## 5 Results Discussion
### Effect of Rayleigh Number  
Figure 4 shows the isosurfaces of the governing variables for different magnitudes of the Ra number at Ha = 0, Da = 0.01, \(\phi\) = 0.02, and \(\varepsilon_{p}=0.4\). The increase in Ra characterizes a start of the temperature contrast between the hot wall and cooler. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(Ra\) & \(\mathbf{10^{3}}\) & \(\mathbf{10^{4}}\) & \(\mathbf{10^{5}}\) & \(\mathbf{10^{6}}\) \\ \hline Present work & 1.1803 & 2.2745 & 4.8759 & 9.6335 \\ \hline [PERSON] et al. & 1.17612 & 2.2679 & 4.8608 & 9.5918 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Validation of the present code, for the Nu\({}_{\text{avg}}\) with Ra at Hartmann number (Ha) = 0, with a cavity by [PERSON] et al. (2018). Figure 3: **(left)** Numerical results from [PERSON] et al. (2018); **(right)** The present study for streamlines at Rayleigh number (Ra) = \(10^{6}\). Figure 4: Variations of the streamlines (**left**), isotherms (**middle**), and general entropy (**right**) with various values of Rayleigh number (Ra),solid volume fraction (\(\phi\)) = 0.02, Ha = 0, Da = 0.01, nanoparticle volume fraction (\(\phi\)) = 0.02, and porosity (\(\epsilon_{p}~{}=~{}0.4\)). The highest value of the stream function increases. This phenomenon means a stronger vortex, and therefore higher velocity of the vortex in the enclosure. The second characteristic is that the density of the streamlines close to the walls, increases. This means that the temperature gradient enhances close to the walls. Hence, the vortex size increases in this area. The Ra number represents the buoyancy force. An augmentation in the Ra results in an increase in the buoyancy force. The buoyancy force is because of the difference between the density of the cold and hot fluid in the area of the cold and hot walls, respectively. A vortex is created as a result of the fluid movement in the enclosure. The vortex velocity increases by increasing the value of the Ra number, this will lead to an enhancement of heat transfer. Entropy generation due to heat transfer accumulates around the upper cold wall and increases in the lower-left corner with increasing values of Ra, due to increased kinetic  
Figure 8: Average Nusselt number (Nu\({}_{\rm ave}\)) as a function of Rayleigh number (Ra) at various values of Darcy number (Da). \(\phi=0.02\) and \(\varepsilon_{p}=0.4\). Figure 7: Average Nusselt number (Nu\({}_{\rm ave}\)) as a function of Rayleigh number (Ra) at various values of solid volume fraction (\(\phi\)). Ha = 0, Da = 0.01 and \(\varepsilon_{p}=0.4\). Figure 9: Total entropy as a function of Rayleigh number (Ra) at various values of Hartmann number (Ha). Da = 0.01, \(\phi=0.02\) and \(\varepsilon_{p}=0.4\). Figure 5 reveals that the Nuavg increases with the Ra, this process is detected for all values of Ha. This is because an augmentation in the buoyancy force by enhancing the Ra results in an augmentation in the velocity of the fluid in the enclosure, hence, the Nu decreases. The Nuavg diminishes with the Ha and the Lorentz force. This is due to the fact that Lorentz force prevents the formation of vortices, which leads to a reduction in the velocity of the fluid in the cavity, resulting in a reduction in the Nu. Variation of Nuavg with Ra for different \(\varepsilon\) values is presented in Figure 6. The Nuavg increases with Ra for all values of \(\varepsilon_{p}\). Additionally, increasing \(\varepsilon\) enhances Nu and heat transfer for higher values of Ra, as fluid motion (and hence convective heat transfer) is hindered for low porosities. The effect is not seen for low values of Ra due to conductive heat transfer dominance. It should be mentioned that the enhancement is most evident when increasing \(\varepsilon\) in the small ranges (e.g., 26% enhancement with an increase from \(\varepsilon_{p}=0.2\) to \(\varepsilon_{p}=0.4\) at Ra = \(10^{6}\) as compared with the enhancement of 10% with an increase from \(\varepsilon_{p}=0.6\) to \(\varepsilon_{p}=0.8\) for the same Ra).