External flow simulations are both ubiquitous in both research and industry, spanning e.g. vehicle aero- or hydrodynamics, biological propulsion, wind energy, civil engineering, etc. The targeted problems remain quite challenging as they usually require an accurate capture of (1) the near-wall region in order to measure accurately the forces at the wall device, and also of (2) the wake when interacting with another device downstream. The research and industry currently offer two solutions to simulate such flows. The first one is a vortex method. This relies on a vorticity-based formulation and proposes several computational assets that are particularly interesting for the wake. Yet, vortex methods are also less suited to capture high Reynolds number boundary because of its isotropic computational elements. The second available solution is an Eulerian body-fitted grid solver, e.g. using finite differences, finite volumes, etc. Thanks to their anisotropic elements, their meshes perfectly fit the body and are then fully suited to capture the boundary layers; yet much less to solve the wake due to their common Eulerian-related dispersion and diffusion errors. This naturally leads to the conclusion that one could exploit the advantages of Lagrangian vortex methods and body fitted-grid solvers in a coupled approach. We present here a weak coupling; i.e, a coupling in which the vortex method solves the flow on the entire domain while the body-fitted Eulerian solver only solves the near-wall region in a much smaller domain. In such coupling, the Lagrangian method drives the Eulerian solver by imposing the boundary condition at the outer boundary of the near-wall domain whereas the Eulerian solution is used to correct the Lagrangian solution in the near-wall region. The weak couplings developed until now suffer from many issues; vorticity noise generation on the outer boundary, circulation not conservation, pressure oscillations at the boundary condition generated by typical Lagrangian distorsion. We here propose novel approaches that addresses all the shortcomings from previous couplings. This new methodology has been developed for two types of body-fitted solver: one based on finite volumes and one based on finite differences. We demonstrate the correct performance of those couplings on the benchmark of an impulsively started cylinder and on the vortex shedding past a cylinder. We also present cases with moving bodies.