## 5 Conclusions  
This study used the Cascaded Lattice Boltzmann Method (CLBM) to provide a complete picture of how to control the temperature of medium- and large-sized heat sinks. The  
Figure 11: Evolution of the average global Nusselt number \((Nu_{g}^{m})\) as a function of \(Re\), \(Ra=5\times 10^{8}\). Figure 12: Evolution of the average global Nusselt number \((Nu_{g}^{m})\) as a function of \(Re\), \(Ra=5\times 10^{7}\). trustworthiness of this method has been further supported by the outstanding agreement between benchmark results from the literature and the built and tested numerical code. Reynolds number values between \(10^{3}\) and \(3\times 10^{3}\) were investigated, along with Rayleigh number values of \(5\times 10^{7}\) and \(5\times 10^{8}\) with an emphasis on mixed convection regimes. Due to these parametric ranges, the interaction between forced and natural convection modes in various flow and temperature conditions could be described. The study found that as the Reynolds number goes up, heat moves much faster along the vertical walls of the heat sinks. This is particularly true up to \(Re=1.5\times 10^{3}\), which is the point where the flow starts to become turbulent (\(Ra=5\times 10^{7}\)). After this point, the forced air jets' impact takes over and more effectively shapes the thermal field. The average Nusselt number varies slightly; its value increased by 8.3%, from 76.94 to 83.3, indicating that considerable cooling improvements are possible with regulated airflow. Localized inefficiencies in heat evacuation result from areas of partial air confinement that continue to exist between the heat sinks. The thermal and dynamic fields become much more complex at higher Rayleigh numbers (\(Ra=5\times 10^{8}\)), which correspond to truly turbulent regimes. Intense mixing and powerful natural convection cells define the flow, greatly increasing the rate of heat transfer. In This case, the average Nusselt number increases significantly in this environment. It went from 199.22 to 152.10, a 27.6% increase, as Re increases. Under very high temperatures, the crucial function of coupled forced and natural convection processes is highlighted by the heat transfer's marked sensitivity to the Reynolds number. This study helps clarify how heat transfer behavior depends on both thermal loading and flow control, providing insight that can inform real-world cooling system design. In particular, since the Rayleigh number inherently includes the characteristic size of the heat sinks, our analysis also captures how the physical scale of the system influences thermal performance. The flow regimes, thresholds, and derived correlations offer valuable guidance for engineering applications, including airflow control and heat sink sizing. Such correlations derived from CLBM simulations can support predictive design approaches without requiring full-scale parametric optimization. This provides a useful framework for practical implementation in applications such as data centers and industrial cooling modules. By showing that the Nusselt number goes up in direct proportion to the Reynolds number, this connection, which was found through simulations using the CLBM, shows how forced flow improves heat transfer in large heat sinks. This connection makes it possible to find the best configurations for systems in places like data centers or tough industrial settings. It also makes it possible to accurately predict the thermal performance of heat sinks on a large scale. This study proposes correlations for such applications. These results have the potential to solve localized inefficiencies and improve thermal management in high-performance systems. Conceptualization, F.Z.L.A. and H.F.; Methodology, E.S., A.G. and A.M.; Software, M.E.A. and E.S. ; Validation, F.Z.L.A., M.E.A. and E.S. ; Formal analysis, F.Z.L.A. and A.G.; Investigation, M.E.A., E.S., H.F., A.G. and A.M.; Resources, A.M.; Writing--Original draft, F.Z.L.A. ; Writing--Review and editing, M.E.A., H.F., A.G. and A.M.; Supervision, M.E.A., E.S.