validation by comparing it with the experimental study of Corvaro and Paroncini [40], as demonstrated in Figure 4.

5. Results and Discussion

The results of the different numerical simulation cases are presented by streamlines, isotherms and isoconcentration contours, as well as variations in the local Nu and Sh, and variations in the average Nu and Sh. All the simulations are performed for the fixed values of n = 0.6, Da = 10⁻³, ϵ = 1.0, Le = 2.5, Pr = 1, N = 0.1, M = 2, Hp = 0.5, A = 0.05, and Ra = 10⁵.

For each case, one parameter is varied. In fact, the results will consecutively present the effect of the variation in the Rayleigh number (Ra), power law index (n), Darcy number (Da), buoyancy number (N), and Lewis number (Le). Furthermore, the effects of the width of the porous media (Hp), the undulation number of interface (K), and its amplitude (A) on the average Nu and Sh are presented for a range of the power law index (n), varied from 0.6 to 1.6.

5.1. Impact of Rayleigh Number

Figure 5 presents the effect of the Rayleigh number on the streamlines (left side), isotherms (center), and isoconcentrations (right side). Due to its great effect on the heat transfer rate, the Rayleigh number is varied from 10³ to 10⁶. For a low Rayleigh number (Ra = 10³), the center of the main vortex is located on the non-Newtonian side of the cavity and rotates counterclockwise. By increasing the Ra number, the rotation of the main vortex is intensified. Remarkable penetration in the direction of the porous layer is detected, due to the elongation of the main vortex.

As can be observed for Ra = 10⁶, the intensity of the streamlines occurs at the core of the cavity, indicating an increase in the heat transport, due to the natural convection. Here, it is noticed that the center of the vortex is still on the non-Newtonian side. By increasing the Ra number, the behavior of the isotherms is modified, due to the enhancement of the temperature gradient. When the Ra number is equal to 10³, the isotherms are parallel to each other, indicating the domination of the conductive heat transfer mode, but as its value rises, an increase in the temperature gradient is observed. The thermal flow passes from a conductive regime to a convective regime by increasing the Ra number. Indeed, the stratification of the isotherms is pronounced when the Ra number passes from 10³ to 10⁶. The thermal gradient is more densely packed close to the downside of the hot wall, indicating an important thermal boundary layer. This fact is confirmed by the coordinates of the height of the local Nusselt numbers in Figure 4. The isoconcentration behavior is greatly affected by the increase in the Ra numbers, which indicates a progressive move from a diffusive to convective mass transfer. When Ra = 10⁶, the solutal gradient near the hot wall is important, which indicates the importance of the convective transfer from the non-Newtonian fluid layer to the porous layer.

Figure 6 presents the variation in the local Nu and Sh for different Ra numbers. The curves prove the high increase in the Nu and Sh values when the Rayleigh number passes from 10⁵ to 10⁶.

Table 3. Average Nu and Sh comparison.

	$\mathit{R}\mathit{a} = {10}^3$	$\mathit{R}\mathit{a} = {10}^4$	$\mathit{R}\mathit{a} = {10}^5$	$\mathit{R}\mathit{a} = {\mathbf{10}}^{\mathbf{6}}$
$N{u}_{avg}$	1.003544	1.294151	3.425027	8.530151
$S{h}_{avg}$	1.021881	2.017548	5.699002	13.07502

presents a comparison between the average Nu and average Sh for various Rayleigh numbers, under the same conditions. For Ra = 10³ and Ra = 10⁴, the Nu values are low compared to the other Rayleigh values. This is due to the lower heat exchange between the porous medium layer and the non-Newtonian layer. The maximum values of the average Nu and Sh numbers confirm the importance of the mass transfer effect on behalf of the heat transfer, especially when the Ra numbers increase. This is due to the current case condition, in which the buoyancy number is equal to 0.1.

5.2. Impact of Power Law Index

Figure 7 presents the streamlines (left side), the isotherms (center), and the isoconcentration (right side). By definition, the power law index affects the heat generation and, consequently, the viscosity. The expected results need to demonstrate the regression of the heat transfer and show a great impact on the mass transfer between the two layers. As the power law index rises, the intensity and movement of the streamlines are slightly affected in the cavity, which means that the energy required to rotate the vortex decreases. This fact is explained further by plotting the variation in the local Nu and Sh, and calculating their mean values at the hot wall (

Table 4. Average Nu and Sh comparison.

	$\mathit{n} = 0.6$	$\mathit{n} = 0.8$	$\mathit{n} = 1$	$\mathit{n} = 1.2$	$\mathit{n} = 1.4$
$N{u}_{avg}$	3.424915	3.338505	3.272834	3.221858	3.181187
$S{h}_{avg}$	5.698709	5.519309	5.383465	5.278273	5.194299

). Regardless of the fact that they exhibit the same behavior, it is clear in Figure 8 that the variation in the local Nu and Sh drops by increasing the n values.

5.3. Impact of Darcy Number

Figure 9 presents the effect of the Darcy number (Da) variation on the streamlines, isotherms, and isoconcentrations, while the other parameters are constant. When the Da is equal to 10⁻⁶ and 10⁻⁵, the streamlines are localized in the non-Newtonian zone, which indicates that no exchange is performed between the porous and non-Newtonian layers. The isothermal and isoconcentration lines are parallel in the porous zone, compared to the non-Newtonian zone. By increasing the Darcy numbers, the spread of the streamline through the wavy interface is increasingly developed. The thermal and solutal gradient near the downside of the hot wall is noticed, due to the rise in the Darcy numbers. The results show that for Da = 10⁻², thermal stratification takes place and solutal distortion appears in the core cavity, since the value of the Darcy number increases to 10⁻³.

The variation in the local Nu and Sh for different Darcy numbers (Da) is shown in Figure 10. For low values of Darcy numbers (10⁻⁶ and 10⁻⁵), the heat and mass transfer are almost constant and at low levels. A slight increase is detected when the Darcy number is equal to 10⁻⁴. A noticeable increase in the heat and mass transfer effect takes place for Da = 10⁻³ and 10⁻², which is described by the rise in the vortex rotation intensity through the two sides of the wavy interface.

Table 5. Average Nu and Sh comparison.

	$\mathit{D}\mathit{a} = {10}^{-6}$	$\mathit{D}\mathit{a} = {10}^{-5}$	$\mathit{D}\mathit{a} = {10}^{-4}$	$\mathit{D}\mathit{a} = {10}^{-3}$	$\mathit{D}\mathit{a} = {10}^{-2}$
$N{u}_{avg}$	1.3556	1.373742	1.652894	3.425139	4.703467
$S{h}_{avg}$	1.537805	1.582851	2.50985	5.699306	7.181993

presents the evolution of the average Nu and Sh for the various values of Da, when Ra = 10⁵, Da = 10⁻³, ϵ = 0.75, and Hp = 0.1. The great increase in the heat and mass transfer values can be deduced by the increase in the diffusivity of the fluid flow through the interface to the porous medium.

5.4. Impact of Buoyancy Ratio

The effect of the variation in the buoyancy number N from 0.1 to 10, when the other values remain constant, is presented in Figure 11. The behavior of the streamlines is similar to the Ra variation cases. The main cell is lengthened to the whole cavity space by increasing the N values, although the center of the vortex is still localized in the non-Newtonian zone. The thermal gradient is densely packed near the downside of the hot wall in the porous media zone by the increase in the N values, which results in intensification of the heat transfer through the interface. Moreover, the solutal gradient is tightened near the downside of the hot wall and the topside of the cold wall, and stratification of the isoconcentrations takes place in the core region of the cavity.

These behaviors could be explained by the profiles of the local Nu and Sh in Figure 12, and their average values in

Table 6. Average Nu and Sh comparison.

	N = 0.1	N = 1	N = 5	N = 10
$N{u}_{avg}$	3.424915	4.102939	5.831249	7.130527
$S{h}_{avg}$	5.698709	7.080791	10.47675	12.8777

. The enhancement of the heat and mass transfer through the interface is measured by increasing the buoyancy effect.

5.5. Impact of Lewis Number

Figure 13 indicates the streamlines at the left side, the isotherms at the center, and the isoconcentration at the right side. It can be observed that there is not a significant change in the streamline (left side), isotherm (center), and isoconcentration (right side) contours by enhancing the Lewis number for both layers. This is confirmed by Figure 14, where the behavior of the local Nu is almost the same for different values of Lewis number. On the other hand, the local Sh is enriched by increasing the Lewis number. The particular reason for this minor change in these results is the small value of the buoyancy ratio (N = 0.1). Furthermore, it is observed that there is a minor effect of concentration in the momentum equation.

Table 7. Average Nu and Sh comparison.

	Le = 1	Le = 2.5	Le = 5	Le = 10
$N{u}_{avg}$	3.483756	3.425136	3.389914	3.366655
$S{h}_{avg}$	3.48395	5.699383	7.703609	10.18238

depicts the trend of the average Nu and Sh for different Lewis numbers (Le), which reflects the minor decline in the local Nusselt number as the Lewis number rises, but the average Sherwood numbers increase considerably.

In the next portion of this discussion, some plots of the average Nu and Sh are drawn on the power law index. Figure 15 indicates that the average Nu and Sh decline with an increase in the width of the porous layer with power law indices. The behavior of the average Nu and Sh with the undulation parameter and amplitude of the interface is shown in Figure 16 and Figure 17, respectively. It is visualized, in both figures, that the results switch towards the decline direction, by increasing the values of K and A, correspondingly.

6. Conclusions

Double-diffusive natural convection has been investigated in a 2D cavity, composed of two layers (non-Newtonian and porous), separated by a wavy interface, using FEM. The effects of the Rayleigh number, power law index, Darcy number, buoyancy ratio, and Lewis number are studied. The main conclusions can be summarized as follows:

• The temperature gradient rises by augmenting the Rayleigh number, as the flow is observed from the vertical to horizontal direction in both layers of the cavity, which seems to be a remarkable change in heat transfer. Similar behavior is observed in the concentration case.
• By increasing the values of the power law indices, a decline in the average Nu and Sh is observed.
• When Da ${10}^{-6}$ to ${10}^{-5}$, the streamlines are localized in the non-Newtonian zone, which may indicate that no exchange is observed between the porous and non-Newtonian layers. By increasing the value of Da, the wavy interface is changed.
• Constant enhancement in heat and mass transfer is noticed in a wavy interface by enriching the buoyancy effect.
• The local Nu is almost the same for different values of Lewis number. On the other hand, the local Sh is enriched by increasing the Lewis number.
• The average Nusselt and Sherwood numbers decline by increasing the width of the porous layer with power law indices.
• It is visualized, in both figures, that the average Nu and Sh move towards the decline direction, by increasing the values of K and A, correspondingly.