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Recently a new procedure, which solves the governing boundary-layer equations with Keller's box method, has been developed for calculating unsteady laminar flows with flow reversal. In this report, we extend this method to turbulent boundary layers with flow reversal. Using the algebraic eddy viscosity formulation of Cebeci and Smith, we consider several test cases to investigate the proposition that unsteady turbulent boundary layers also remain free of singularities. We also perform turbulent flow calculations by using the turbulence model of Bradshaw, Ferriss, and Atwell; we solve the governing equations for both models by using the same numerical scheme and compare the predictions with each other. The study reveals that, as in laminar flows, the unsteady turbulent boundary layers are free from singularities, but there is a clear indication of rapid thickening of the boundary layer with increasing flow reversal. The study also reveals that the predictions of both turbulence models are the same for all practical purposes. (Author)