Two broad categories of methods are used in the computational fluid dynamics community: Eulerian and Lagrangian methods. Eulerian methods are well-suited to capture sharp gradients, such as boundary layers, as the mesh can be body-fitted and anisotropic. On the other hand, Lagrangian methods are typically isotropic (as using particles) and have low numerical dispersion and diffusion. In particular, Vortex Particle-Mesh methods (i.e. with regular redistribution of the particle using a background mesh) have shown to be very efficient and accurate in the simulation of wake flows over long distances. One could then want to couple a Eulerian method to a vortex method, in order to achieve a high resolution of the flow near the solid boundaries, while being able to simulate the wake far downstream in an efficient way. We present the weak coupling between a Eulerian, Finite Difference (FD) solver, and a hybrid Vortex Particle-Mesh (VPM) method to perform Direct Numerical Simulation (DNS) of incompressible 2D flows past bodies, as validated by Billuart et al. [1]. The FD solver uses the staggered grid method for the velocity-pressure formulation of the Navier-Stokes equations, while the VPM solver uses the vorticity-velocity formulation of those equations. As such, the methodology can be used to investigate the flow past impulsively started (or stopped) bodies at various Reynolds numbers. Thanks to the body-fitted grid, we capture accurately the flow details in the near-wall region (the development of the boundary layers near the wall, their separation, the near wake-development) and the wall-resolved quantities (pressure and shear stress) used to evaluate the aerodynamic forces and moments. We further extended the methodology to handle flow past arbitrarily moving bodies: the FD mesh, attached to the body, now moves relatively to static background VPM mesh. We will present some applications of this methodology to moving cylinders and airfoils.