The use of additive manufacturing (AM) processes using Direct Energy Deposition methods is expanding as they offer new possibilities for manufacturing metallic components. The numerical modeling of these processes is of interest to manufacturers seeking faster design methodologies and/or improved control over weld or component quality. However, the computational cost of such models remains very high, making the development of more efficient numerical approaches a major challenge. This high cost arises from the multiphysical nature of the problem, involving electromagnetic, thermohydraulic, mechanical, and metallurgical phenomena. In AM processes, the fluidic phenomena dominate the thermal effects, influencing both the final geometry and mechanical state of the components. The mathematical modeling of the dominant fluidic effects at the process scale is the focus of this work. Indeed, the fluid behavior of the melt pool depends on the surface tension, inertia, viscous effects, and external forces. The interactions between shielding gas, added droplets, and the melt pool thermally affect the final geometry, mechanical stresses, and metallurgical states, impacting component quality. Insufficient control of these fluidic interactions may lead to defects such as the lack of fusion between layers, porosity, and/or cracking. To achieve a robust, efficient and accurate description of the fluidic phenomena, a dedicated numerical strategy is under development. The system is decomposed into two computational domains evolving at distinct spatiotemporal scales. The first describes a droplet falling through a gas jet, while the second represents the pool receiving the droplet. As the dynamics of the first domain are faster than the dynamics of the second domain, this decomposition enables the use of two distinct computational methods. The Level Set method is an interface capturing method, used to capture the rapidly moving droplet whereas the Arbitrary Lagrangian Eulerian method is an interface tracking method, used to track the melt pool, where higher precision in surface displacement is required. The two domains are finally coupled through spatiotemporally consistent boundary conditions.