Computing Linear Harmonic Unsteady Flows in Turbomachines with Complex Iterative Solvers

The linear flow analysis of turbomachinery aeroelasticity views the unsteady flow as the sum of a background nonlinear flow field 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 instead given by the solution of a large complex linear system which 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 which 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. This paper summarizes the implementation of the complex GMRES and RPM algorithms which 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 paper also reports the results of a nonlinear unsteady calculation which further emphasizes the physical origin of the numerical instabilities.