## 4 Results and Discussion
### Heat Transfer
This supports the validity of modeling the variation as a linear function of Reynolds number. The resulting correlation is given by  
\[Nu_{g}^{m}=0.0031\times Re+74.1\Big{(}\text{for }Ra=5\times 10^{7}\Big{)} \tag{37}\]  
With  
* regression coefficient \(R^{2}=0.985\)
* confidence interval: [0.00274, 0.00358]  
For \(Ra=5\times 10^{8}\), in the same way, the \(Nu_{g}^{m}\) was studied in the turbulent regime zone, as a function of \(Re\). Table 5 presents \(Nu_{g}^{m}\) values for different values of \(Re\) considered in this study. As indicated, it varies from 119.22 for \(Re=10^{3}\) till 152.10 for \(Re=2.5\times 10^{3}\), with a variable growth rate (varying from 0.560 till 0.895), much higher than that relating to the case of \(Ra=5\times 10^{7}\). The confidence interval for the slope, [0.01941-0.02656], remains relatively tight despite the higher variability, which confirms the robustness and statistical significance of the correlation in this regime. The evolution of the average value of the global Nusselt number is also linear in this case too, as presented in Figure 11. A correlation is proposed for this relation as follows:  
\[Nu_{g}^{m}=0.0232\times Re+96.9 \tag{38}\]  
With  
* regression coefficient \(\text{R}^{2}=0.9995\)
* confidence interval: [0.01941-0.02656]. Figures 11 and 12 illustrate the relationship between the average global Nusselt number \((Nu_{g}^{m})\) and Reynolds number \((Re)\) at different Rayleigh number values \((Ra)\). Figure 12 shows that for \(Ra=5\times 10^{8}\), the effect of turbulence on heat transfer is moderate, and \(Nu_{g}^{m}\) increases steadily with \(Re\). In contrast, Figure 11, which corresponds to \(Ra=5\times 10^{8}\), highlights a more pronounced influence of convection, leading to a stronger increase in the \(Nu_{g}^{m}\). This indicates that at higher Rayleigh numbers, natural convection becomes more \begin{table}
\begin{tabular}{c c c c c c} \hline \(Re\) & **1000** & **1500** & **2000** & **2500** & **3000** \\ \hline \(Nu_{g}^{m}\) & 76.94 & 78.8 & 80.54 & 81.87 & 83.3 \\ \hline \end{tabular}
\end{table} 
Table 4: Values of \(Nu_{g}^{m}\) for different values of \(Re\), \(Ra=5\times 10^{7}\). \begin{table}
\begin{tabular}{c c c c c} \hline \(Re\) & **1000** & **1500** & **2000** & **2500** \\ \hline \(Nu_{g}^{m}\) & 120.1 & 133.25 & 143.19 & 155.09 \\ \hline \end{tabular}
\end{table} 
Table 5: Values of \(Nu_{g}^{m}\) for different values of \(Re\), \(Ra=5\times 10^{8}\). dominant, and more advanced cooling solutions might be required to maintain optimal temperature conditions in heat sinks. These findings are particularly relevant for thermal management in large-scale systems where both temperature gradients and flow control play a critical role. The correlations between \(Nu_{g}^{m}\) and \(Re\) shown in these graphs tell us a lot about how turbulence and thermal gradients affect heat transfer. They also provide useful predictive tools for engineers to anticipate the thermal behavior under different operating conditions. This knowledge can be used to enhance the design of cooling systems in heat sinks (data centers, electronic components...), leading to improved performance and reduced energy consumption. In particular, these correlations allow for faster thermal assessments during the early design stages, help avoid overdesign, and support decisions on airflow rate and geometry sizing without relying exclusively on costly experimental or high-fidelity simulations.