Authors’ summary: The unsteady compressible Reynolds-averaged Navier-Stokes equations are discretized using the Osher approximate Riemann solver with fully implicit time stepping. The resulting nonlinear system at each time step is solved iteratively using a Newton/GMRES method. In the solution process, the Jacobian matrix-vector products are replaced by directional derivatives, so that the evaluation and storage of the Jacobian matrix is removed from the procedure. An effective matrix-free preconditioner is proposed to fully avoid matrix storage. Convergence rates, computational costs and computer memory requirements of the present method are compared with those of a matrix Newton/GMRES method, a four stage Runge-Kutta explicit method, and an approximate factorization sub-iteration method. Effects of convergence tolerances for the GMRES linear solver on the convergence and efficiency of Newton iteration for nonlinear systems at each time step are analysed for both matrix-free and matrix methods. Differences in the performance of the matrix-free method for laminar and turbulent flows are highlighted and analysed. Finally, we present unsteady turbulent Navier-Stokes solutions of pitching and combined translation-pitching aerofoil oscillations for unsteady shock-induced separation problems associated with rotor blade flows of forward flying helicopters.