This paper studies the application of two-level Schwarz algorithms to several models of Computational Fluid Dynamics. The purpose is to build an algorithm suitable for elliptic and convective models. The sub-domain approximated solution relies on the incomplete lower-upper factorisation (ILU). The algebraic coupling between the coarse grid and the Schwarz preconditioner is discussed. The Deflation Method (DM) and the Balancing Domain Decomposition (BDD) Method are studied for introducing the coarse grid correction as a preconditioner. Standard coarse grids are built with the characteristic or indicator functions of the sub-domains. The building of a set of smooth basis functions (analogous to smoothed-aggregation methods) is considered. A first test problem is the Poisson problem with a discontinuous coefficient. The two options are compared for the standpoint of coarse-grid consistency and for the gain in scalability of the global Schwarz iteration. The advection-diffusion model is then considered as a second test problem. Extensions to compressible flows (together with incompressible flows for comparison) are then proposed. Parallel applications are presented and their performance measured.