The finite element method in the form of the weak Galerkin formulation with the penalty function method was applied to several problems of axisymmetric turbulent flows including flow through a sudden pipe expansion, the stern region flow of a slender body, and flows past ducted and nonducted propellers in action. The coupled set of the Reynolds time-averaged Navier-Stokes equations and two turbulence transport equations for the turbulent kinetic energy and its rate of dissipation was solved by L/U decomposition and successive substitution with relaxation. An existing finite element code was modified with a low Reynolds number form for an appropriate treatment of wall influences on turbulence transport, which produces a better solution and provides an easier imposition of boundary conditions by solving up to wall with no slip boundary conditions. The two-equation turbulence model with the wall modification was first successfully tested by solving the turbulent flow through a sudden pipe expansion. The numerical simulation of the stern region flow of a streamlined body resulted in an excellent agreement with the measured data in terms of the mean-flow and turbulence quantities. Turbulent shear flows past a propeller at the rear end of the same slender body, modeled by an actuator disk, were successfully solved at two rotational speeds, self-propelled and 100% over-thrusted, using the same two-equation model. And finally, comparisons of the wake behind a propeller were made for the self-propelled conditions of a ducted and nonducted propeller on the same streamlined body.