A study of the flowfield around axisymmetric launch vehicles in different flight conditions is performed. In particular, the first Brazilian Satellite Launcher second-stage configuration is analyzed considering the case with and without a propulsive jet in the vehicle base. The study is performed using a cell-centered finite volume formulation on unstructured grids. Different spatial discretization schemes are compared, including a centered and an upwind scheme. The upwind scheme is a second-order version of the Liou flux vector-splitting scheme. Turbulence effects are accounted for using two one-equation turbulence closure models, namely, the Baldwin and Barth and the Spalart and Allmaras models. An agglomeration multigrid algorithm is used to accelerate the converge to steady state of the numerical solutions. The numerical results obtained are compared with experimental data, as well as with previous structured grid simulations.