This paper presents an adaptive finite element method for solving incompressible turbulent flows using the k-e model of turbulence. Solutions are obtained in primitive variables using a highly accurate quadratic finite element on unstructured grids. Turbulence is modeled using the k-e model. A projection error estimator is presented that takes into account the relative importance of the errors in velocity, pressure, turbulence variables and eddy viscosity. The methodology is verified by solving problems with known analytical solutions. The method is then applied to turbulent free shear flows and predictions are compared to experiment.