An efficient harmonic balance method for unsteady flows in cascades

In this paper, nonlinear unsteady flows in cascades arising from self-excited blade vibrations are investigated using a mixed time/frequency domain harmonic balance technique. The time-periodic flow is computed at several sub-time levels that are equally spaced over a single period. The time derivative term in the unsteady Navier-Stokes equations is approximated by a pseudo-spectral operator, which couples solutions at different sub-time levels. Compared to the classical time-accurate approach, the present technique enjoys an improved computational efficiency. In addition, using complex periodic boundary conditions, the computational domain can be reduced to a grid spanning only a single blade in the cascade, which further improves the computational efficiency. With the use of the pseudo-spectral operator and pseudo-time marching, the problem can be treated as steady state and convergence acceleration techniques such as local time stepping, implicit residual smoothing and multigrid can be used. To demonstrate the efficiency and accuracy of the technique, we present results for the Tenth Standard and the Eleventh Standard Configurations.