Harmonic Solution of Unsteady Flow Around Blades with Separation

Conventional time-linearized harmonic methods using a pseudo-time-marching technique often exhibit solution divergence behavior for highly loaded conditions with flow separation. This is attributed to the growth of linear instabilities associated with small-scale limit-cycle oscillations of the base-flow state. In the present work, an alternative harmonic formulation is adopted for modeling harmonic disturbances with an intention to harness nonlinearity as a means of stabilization while achieving the same computational efficiency as a conventional time-linearized model. One unsteady complex-number harmonic solution is made equivalent to two real-number steady-flow solutions with simple source terms, whereas nonlinearity is retained in the convective fluxes of discretized flow equations. The validity and effectiveness of the method has been demonstrated in comparisons with semi-analytical results for a flat plate cascade and experimental data for low- and high-speed turbine cascades. The computational results show that the present harmonic solver is robust with much improved convergence over the conventional time-linearized model. It is shown that converged solutions can be effectively obtained using the present method for unsteady flows with large-scale separation.