The practical use of automated CFD-based design tools in ship-building industry requires powerful flow solvers, able to take into account realistic geometries as well as complex physical phenomena, such as turbulence. A shape optimization tool is developed in this framework. A derivative-free optimizer, yielding both flexibility and robustness, is preferred to the classical gradient-based method, more fastidious to implement and still limited to moderately complex problems. The flow solver included in the design procedure solves the incompressible Reynolds-averaged Navier-Stokes equations on unstructured grids using a finite-volume formulation, involving several near-wall low-Reynolds number turbulence models. The design tool is then applied to optimize the stern of a modern hull shape at model and full scale, different purposes being considered. Our interest is particularly focused on the influence of turbulence modeling in the design process. The effects of a two-equation model based on the eddy-viscosity assumption and a second-order closure relying on the Reynolds stress transport equations are compared.