Stabilizing Linear Harmonic Flow Solvers for Turbomachinery Aeroelasicity with Complex Interative Algorithms

The linear flow analysis of turbomachinery aeroelasticity views the unsteady flow as the sum of a background nonlinear flowfield and a linear harmonic perturbation. The background state is usually determined by solving the nonlinear steady-flow equations. The flow solution representing the amplitude and phase of the unsteady perturbation is given by the solution of a large complex linear system that results from the linearization of the time-dependent nonlinear equations about the background state. The solution procedure of the linear harmonic Euler/Navier-Stokes solver of the HYDRA suite of parallel FORTRAN codes consists of a preconditioned multigrid iteration that in some circumstances becomes numerically unstable. The results of previous investigations have already pointed at the physical origin of these numerical instabilities and have also demonstrated the code stabilization achieved by using the real GMRES and RPM algorithms to stabilize the existing preconditioned multigrid iteration. This approach considered an equivalent augmented real form of the original complex system of equations. We summarize the implementation of the complex GMRES and RPM algorithms that are applied directly to the solution of the complex harmonic equations. Results show that the complex solvers not only stabilize the code, but also lead to a substantial enhancement of the computational performance with respect to their real counterparts. The results of a nonlinear unsteady calculation that further emphasize the physical origin of the numerical instabilities are also reported.